MICROWAVE IMAGING USING A TUNABLE REFLECTARRAY ANTENNA
AND
SUPERRADIANCE IN OPEN QUANTUM SYSTEMS
By
Amin Tayebi

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering—Doctor of Philosophy
Physics—Doctor of Philosophy
2017

ABSTRACT
MICROWAVE IMAGING USING A TUNABLE REFLECTARRAY
ANTENNA
AND
SUPERRADIANCE IN OPEN QUANTUM SYSTEMS
By
Amin Tayebi
Theory, experiment, and computation are the three paradigms for scientific discoveries.
This dissertation includes work in all three areas. The first part is dedicated to the practical
design and development of a microwave imaging system, a problem mostly experimental
and computational in nature. The second part discusses theoretical foundations of possible
future advances in quantum signal transmission.
In part one, a new active microwave imaging system is proposed. At the heart of this novel
system lies an electronically reconfigurable beam-scanning reflectarray antenna. The high tuning capability of the reflectarray provides a broad steering range of ±60◦ in two distinct frequency bands: S and F. The array, combined with an external source, dynamically steers the
incoming beam across this range in order to generate multi-angle projection data for target
detection. The collected data is then used for image reconstruction by means of time reversal
signal processing technique. Our design significantly reduces cost and operational complexities compared to traditional imaging systems. The inclusion of the beam scanning array
and the utilization of a single source, eliminates the need for a costly array of transceiver
antennas and the involved switching circuitry. In addition, unlike conventional setups, this
system is not constrained by the dimensions of the object under test. Thus the inspection of
large objects, such as extended laminate structures, composite airplane wings and wind turbine blades becomes possible. Experimental results of detection of various dielectric targets
as well as anomalies within them, such as defects and metallic impurities, are presented.

The second part includes the theoretical consideration of three different problems: quantum transport through two different nanostructures, a solid state device suitable for quantum
computing and spherical plasmonic nanoantennas and waveguides. These three physically
different systems are all investigated within a single quantum theory; the effective nonHermitian Hamiltonian framework. The non-Hermitian Hamiltonian approach is a convenient mathematical formalism for the description of open quantum systems. This method
based on the Feshbach projection formalism provides an alternative to popular methods
such as the Feynman diagrammatic techniques and the master equation approach that are
commonly used for studying open quantum systems. It is formally exact but very flexible
and can be adjusted to many specific situations.
One bright phenomenon emerging in the situation with a sufficiently strong continuum
coupling in the case when the number of open channels is relatively small compared to
the number of involved intrinsic states is the so-called superradiance. Being an analog of
superradiance in quantum optics, this term stands for the formation in the system of a
collective superposition of the intrinsic states coherently coupled to the same decay channel.
The footprint of superradiance in each system is investigated in detail. In the quantum
transport problem, signal transmission is greatly enhanced at the transition to superradiance. In the proposed solid state charge qubit, the superradiant states effectively protect
the remaining internal states from decaying into the continuum and hence increase the lifetime of the device. Finally, superradiance provides us a tool to manipulate light at the
nanoscale. It is responsible for the existence of modes with distinct radiation properties in
a system of coupled plasmonic nanoantennas: superradiant states with enhanced and dark
modes with extremely damped radiation. Furthermore, similar to the quantum case, energy
transport through a plasmonic waveguide is greatly enhanced.

Copyright by
AMIN TAYEBI
2017

To my parents: Mo and Feri.

v

ACKNOWLEDGMENTS
Progress is impossible without change, and those who cannot
change their minds cannot change anything.
George Bernard Shaw

To all those who have helped shape and develop my thinking, keeping my mind open to
new ideas and approaches, you have my deepest appreciation.
I must first thank Michigan State University for its commitment to quality education
and state of the art, nationally recognized science programs. MSU’s campus was my first
experience of life in the U.S. and I’ll always remember a sense of acceptance and respect for
all students, no matter their origin. My advisors and committee members have been key
sounding boards throughout my graduate studies and I want to thank them for their distinct
roles in my development.
Professors Satish and Lalita Udpa, you were the inception of making my dream of pursuing a PhD a reality. I must first and foremost thank you for accepting me into your program
and making me a part of your ever-growing NDE family, of which I know I’ll feel included in
for many years to come. Despite your many endeavors and administrative duties, you both
always made time to meet with me and guide me along the path to success.
To Professor Vladimir Zelevinksy, I will be forever grateful that you accepted me as a
dual-PhD student, allowing me to enter into an entirely new world and develop a fresh passion
for physics. Not only have you been an inspiring mentor, you have become a lifelong friend.
Your guidance in the field of physics over the last six years has permanently transformed my
way of thinking. I thank you and your wife, Vera for your continual support, companionship
and endless supply of chocolate.
Professors Edward J. Rothwell and Prem Chahal, I cannot thank you both enough for
vi

your open availability and willingness to discuss any problems I ran into. Your guidance
and counsel has been invaluable to me and instrumental in my growth as an engineer. To
Professor Alexander Volya, thank you for providing me with your notes and allowing me to
use your slides in carrying on your work. To my labmates in the NDE lab, I will greatly miss
our comradery and daily interactions. I have enjoyed getting to know and work with each of
you in and outside of the lab. Scott Rice, thank you for the late nights and many hours you
have put in helping me on the Plasmonics project. I truly appreciate your distinct approach
and persistence when working on the problem.
Finally, to my family, I owe everything to you for where I am today. To my mother,
Feri, and my father, Mo, thank you for the many sacrifices you made on my behalf. Mom,
you spent countless hours making sure I could focus solely on school, which set me on this
path toward attaining my PhD. Dad, I will always admire and aspire to have your courage
and vision in life. To my brother, Ali, thank you for teaching me high school math when I
was in elementary school and initiating my love of numbers, also for being everything a big
brother should be. To my sister, Asal, who has always been a loving big sister, thank you
for sharing your outlook on life and your continued support.
To my best friend and partner, Alicia, thank you for your love and encouragement despite
all the stress of graduate school. I am excited for the next chapter of our lives and the many
happy years we have to come.

vii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

Part 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Part 2 Microwave Imaging Using an Electronically Tunable
Reflectarray Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Chapter 1 Introduction . . . . . . . . . . . . . . . .
1.1 A Brief History of Imaging . . . . . . . . . . . . .
1.2 Microwave and Millimeter Wave Imaging . . . . .
1.3 A New Design: Mirror-Based Microwave Imaging
1.3.1 Deformable Mirror . . . . . . . . . . . . .
1.3.2 Image Reconstruction . . . . . . . . . . . .
1.4 Organization . . . . . . . . . . . . . . . . . . . .

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5
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12
13

Chapter 2 The Reflectarray Antenna
2.1 Introduction . . . . . . . . . . . . .
2.2 The Theory of Reflectarrays . . . .
2.3 Analysis of The Unit Cell . . . . .
2.3.1 Full Wave Simulation . . . .
2.3.2 Equivalent Circuit Model . .
2.3.3 Measurements . . . . . . . .
2.4 Fabricated Reflectarray . . . . . . .
2.5 Discussion . . . . . . . . . . . . . .

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15
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46

Chapter 3 Time Reversal Imaging . . . . . . . . .
3.1 Inverse Problems and Reconstruction Methods . .
3.2 Fundamentals of Time Reversal Signal Processing
3.3 Finite Difference Time Domain (FDTD) . . . . .
3.3.1 Maxwell’s Equations . . . . . . . . . . . .
3.3.2 Finite Difference Approximation . . . . . .
3.3.2.1 Reduction to Two Dimensions . .
3.3.2.2 Source . . . . . . . . . . . . . . .
3.3.2.3 Boundary Conditions . . . . . . .
3.4 Computer Simulation of Time Reversal Imaging .
3.4.1 Case I: Free Space - Focusing on Source .
3.4.2 Case II: Single Scatterer . . . . . . . . . .
3.4.3 Case III: Two Scatterers . . . . . . . . . .

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50
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76

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viii

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Chapter 4 Experimental Setup and Results . . . . . . . . . . . . . . . .
4.1 Experiment I: Single Scatterer . . . . . . . . . . . . . . . . . . . . . . .
4.2 Experiment II: Two Scatterers - Beam Scanning . . . . . . . . . . . . .
4.3 Experiment III: Dielectric Anomalies - Defects and Metallic Impurities

Part 3

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82
85
87
88

Superradiance in Open Quantum Systems . . . . . . . . . 96

Chapter 5 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 6 The Effective non-Hermitian Hamiltonian
6.1 Derivation-Feshbach Projection Formalism . . . . . .
6.2 The Superradiance Phase Transition . . . . . . . . .
6.2.1 Dicke Superradiance in Quantum Optics . . .
6.2.2 Superradiance in Open Systems . . . . . . . .
6.3 Scattering Matrix and Transmission . . . . . . . . . .

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102
102
105
105
107
108

Chapter 7 Quantum Transport . . . . . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Quantum Signal Transmission Through a Single-Qubit Chain .
7.2.1 Closed Chain . . . . . . . . . . . . . . . . . . . . . . .
7.2.1.1 Symmetric and Antisymmetric Modes . . . .
7.2.1.2 Case ∆ > 2v . . . . . . . . . . . . . . . . . .
7.2.1.3 Case ∆ < 2v . . . . . . . . . . . . . . . . . .
7.2.1.4 Large N Chain . . . . . . . . . . . . . . . . .
7.2.2 Open Chain . . . . . . . . . . . . . . . . . . . . . . . .
7.2.3 Transmission Through The Chain . . . . . . . . . . . .
7.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Two-Channel Quantum Wire . . . . . . . . . . . . . . . . . .
7.3.1 The Isolated Two-Channel Wire . . . . . . . . . . . . .
7.3.2 Coupling to the Continuum and Superradiance . . . . .
7.3.3 Transmission Through The Wire . . . . . . . . . . . .
7.3.4 Energy-Dependent Coupling . . . . . . . . . . . . . . .
7.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .

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110
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144
146

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Edge States
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148
148
151
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167
172

Chapter 8 Solid State Quantum Computing . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . .
8.2 Closed System . . . . . . . . . . . . . . . . . . .
8.2.1 Case λ → 0 . . . . . . . . . . . . . . . .
8.2.2 Case Îť2 = Îş . . . . . . . . . . . . . . . .
8.3 Open System - Superradiance and Emergence of
8.4 Noise and Decoherence . . . . . . . . . . . . . .
8.5 Discussion . . . . . . . . . . . . . . . . . . . . .

ix

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Protecting
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97
97
99

Chapter 9 Non-Hermitian Plasmonics Antennas and Waveguides
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Coupled Mode Theory and The Non-Hermitian Hamiltonian . . . .
9.3 A Single Metallic Sphere . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Superradiant and Dark States in System of Two Coupled Spheres .
9.5 Plasmonic Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part 4

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174
174
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191
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203

Conclusion and Future Work . . . . . . . . . . . . . . . . . . . 206

Chapter 10 Conclusion . . . . . . . . . . . . . . . . .
10.1 Summary . . . . . . . . . . . . . . . . . . . . . .
10.1.1 The Novel Microwave Imaging System . .
10.1.2 Superradiance in Open Quantum Systems
10.2 Future Work . . . . . . . . . . . . . . . . . . . . .
10.2.1 The Novel Microwave Imaging System . .
10.2.2 Superradiance in Open Quantum Systems

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207
207
207
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209
210

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A The Stochastic Liouville Equation . . . . . . . . . . .
Appendix B Quasi Normal Modes Normalization . . . . . . . . . .
Appendix C Tesseral Harmonics and Spherical Functions Identities

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212
213
216
219

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

x

LIST OF TABLES

Table 2.1:

Voltage applied to each column for various steering angles . . . . . .

45

Table 4.1:

Detection error calculated for different positions of the dielectric cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

Detection error calculated for different steering angles. ND indicates
that the target was not detectable at the particular steering angle.

90

Table 4.2:

Table 4.3:

Detection error calculated for the two dielectric samples with anomalies. 94

xi

LIST OF FIGURES

Figure 1.1:

Schematic of the proposed microwave imaging system [19]. . . . . .

11

Figure 2.1:

Reflectarray geometry.

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19

Figure 2.2:

Geometry of the unit cell. . . . . . . . . . . . . . . . . . . . . . . . .

22

Figure 2.3:

Reflection phase for different varactor capacitance values, Cv (pF). .

24

Figure 2.4:

Reflection phase at different frequencies in (a) the higher frequency
band and (b) the lower frequency band. . . . . . . . . . . . . . . . .

25

Magnitude and phase of the reflection coefficient of the y component
of the electric field on a plane 0.1 mm above the surface of the unit
cell at (a) 8.0 GHz and Cv =0.2 pF and (b) 4.25 GHz and Cv =0.2 pF.
The layout of the unit cell is shown with white lines. . . . . . . . . .

27

(a) Geometry of a double square loop and the equivalent circuit of an
infinite array of double square loops. (b) Geometry of the proposed
unit cell derived from a double square loop with w1 = 1 /2 and the
corresponding equivalent circuit. . . . . . . . . . . . . . . . . . . . .

30

Reflection phase curves for different capacitance values, Cv (pF), predicted by the equivalent circuit model. . . . . . . . . . . . . . . . . .

32

(a) Unit cells. Left: S-band. Right: F-band. (b) Side-view schematic
of the biasing circuit. . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Reflection phase of the F-band unit cell as a function of frequency for
different voltages (or equivalently Cv ) (a) experiment (b) simulation.

36

Figure 2.10: Magnitude of the reflection coefficient of the F-band unit cell as a
function of frequency for different voltages (a) experiment (b) simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Figure 2.11: Reflection phase of the S-band unit cell as a function of frequency for
different voltages (or equivalently Cv ) (a) experiment (b) simulation.

38

Figure 2.5:

Figure 2.6:

Figure 2.7:

Figure 2.8:

Figure 2.9:

xii

Figure 2.12: Magnitude of the reflection coefficient of the S-band unit cell as a
function of freqeuncy for different voltages (a) experiment (b) simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

Figure 2.13: (a) Close-up view of the reflectarray (b) Experimental setup for bistatic
measurement with reflectarray at the center of the arch. . . . . . . .

41

Figure 2.14: Beam steering into (a) 0◦ , (b) 30◦ , (c) 45◦ and (d) 60◦ at 6.12 GHz.
The red and black curves correspond to beam steering using the reflectarray and metal plate, respectively. . . . . . . . . . . . . . . . .

43

Figure 2.15: Beam steering into (a) 0◦ , (b) 30◦ , (c) 45◦ and (d) 60◦ at 6.12 GHz
with the cross polarized component. The red and black curves correspond to beam steering using the reflectarray and metal plate, respectively. The blue curve is the cross polarization component. . . .

44

Figure 2.16: Beam steering into (a) 0◦ , (b) 30◦ , (c) 45◦ and (d) 60◦ at 3.38 GHz.
The red and black curves correspond to beam steering using the reflectarray and metal plate, respectively. . . . . . . . . . . . . . . . .

46

Figure 2.17: Beam steering into 0◦ , 30◦ , 45◦ and 60◦ at 3.38 GHz with the cross
polarized component. The red and black curves correspond to beam
steering using the reflectarray and metal plate, respectively. The blue
curve is the cross polarization component. . . . . . . . . . . . . . . .

47

Figure 2.18: Reflection phase curves for a unit cell scaled by a facotor of 10 for different capacitance values, Cv (pF), predicted by the equivalent circuit
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Figure 3.1:

Yee grid cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Figure 3.2:

Forward propagation of the electric field at t = 1.17 ns. The eight
black dots represent the location of the recievers. . . . . . . . . . . .

68

Figure 3.3:

Forward propagation of the electric field at t = 1.94 ns. . . . . . . .

68

Figure 3.4:

Received signals at the recievers. The numbering corresponds to the
eight receivers shown in Fig. 3.2. . . . . . . . . . . . . . . . . . . . .

69

Figure 3.5:

Back propagation of time reversed fields at t = 0.65 ns. . . . . . . .

70

Figure 3.6:

Entropy of the electric field calculated in accordance to eqn. 3.68. . .

70

xiii

Figure 3.7:

Focusing of time-reversed back propagated waves on the location of
the initial source at t = 1.75 ns. This time instance correspond to
the minimum of the entropy plot in Fig. 3.6. . . . . . . . . . . . . .

71

Energy plot of the backpropagated waves calculated accoridng to
eqn. 3.69. The eight bright points in the figure correspond to the
sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

Energy plot of the back propagated waves zoomed in the location of
the intial source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Figure 3.10: Forward propagation of the electric field at t = 0.67 ns. The presence
of the scatterer, shown with the black circle, causes distortion in the
propagating wave. The relative permittivity of the target is 3. . . .

74

Figure 3.11: Entropy of the electric field as a function of time. The local minimum
time instance corresponds to the focusing of the waves. . . . . . . .

75

Figure 3.12: Focusing of time-reversed back propagated waves on the location of
the scatterer at t = 2.03 ns. This time instance correspond to the
minimum of the entropy plot in Fig. 3.11. . . . . . . . . . . . . . . .

75

Figure 3.13: Energy plot of the back propagated waves zoomed in the location of
the scatterer. The black circle corresponds to the true location of the
scatterer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Figure 3.14: Forward propagation of the electric field at t = 0.56 ns when the wave
front reaches the first scatterer. . . . . . . . . . . . . . . . . . . . . .

77

Figure 3.15: Forward propagation of the electric field at t = 0.99 ns when the wave
front reaches the second scatterer. . . . . . . . . . . . . . . . . . . .

78

Figure 3.16: Entropy of the electric field as a function of time. The two local
minima correspond to the time instances that the field focuses on the
location of the two scatterers in the medium. . . . . . . . . . . . . .

79

Figure 3.17: Focusing of time-reversed back propagated waves on the location of
scatterer two at t = 1.44 ns. This time instance corresponds to the
first local minima of the entropy plot in Fig. 3.16. . . . . . . . . . .

79

Figure 3.18: Focusing of time-reversed back propagated waves on the location of
scatterer one at t = 2.2 ns. This time instance corresponds to the
second local minima of the entropy plot in Fig. 3.16. . . . . . . . . .

80

Figure 3.8:

Figure 3.9:

xiv

Figure 3.19: Energy plot of the back propagated waves zoomed in the location
of the two scatterers. The black circle and ellipse in the figure correspond to the true location of the circular and elliptical scatterers,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Figure 4.1:

Schematic of the microwave imaging setup. . . . . . . . . . . . . . .

83

Figure 4.2:

Energy image of the three configurations in experiment one. The
target was placed at three different distances from the center of the
array. The black circle corresponds to the actual position of the
scatterer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

Energy image for different steering angles in experiment two. The
black circles correspond to the actual positions of the scatterers. . .

89

Figure 4.3:

Figure 4.4:

Dielectric samples with different types of anomalies. The left sample
contains a 1.75-inch-wide and 1-inch-deep axial defect. The second
sample contains a 2-inch-wide piece of copper tape along its main axis. 91

Figure 4.5:

(a) Calculated energy image of the backpropagating waves for the
dielectric sample with defect facing the arch rail (b) close-up view.
The red arrows point to the location of the defect. . . . . . . . . . .

92

(a) Calculated energy image of the backpropagating waves for the
dielectric sample with defect facing the beam scanning array (b) closeup view. The red arrows point to the location of the defect. . . . . .

93

(a) Calculated energy image of the backpropagating waves for the
sample with copper anomaly when the copper tape is facing the arch
rail (b) close-up view. The red arrows point to the location of the
copper anomaly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

(a) Calculated energy image of the backpropagating waves for the
sample with copper anomaly when the copper tape is facing the arch
rail (b) close-up view. The red arrows point to the location of the
copper anomaly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Examples of open quantum systems: (a) an unstable nucleus breaking
apart and undergoing nuclear decay and (b) a quantum wire attached
to leads acting as electron reservoirs. . . . . . . . . . . . . . . . . . .

99

Figure 4.6:

Figure 4.7:

Figure 4.8:

Figure 5.1:

xv

Figure 6.1:

Spontanous emission of a dilute classical gas initially prepared in the
excited state. The radiation is isotropic and the intensity is proportional to the number of atoms. . . . . . . . . . . . . . . . . . . . . . 106

Figure 6.2:

Dicke coherent spontanous radiation. The radiation is directional and
the intensity is proportional to the squared of the number of atoms.

107

Figure 7.1:

Energy levels for a system of the closed chain of eleven sites and the
excited qubit state in the middle as a function of the qubit coupling
strength Îť; the hopping amplitude is set to v = 1, and the excitation
energy of the qubit ∆ = 2.5. . . . . . . . . . . . . . . . . . . . . . . 116

Figure 7.2:

Squared components of the lower (left column) and upper (right column) eigenstates as a function of the qubit excitation strength Îť, for
a chain N = 5 and v = 1. . . . . . . . . . . . . . . . . . . . . . . . . 116

Figure 7.3:

Energy levels for a system of the closed chain of eleven sites and the
excited qubit state as a function of the qubit coupling strength Îť; the
hopping amplitude is set to v = 1, and the excitation energy of the
qubit ∆ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 7.4:

Squared components of the lower (left column) and upper (right column) of the wave functions as a function of the qubit excitation
strength λ, for a chain N = 5, ∆ = 0.5 and v = 1. . . . . . . . . . . 118

Figure 7.5:

Squared components of two wave functions for degenerate level as
a function of the qubit excitation strength Îť, for a chain N = 5,
√
∆ = 3 and v = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 7.6:

Resonance complex energies (eigenvalues of the effective Hamiltonian
for weak continuum coupling, Îł = 0.1) evolve as a function of the
qubit excitation strength Îť. Two qubit states are effectively decoupled from the chain and move along the real energy axis. . . . . . . 122

Figure 7.7:

Resonance complex energies (eigenvalues of the effective Hamiltonian
for strong continuum coupling, Îł = 20) show segregation of two superradiant states in the middle of the band from ten trapped states which
include the strongly localized states of the qubit. . . . . . . . . . . . 123

xvi

Figure 7.8:

Complex-plane trajectories of eigenstates of the effective Hamiltonian; the parameter values are v = 1, λ = 0.8, ∆ = 2.5. The arrows
show the direction of the evolution as Îł changes from 0 to 10. Panel
(a) shows the behavior of 12 states in the chain while panel (b) singles
out the state genetically connected to the excited qubit state located
outside the band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Figure 7.9:

Complex-plane trajectories of eigenstates of the effective Hamiltonian; the parameters values are v = 1, λ = 2.1, ∆ = 2.5. Ten states,
including the super-radiant at the edges of the chain, are moved closer
inside the band, panel (a), two states mainly localized at the qubit
have energies outside the band and essentially interact only with each
other having a large lifetime with respect to tunneling through the
chain, panels (b) and (c) . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 7.10: Complex-plane trajectories of eigenstates of the effective Hamiltonian; the parameters values are v = 1, λ = 8, ∆ = 2.5. Ten states,
including the super-radiant at the edges of the chain, are moved closer
inside the band, panel (a). The qubit states, the ground state, (b),
and the excited state, (c), are essentially decoupled. . . . . . . . . . 126
Figure 7.11: Resonance complex energies (eigenvalues of the effective Hamiltonian)
for ∆ = 0.5 and for weak continuum coupling, γ = 0.1 as a function
of the qubit excitation strength Îť. . . . . . . . . . . . . . . . . . . . 127
Figure 7.12: Resonance complex energies (eigenvalues of the effective Hamiltonian)
for ∆ = 0.5 and for strong continuum coupling, γ = 20 as a function
of the qubit excitation strength Îť. . . . . . . . . . . . . . . . . . . . 128
Figure 7.13: Evolution of the transmission for small Îť = 0.1 as a function of the
continuum coupling parameter γL = γR = γ, from 12 isolated resonances (one outside the energy band) through overlap and superradiance to ten resonances corresponding to trapped states. As earlier, ∆ = 2.5, v = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Figure 7.14: The same as Fig. 7.13, with Îť = 2. . . . . . . . . . . . . . . . . . . . 130
Figure 7.15: The same as Fig. 7.13, with Îť = 5. . . . . . . . . . . . . . . . . . . . 131
Figure 7.16: Transmission through the chain for N = 5, λ = 0.1, ∆ = 0.5 as
function of continuum coupling Îł . . . . . . . . . . . . . . . . . . . . 132
Figure 7.17: Transmission through the chain for N = 20, λ = 0.1, ∆ = 2.5 as
function of continuum coupling Îł . . . . . . . . . . . . . . . . . . . . 132

xvii

Figure 7.18: Schematic of the two-channel wire. . . . . . . . . . . . . . . . . . . . 136
Figure 7.19: Energy levels of a two-band wire as a function of the coupling between
the ground and excited states of each cell, the parameters of the wire
are: N = 5, = 0, ∆ = 5, v1 = 1 and v2 = 2. . . . . . . . . . . . . . . 139
Figure 7.20: Complex eigenenergies for different values of the coupling strength to
the continuum, γ. The parameters of the wire are: N = 5, = 0, ∆ =
5, Îť = 3, v1 = 1 and v2 = 2. . . . . . . . . . . . . . . . . . . . . . . . 142
Figure 7.21: Evolution of the energies in the complex plane. Up to the critical
point all the decay widths increase as Îł increases. A superradiant
transition occurs as Îł exceeds the critical value. After passing the
critical point, the decay width of the two superradiant states increases
with the increase of Îł, whereas the decay width of the other eight
states decreases. The parameters of the wire are similar to the one in
Fig . 7.20. The arrows show the direction of the evolution as Îł varies
from 0 to 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Figure 7.22: Transmission through the two-channel wire for different values of Îł.
The parameters of the wire are similar to those in Fig . 7.20. . . . . 144
Figure 7.23: Complex energies in the case of energy-dependent couplings for different values of the coupling strength, Îą. The cutoff threshold is shown
with a vertical line, Ec =0. The wire parameters are the same as in
Fig . 7.20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Figure 7.24: Transmission through the two-channel wire with energy-dependent
couplings for different values of the coupling strength, Îą. The cutoff
threshold located between the two bands is shown with a red vertical
line, Ec =0. The wire parameters are the same as in Fig. 7.20. . . . 147
Figure 8.1:

Graphical presentation of qubit rotation on the Bloch sphere. Gate
operations can be performed by varying the value of λ.√ The initial
state at Îť = 0 is localized in the left side, |L0 . At Îť = Îş the state
is 90◦ rotated becoming an equal superposition of the left and right
states. When λ → ∞, the state becomes localized on the right. . . . 157

Figure 8.2:

(a) Upper half of the band structure as a function of Îť for a system
with N = 10, 0 = 0, t = 1, δ = 2.5 and κ = 4. The black curves
correspond to states localized in the central qubits and the red curves
are pairs of Bloch waves. (b) A closer view of pair I as a function of Îť.158

xviii

Figure 8.3:

Evolution of the squared components of the (a) upper state and (b)
lower state wave functions in pair one indicated in Fig. 8.2. Only
the components in the chain are considered in the figure: amplitudes
a(n) in the wave function (8.5). . . . . . . . . . . . . . . . . . . . . 159

Figure 8.4:

Evolution of the squared components of the upper state wave function
in pair three indicated in Fig. 8.2. Only the components inside the
chain are considered in this figure. . . . . . . . . . . . . . . . . . . . 160

Figure 8.5:

Evolution of complex eigenenergies as a function of coupling to the
continuum for (a) states within the Bloch band and (b) states outside
the band. The arrows indicate the direction in which Îł evolves from
0.05 to 20 with 0.01 increments. The parameter values are δ = 2.5,
Îť = 0.01 and Îş = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure 8.6:

Evolution of the complex eigenenergies as a function of coupling to
the continuum for (a) states within the Bloch band and (b) states
outside the band. Îł varies from 0.05 to 20 with 0.01 increments. The
parameter values are δ = 2.5, Ν = 2 and κ = 4. . . . . . . . . . . . . 164

Figure 8.7:

Lifetime for different values of coupling to the continuum Îł for a
particle initialized in (a) the upper state of pair I and (b) the upper
state of pair V (superradiant pair) in Fig. 8.6. The parameter values
are δ = 2.5, Ν = 2 and κ = 4. . . . . . . . . . . . . . . . . . . . . . . 166

Figure 8.8:

Upper half of the band structure as a function of Îť for the open
system with δ = 2.5, κ = 4 and γ = 3. The black curves correspond
to states localized in the central qubits and the red curves are pairs
of Bloch waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Figure 8.9:

Evolution of the squared components of the wave functions for (a)
upper and (b) lower superradiant states. The parameters are δ = 2.5,
Îş = 4 and Îť = 0.01. Only the components inside the chain are
considered in the figure. . . . . . . . . . . . . . . . . . . . . . . . . . 168

Figure 8.10: Evolution of the squared components of the wave functions for (a)
upper and (b) lower superradiant states. The parameters are δ =
2.5, Îş = 4 and Îť = 2. Only the components inside the chain are
considered in the figure. . . . . . . . . . . . . . . . . . . . . . . . . . 169
Figure 8.11: Coherence time as defined in (8.31) for systems with different numbers
of cells when αφ = 10−3 . The initial state is the upper state√of the
highest pair inside the Bloch band. Other parameters are Îť = Îş = 2
and δ = 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

xix

Figure 8.12: Coherence time of systems with different numbers of cells when αφ =
10−2 . The initial state is the upper state of the highest pair inside
the Bloch band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Figure 8.13: Coherence time of systems with different numbers of cells when αφ =
10−1 . The initial state is the upper state of the highest pair inside
the Bloch band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Figure 9.1:

Eigenfrequencies for a metallic silver sphere with free space background as a function of radius for various values (a) real part and
(b) imaginary part. The left and right axes represent values in units
of eV and THz, respectively. . . . . . . . . . . . . . . . . . . . . . . 189

Figure 9.2:

Electric field patterns of a silver sphere with a radius of 40 nm for (a)
dipole mode, = 1, (b) quadrupole mode, = 2, and (c) octupole
mode, = 3. The values are normalized to the maximum of the
electric field. The projection m is equal to zero for all three plots. . 190

Figure 9.3:

Electric field pattern for (a) r/a = 1, (b) r/a = 3, (c) r/a = 5, (d)
r/a = 10, (e) r/a = 20, and (f) r/a = 30. ρ is the radial distance and
z is the direction along the dipole. The values are normalized with
respect to the maximum of the electric field. . . . . . . . . . . . . . 192

Figure 9.4:

Real and imaginary components of the eigenfrequencies of a system
of two coupled identical silver spheres with (a) vertical and (b) horizontal dipole orientations. The radii of the spheres are 10 nm. d
is the center-to-center separation and a is the radius of the spheres.
The black dotted lines represent the unperturbed eigenfrequency of
a single sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Figure 9.5:

Real and imaginary components of the eigenfrequencies of a system of
two coupled identical silver spheres with horizontal dipole orientation.
The radii of the spheres are 40 nm. d is the center-to-center separation
and a is the radius of the spheres. The black dotted lines represent
the unperturbed eigenfrequency of a single sphere. . . . . . . . . . . 198

Figure 9.6:

Real and imaginary components of the eigenfrequencies of a system
of two coupled identical spheres with horizontal dipole orientation.
The material properties of the sphere are: ∞ = 1, ωp = 10.918 eV
and Îłs = 0. The radii of the spheres are 20 nm. d is the center to
center separation and a is the radii of the spheres. The black dotted
lines represent the unperturbed eigenfrequency of a single sphere. . . 199

xx

Figure 9.7:

Schematics of a plasmonic waveguide; a one-dimensional chain of five
silver spheres with nearest neighbor coupling Îş. The two edges are
symmetrically coupled to continuum, the excitation source with frequency ωe , with coupling coefficient γe . . . . . . . . . . . . . . . . . 200

Figure 9.8:

Complex-plane trajetories of the effective Hamiltonian for a onedimensional chain of five identical silver spheres with radii of (a) 10
nm and (b) 40 nm. The spheres are in contact with one another and
the dipoles are oriented along the waveguide. The arrows show the
direction of the evolution as Îłe changes from 0.01 eV to 10 eV. . . . 202

Figure 9.9:

Transmission through a one-dimensional chain of five identical spheres
with radii of 10 nm as a function of excitation source frequency ωe
for (a) Îłe =0.03 eV, (b) Îłe =0.55 eV, and (c) Îłe =10 eV. . . . . . . 204

Figure 9.10: Transmission through a one-dimensional chain of five identical spheres
with radii of 40 nm as a function of excitation source frequency ωe
for (a) Îłe =0.05 eV, (b) Îłe =0.7 eV, and (c) Îłe =10 eV. . . . . . . . 205

xxi

Part 1
Introduction

1

Theory, experiment, and computation are the three paradigms for scientific discoveries.
This dissertation includes work in all three areas. In the first part of this thesis, a problem
with mostly experimental and computational nature is considered. Part one is dedicated
to the design and development of a novel microwave imaging system. The applications of
microwave imaging range from breast cancer detection to non-destructive testing of dielectric
materials. Medical imaging is one field in which microwaves have gained significant attention,
particularly due to the ever increasing number of breast cancer diagnoses around the world.
Microwave imaging techniques are known to be a promising candidate for early breast cancer
detection due to the high contrast in dielectric properties of malignant and healthy breast
tissues at microwave frequencies. Another application in which microwave imaging plays
an important role is in the non-destructive evaluation of composite materials and laminate
structures. Such materials are gradually replacing traditional metallic counterparts due
to their versatility and superior mechanical properties such as light weight and superior
strength. With their increasing use in structures, comes the necessity of a robust method for
inspection. Microwaves have proven to be a successful candidate. In light of this, the first
part of the thesis presents a novel microwave imaging system suitable for medical applications
and the inspection and detection of anomalies in dielectric materials. The design, simulation
and fabrication process of the components of the system are described in detail in this part.
The second part of the thesis includes the theoretical consideration of various quantum
systems. With recent advances in nanotechnology, quantum mechanics has come more closer
than ever to problems with real-world applications. One ultimate driving force in this area is
evidently the quest for the realization of quantum computers. Quantum computers, are believed to have a significant impact on our lives. Their computing power is expected to greatly
exceed that of classical computers, network security will vastly improve and simulation of
2

physical systems with much larger scales would become possible. Therefore there has been
a significant effort over the last two decades to design, simulate and fabricate components
of quantum computers and develop theories for better describing such devices. In light of
this developments, the second part of the thesis is dedicated to a quantum theory that is
convenient for studying such devices. The effective non-Hermitian Hamiltonian method is
formally exact but very flexible and can be adjusted to many specific situations. In part
two, this framework is utilized to consider nanodevices from three different fields: quantum
transport, solid state quantum computing and plasmonic nanoantennas and waveguides. The
derivation of the theory and its applications to the three problems are discussed in detail in
the second part .

3

Part 2
Microwave Imaging Using an
Electronically Tunable Reflectarray
Antenna

4

Light brings us the news of the Universe.

William Henry Bragg

Chapter 1
Introduction

1.1

A Brief History of Imaging

Human curiosity and want for a greater understanding of the universe have led to the invention of a great many imaging instruments used for observation of the unknown. Optical
devices were the first type of instrument used by humans in order to enhance observation.
Even though it is believed that the oldest optical lens, a crystal called the Nimrud lens, may
have been used as a magnifying glass about 2700 years ago, it wasn’t until the 10th century
that we studied lenses in a scientific context. Ibn Sahl (c. 940-1000), a Persian mathematician, physicist and engineer, was one of the early leaders in the field of optics. His theoretical
and experimental work, such as the discovery of the law of refraction, now commonly known
as “Snell’s law”, formed the basis for the study of optical devices for scientists to come. An
important breakthrough in the optical industry was made by the famous Dutch spectacle
makers, Zacharias Jansen and his father Hans. Their experiment of placing several lenses
in a tube resulted in one of the great discoveries of the 1590’s, the compound microscope.

5

In parallel, instruments were being developed to better observe the night sky. The first
telescope was built in 1608 by Hans Lippershey, a German-Dutch glass maker. Over many
years, both microscopes and telescopes significantly improved. Electron microscopes being
used today have achieved resolutions of up to 50 pm and magnifications of up to 10 million.
Likewise, telescopes have greatly increased their capacity and have made imaging of distant
galaxies possible. Recently, a galaxy 13.1 billion light-years from earth was imaged by the
Hubble telescope.
The discovery of X-rays in 1895, by German physicist Wilhelm R¨ontgen, began a new era
in imaging technology for non-destructive evaluation (NDE) applications. Physicians quickly
realized the power of X-ray imaging in medical applications. Radiographs, images produced
on a sensitive film utilizing X-rays, provided means of skeletal evaluation of patients. Later, a
new imaging technique exploiting X-rays, called computed tomography (CT) was developed.
CT is capable of producing detailed cross-sectional (tomographic) images of the human body.
Yet another advancement of significant impact was the development of magnetic resonance
imaging (MRI). Because of the extraordinary contributions to the field of medical imaging
by both CT and MRI, each received the Nobel prize in Medicine and Physiology, in 1979
and 2003 respectively. It is important to mention that the application of X-ray is not limited
to medical imaging; X-ray imaging is widely used in the non-destructive testing (NDT) of
various industrial components and structures such as circuit boards, concrete, metal parts,
et cetera.
Despite the rapid speed in the development and wide-spread use of X-ray imaging after
its discovery, the long-term risks of repeated exposure to radiation only began to be studied
in the late 1940’s [1]. X-ray machines were popular in many industries and were occasionally
used as a gimmick to attract customers. Many American shoe stores provided X-ray machines
6

for their customers, enabling them to see the skeletal structure of their feet. This practice
was determined to be extremely risky in the 1950’s and began to be phased out. The danger
of prolonged radiation exposure is now well understood. Due to the harmful nature of
X-ray ionizing radiation, many scientists started to search for a less hazardous alternative
imaging technique. English physicist and physician John Wild, commonly known as the
father of medical ultrasound, made an important breakthrough in 1949 when he utilized
ultrasonic energy to assess the thickness of bowel tissue. As in the case of X-rays, ultrasound
imaging quickly became widely popular. Today, physicians utilize ultrasound technology to
visualize various tissues, listen and look at fetuses in utero, and to view heart function
(echocardiogram). Ultrasound imaging is also being vastly used for NDE purposes such
as structure health monitoring, material characterization and flaw detection. Despite its
popularity, ultrasonic technology has one main disadvantage, the requirement of a couplant
in order to effectively couple sound waves to the desired medium. In addition, porous
materials do not allow for an efficient propagation of acoustic signals.
Next we consider an imaging technique utilizing electromagnetic microwaves, which has
the benefit of being harmless while also outperforming X-ray and ultrasonic approaches in
certain applications.

1.2

Microwave and Millimeter Wave Imaging

Generally, Microwaves are defined as electromagnetic waves with wavelengths ranging from
one millimeter to one meter, corresponding to frequencies from 300 GHz to 300 MHz, respectively. A more precise definition was proposed by IEEE; according to IEEE standard
521, microwaves cover a frequency range from 1 GHz to 110 GHz and millimeter waves

7

cover the range of 110 GHz to 300 GHz. Nevertheless, compared to other frequency bands,
microwaves have properties such as propagation in air with very low attenuation. In general, atmospheric attenuation of electromagnetic waves above 1 GHz increases rapidly with
increase in frequency with isolated resonances due to the presence of oxygen and water
molecules in air [2, 3]. This makes microwaves suitable for long distance applications such
as communications and the detection of flying objects. Similarly, in dielectric materials microwaves have a larger penetration depth compared to higher frequencies, such as terahertz,
and provide better resolution compared to lower frequency bands. It is important to mention
that the microwave frequency band corresponds to photons with energies ranging from 1.24
ÂľeV at 300 Mhz to 1.24 meV at 300 GHz, which when compared to X-ray radiation, makes
them virtually harmless and safe for a greater number of applications.
The properties mentioned above make microwaves an invaluable tool for imaging, with
many applications varying from breast cancer detection [4] to machine vision and robotics [5].
Medical imaging is one field in which microwave imaging has gained significant attention,
particularly due to the ever increasing number of breast cancer diagnoses around the world.
In 2015, in the United States alone, it was estimated that 292,130 women were diagnosed
with invasive or carcinoma in situ types of breast cancer [6]. Thus, there is an urgent need
to develop a more reliable and effective detection system. Microwave imaging is known to be
a promising candidate for early breast cancer detection [7] due to its ability to differentiate
with high contrast, the dielectric properties of malignant and healthy breast tissue. Normal
breast tissue has a relative dielectric constant of about 4.5 at 10 GHz, whereas the dielectric
constant of cancerous tissue is approximately 10 times more [8]. For X-rays however, the
contrast between tumor and normal tissue is low. A maximum ratio of 1.7 between the
attenuation coefficients of breast tumors and healthy tissues occurs at photon energies of 20
8

keV [9, 10]. Similarly, ultrasound suffers from poor contrast ratio. The maximum ratio of
attenuation coefficients between cancerous and normal breast tissues is only about 2.5 [11].
This allows microwave imaging to potentially surpass X-ray mammography and ultrasonic
instruments in breast cancer detection. Another area in which microwave imaging has been
widely applied is in the NDT of low loss dielectrics and composite materials, where traditional
eddy current based approaches fail. This is because Eddy currents can be induced only in
materials with high conductivity and therefore are only useful in the inspection of metallic
materials. Moreover, unlike Eddy current testing, which can only be used for surface and
near surface flaw detection, microwaves are capable of propagating deep into dielectrics.
Therefore the detection of anomalies deep inside materials is possible with microwaves. A
detailed overview and discussion of the advances in microwave NDT can be found in [12].
Next, some of the drawbacks of traditional microwave imaging systems are discussed
along with a proposal for an alternative setup.

1.3

A New Design: Mirror-Based Microwave Imaging

Traditional microwave imaging using tomographic reconstruction methods, involves the illumination of a region of interest (ROI) at different angles from 0◦ to 360◦ and collection of
scattered field data around the test object. These projections are then used to reconstruct
the ROI by means of an appropriate inverse problem method such as full-wave inverse scattering and diffraction tomography based techniques. However, the conventional microwave
setup has a number of drawbacks in practical implementation. A traditional system needs
an array of costly transceivers to be placed all around the object [13–18]. This requires
complicated circuitry to accommodate transceiver switching for projection data acquisition.

9

The conventional system also requires antenna compensation algorithms and an increased of
number of antennas in order to obtain higher resolution.
A new prototype for an active microwave imaging system is presented in this thesis
that greatly reduces design and operational complexities compared to traditional systems.
This work was inspired by a number of papers from Arunachalam et al. [19–21], where a
system employing a continuous deformable reflective surface, or mirror, was investigated in
a theoretical framework. Fig. 1.1 shows the schematic of the mirror-based imaging system.
The setup uses a single source instead of a transceiver array of source elements, which
greatly reduces the overall cost. The source is coupled to a continuously conformable mirror
that is used to generate meaningful projections by steering the incident beam into various
angles. The scattered field is then measured by a receiver antenna array and used for image
reconstruction. This design eliminates the need for complicated antenna switching circuitry.
Moreover, the test object need not be enclosed in a gantry to generate a 360◦ projection
data set and hence there is no constraint on object dimensions. Therefore, imaging of larger
dielectric materials such as laminate structures and airplane wings would be possible. The
proposed system is highly scalable and also has potential for various medical applications
including breast cancer detection.
Although the theoretical studies demonstrated the feasibility of the concept, there were
two main challenges that must be overcome for practical implementation. The first challenge
is the realization of the deformable mirror. The second is image reconstruction from limited
view projections. Since projections are not taken from all angles, reconstruction becomes a
more challenging task. These key issues are briefly addressed here and again in more detail
in the following chapters of the thesis.

10

Figure 1.1: Schematic of the proposed microwave imaging system [19].

1.3.1

Deformable Mirror

In the theoretical study by Arunachalam et al. [21] several options were suggested for realizing the active mirror. Some of the designs include the use of segmented, continuous
thin-plate, monolithic and membrane mirrors. The advantages and disadvantages of each
of the mirror designs were discussed in [21]. It is important to ensure that the mirror is
capable of steering the incident beam into various angles in order to generate information
rich, unique measurements. A thin plate conformable mirror was chosen in [21] and optimal
mirror shapes for beam steering were studied.
In reality however, using a thin metal plate as the mirror proved to be extremely challenging. In this case, mechanical actuators with translational and rotational degrees of freedom
are required in order to alter the mirror shape. Also, the mirror has to be under constant
tension in order to maintain surface flatness. This makes precise shaping (deformation) of

11

the thin metal plate difficult. In addition, changing the configuration of the mirror in order
to steer the beam into different angles using mechanical motion would be slow. For these
reasons we have adopted a fully electronically controllable mirror; a reflectarray antenna.
A reflectarray antenna is an array of radiating elements illuminated by an external source.
The elements of the array possess distinct radiation characteristics. The properties of the
individual cells are determined in such a way that the field reflected from the surface of
the array has a desired radiation pattern. The properties of the elements of the array
can be dynamically manipulated by electric means. This yields the so-called reconfigurable
reflectarray antenna, where one can exploit features such as beam steering.
Tunable reflectarrays are excellent candidates for implementing the proposed mirror. The
mirror utilized in our system is a dual-band tunbale reflectarray antenna with a high tuning
capability. The array provides a broad steering range of ±60◦ . The beam is steered across
this range and the scattered field is recorded to be used for image reconstruction. The electric
mirror not only eliminates all mechanical parts, it also significantly reduces the time needed
to switch between steering angles. The design and fabrication of the array is discussed in
Chapter 2.

1.3.2

Image Reconstruction

Due to the interaction of the electromagnetic field with the test object, the scattered filed
contains information about the object. Therefore the projection data collected by the receiver
array is used to solve the inverse scattering problem i.e. reconstructing the image of the
object being tested. The inverse scattering problem is in principle ill-posed. In our case, the
situation becomes even more challenging because projection measurements do not provide a
full 360◦ view around the test object. With limited access to the object and decreased angular
12

coverage, advanced image reconstruction schemes are required. Several techniques have been
developed in order to handle the so-called limited angle scenario. The most popular methods
are model-based (full-wave) approaches, diffraction tomography approaches such as filtered
back-propagation and Fourier inversion algorithms, and the time reversal signal processing
technique. In this thesis, in order to realize the first prototype of the new imaging setup,
the use of the time reversal technique is chosen. This method, which relies on the time
reversal invariance property of the wave equation, allows us to detect and pinpoint any
scatterer located in the region of interest (ROI). Details of the image reconstruction process
are discussed in Chapter 3.

1.4

Organization

The design, simulation, fabrication and measurement of the reconfigurable reflectarray antenna is explained in Chapter 2. The concept of reflectarrays is introduced in the beginning
of this chapter. Various means of achieving tunability are explained. Next, a brief summary
of the theory of reflectarrays is provided. Since the most important component of any reflectarray is its unit cell, the main section of this chapter is on the design, measurement and
testing of a single element of the array. Finally, a 10x10 array is fabricated and experimental
results of the measurement of radiation patterns from the array are presented. Chapter 3 describes the time reversal image reconstruction technique in detail. The chapter begins with a
brief introduction to inverse problems and various methods of image reconstruction. A finite
difference time domain (FDTD) code is developed to solve Maxwell’s equations numerically.
The concept of time reversal is then validated using computer simulation. Chapter 4 presents
experimental results obtained using the prototype of the imaging setup. Two experiments

13

conducted in order to evaluate the detection capabilities of the system are discussed. The
capability of the system to detect anomalies within dielectric materials is also demonstrated.
Specifically two types of anomalies are considered in this chapter: defects and metallic impurities. It is shown that the system can accurately detect both anomaly types. Part 4
includes the conclusion and future directions.

14

It’s of no use whatsoever ... this is just an experiment that
proves Maestro Maxwell was right − we just have these
mysterious electromagnetic waves that we cannot see with the
naked eye. But they are there.
I do not think that the wireless waves I have discovered will
have any practical application.
Heinrich Hertz

Chapter 2
The Reflectarray Antenna

2.1

Introduction

The reflectarray antenna, first introduced in 1963 by Berry et al., can be thought of as
a reflecting surface where elements in the array determine the reflection properties of the
surface [22]. The amplitude and phase change between the incident and the reflected fields,
at any point on the surface of the reflectarray, are determined by the elements of the array.
Similar to antenna arrays, in order to achieve the desired radiation pattern, proper phase
shift has to be assigned to each element of the array, which can be done by appropriately
choosing the parameters of individual elements. An array of variable-length, open-ended
waveguides was used by Berry et al. in [22] to demonstrate the reflectarray principle.
Later, with the advent of microstrip technology and the extension to microstrip arrays,
reflectarrays became easier to build. The behavior of a microstrip array is defined by its

15

printed resonating elements. First generation microstrip reflectarrays as seen in the work of
Javor et al., comprised patch elements with variable-length transmission lines attached to
the patches where the reflection from the open end of the line determined the phase shift
of the element [23]. Rectangular patches of variable lengths are used in the paper by Pozar
and Metzler [24] since the reflection phase changes with the length of the patch. A variety of
microstrip reflectarrays have been developed using other radiating elements, such as variablelength crossed dipoles by Kelkar [25] and rotated patches by Huang and Pogorzelski [26],
that lead to circularly-polarized reflectarrays.
Research on reflectarrays continued to advance with the introduction of dual-band reflectarrays, in which each unit cell comprises two different elements, each accountable for
radiating in one of the bands. Crossed-dipoles of variable sizes printed on a single layer in
the work of Kelkar [25], patches of variable sizes on two stacked layers in the paper by Encinar [27] and double ring elements used by Huang et al. in [28] are all examples of dual-band
reflectarray antennas.
An extra dimension of versatility is added to this group of antennas by their ability
to dynamically adjust element scattering properties in order to alter the radiation pattern,
and thereby produce tunable reflectarrays. Reconfigurable elements are achieved via several
techniques. In the paper by Inam and Ismail [29] a pin diode is incorporated in the unit
cell; the ON and OFF states of the diode alter the electrical length of the element and
hence provide tunability. Beam shaping is achieved by means of RF MEMS in the work of
Bayraktar et al. [30]. Technologies using liquid crystals [31], ferroelectric materials [32] and
graphene [33] have been used as well. A recent review on the advantages and disadvantages
of different technologies can be found in the paper of Hum and Perruisseau-Carrier [34].
These techniques can be combined with dual-band unit cells to produce tunable dual-band
16

reflectarrays. In [35] by Guclu et al. and in [36] by Moghadas et al. two tunable dual-band
unit cells based on RF MEMS are proposed as reflectarray antennas, though the array has
not yet been built.
Beam steering using high impedance surfaces (HIS) is a relatively new and fascinating
area in the field of reflectarrays. A varactor tunable HIS element is presented in the paper
by Mias and Yap [37]. Despite their promising performance, the size of the periodic element
is often much smaller than the operating wavelength, therefore more elements are required
on the surface of the array. This leads to an increase in the cost and complexity of the array.
In this thesis varactor diodes are used to create tunable array elements, since they can
be easily incorporated into the unit cell and are more cost-effective, and more reliable,
than the previously mentioned techniques. Recent developments in unit cells that exploit
varactor diodes for phase adjustability have shown promising results [38], [39]. The unit
cell introduced by Hum et al. [38], which uses two varactor diodes, is capable of providing
a phase shift of up to 320◦ . Using two varactors per unit cell is not only costly, but the
biasing circuit fabricated on the reflecting surface of the antenna increases loss and lowers
the scattering amplitude. In the paper by Rodrigo et al. [39], frequency tunability is achieved
using a single varactor, but at the cost of a lower phase change of 270◦ , and the additional
need for two RF switches and corresponding complexity of fabrication.
In the following sections, a dual-band metamaterial-inspired unit cell is introduced that
uses a single varactor diode to dynamically adjust the phase of the reflected electric field in
the two frequency bands [40–42]. The unit cell is modeled and analyzed, and the results are
used to design a reflectarray antenna. Finally, a prototype of the antenna is constructed,
and the subsequent measurements are used to verify the predicted properties of the proposed
antenna.
17

2.2

The Theory of Reflectarrays

Several techniques have been used for the analysis of reflectarrays. Numerical methods,
such as finite difference time domain (FDTD) and finite element method (FEM) modeling,
described in the works of Cadoret et al. [43] and Bradi et al. [44] respectively, provide
accurate results, although they are time and computation expensive. In this thesis, similar
to the paper by Berry et al. [22], a simple yet effective theory based on the concept of
impedance surfaces is utilized.
Consider the infinitely large reflectarray shown in Fig. 2.1. For simplicity, it is assumed
that the incident field is a plane wave normal to the reflectarray and linearly polarized
along the y-axis. This is a good approximation when the reflectarray feed is far away from
the surface of the array or is capable of generating plane waves, such as with lensed horn
antennas. Under this assumption the electric field on the surface of the array is expressed as

Ei = E0 e−jkd yˆ,

(2.1)

where E0 is the amplitude of the wave, k = 2π/λ0 is the wavenumber corresponding to the
free space wavelength Îť0 , and d is the distance of the source from the array.
Exploiting the concept of surface impedance, introduced by Schelkunoff [45], the reflected
electric field is
Er = Γ(x, y)Ei ,

(2.2)

where Γ is the reflection coefficient at (x, y) on the surface of the array as given by

Γ(x, y) =

Zs (x, y) − η0
.
Zs (x, y) + Ρ0

18

(2.3)

Here Ρ0 is the free space impedance of plane waves and Zs is the surface impedance at point
(x, y). The parameter Zs projects the effects of various phenomena, such as absorption,
radiation, and scattering that occur in the region z ≤ 0, and projects them onto the surface
z = 0.

𝑦

Plane wave

𝜃

(𝑥, 𝑦)

𝜙

𝑥
𝜃

Source at z=d

𝑧

Figure 2.1: Reflectarray geometry.

Assuming the reflectarray is lossless and passive, the surface impedance is purely reactive
such that
Zs = jXs

(2.4)

An appropriate expression for Xs is derived in section 2.3.2, where the equivalent circuit of
the unit cell is introduced. For now, it is sufficient to assume that Xs ≥ 0; Xs = 0 occurs in
the case of a perfectly reflecting surface (short circuit).
Using (2.3) and (2.4), the reflected electric field on a plane infinitesimally close to the
surface of the array is
Er =

jXs (x, y) − η0
E0 e−jkd yˆ.
jXs (x, y) + Ρ0

Two limiting cases are of practical interest. First, when Xs

(2.5)
Ρ0 a surface with properties

similar to a perfect electrical conductor (PEC) is produced, and the surface reflects out of

19

phase (Γ ≈ ejπ ). Second, when Xs

Ρ0 a high impedance surface is produced. This is also

called an artificial magnetic conductor since the surface acts similar to a perfect magnetic
conductor (PMC), reflecting the wave back in phase (Γ ≈ ej2π ).
In general, (2.5) can be written as

Er = E0

X (x,y)
−j{(2 tan( sη
)+kd}
0
e
yˆ,

(2.6)

and thus the phase of the reflected field is

Φ(x, y) = −2 tan

Xs (x, y)
Ρ0

− kd.

(2.7)

The first term in (2.7), the so-called “reflection phase”, is the phase difference between the
incident and reflected fields at the surface of the array, and is solely due to the presence of
the reflectarray. The second term is due to the propagation of the wave from the source to
the reflecting surface. With other kinds of incident waves, such as spherical waves, this term
requires a different expression, which is of course a function of position on the reflectarray.
It is obvious from (2.7) that the properties of the surface, and hence Xs , can be changed
in order to assign different phase shifts to different points on the array. As a result, one
can form a desired radiation pattern by properly selecting the surface impedance at each
point on the array. For instance, in order to obtain the maximum of the scattered field in
the direction (θ,φ) shown in Fig. 2.1, the reflected waves should constructively interfere at
a point far away from the surface along those angles. This criterion leads to

Φ(x, y) + k sin θ(x sin φ + y cos φ) − Φ(0, 0) = 2nπ,

20

(2.8)

where n is an integer number. Using (2.7), the phase criterion across the array can be written
in terms of Xs as

−2 tan

Xs (x, y)
Ρ0

Xs (0, 0)
Ρ0

− tan

+ k sin θ = 2nπ.

(2.9)

In microstrip reflectarrays, printed radiating elements are used to manipulate the surface
impedance and hence generate the proper phase shift across the array needed to produce the
desired beam shape. It is not possible to practically measure the reflection phase at each
point on the surface of the reflectarray, so the average phase over a unit cell is used as a
design parameter. The design of a single unit of the array is explained in the next section.

2.3

Analysis of The Unit Cell

Reflection phase, reflection efficiency, and the bandwidth of the element are the most important features of a unit cell used in a reflectarray antenna [46]. In this thesis, the focus is
mainly on the reflection phase properties of the unit cell. However, the reflection efficiency
and the bandwidth will be briefly addressed.
In order to achieve a dual-band, phase-adjustable element with a wide reflection phase
range and a high reflection efficiency, an appropriate geometry must be considered. The
proposed unit cell is shown in Fig. 2.2. The geometry of the unit cell was inspired by a
double square loop frequency selective surface (FSS), which perfectly blocks the incident
electromagnetic wave at its two resonance frequencies [49]. In this work, the inner loop is
replaced by a square patch since the square patch provides wider bandwidth. This improves
the scattering properties of the unit cell. Hence, the proposed unit cell consists of a square

21

patch centered in a square ring along with a varactor diode placed across the gap between the
ring and the square patch to provide phase adjustability. A tunable dual-band unit cell based
on square loops is studied in [50] in the context of electromagnetic band-gap structures. The
unit cell has the advantage of tuning each band independently. However, using six capacitors
and inductors per unit cell increases the complexity of the structure.

𝓁2

𝑔

𝓁1

Figure 2.2: Geometry of the unit cell.

The unit cell is analyzed in two ways – with the aid of a commercial full wave solver
(ANSYS HFSS), and by using an equivalent circuit model. In both cases it is assumed that
the unit cell is printed on a square substrate of side p (the periodicity of the unit cell) with
a conducting ground on the reverse side. Although it is assumed here that the incident wave
is a plane wave normal to the unit cell, so as to simplify the derivation of the equivalent
circuit parameters, in principle the problem can be solved for TE or TM fields at arbitrary
incidence.

22

2.3.1

Full Wave Simulation

The commercial full wave simulator ANSYS HFSS was used to analyze the scattering properties of the unit cell. Since waveguides are convenient for measuring the reflection and
transmission coefficients, the unit cell was located inside a fictitious square waveguide and
periodic boundary conditions used to model the array by infinitely replicating the unit cell.
As a result, mutual interactions between the neighboring unit cells are taken into account.
The unit cell was designed to operate within S band (2.6-3.95 GHz) and F band (4.97.05 GHz). The geometry parameters were optimized in order to maximize the reflection
coefficient and the phase change over the two bands. The periodicity of the unit cell, and
thus the dimension of the square waveguide, was chosen to be 22 mm. The other parameters
are 2 = 18 mm, 1 = 12 mm and g = 1 mm. The substrate is 1.575 mm thick with a metal
plane on the back, and the dielectric constant is set to unity. This last assumption is made
in order to simplify the equations derived in the next section.
A varactor diode with a dynamic range of 0.1-0.6 pF is placed across the gap to achieve
tunability. Two key points have to be considered in order to properly incorporate the varactor
diode into the unit cell. First, it is crucial that the diode offers an appropriate range of
capacitances, otherwise it will dominate or be dominated by the intrinsic parameters of
the unit cell. Second, the diode should be connected in a way that the polarity of the
diode is aligned with the polarization of the incident electric field. The presence of the
diode breaks the symmetry of the structure. Therefore, unlike conventional double-squareloop elements, the unit cell no longer provides dual-polarized operations. However, the
symmetrical structure of the unit cell does suppress cross polarization effects [38]. These
two points will be visited in the investigation of the equivalent circuit model.

23

Fig. 2.3 shows the reflection phase (2.7) as a function of frequency for different values of
the varactor capacitance. The reflection phase has a range of almost 360◦ , indicating high
tuning ability. Moreover, two resonant frequencies can be observed. The higher resonant
frequency is mainly due to the square patch, and the lower resonant frequency is largely
due to the ring. This indicates that the array has the potential to steer the beam within
two distinct frequency bands. Furthermore, the unit cell has the advantage of having stable

Reflection Phase (degree)

performance over a wide range of incident angles 0-45◦ [53].

200

𝑪𝒗 = 𝟎. 𝟏

0

𝑪𝒗 =
𝟎. 𝟐
𝑪𝒗 = 𝟎. 𝟑

-200

𝑪𝒗 = 𝟎. 𝟒

2

4
6
Frequency (GHz)

8

Figure 2.3: Reflection phase for different varactor capacitance values, Cv (pF).

Plots of the reflection phase as a function of the capacitance for both frequency bands
are shown in Fig. 2.4. It is clear that the phase reaches its maximum and minimum at
the low and high values of the varactor capacitance (Cv ) respectively. This suggests that
the dynamic capacitance range of the diode has to be carefully chosen in order to maximize
the phase shift produced by the unit cell. Notice that the parameters of the unit cell are
designed in such a way that a single diode is capable of changing the reflection phase in the
two bands simultaneously.

24

Reflection Phase (degree)

200

𝟖. 𝟎 𝑮𝑯𝒛

0

𝒇 = 𝟕. 𝟎 𝑮𝑯𝒛

𝟖. 𝟓 𝑮𝑯𝒛

𝟕. 𝟓 𝑮𝑯𝒛

𝒇 = 𝟗. 𝟎 𝑮𝑯𝒛

-200
0.1

0.2

0.3
0.4
Capacitance (pF)

0.5

0.6

Reflection Phase (degree)

(a)

200

𝒇 = 𝟑. 𝟕𝟓 𝑮𝑯𝒛

𝟑. 𝟖𝟕 𝑮𝑯𝒛

0
𝒇 = 𝟒. 𝟐𝟓 𝑮𝑯𝒛

-200
0.1

0.2

0.3
0.4
Capacitance (pF)

0.5

(b)

Figure 2.4: Reflection phase at different frequencies in (a) the higher frequency band and
(b) the lower frequency band.
A good way to envision the role of the diode in changing the reflection phase is the 2D
plot of the reflection coefficient on a surface near the unit cell [47]. The magnitude and
phase of the reflection coefficient of the electric field on a plane located 0.1 mm above the
surface of the unit cell is shown in Fig. 2.5. Note that the reflection phase given in Fig.
2.3 and Fig. 2.4 can be interpreted as the phase of the average reflection coefficient at each

25

point on this surface. To obtain Γ on the surface, two simulations are required, one with the
unit cell inside the waveguide in order to calculate the total electric field, and one without
the unit cell in order to calculate the incident field. The scattered field is calculated in the
post processing stage by subtracting the incident field from the total field, and finally Γ is
calculated with the aid of (2.2). The incoming wave is polarized along the y-axis (parallel to
the diode polarity), and hence the magnitude and phase of the x and z components of the
electric field are not considered. The magnitude and phase of the reflection coefficient of the
y component of the electric field at 8.0 GHz with Cv = 0.2 pF are shown in Fig. 2.5(a). As
expected, the main contribution to the scattered field is from the two radiating edges of the
square patch. Consequently, the reflection phase is greatly influenced by the phase at the
radiating edges, and is approximately equal to it. In Fig. 2.5(a) this phase is equal to 62◦ ,
which matches with the corresponding curve in Fig. 2.4(a). Fig. 2.5(b) shows the magnitude
and phase of the reflection coefficient of the y component electric field at 4.25 GHz with
Cv = 0.2 pF. It is obvious that the main contribution to the scattered field is from the two
edges of the ring, so that again the phase of the two edges of the ring defines the final phase
of the reflection coefficient. This phase is equal to 75◦ and matches with Fig. 2.4(b). Note
that for both frequencies the phase of the reflection coefficient at the center of the unit cell,
is equal to 180◦ . This is because the edge effects are minimal at the center, thus the center
behaves similar to a PEC.

2.3.2

Equivalent Circuit Model

One of the objectives of studying an FSS is the evaluation and optimization of the transmission and reflection coefficients of the incident wave with the help of an appropriate model [48].
Modeling an FSS using equivalent circuits is a well-known technique, which not only provides
26

Magnitude dB

22

20

11

10

0

0

11
22
x (mm)

y (mm)

y (mm)

22

Phase

degree

170

11
0

120

11
22
x (mm)

70

(a)

22
22
22

Magnitude dB

200
150

11
11
11

10
10

000

00

degree

000

200
230

20

150
180

11
11
11

100

11
22
11
22
x
(mm)
x
(mm)
x (mm)

Phase

yy (mm)
(mm)
(mm)

20
20

(mm)
yy (mm)
(mm)

22
22
22

10
130

100

11
22
11
22
11
22
x
(mm)
x
(mm)
x (mm)

080

(b)

Figure 2.5: Magnitude and phase of the reflection coefficient of the y component of the
electric field on a plane 0.1 mm above the surface of the unit cell at (a) 8.0 GHz and
Cv =0.2 pF and (b) 4.25 GHz and Cv =0.2 pF. The layout of the unit cell is shown with
white lines.
fast and accurate results but also reveals the physics of the problem. The first attempt in
modeling an FSS with equivalent circuits can be found in the work of Marcuvitz [51] where
two simple cases of a field incident on an infinite array of metallic strips is considered. If
the strips are parallel to the electric field, the surface can be modeled with a single inductor;
if the strips are parallel to the magnetic field, the surface can be modeled with a single
capacitor. Equivalent circuits of more complicated geometries such as square loops [52],
gridded-square elements [53], double-square loops [53]- [54] and Jerusalem crosses [55] can

27

be found by combining these two type of strips.
Fig. 2.6(a) shows the double loop geometry and its corresponding equivalent circuit [54].
The model consists of two shunt LC circuits representing the outer and inner loops. It is
easy to see that the limiting case of the double loop, described by w1 = 1 /2, lends itself
well to the proposed unit cell, Fig. 2.6(b). One can modify the equivalent circuit of a double
loop structure to achieve a model for the proposed unit cell as shown in the right panel of
Fig. 2.6(b). For a TEM incident wave, the normalized values of the inductances and the
capacitances of the equivalent circuit shown in Fig. 2.6(a) are given by [54]
ωLw1
Xw 1
=
= X1 1 ,
Ρ0
Ρ0
p
Xw 2
ωLw2
X2 X3
=
=2
Ρ0
Ρ0
X2 + X 3
B1 B2
Bg1 η0 = ωCg1 η0 =
B1 + B2
3
Bg2 η0 = ωCg2 η0 = B2 2 .
4
p

2

p
1

p

,
(2.10)
,

Here X1 , X2 , X3 , B1 and B2 in (2.10) are given by

X1 = F (p, 2w1 , Îť),
X2 = F (p, w2 , Îť),
X3 = F (p, w1 , Îť),
B1 = 4F (p, g1 , Îť),
B2 = 4F (p, g2 , Îť),

28

(2.11)

where F (p, x, Îť) has a general form of

F (p, x, Îť) =

πx
2p

p
ln csc
Îť

+ G(p, x, Îť) .

(2.12)

Here G(p, x, Îť) is a correction factor given by

G(p, x, Îť) =
1
(1 − β 2 )2 [2C(1 − β 2 /4) + 4C 2 β 2 ]
2 (1 − β 2 /4) + 2Cβ 2 (1 + β 2 /2 − β 4 /8) + 2C 2 β 6

(2.13)

where
β = sin

πx
2p

(2.14)

and
C=

1
1 − (p/λ)2

−1

(2.15)

The model accurately predicts the behavior of a double loop as long as w1 , w2 , g1 , and g2
are all much less than p, and p < Îť.
The values of the inductances and capacitances of the unit cell equivalent circuit can be
found using (2.10)-(2.15). In addition, the effect of the substrate, the metal plate on the
back, and the varactor diode should be taken into account. The substrate is treated as a
transmission line with a length equal to the thickness of the substrate, while the ground
plate provides a short circuit at the end of the line. Hence the impedance seen at the surface
due to the substrate is
Z1 = jΡ0 tan(βd),

(2.16)

√
where η0 is the free space impedance, β = 2πf µ0 0 is the phase constant at frequency f , 0

29

𝒈𝟏

𝓵𝟐 𝓵𝟏

0
𝒑

𝒘𝟏
𝒈𝟐

𝑿𝒘𝟏

𝑿𝒘𝟐

𝑩 𝒈𝟏

𝑩 𝒈𝟐

0

𝒘𝟐
(a)

𝒈𝟏

𝓵𝟐 𝓵𝟏

𝒅

0
𝑩𝒈𝟏

𝒑

𝒘𝟏 =

𝓵𝟏
𝟐

𝒈𝟐

𝑿𝒘𝟏
𝑿𝒘𝟐
𝒄𝒗
𝒄𝒗
𝟐 𝑩 𝒈𝟐 𝟐

0

𝒘𝟐
(b)

Figure 2.6: (a) Geometry of a double square loop and the equivalent circuit of an infinite
array of double square loops. (b) Geometry of the proposed unit cell derived from a double
square loop with w1 = 1 /2 and the corresponding equivalent circuit.
and Âľ0 are the free space permittivity and permeability respectively, and d is the thickness
of the substrate. The effect of the varactor diode is taken into account by assuming that
the diode almost equally shifts the two resonant frequencies, and thus it equally affects the
outer loop and the patch. It is clear that the diode polarity should be along the incident
electric field. This configuration minimally perturb the intrinsic inductances of the unit cell.

30

The surface impedance (see eqs. 2.3 and 2.4) of the array can be calculated as

Zs = Ρ0 /Ys = Ρ0 /(Y1 + Y2 + Y3 )

(2.17)

The quantity Ys is the normalized surface admittance. Y1 , Y2 and Y3 are the normalized
admittances corresponding to the substrate (eqn. 2.16), the ring and the patch respectively,
and are given by
Y1 = 1/Z1 ,
Y2 =

j(Bg2 + BCv /2)
,
1 − Xw2 (Bg2 + BCv /2)

Y3 =

j(Bg1 + BCv /2)
,
1 − Xw1 (Bg1 + BCv /2)

(2.18)

where BCv = Ρ0 wCv is the normalized reactance of the capacitor.
Notice that since there is no lossy element in the circuit, Zs is purely reactive. Once
the surface impedance is known, the phase of the reflection coefficient can be calculated as
shown in (2.3),
φ=

ln

Zs − η0
Zs + Ρ0

(2.19)

The reflection phase predicted by the circuit is shown in Fig. 2.7. The parameters are the
same as in the previous section. A good agreement can be seen between the phase predicted
by the circuit (Fig. 2.7) and predicted by HFSS (Fig. 2.3). Note that with the dimensions
mentioned, the condition w1

p still holds. It is seen that the addition of Cv shifts the

resonance frequency of the structure.
It is worth mentioning that, in principle, the effect of dielectric substrates can be taken
√
into account in various ways. In [56]- [57] the frequency axis is scaled by a factor of 1/ r .
In [49] the effect of the dielectric properties is incorporated in the equivalent susceptances

31

Reflection Phase (degree)

200

𝑪𝒗 = 𝟎. 𝟏
𝑪𝒗 =
𝟎. 𝟐

0
𝑪𝒗 = 𝟎. 𝟑

-200

𝑪𝒗 = 𝟎. 𝟒

2

4
6
Frequency (GHz)

8

Figure 2.7: Reflection phase curves for different capacitance values, Cv (pF), predicted by
the equivalent circuit model.

given in (2.11). Although the dielectric properties have not been considered so far, in the next
section where simulation and experimental results are compared the substrate properties are
taken into account.

2.3.3

Measurements

Since the higher and lower resonance frequencies of the structure lie in the F (4.9-7.05 GHz)
and S (2.6-3.95 GHz) frequency bands respectively, two waveguides were used to experimentally validate the simulated results. Two prototypes based on the aperture sizes of the
waveguides were fabricated to completely fill the waveguide cross sections; see Fig. 2.8(a).
The unit cells with the dimensions provided in section 2.3.1, were fabricated on a 1.575 mm
thick RT/Duroid 5880 substrate with r of 2.2 and tan δ = 0.0009. The biasing circuit for
the varactor diode, shown in Fig. 2.8(b), consists of a resistor R=10 MΩ and a capacitor
C=1000 pF, and prevents the structure from coupling to the bias line, bypassing the RF
noise from the power source. Vias are used in order to preserve a high reflection efficiency.
32

This eliminates the need to use wires and circuitry on the front side of the unit cell, which
would perturb the scattered field. The appropriate placement of the vias and the biasing
components was determined by investigating the surface current distribution on the structure
using HFSS. The regions with relatively low surface currents were considered and different
combinations of the vias located in these areas were implemented and compared. The best
configuration was chosen to produce minimum impact on the performance of the resonant
structure. One via close to the center of the square patch is connected to the ground plane,
while another via is placed slightly below the bottom edge of the square ring, thus creating
enough space for mounting the resistor on the ring.

20.193 𝑚𝑚

34.036 𝑚𝑚

Varactor diode
Via to GND
Via to 𝑉𝐷𝐷

Resistor

40.386 𝑚𝑚

72.136 𝑚𝑚

(a)

R

Patch

Varactor

Ring

Vias

𝑉𝐷𝐷

C

GND
(b)

Figure 2.8: (a) Unit cells. Left: S-band. Right: F-band. (b) Side-view schematic of the
biasing circuit.

A GaAs hyperabrupt varactor diode (Aeroflex Metelics MGV100-20) with a practical
dynamic tuning range of Cv = 0.18 − 2.0 pF (corresponding to the voltage range 28 − 0.5 V)

33

is used, since the diodes have both low parasitic parameters and low power dissipation. The
series resistance of the diode is 3 Ω and the parasitic inductance is 0.4 nH.
In order to properly analyze the behavior of the unit cell inside the two waveguides, one
has to consider two key differences from the model described in the previous sections. First,
since the S-band and F-band unit cells have different periodicities, the equivalent infinite
arrays generated by imaging into the waveguide walls have different unit cell spacings. This
spacing is also different from the initial design (p in sections 2.3.1 and 2.3.2). This alters the
scattering properties, shifts the resonance frequencies, and as a result the reflection phase
curves are shifted. Second, real waveguides do not support TEM modes. The dominant
propagating mode is TE10 , which is a superposition of two plane waves bouncing between
the waveguide walls, at angle θ with respect to the main symmetry axis of the waveguide.
The angle θ for the TE10 mode is given as [58]

θ = sin−1

Îť0
2a

(2.20)

where Îť0 is the free space wavelength and 2a is the cutoff wavelength of the TE10 mode,
where a is the dimension of the longer side of the rectangular waveguide cross section. For
the S-band and F-band waveguides, at their central frequencies, θ = 39.38◦ and θ = 38.40◦ ,
respectively. These deviations from the original model considered in the previous section
have to be incorporated and the simulation model has to be altered correspondingly before
comparing the simulation results with the measurements.
Fig. 2.9(a) shows the measured reflection phase of the F-band unit cell with a varactor
diode biased under various voltages. The applied voltage varies from 0.5 V to 28 V (with
0.5 V increments) corresponding to a capacitance range of 2.0 pF to 0.18 pF. Simulations

34

of the unit cell under the same capacitance conditions as the experiment were performed,
and the phase responses are shown in Fig. 2.9(b). It is seen that good agreement of the
phase response between measurement and simulation is achieved, and the unit cell is able
to provide excellent phase shift capability over a wide frequency range. A maximum phase
range of 335◦ is achieved experimentally at 5.30 GHz, and acceptable phase tuning ability
over a wide frequency band of 5.0 GHz to 6.25 GHz can still be obtained. Note that the
reflection phases are different from the unit cell with a TEM incident wave, which are shown
in Fig. 2.3 and Fig. 2.7. This suggests that the oblique incidence together with larger
periodicity has shifted the resonance frequency.
Fig. 2.10(a) and Fig. 2.10(b) show the measured and simulated magnitude of the reflection coefficient respectively, with the same biasing configuration as in Fig. 2.9. As the
simulation predicts, the measured return loss of the unit cell decreases as the voltage increases. For an applied voltage larger than 8 V, a return loss that is less than 5 dB can be
achieved, which implies a potential high reflection from the unit cell.
Both experimental and simulated results of phase and return loss of the S-band unit cell
are provided in Fig. 2.11 and Fig. 2.12. It can be seen from Fig. 2.11 that parallel results
between simulation and experiment are achieved and a maximum phase range of 340◦ is
reached at 3.49 GHz . Note that the phase responses are for a voltage range of 5 V to 28 V
(with 1 V increments) or an equivalent capacitance range of 0.38 pF to 0.18 pF. Reflection
phase for a lower biasing voltage or higher capacitance can still be obtained. However,
discontinuous variations of the phase responses are observed, which make the unit cell under
these configurations not suitable for beam steering purposes. This issue can be explained as
follows. From (2.12), the value of different elements in the equivalent circuit of Fig. 2.6(b)
are proportional to 1/Îť, and thus the intrinsic capacitance of the unit cell is lower in the
35

Reflection Phase (degree)

200

𝑪𝒗 = 𝟎. 𝟏𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟐𝟖. 𝟎 𝐕

100
0
-100
-200

𝑪𝒗 = 𝟐𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑 = 𝟎. 𝟓 𝑽

5

5.5
6
6.5
Frequency (GHz)

7

(a)

Reflection Phase (degree)

200

𝑪𝒗 = 𝟎. 𝟏𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟐𝟖. 𝟎 𝑽

100
0

-100
-200

𝑪𝒗 = 𝟐𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑 = 𝟎. 𝟓 𝑽

5

5.5
6
6.5
Frequency (GHz)

7

(b)

Figure 2.9: Reflection phase of the F-band unit cell as a function of frequency for different
voltages (or equivalently Cv ) (a) experiment (b) simulation.
S-band. As a result, for voltages lower than 5 V, the capacitance of the varactor diode
dominates the intrinsic capacitance of the unit cell, and therefore interrupts the continuous
phase variations of the unit cell. In addition, in contrast to the wide frequency range at
F-band, the operating bandwidth for this unit cell is relatively narrow. This is due to the

36

|S11| (dB)

0
-5

𝑪𝒗 = 𝟎. 𝟏𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟐𝟖. 𝟎 𝐕

-10
𝑪𝒗 = 𝟐𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑 = 𝟎. 𝟓 𝑽

5

5.5
6
6.5
Frequency (GHz)

7

(a)

|S11| (dB)

0
𝑪𝒗 = 𝟎. 𝟏𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟐𝟖. 𝟎 𝐕

-5
-10
𝑪𝒗 = 𝟐𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑 = 𝟎. 𝟓 𝑽

5

5.5
6
6.5
Frequency (GHz)

7

(b)

Figure 2.10: Magnitude of the reflection coefficient of the F-band unit cell as a function of
frequency for different voltages (a) experiment (b) simulation.
narrow bandwidth nature of the square ring structure.
Fig. 2.12 shows the magnitude of the reflection coefficient. Reasonable agreement is
seen between simulation and measurement. Note that the S-band unit cell exhibits a lower
reflection than the F-band unit cell. This is because of the higher loss of the ring compared

37

Reflection Phase (degree)

200

𝑪𝒗 = 𝟎. 𝟏𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟐𝟖. 𝟎 𝑽

100
0
-100
-200

𝑪𝒗 = 𝟎. 𝟑𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟓. 𝟎 𝐕

2.8

3.2
3.6
Frequency (GHz)

Reflection Phase (degree)

(a)

200

𝑪𝒗 = 𝟎. 𝟏𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟐𝟖. 𝟎 𝑽

100
0
-100
-200

𝑪𝒗 = 𝟎. 𝟑𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟓. 𝟎 𝐕

2.8

3.2
3.6
Frequency (GHz)
(b)

Figure 2.11: Reflection phase of the S-band unit cell as a function of frequency for different
voltages (or equivalently Cv ) (a) experiment (b) simulation.
to that of the square patch. In addition, the larger periodicity of the S-band unit cell affects
the efficiency negatively.

38

|S11| (dB)

0
-10

𝑪𝒗 = 𝟎. 𝟏𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟐𝟖. 𝟎 𝑽

-20
-30
-40

𝑪𝒗 = 𝟎. 𝟑𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟓. 𝟎 𝐕

2.8

3.2
3.6
Frequency (GHz)
(a)

0

|S11| (dB)

-10

𝑪𝒗 = 𝟎. 𝟏𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟐𝟖. 𝟎 𝑽

-20
-30
-40

𝑪𝒗 = 𝟎. 𝟑𝟖 𝒑𝑭 𝒐𝒓 𝑽𝒂𝒑𝒑
= 𝟓. 𝟎 𝐕

2.8

3.2
3.6
Frequency (GHz)
(b)

Figure 2.12: Magnitude of the reflection coefficient of the S-band unit cell as a function of
freqeuncy for different voltages (a) experiment (b) simulation.

2.4

Fabricated Reflectarray

A 10x10 element reflectarray antenna was fabricated based on the proposed unit cell design.
See Fig. 2.13. The dimensions and periodicity of the unit cell, described in section 2.3.1,

39

correspond to a square array of 220 mm x 220 mm. The substrate and the biasing circuit
used in the full array are the same as that used in the single unit cell prototype described in
section 2.3.3. The biasing network for the varactor was fabricated on an additional substrate
and glued behind the ground plane of the array. With this prototype, instead of controlling
each unit cell individually to achieve a 3-D steerable beam, only the 2-D beam steering ability
is investigated. Therefore, each column of the reflectarray is biased with the same voltage.
Hence, ten digital voltage regulators were used to accurately bias the array elements.
A bistatic measurement was performed using an arch range at Michigan State University
[59] in order to measure the radiation pattern. TEM-horn antennas were connected to an
Agilent E5071c network analyzer to measure the transmission coefficients S21 . Dielectric
lenses were placed in front of the horn antennas to generate a focused beam with uniform
phase. At 3 GHz and 6 GHz, the diameters of the beam are approximately 43 cm and 31 cm
respectively. Note that in the lower frequency band, the beam size is larger than the array,
which may cause a reduction in the measured S21 . Moreover, diffraction at the edge of the
antenna will also bring a further reduction in the total transmitted power.
The VNA was calibrated from 1.5 GHz to 7.5 GHz with 1601 frequency points. The
calibration of the measurement system was then conducted by finding the system response.
The response of the system was calculated by considering the canonical problem of scattering
off a metal sphere. The response function was obtained by comparing the experimental
results with the analytical solution (Mie theory) of the scattering problem. The antenna
array was mounted on a sheet of Styrofoam and placed at the center of the arch range. The
arch range has a radius of 3.53 m. The center of the array was adjusted to be at the same
height as the horn antennas. The array and the transmitter were kept stationary and the
direction of the incident wave was adjusted to be perpendicular to the surface of the array.
40

Fig. 2.13 shows the experiment setup with the antenna in place. The full radiation pattern
of the antenna can be measured by moving the receiving antenna (the antenna on the right
in Fig. 2.13) along the arch rail. It is important to mention that, due to the large profile of
the dielectric lens, the receiving and transmitting antennas cannot be co-located. Therefore,
a gap in the antenna pattern appears from approximately -15◦ to 15◦ . However, this gap
does not affect the measurement at 0◦ for the co-polarization component.

(a)

(b)

Figure 2.13: (a) Close-up view of the reflectarray (b) Experimental setup for bistatic
measurement with reflectarray at the center of the arch.

41

Different voltage configurations were applied to the reflectarray in order to steer the main
beam in multiple directions. Fig. 2.14 shows four different cases were the beam is steered
onto the angles 0◦ , 30◦ , 45◦ and 60◦ at 6.12 GHz (in the higher frequency band). The values
in each plot are normalized so that the maximum of the transmission coefficient, S21, is equal
to 1 or 0 dB. We define the reflection efficiency of the array as the ratio of the maximum
of the transmission coefficient of the array to that of a metal plate. The metal plate of the
same dimensions as the reflectarray was manually rotated in order to steer the beam in the
same angle as the array.
When the array is not biased, the measured pattern is similar as that of the metal plate,
with the main beam reflected at 0◦ . The triangular shape of the main beam is due to
the described large profile of the receiving and transmitting antennas. At 30◦ , reflection
efficiency, the gain difference between the beam reflected by the array (red curve) and the
one reflected by the metal plate (black curve) is -2.9 dB. At 45◦ , the gain difference between
the main lobes is -3.2 dB and at 60◦ is -4.2 dB. The increase of the power loss for beams
steered at large angles is expected and is observed in [38]. This is because the overall return
loss of the unit cells under the voltage combination for large steering angles is greater than
that of smaller steering angles. In addition, It is obvious that the beam can be deflected into
negative angles by simply reversing the applied voltage. This results in an effective beam
tuning range of 120◦ . It was observed that for the voltage configuration applied, within 6-6.2
GHz, the gain reduction is less than 1.5 dB for any steering angles below 60◦ . This behavior
indicates that the phase relationships are maintained in a narrow-band interval. Therefore,
a new voltage configuration should be applied to steer the beam in a different frequency
range.
In addition to the co-polarization radiation pattern (when the transmitter and the receiver
42

(a)

(b)

(c)

(d)

Figure 2.14: Beam steering into (a) 0◦ , (b) 30◦ , (c) 45◦ and (d) 60◦ at 6.12 GHz. The red
and black curves correspond to beam steering using the reflectarray and metal plate,
respectively.
have the same polarization), it is important to measure the cross-polarization pattern as well.
The cross-polarization component is the component of the electric field orthogonal to the
main polarization. The cross-polarization pattern was measured by rotating the receiver
horn antenna by 90◦ . The normalized radiation pattern is shown in Fig. 2.15. The cross
polarized component is expected to be low as explained in section 2.3.1. At the beam peak,
the cross polarized component is at least 19 dB lower for all steering angles.
Fig. 2.16 shows beam steering at 3.38 GHz (in the lower frequency band). Similar to the

43

(a)

(b)

(c)

(d)

Figure 2.15: Beam steering into (a) 0◦ , (b) 30◦ , (c) 45◦ and (d) 60◦ at 6.12 GHz with the
cross polarized component. The red and black curves correspond to beam steering using
the reflectarray and metal plate, respectively. The blue curve is the cross polarization
component.
previous case, the values in each plot are normalized to the maximum. Since the unit cell is
more lossy (see section 2.3.3) the efficiency of the array is in general lower in this frequency
band. However, the beam could still be deflected up to almost 60◦ .
As in the case of steering at higher frequency, when the array is turned off it behaves
similarly to a metal plate. Note that the maximum of the main beam has been decreased in
comparison to the measurements at 6.12 GHz. This is due to the fact that the incident beam

44

is more focused at higher frequencies and hence a larger portion of the power is reflected
by both the array and the metal plate. The key element in evaluating efficiency of the
array is the relative difference between the metal plate and array. The efficiency is -4.15
dB, -4.25 dB and -4.8 dB at 30◦ , 45◦ and 60◦ , respectively. To the best of our knowledge,
such high beam steering in a dual-band tunable reflectarray has not yet been reported in
literature. In general, a higher cross polarization component is observed in the S-band (see
Fig. 2.17). A difference of at least 8 dB between the two polarizations is observed at the
beam peak at all steering angles. Compared to the F-band, the operating frequency range is
relatively narrow. This is expected as seen in the measurement of the unit cell in the S-band
which indicates the narrow-band nature of such a structure.
Voltage configurations for steering the beam into different angles are given in Table 2.1.
Notice the sawtooth shape of the applied voltage for all angles. This voltage variation helps
to apply a gradient of the reflection phase across the array. For larger steering angles the
phase gradient is larger as expected.

column
1
2
3
4
5
6
7
8
9
10

Voltage configurations for beam steering (Volts)
30◦ F-band 45◦ F-band 60◦ F-band 30◦ S-band 45◦ S-band
20.2
15.0
14.5
19.1
17.5
13
5.9
11.8
14.2
13.6
9.3
5
26.5
13.8
13.5
6.5
15.3
12.9
13.6
13.0
20.4
12.1
6.0
13.3
5.8
14.2
5.0
15.0
5.2
18.0
11
15.8
11.7
17.8
13.7
6.6
11.5
26.4
17.9
13.4
20.8
6.6
13.9
17.8
12.7
15.2
16.9
7.2
17.7
6.1
Table 2.1: Voltage applied to each column for various steering angles

45

60◦ S-band
20.9
16.9
15.0
5.1
20.5
16.6
14.0
7.0
20.4
14.2

(a)

(b)

(c)

(d)

Figure 2.16: Beam steering into (a) 0◦ , (b) 30◦ , (c) 45◦ and (d) 60◦ at 3.38 GHz. The red
and black curves correspond to beam steering using the reflectarray and metal plate,
respectively.

2.5

Discussion

This chapter presents the design of a unit cell with tuning ability over a wide range of phase.
The unit cell operates at two frequency bands and requires only one varactor diode to dynamically alter the phase of the scattered field. This significantly reduces the manufacturing
cost of the array compared to more complicated systems that require multiple varactors. The
unit cell was evaluated using both full-wave simulations and equivalent circuit modeling. The
equivalent circuit provides a simple description of the unit cell in terms of passive circuit
46

(a)

(b)

(c)

(d)

Figure 2.17: Beam steering into 0◦ , 30◦ , 45◦ and 60◦ at 3.38 GHz with the cross polarized
component. The red and black curves correspond to beam steering using the reflectarray
and metal plate, respectively. The blue curve is the cross polarization component.
elements, and requires far less computation time compared to the full wave simulations. The
unit cell was evaluated experimentally by placing it into a waveguide and measuring the
reflection coefficient. A maximum phase shift of 335◦ was achieved in the upper band, and
340◦ in the lower band.
The proposed unit cell is scalable and it can operate at other frequency bands by simply
scaling its dimensions. In order to demonstrate this point, we scale the unit cell (see Fig.
2.2) by a factor of 10. Thus, the parameters of the scaled unit cell are: p = 220 mm,
47

2

= 180 mm, 1 = 120 mm and g = 10 mm. The distance between the unit cell and the

ground plane is also scaled by a factor of 10. Therefore the thickness of the substrate is
d = 15.75 mm. To only consider the phase response of the unit cell and not dielectric effects,
similar to the unit cell discussion in sections 2.3.1 and 2.3.2, the dielectric constant of the
substrate is set to unity. The equivalent model in section 2.3.2 is used (see Fig. 2.6) to
predict the behavior of the scaled unit cell as a function of frequency. Fig. 2.18 shows the
reflection phase of the unit cell for different capacitance values. As expected, the higher and
lower resonant frequencies are both scaled by a factor of 0.1 (compare with Fig. 2.7) and
a large phase range is provided in both bands. Consequently the unit cell can be properly

Reflection Phase (degree)

scaled in order to operate at the desired frequency band.

200

𝑪𝒗 = 𝟏

𝑪𝒗 =
𝟐

0
𝑪𝒗 = 𝟑

-200

𝑪𝒗 = 𝟒

200

400
600
800
Frequency (MHz)

Figure 2.18: Reflection phase curves for a unit cell scaled by a facotor of 10 for different
capacitance values, Cv (pF), predicted by the equivalent circuit model.

The 10x10 reflectarray built using the proposed unit cell shows excellent steering capabilities. The beam can be deflected up to ±60◦ in both frequency bands. This steering range
has not yet been reported in literature. This unit cell design can be expanded to a larger
48

scale to allow for a greater range of steering capabilities. Moreover, by individually biasing
each unit cell, dynamic 3-D beam steering should be achievable. This is left for future study.

49

Symmetry, as wide or as narrow as you may define its meaning,
is one idea by which man through the ages has tried to
comprehend and create order, beauty and perfection.
Hermann Weyl

Chapter 3
Time Reversal Imaging

3.1

Inverse Problems and Reconstruction Methods

An inverse problem is one in which a set of measurements is used in order to predict the causes
that produced an observed outcome. Some examples of this method are the calculation of
Earth’s density using the measured strength of its gravitational field, X-ray imaging using
projection data, and flaw profiling using the measured voltage of eddy current probes in
non-destructive testing.
Inverse problems are generally ill-posed i.e. they are not well-posed problems [60]. A
well-posed problem satisfies the following requirements:
• Existence of a solution
• Uniqueness of the solution

50

• Continuity of the solution
The last condition means that the function should change continuously with a change in the
initial conditions. Inverse problems are typically ill-posed [61] and therefore special attention
is required for solving this class of problems.
In this work, for the purpose of imaging, we are interested in the inverse electromagnetic
scattering problem. The problem is to find the spatial distribution of the scatterers and
their electrical properties ( and Âľ) using the measured scattered field. This is known as
image reconstruction. The inverse scattering problem is ill-posed mainly due to the lack of
measured data. Often, the electric field is measured on 1D or 2D scan at discrete intervals.
However, in order to have a well-posed problem, both the magnitude and phase of the
electric field are required over the continuum of the domain. Many methods have been
developed in order to counter the ill-posed aspects of the problem. One such instance is
image reconstruction using diffraction tomography [62–67]. This method is even capable of
reconstructing electrical properties of scatterers in highly ill-posed conditions due to limited
angular coverage of the object [68–71].
In this thesis, we use the time reversal signal processing method for reconstruction purposes. This approach is model based and therefore requires a forward solver. Because we
are only interested in detecting targets and not extracting their properties, no iteration is
required. In what follows, time reversal imaging is explained in detail. Additionally, the
forward FDTD solver is developed and used for the demonstration of the concept of time
reversal.

51

3.2

Fundamentals of Time Reversal Signal Processing

It is known that the classical electromagnetic theory is invariant under the time reversal
transformation. In order to see this, consider the electromagnetic wave equation describing
the propagation of the electric field in a loss-less medium

∇2

1 ∂2
− 2 2 E(r, t) = 0,
c ∂t

(3.1)

where ∇ is the vector Laplacian operator, c is the speed of light in the medium and E(r, t)
is the electric field. Under the time reversal transformation, i.e. t → −t, the wave equation
(3.1) stays invariant and holds its form. Consequently, if E(r, t) is a solution to the equation,
E(r, −t) is a solution as well. Notice, the symmetry is violated if the medium is lossy. This
is because the presence of loss introduces a first order time derivative in (3.1) and hence the
wave equation is no longer invariant under the time reversal transformation.
The time reversal symmetry has a significant practical implication. If one measures the
electric field emitted from a source via an antenna array, time reverses the received signals
and re-emits them back, the electric field focuses at the initial source of radiation. In a case
where there are scatterers present in the medium, they act as secondary sources and the reemitted time-reversed signals focus at the locations of the scatterers. Therefore, this property
can be successfully used as a reconstruction scheme in order to detect and pinpoint scatters
in a medium. This approach is known as time reversal imaging and it has been vastly used as
a powerful imaging technique. In [72], Bellomo et al. introduce a new radar system utilizing
time reversal imaging. It has been experimentally shown that the time-reversed signals focus
on the target. Through-the-wall target localization has been demonstrated using the time
reversal MUSIC method in the work of Zhang et al. [73]. Time reversal microwave imaging
52

with random configurations of transmitters or receivers is studied by Razavian et al. in [74].
In [75] by Kosmas et al., the feasibility of microwave breast cancer detection using time
reversal algorithms is investigated. Time reversal microwave techniques were also utilized
for NDT of laminates and composite materials in the work of Reyes-Rodrigueaz et al. [76].
The application of the time reversal symmetry is not limited to imaging however; in the
paper by El Sahmarany et al. [77], time reversal is used for fault detection in wire networks.
In this work, time reversal imaging is the main reconstruction scheme used in order to
detect and identify the location of scatterers. Next, we develop a forward solver, i.e. a
scheme to numerically solve Maxwell’s equations. The solver is then used in order to prove
and demonstrate the time reversal concept using computer simulations.

3.3

Finite Difference Time Domain (FDTD)

Over the past few decades many numerical methods have been developed in order to solve
Maxwell’s equations in different media and for various geometries. Finite Element Method
(FEM), Method of Moments (MoM) and Finite Difference Time Domain (FDTD) are among
the most popular numerical techniques. A natural choice here would be FDTD because
the reconstruction technique is essentially performed in the time domain. Starting with
Maxwell’s equations, and applying the finite difference approximation, the main equations
used for the FDTD simulations are derived in this section.

3.3.1

Maxwell’s Equations

Maxwell’s equations are partial differential equations that describe the behavior of electric
and magnetic fields. Assuming there are no sources in the medium, in the SI units they have

53

the following form

∇ · D(t) = 0,

(3.2)

∇ · B(t) = 0,

(3.3)

∂ B(t)
,
∂t
∂ D(t)
∇ × H(t) =
,
∂t
∇ × E(t) = −

(3.4)
(3.5)

where E is the electric field intensity (units of V/m), D is the electric flux density (units
of C/m2 ), H is the magnetic field intensity (units of A/m) and B is the magnetic flux
density (units of Wb/m2 ). Furthermore E and D and also H and B are related through the
constitutive equations

D(t) = 0 r E(t),

(3.6)

B(t) = Âľ0 Âľr H(t),

(3.7)

where 0 (Âľ0 ) is the permittivity (permeability) of free space and r (Âľr ) is the relative
permittivity (permeability) of the medium.
For convenience, we normalize the electric field by defining a new variable E(t)

E(t) =

0

Âľ0

E(t) =

1
E(t)
Ρ0

(3.8)

where Ρ0 is the free space impedance. Accordingly, we modify D by defining a new variable

D(t) = √

1
0 Âľ0

D(t) = cD(t)

54

(3.9)

here c is the speed of light in free space. Now, rewriting the two curl equations in (3.4) and
(3.5) in terms of E and D, we have

µr ∂ H(t)
,
c ∂t
1 ∂ D(t)
∇ × H(t) =
,
c ∂t
∇ × E(t) = −

(3.10)
(3.11)

Notice that these equations are independent of r . Moreover, the materials used for imaging
in this thesis are not magnetic materials, i.e. Âľr =1. The independence of these two equations
from material properties simplify our numerical calculations discussed in the next section.
The new constitutive relation is

D(t) = r E(t)

(3.12)

Next, we write the explicit equations for the two curl equations in (3.10) and (3.11) and
the modified constitutive relation (3.12) using the Cartesian coordinate system. Furthermore,
we assume that the material has a diagonal relative permittivity and permeability tensor.
Therefore the only non-zero components of Âľr are Âľxx , Âľyy and Âľzz and similarly for r .
We have for equation (3.10)

µxx ∂Hx
,
c ∂t
µyy ∂Hy
=−
,
c ∂t
µzz ∂Hz
=−
,
c ∂t

CxE = −

(3.13)

CyE

(3.14)

CzE

55

(3.15)

where CiE is the i’th component of the curl of the electric field i.e.
∂Ey
∂Ez
−
,
∂y
∂z
∂Ex ∂Ez
CxE =
−
,
∂z
∂x
∂Ey ∂Ex
−
.
CxE =
∂x
∂y
CxE =

(3.16)
(3.17)
(3.18)

Similarly for equation (3.11) we have

1 ∂Dx
,
c ∂t
1 ∂Dy
CyH =
,
c ∂t
1 ∂Dz
CzH =
,
c ∂t

CxH =

(3.19)
(3.20)
(3.21)

where

∂Hy
∂Hz
−
,
∂y
∂z
∂Hx ∂Hz
CxH =
−
,
∂z
∂x
∂Hy ∂Hx
CxH =
−
,
∂x
∂y
CxH =

(3.22)
(3.23)
(3.24)

Finally, for (3.12) we have

Dx = xx Ex ,

(3.25)

Dy = yy Ey ,

(3.26)

Dz = zz Ez .

(3.27)

56

Next we exploit the finite difference approximation in order to numerically solve the
equations (3.13)-(3.15), (3.19)-(3.21) and (3.25)-(3.27).

3.3.2

Finite Difference Approximation

In order to calculate the electric and magnetic fields in Maxwell’s equations, the derivatives
in the differential equations are approximated by a linear difference equation. In this thesis,
the so called central difference approximation is used. Consider a real one dimensional
function, f (x), which is analytic over Ω = (a, b). f (x) can be approximated by a discrete
valued function
fi = f (xi ) i = 1, 2, ..., N,

(3.28)

where xi ’s are the grid points and N is the number of the grids. The grid is assumed to be
uniform with the sampling interval represented by ∆x = (b − a)/N .
We now introduce the forward and backward differences as approximations for the derivative of f (x) at grid point xi . The forward difference is
f
− fi
df F
≈ i+1
,
dx i
∆x

(3.29)

df B
f − fi−1
≈ i
.
dx i
∆x

(3.30)

and the backward difference is

Both approximations in (3.29) and (3.30) are accurate only up to the first order in ∆x. In
order to come up with a better approximation, we first write the complete expansion for

57

both the forward and backward differences. For the forward difference we have
df F
f
− fi ∆x d2 f
∆x2 d3 f
−
= i+1
−
+ ....,
dx i
∆x
2 dx2 i
6 dx3 i

(3.31)

and for the backward difference we have
df B
∆x2 d3 f
f − fi−1 ∆x d2 f
+
−
+ .....
= i
dx i
∆x
2 dx2 i
6 dx3 i

(3.32)

Now we add (3.31) and (3.32) together. Consequently
f
− fi−1 ∆x2 d3 f
df C
+ .....
= i+1
−
dx i
2∆x
6 dx3 i

(3.33)

From this, one can approximate the derivative with a truncation error of O(∆x)2 . Therefore,
we define the central difference approximation as

f
− fi−1
df C
≈ i+1
.
dx i
2∆x

(3.34)

This approximation is used in order to numerically evaluate the time derivatives and curl
equations derived in the previous section. In addition, in order to discretize the continuous
electric and magnetic fields, we adopt the popular Yee grid scheme [78]. Fig. 3.1 shows the
schematic of a grid cell. This specific configuration of the electric and magnetic fields in the
cell has many advantages. It can be shown that the electric and magnetic fields are both
divergenceless. Therefore equations (3.2) and (3.3) are naturally satisfied. Moreover, using
the central difference approximation becomes easier with this configuration.
In addition to space, time has to be descritized as well. We assume that the electric

58

Figure 3.1: Yee grid cell.
field exists in integer time steps i.e. 0, ∆t, 2∆t, ..., where ∆t is the time step. However, the
magnetic field is assumed to exist on half-integer time steps i.e. ∆t/2, t + ∆t/2, 2t + ∆t/2....
Consequently, each component of the fields requires four indices, (i, j, k) representing the
cell location and t representing time.
We now write the descritized expressions for the curl terms in (3.16)-(3.18)

i,j,k
CxE
t

CyE

CzE

i,j,k
t

i,j,k
t

Ez
=

t

i,j,k

− Ez

t

∆y
Ex

=

i,j,k+1
t

− Ex

i+1,j,k
t

− Ey

Ey
−

i,j,k
t

∆z
Ey

=

i,j+1,k

t

∆x

−

i,j,k
t

i+1,j,k
t

− Ez

i,j+1,k
t

− Ex

∆y

,

(3.35)

,

(3.36)

.

(3.37)

i,j,k
t

∆x
Ex

−

t

− Ey

∆z
Ez

i,j,k

i,j,k+1

i,j,k
t

Using (3.35)-(3.37), the finite difference approximation of Faraday’s equation in (3.13)-

59

(3.15) is then

CxE

i,j,k
t

i,j,k
CyE
t

CzE

i,j,k
t

=−

=−

=−

i,j,k

Âľxx

Hx

i,j,k
t+∆t/2

c
Âľyy

i,j,k

Âľzz

t−∆t/2

∆t
Hy

i,j,k
t+∆t/2

c
i,j,k

i,j,k

− Hx

i,j,k
t+∆t/2

c

− Hz

(3.38)

,

(3.39)

.

(3.40)

i,j,k

− Hy

t−∆t/2

∆t
Hz

,

i,j,k
t−∆t/2

∆t

In a similar way, we can approximate Ampere’s law. We first calculate the curl expressions
in (3.22)-(3.24)

CxH

CyH

CzH

i,j,k
t+∆t/2

i,j,k
t+∆t/2

i,j,k
t+∆t/2

Hz
=

− Hz

t+∆t/2

i,j−1,k
t+∆t/2

∆y
Hx

=

i,j,k
t+∆t/2

− Hx

i,j,k
t+∆t/2

− Hy

Hy
−

i,j,k−1
t+∆t/2

∆z
Hy

=

i,j,k

t+∆t/2

∆x

−

t+∆t/2

i,j,k−1
t+∆t/2

i,j,k
t+∆t/2

− Hz

i,j,k
t+∆t/2

− Hx

∆y

,

(3.41)

,

(3.42)

.

(3.43)

i−1,j,k
t+∆t/2

∆x
Hx

−

− Hy

∆z
Hz

i−1,j,k

i,j,k

i,j−1,k
t+∆t/2

(3.44)

60

Ampere’s law in (3.19)-(3.21) becomes
i,j,k

i,j,k

i,j,k
CxH
t+∆t/2

1 Dx t+∆t − Dx t
=
c
∆t

i,j,k
CyH
t+∆t/2

1 Dy t+∆t − Dy t
=
c
∆t

i,j,k
CzH
t+∆t/2

1 Dz t+∆t − Dz t
=
c
∆t

i,j,k

,

(3.45)

,

(3.46)

.

(3.47)

i,j,k

i,j,k

i,j,k

Finally, the descritized constitutive relation (3.25)-(3.27) is

Dx
Dy
Dz

3.3.2.1

i,j,k
t
i,j,k
t
i,j,k
t

i,j,k

=

xx

=

yy

=

zz

i,j,k

i,j,k

Ex
Ey
Ez

i,j,k
t
i,j,k
t
i,j,k
t

,

(3.48)

,

(3.49)

.

(3.50)

Reduction to Two Dimensions

For simplicity, we assume a two dimensional scenario i.e. all the physical objects are infinitely
extended and uniform in one dimension (here, we choose the z-direction). Mathematically,
∂ = 0. Moreover, we assume that the wave produced by the source is of TM
we require ∂z
z

type. This means that there is no magnetic field in the z-direction, or H.ˆ
z = 0. As we will
see in the next chapter, these are realistic assumptions seeing that the targets used in the
experiment are uniform and long with respect to the wavelength of the wave. Also, the horn
antennas used in the experimental setup are linearly polarized, therefore we simply choose
our coordinate system so that the z-direction coincides with the generated electric field.
61

Consequently, the condition for H.ˆ
z = 0 is satisfied. With these assumptions we can greatly
simplify the equations derived in the previous section. For a two dimensional problem and
a TMz source, the curl expressions (3.35)-(3.37) in Faraday’s law become

Ez

i,j
CxE
t

=

i,j+1
t

i,j
t

∆y
Ez

i,j
CyE
t

− Ez

=−

i+1,j
t

− Ez

,

(3.51)

i,j
t

∆x

.

(3.52)

Therefore, for Faraday’s law (3.38)-(3.40) we have

CxE

CyE

i,j
t

i,j
t

=−

=−

Âľxx

i,j

i,j

Hx

t+∆t/2

c
Âľyy

− Hx

i,j
t−∆t/2

∆t
i,j

Hy

i,j
t+∆t/2

c

− Hy

,

(3.53)

.

(3.54)

i,j
t−∆t/2

∆t

Similarly, the curl equations in (3.41)-(3.43) become

CzH

i,j
t+∆t/2

Hy
=

i,j
t+∆t/2

− Hy

i−1,j
t+∆t/2

∆x

Hx
−

i,j
t+∆t/2

− Hx

∆y

i,j−1
t+∆t/2

.

(3.55)

Consequently for Ampere’s law in (3.45)-(3.47), we have
i,j

i,j
CzH
t+∆t/2

i,j

1 Dz t+∆t − Dz t
=
.
c
∆t

62

(3.56)

Finally, the constitutive relation in equations (3.48)-(3.50) reduces to

Dz

i,j
t

i,j

=

zz

Ez

i,j
t

.

(3.57)

Equations (3.51)-(3.57) are our main FDTD expressions. The expressions can be easily
implemented in order to simulate wave propagation in different media. It is important to
mention that in order to avoid non physical propagation of the wave, the so called Courant
stability condition should be satisfied. The stability condition is

c ∆t <

1
1
∆x2

+ 12 + 12
∆y
∆z

.

(3.58)

The condition assures that the wave cannot propagate faster than the speed of light.

3.3.2.2

Source

There are different ways to introduce a source in the computational domain. For instance,
there are hard and soft sources. A hard source is simply enforcing a point in the grid to have
a particular value at each time interval i.e.

Ez

i,j
t

= S(t),

(3.59)

where S(t) is the desired source as a function of time. Even though this is easy to implement,
it has the disadvantage of not being transparent to waves in the medium. The hard source
acts like a perfect electric conductor and causes scattering of other incoming waves. In this

63

thesis, a so called soft source is used. A soft additive source is defined as

Ez

i,j
t+∆t

= Ez

i,j
t

+ S(t).

(3.60)

Contrary to hard sources, soft sources are transparent and therefore incoming waves can
pass through the source without any scattering.
For simulation purposes we use a Gaussian pulse, defined as

S(t) = A0 exp −

(t − t0 )2
.
τ2

(3.61)

where A0 is the amplitude, t0 is associated with delay, and τ is a parameter related to the
bandwidth of the source. In order to see this, we compute the Fourier transform of (3.61)
∞

S(f ) =

dtS(t)e−i2πf t =

−∞

√
π2f 2
πA0 B e−i2πf t0 exp −
,
B2

(3.62)

where B = 1/τ is the bandwidth of the input signal.

3.3.2.3

Boundary Conditions

Because the boundaries of the computational domain are truncated, an artificial reflection of
the waves at the boundaries is created. In order to prevent this phenomena, we can introduce
loss at the boundaries of the computational domain. Moreover, the absorbing boundaries
are impedance matched at all frequencies and for any angle of incidence. Following [80] we
implement the uniaxial perfectly matched layer (PML) by modifying the permittivity and
permeability matrices in the absorbing boundaries. This is achieved by introducing a new

64

matrix, M . The modified parameters, N and ÂľN are

= 0M

(3.63)

ÂľN = Âľ0 M

(3.64)

N

where M is




M y Mz
 Mx







0

0

0

Mx Mz
My

0

0

0

Mx M y
Mz








here Mx , My and Mz are defined as follows

x 3
,
2∆t Lx
y 3
,
σ(y) = 0
2∆t Ly

σ(x)
;
iω 0
σ(y)
My (y) = 1 +
;
iω 0

Mx (x) = 1 +

Mz (z) = 1 +

σ(x) =

σ(z)
;
iω 0

σ(z) =

0

z 3
.
2∆t Lz
0

(3.65)
(3.66)
(3.67)

Lx , Ly and Lz are the thickness of the PML boundaries in the x, y and z directions.
It is clear from (3.65)-(3.67) that the conductivity terms are responsible for the loss in the
boundaries. Moreover, these terms increase gradually, resulting in the further reduction of
the reflection.

65

3.4

Computer Simulation of Time Reversal Imaging

In this section it is shown how time reversal imaging works using the FDTD solver. The
goal is to demonstrate how the properties of the time reversal technique allow us to detect
scattering objects in a medium. In general, the demonstration can be divided into four
steps. First is to define the geometry of the problem i.e. the properties of the medium
and scattering objects, the location of the source and the receivers, and the dimensions of
the computational domain. The second step is to measure the so called background signal.
For this, all the scattering objects are removed from the medium. The source radiates and
the radiated field is measured by the receivers. In the third step, the scattering objects
are present. In this case, the measured field radiated by the source, contains information
about the scattering objects. The background signal is then subtracted from this field in
order to obtain the perturbation fields due to the scatterer. In the final step, this field is
time reversed and re-radiated back into the medium. The re-emitted signal focuses on the
scattering objects in the medium. Detection is possible through this focusing. There are
several ways to find the location of the target. Here we use two approaches. One is to
calculate the “Entropy” of the waves. The minimum of the entropy provides us the exact
moment of focusing. The entropy is defined as [79]
i,j 2 2

H(t) =

i,j

Et

i,j 4

,

(3.68)

i,j Et

where t is the time frame and i, j are grid numbers. Here, the summation is over the entire
computational domain. Another way to pinpoint the object is to calculate the energy that

66

passes through each point over the entire computational time T . The energy is defined as
T

W (i, j) =

i,j 2

Et

.

(3.69)

t=0

In what follows we consider three cases. First, we demonstrate that in the absence of
scatterers, the time-reversed waves focus on the source. In the second and third cases, we
demonstrate detection of one and two scattering objects, respectively. For all three cases,
the computational domain is 0.5 m x 0.5 m and the medium is assumed to be air, therefore
r

= Âľr = 1.

3.4.1

Case I: Free Space - Focusing on Source

In this case, there is no scattering object present in the medium. Therefore, the time-reversed
background signal should focus on the source location. The input to the source is a Gaussian
wave function defined in (3.61) with amplitude A0 = 1, bandwidth B = 1/τ = 10π GHz and
delay time t0 = 30τ . The source is located at x = 0.1250 m and y = 0.1250 m as shown in
Fig. (3.2). Eight receivers are located on an arc of a circle centered at x = 0 and y = 0 with
radius 0.5 m. The angular separation between the receivers is 6◦ . The total computational
time is 2.36 ns. This is the time required for the wave to traverse the longest path, the
diagonal, of the computational domain. Fig. (3.2) shows the amplitude of the electric field,
Ez , propagating towards the receivers at t = 1.17 ns.

67

Distance (m)

0
0.04

0.1

Source

1

0.2
Receivers

0.02

2
3

0.3

0

4
5

0.4

6
8

0.5
0

-0.02

7

0.1 0.2 0.3 0.4
Distance (m)

0.5

Figure 3.2: Forward propagation of the electric field at t = 1.17 ns. The eight black dots
represent the location of the recievers.

The electric field is depicted at a later time instance in Fig. 3.3. Notice, that due to
absorption in the PML region, there is no artificial reflection from the boundaries of the
computational domain. Also, the amplitude of the wave has decreased due to 1/r2 propagation loss.

Distance (m)

0
0.1

0.02

Source

0.01
0.2
0

0.3
Receivers

0.4
0.5
0

0.1 0.2 0.3 0.4
Distance (m)

-0.01

0.5

Figure 3.3: Forward propagation of the electric field at t = 1.94 ns.

68

We let the simulation run for a sufficiently long time i.e. until most of the wave is
absorbed by the PML boundaries. The measured electric field at the receivers are shown in

Received signals

Fig. 3.4.

1

8
7
6
5
4
3
2
1

1.2

1.4

1.6 1.8 2
Time (ns)

2.2
x 10-9

Figure 3.4: Received signals at the recievers. The numbering corresponds to the eight
receivers shown in Fig. 3.2.

The received signals are then time reversed and re-emitted into the medium. Now the
receivers act like new sources. The back propagation of the time reversed signals at time
instance t = 0.65 ns is shown in Fig. 3.5.

69

x 10-3

Distance (m)

0
0.1

4

0.2

2

0.3

0
New
Sources

0.4

-2
-4

0.5
0

0.1 0.2 0.3 0.4
Distance (m)

0.5

Figure 3.5: Back propagation of time reversed fields at t = 0.65 ns.

Again we let the simulation run for a adequately long time. In order to pinpoint the
location of any scattering object we use the entropy defined in (3.68). The entropy plot is
shown in Fig. 3.6.

Entropy

8000
6000
Local
Minima

4000
2000
00

0.5

1 1.5
Time (ns)

2

2.5
x 10-9

Figure 3.6: Entropy of the electric field calculated in accordance to eqn. 3.68.

The minimum in the entropy corresponds to the time instance where the back propagating
70

waves focus on the location of the scatterer. Since there was no scattering object present in
this case, the time-reversed waves focus on the location of the initial source. The electric
field is shown at t = 1.75 ns in Fig. 3.7. It is apparent that the fields are constructively
interfering at the location of the initial source, shown with the black dot.

x 10-3

Distance (m)

0
0.1
0.2

5

Initial
Source

0.3

0

0.4
0.5
0

-5
0.1 0.2 0.3 0.4
Distance (m)

0.5

Figure 3.7: Focusing of time-reversed back propagated waves on the location of the initial
source at t = 1.75 ns. This time instance correspond to the minimum of the entropy plot in
Fig. 3.6.

As mentioned, another way to pinpoint the location of targets is to use the energy plot.
The energy was calculated according to eqn. 3.69 and is shown in Fig .3.8.

71

x 10-3

Distance (m)

0

6

0.1
0.2

4

0.3
2
0.4
0.5
0

0.1 0.2 0.3 0.4
Distance (m)

0.5

Figure 3.8: Energy plot of the backpropagated waves calculated accoridng to eqn. 3.69.
The eight bright points in the figure correspond to the sources.

However, since the maximum of the energy plot occurs at the location of the sources,
it is hard to see the focused energy on the initial source location. This focusing is due to
the convergence of time-reversed signal at the initial source. In Fig. 3.9 the energy of a
smaller region of the computational domain is shown. The bright points are excluded from
the region and therefore the focused energy on the source can be easily observed. The black
dot in the figure corresponds to the real location of the initial source.

72

x 10-4
15

Distance (m)

0
0.1

10
Source

0.2
0.3
0

5

0.1
0.2
Distance (m)

0.3

Figure 3.9: Energy plot of the back propagated waves zoomed in the location of the intial
source.

The stretched out pattern of the focused power in the figure is due to the limited angle
coverage of the receivers. If the receivers were located along a full circle, the focused energy
would be circular rather than elliptical pattern.

3.4.2

Case II: Single Scatterer

Now we consider the case of a single scatterer in the medium. The scatterer is assumed to be
a circular dielectric of radius 1 cm and relative dielectric constant of 3. The target is located
at x = 0.09 m and y = 0.15 m and the source is located at x = 0.075 m and y = 0.075 m.
Similar to the previous case, the bandwidth of the input Gaussian signal is B = 1/τ = 10π
GHz. The delay parameter t0 is set to 6τ . The receivers are located in the same position
as in the previous case. And as before, the first step is to store the background signal. The
second step is to measure the field in the presence of the target. Fig. 3.10 shows the electric
field propagating from the source towards the receivers. The distortion in the wave front is
73

due to the interaction of the field with the scatterer (black circle in the figure).

Distance (m)

0

0.03

0.1 Scatterer

0.02

0.2

0.01
0

0.3

-0.01
0.4
0.5
0

-0.02
0.1 0.2 0.3 0.4
Distance (m)

0.5

Figure 3.10: Forward propagation of the electric field at t = 0.67 ns. The presence of the
scatterer, shown with the black circle, causes distortion in the propagating wave. The
relative permittivity of the target is 3.

The scattered field is then calculated, time-reversed and re-emitted back through the
medium. The entropy of the wave at each time instance is shown in Fig. 3.11. The local
minimum indicated in the figure corresponds to the time instance of focusing.

74

8000

Entropy

6000
4000
Local
Minima

2000
00

0.5

1
1.5
Time (ns)

2

2.5
x 10-9

Figure 3.11: Entropy of the electric field as a function of time. The local minimum time
instance corresponds to the focusing of the waves.

Fig. 3.12 shows the electric field at the time instance of minimum entropy, t = 2.03 ns.
As expected the re-radiated fields are focused on the location of the scatterer.

hh (m)
Distance

0
0.1
50 Scatterer
0.2
100
0.3
150
0.4
0.5
200
0

0.150 0.2 100 0.3 1500.4
Distance
vv (m)

0.5
200

x 10-3
30.03
0.02
2
0.01
1
0
0
-0.01
-1
-0.02
-2

Figure 3.12: Focusing of time-reversed back propagated waves on the location of the
scatterer at t = 2.03 ns. This time instance correspond to the minimum of the entropy plot
in Fig. 3.11.

75

Using the aforementioned energy plot is an alternative approach for detecting the scatterer. The energy plot is shown in Fig. 3.13. The sources are excluded from the plotted
region due to their high power intensity. The black circle in the figure corresponds to the
true location of the target. As it can be seen, the energy is focused in the vicinity of the
scatterer.

x 10-5

Distance (m)

0

15
0.1
10
0.2

Scatterer

5
0.3
0

0.1
0.2
Distance (m)

0.3

Figure 3.13: Energy plot of the back propagated waves zoomed in the location of the
scatterer. The black circle corresponds to the true location of the scatterer.

3.4.3

Case III: Two Scatterers

In the final case we consider the presence of more than one scatterer. Target one is a circular
dielectric with a radius of 0.5 cm, positioned at x = 0.09 m and y = 0.15 m. Target two is
an elliptical dielectric with semi-major axis of 1 cm and semi-minor axis of 0.5 cm, located
at x = 0.25 m and y = 0.20 m. Both targets have a relative dielectric constant of 4.1. The
configuration of the receivers, along with the source parameters and location, is the same as
cases I and II. We first need to calculate the scattered field. For this, the background signal

76

has to be subtracted from the total field in the presence of scatterers. Fig. 3.14 shows the
electric field interacting with the first scatterer. The disturbance in the field is due to this
interaction. Fig. 3.15 shows the field a few moments later when the wave front has reached
the second scatterer. The field finally reaches the receivers and the scattered field can be
easily calculated.

Distance
hh (m)

0

0.03

Scatterer I
0.1
50

0.02
0.02
0.01

0.2
100
0.3
150
0.4

200
0.5
0

Scatterer II

0
-0.01
-0.02
0.150 0.2 100 0.3 1500.4
Distance
hh (m)

0.5
200

Figure 3.14: Forward propagation of the electric field at t = 0.56 ns when the wave front
reaches the first scatterer.

77

Distance
jkh (m)

0

0.03
0.02
0.02
0.01
0.01

Scatterer I
0.1
50

0.2
100
0.3
150
0.4

200
0.5
0

Scatterer II

0
-0.01
-0.01
-0.02
-0.02
0.150 0.2 100 0.3 1500.4
kjh (m)
Distance

200
0.5

Figure 3.15: Forward propagation of the electric field at t = 0.99 ns when the wave front
reaches the second scatterer.

We again calculate the entropy of the re-emitted waves.

The entropy is shown in

Fig. (3.18). The two local minima correspond to the time instances that the wave is focused
on the location of the two scatterers. Clearly the first minima corresponds to the focusing
on the elliptical scatterer and the second minima corresponds to the circular scatterer.
Fig. (3.17) corresponds to the time instance of the first local minimum in the entropy
plot (t = 1.44 ns). The waves are clearly focused on scatterer two.

78

5000

Entropy

4000
3000
2000
Local
Minima

1000
00

0.5

1
1.5
Time (ns)

2

2.5
x 10-9

Figure 3.16: Entropy of the electric field as a function of time. The two local minima
correspond to the time instances that the field focuses on the location of the two scatterers
in the medium.

x 10-3
0.03
3
0.02
2
0.01
01

Distance
hjg (m)

0
Scatterer I
0.1
50
0.2
100
0.3
150
0.4

200
0.5
0

Scatterer II

-0.01
0
-0.02
-1

0.150 0.2 100 0.3 1500.4
Distance
jhg (m)

0.5
200

Figure 3.17: Focusing of time-reversed back propagated waves on the location of scatterer
two at t = 1.44 ns. This time instance corresponds to the first local minima of the entropy
plot in Fig. 3.16.

Fig. (3.17) shows the back propagated electric field a few moments later, at t = 2.2. The
time instance corresponds to the time instance of the second local minimum in the entropy.
79

The waves are focused on scatterer one.

x 10-3
0.03
1.5
0.02
1
0.01
0.5
0
0
-0.01
-0.5
-0.02
-1

Distance
hh (m)

0
Scatterer I
0.1
50
0.2
100
0.3
150
0.4

200
0.5
0

Scatterer II

0.150 0.2 100 0.3 1500.4
Distance
hh (m)

0.5
200

Figure 3.18: Focusing of time-reversed back propagated waves on the location of scatterer
one at t = 2.2 ns. This time instance corresponds to the second local minima of the
entropy plot in Fig. 3.16.

Again, the alternative approach for detection is using the energy plot. The energy plot is
shown in Fig. 3.19. Because the elliptical scatterer is larger, the scattered field associated to
this target is larger compared to the field associated with the smaller scatterer. Consequently,
the indication corresponding to the elliptical scatterer has higher intensity.

80

x1510-5

Distance (m)

0
0.1 Scatterer I

10

Scatterer II

0.2
0.3
0

5

0.1
0.2
Distance (m)

0.3

Figure 3.19: Energy plot of the back propagated waves zoomed in the location of the two
scatterers. The black circle and ellipse in the figure correspond to the true location of the
circular and elliptical scatterers, respectively.

81

An experiment is a question which science poses to Nature, and
a measurement is the recording of Nature’s answer.
Max Karl Ernst Ludwig Planck

Chapter 4
Experimental Setup and Results
A prototype of the proposed microwave imaging system was built using the reflectarray
antenna and the arch range at Michigan State University [59, 81]. The setup is very similar
to the one used for measuring the radiation pattern of the reflectarray described in Chapter
2. Here we briefly describe the experimental setup. Fig. 4.1 depicts the schematic of the
system. The reflectarray antenna, mounted on a Styrofoam sheet, was located at the center
of the arch range, which has a radius of 3.53 m. The center of the arch range is also
chosen to be the origin of the polar coordinate system used throughout the experiment (see
Fig. 4.1). The line normal to the array corresponds to θ = 0 and the curved arrow shows
the direction in which θ increases. An H-1498 series broadband horn antenna was placed
on the arch rail facing the reflectarray and was used as the source. A dielectric lens was
placed in front of the antenna in order to create a focused plane wave beam. Moreover,
the transmitter was aligned with the center of the reflectarray and the incident beam was

82

directed to be perpendicular to the surface of the array. Both the array and the transmitter
were kept stationary throughout the experiment. The reflectarray antenna steered the beam
into different angles, sweeping the region of interest, and the scattered field was measured by
a receiver antenna. An identical horn antenna was used as the receiver. The transmitter and
the receiver were both connected to an Agilent E5071c network analyzer, calibrated from
1.5 GHz to 7.5 GHz with 1601 frequency points. The receiver antenna was moved along the
arch rail in order to measure the transmission scattering parameter S21 . The white dots in
Fig. 4.1 represent locations were a reading was taken. Measurements were taken every two
degrees from θ = −74◦ , represented by the left-most dot, to θ = 74◦ , represented by the
right-most dot. Due to the large size of the focusing lens of both the transmitter and the
receiver, measurements could not be taken from θ = −14◦ to θ = 14◦ . The measured signals
at each position were calibrated using the response of the system. The system response was
calculated in accordance to [59], by considering the canonical problem of scattering off a
metal sphere.

Distance (m)

0
Reflectarray

1
Scatterer

Region of
Interest

2
3

Rx

Rx
Tx

-3

-2

-1

0

1

Distance (m)

2

Figure 4.1: Schematic of the microwave imaging setup.

83

3

For each angle at which the beam was steered, the background (free space) field was
measured to be used as a reference. This reference was later subtracted from field measurements of different scatterer configurations in order to obtain the respective scattered fields.
Since the measurements were performed in the frequency domain, the resulting scattered
field was mapped back to the time domain using Fourier transform. Next, these signals were
time reversed and re-radiated back using in-house finite difference time domain (FDTD)
simulation software. As explained in Chapter 3, the propagating time-reversed signals focus
on the scatterers in the medium and hence detect and locate the targets. Similar to [76, 82],
the electric field energy density (in units of 12 0 ) was calculated using the expression
T

|E(r, t)|2 ,

W (r) =

(4.1)

t=0

where E is the electric field at point r and time t, and T is the total simulation time. W (r)
can be interpreted as the accumulated energy at each point r during backpropagation. In
all experiments performed, the scattering objects were sufficiently tall so that the problem
was considered to be two dimensional. Also, the region of interest where the image is
reconstructed is a 100-inch radius circular sector shown in Fig. 4.1. The area near the
sources has been excluded from the ROI because of the high energy concentration and
possible masking of target indications.
Two experiments were performed in order to demonstrate the feasibility of the proposed
system.

84

4.1

Experiment I: Single Scatterer

In the first experiment, the reflectarray was set to deflect the beam at the fixed angle of
45◦ . A cylindrical acetal (polyoxymethylene) with dielectric constant of 2.74 and dissipation
factor of tanδ= 0.006 at room temperature was used as a target [83]. The cylinder had a
diameter of 4 inches and a height of 12 inches and was located at θ = 45◦ . Measurements
were taken of the cylinder placed at three different distances from the center of the array; 40,
60 and 80 inches. For each case, the energy image was reconstructed according to equation
4.1. Fig. 4.2 shows the reconstructed image without any further processing for the three
different cases. As expected, the indication associated with the target is stretched out. This
is because the measurements of the scattered field were not taken from all possible angles.
This pattern was also observed in the paper by Rodriguez et al. [74,76]. The figure indicates
that the intensity of the scattered field reduces as the object is placed farther away from
the reflectarray antenna. This is due to path loss and divergence of the beam as the field
propagates through space. The maximum intensity in Fig. 4.2(a), when the target 40 is
inches from the array, is almost an order of magnitude greater than the maximum intensity
in Fig. 4.2(c), where the target is located 80 inches away from the array. A similar experiment
was performed using a metal plate. The plate was rotated manually to deflect the beam to
45° . It was observed that the artifacts seen in the energy figures were removed. This suggests
that the artifacts in the images are due to the limited bandwidth of the steered electric field.
A metal plate would reflect the entire frequency band of 1.5-7.5 GHz. Compared to the
metal plate, the reflectarray has a relatively narrow bandwidth and therefore it generates
more artifacts in the images. This hypothesis is under further investigation.
In order to evaluate the accuracy of our system, it is assumed that the maximum intensity

85

−4
x10
x 10

−5
x10
x 10

-4

1.5
1

1

0.5

2
-2

-1

0

1

8
6

1
4
2

2

0

2

Distance (m)

-5

0
Distance (m)

Distance (m)

0

-2

(a) Target at 40 inches from the origin

-1

0

1

Distance (m)

2

0

(b) Target at 60 inches from the origin
−5
x10
x 10

-5

Distance (m)

0

3

1

2

2

1

-2

-1

0

1

Distance (m)

2

0

(c) Target at 80 inches from the origin

Figure 4.2: Energy image of the three configurations in experiment one. The target was
placed at three different distances from the center of the array. The black circle
corresponds to the actual position of the scatterer.
point in the image corresponds to the center of the cylindrical scatterer. Then the relative
error in detection is defined as:
=

|rdet − ract |
,
rroi

(4.2)

where rdet is the highest intensity point in the image, ract is the actual location of the
dielectric center and rroi is the radius of the ROI which is equal to 100 inches. Table 4.1
shows the detection error calculated for different target positions. The results indicate that
the error increases as the target moves away from the reflectarray. When the target is close
to the array the error is less than 1%, however, it increases to about 9% when the target is
located at 80” from the center. The results match with the theoretical predictions by Born

86

and Wolf [84]. They have shown that the radial resolution is proportional to Îť a
Îť is the central frequency of the pulse,

2

, where

is the distance of the object from the source and a

is the length of the receiver array. Radial resolution is proportional to power, therefore for
objects placed far from the source, as the power decreases, detection error increases.

Target distance from

Detection error

reflectarray (inch)

( )

40

9.4e-04

60

0.0471

80

0.0933

Table 4.1: Detection error calculated for different positions of the dielectric cylinder.

4.2

Experiment II: Two Scatterers - Beam Scanning

The second experiment was designed to demonstrate the beam scanning ability of the system.
Two cylindrical dielectrics of diameters 2.25 and 6 inches were located at (r, θ) = (40”, −30◦ )
and (r, θ) = (60”, 45◦ ), respectively. The cylinders were both 3 feet long and made of acetal.
In order to scan the region of interest, the beam was steered into various angles, starting
from −60◦ to +60◦ in 15-degree increments, excluding ±15◦ due to the large profile of the
lenses (see Chapter 2). At each steering angle the scattered field was measured along the
arch rail. Similar to the previous experiment, the field was time-reversed and re-radiated
back into the medium using computer simulation. Finally the energy image was plotted in
order to detect and locate the scatterers in the ROI. The results are shown in Fig. 4.3. The
beam is steered to −60◦ and −45◦ , corresponding to Fig. 4.3(a) and Fig. 4.3(b), respectively.
In both cases, the target located at (r, θ) = (40”, −30◦ ) is detected. This is due to the side
87

lobes of the reflectarray (see Fig. 2.14 in Chapter 2). Fig. 4.3(c) shows the case where the
main beam is steered to −30◦ and consequently passes through the target. In this case the
interaction is maximized and hence, compared to the two previous cases, the target has a
much stronger indication in the energy image. Similar to the first experiment, when there
is maximum interaction, artifacts are more visible in the image. Fig. 4.3(d) corresponds to
the case where the beam is reflected to 0◦ . In this case, both scatterers are detected. In
the last three cases, where the main beam is deflected into +30◦ , +45◦ and +60◦ , only the
bigger dielectric cylinder located at (r, θ) = (60”, +45◦ ) is detected. However, the strongest
indication of the target appears when the beam is steered to +45◦ which also results in
stronger artifacts in the energy image.
In order to evaluate the accuracy of detection when the beam is steered into different
angles, the relative error is calculated (see Table 4.2). As expected, the error is at a minimum
(equal or below 5%) when the main beam passes through the object and increases otherwise.
Tables 4.1 and 4.2 indicate that detection error is smaller for larger scatterers due to a
bigger scattered field. In addition, the error decreases for targets closer to the reflectarray.
This is due to a more focused beam in closer proximity to the array.

4.3

Experiment III: Dielectric Anomalies - Defects and
Metallic Impurities

This section demonstrates the capability of the proposed microwave imaging system to detect
anomalies within dielectrics. Two types of anomalies are considered: defects and metallic
impurities. For each type, a cylindrical dielectric sample was prepared. The samples both
have a radius of 2 inches and are 12 inches tall (see Fig. 4.4). The first sample contains an
88

−5
x 10
x10

-5

4

1
2

2
-2

-1

0

1

-5

4

1
2

2

0

2

Distance (m)

−5
x10
x 10

0
Distance (m)

Distance (m)

0

-2

(a) Beam steered to −60◦

-1

0

1

Distance (m)

2

(b) Beam steered to −45◦
−4
x10
x 10

−5
x10
x 10

-4

1

1

0.5

2
-1

0

1

3

1

2
1

2

0

2

Distance (m)

-5

0
Distance (m)

Distance (m)

0

-2

-2

(c) Beam steered to −30◦

-1

0

1

Distance (m)

2

−5
x10
x 10

−5
x10
x 10

-5

-5

1

2
1

2
0

1

8
6

1

4

2

2

0

2

Distance (m)

0
Distance (m)

Distance (m)

3

-1

0

(d) Beam steered to 0◦

0

-2

0

-2

(e) Beam steered to +30◦

-1

0

1

Distance (m)

2

0

(f) Beam steered to +45◦
−5
x10
x 10

-5

Distance (m)

0

3
2

1

1

2
-2

-1

0

1

Distance (m)

2

0

(g) Beam steered to +60◦

Figure 4.3: Energy image for different steering angles in experiment two. The black circles
correspond to the actual positions of the scatterers.
89

Steering angle

Detection error ( )

Detection error ( )

(degree)

2.25” diameter cylinder

6.0” diameter cylinder

-60

0.0535

ND

-45

0.0592

ND

-30

0.0505

ND

0

0.0699

0.1046

30

ND

0.0565

45

ND

0.0401

60

ND

0.1141

Table 4.2: Detection error calculated for different steering angles. ND indicates that the
target was not detectable at the particular steering angle.
axial defect along the axis of the cylinder. The defect is 1-inch-deep and 1.75 inches wide.
The second sample, resembling cases of metallic impurities within dielectric materials [85],
was affixed with a 2-inch-wide thin layer of copper tape along its main axis. The two samples
were examined one at a time, each located at (r, θ) = (40”, 45◦ ). The main beam was also
steered to 45◦ in order to achieve maximum interaction. To find the defect contribution, the
signal from a healthy sample was recorded and subtracted from the signals of the targets
with anomalies. This ensures that the resulting scattered field contains only contributions
from the anomaly.
The calculated energy images for the cylinder with defect are shown in Fig. 4.5 and Fig.
4.6. Two configurations were considered with the defect centered along θ = 45◦ . In the first
case the defect was facing the arch rail, Fig. 4.5(a), and in the second case, the defect was
180◦ -rotated to face the beam scanning array, Fig. 4.6(a). The red arrows in the figures
indicate the location of the defect. It is apparent that the location of maximum intensity in
the energy image changes as the cylinder is rotated, implying high detection accuracy of the
system.
90

Figure 4.4: Dielectric samples with different types of anomalies. The left sample contains a
1.75-inch-wide and 1-inch-deep axial defect. The second sample contains a 2-inch-wide
piece of copper tape along its main axis.
The calculated energy images for the cylinder with copper anomaly are shown in Fig. 4.7
and Fig. 4.8. Two cases were considered again; the copper tape facing the arch rail in the
first case and the beam scanning array in the second case (sample was 180 degrees rotated).
The copper indication in the energy images moves clearly as the sample is rotated indicating
the sensitivity of the system with respect to the location of the anomaly.
Similar to the previous experiments, we calculate the detection error given by (4.2) for
Fig. 4.5, Fig. 4.6, Fig. 4.7 and Fig. 4.8. For the sample with defect, ract is the centroid or
the geometric center of the defect and for the other sample, the center of the copper tape
is considered as ract . For all cases rdet is the pixel with maximum intensity in the energy
image and rroi is 100 inches. The results are given in Table 4.3.

91

(a)

(b)

Figure 4.5: (a) Calculated energy image of the backpropagating waves for the dielectric
sample with defect facing the arch rail (b) close-up view. The red arrows point to the
location of the defect.
Table 4.3 indicates that both anomalies can be detected with high accuracy. However,
the defected cylinder generates a stronger scattered field and therefore is detected more
accurately. It is interesting that compared to Table 4.2, the error values shown in Table 4.3,
are in general smaller. This is because the time reversal signal processing approach is mainly

92

(a)

(b)

Figure 4.6: (a) Calculated energy image of the backpropagating waves for the dielectric
sample with defect facing the beam scanning array (b) close-up view. The red arrows point
to the location of the defect.
for detecting point-like scatterers [86]. Indeed, the anomalies considered in this work, both
the defect and copper strip, are more representative of a point source than the dielectric
cylinders in the previous section.

93

(a)

(b)

Figure 4.7: (a) Calculated energy image of the backpropagating waves for the sample with
copper anomaly when the copper tape is facing the arch rail (b) close-up view. The red
arrows point to the location of the copper anomaly.

Defect

Detection error ( )

Detection error ( )

orientation

Sample with defect

Sample with copper

Facing receiver

0.012

0.015

Facing array

0.015

0.045

Table 4.3: Detection error calculated for the two dielectric samples with anomalies.
94

(a)

(b)

Figure 4.8: (a) Calculated energy image of the backpropagating waves for the sample with
copper anomaly when the copper tape is facing the arch rail (b) close-up view. The red
arrows point to the location of the copper anomaly.

95

Part 3
Superradiance in Open Quantum
Systems

96

Physics is like sex: sure, it may give some practical results, but
that’s not why we do it.
Richard Phillips Feynman

Chapter 5
Introduction

5.1

Open Quantum Systems

Open quantum systems are currently the research focus of many physicists in various subfields. The main driving force in this direction is evidently the quest for new developments in
quantum informatics. Another area with significant recent achievements related to openness
of a quantum system is nuclear physics, where the understanding of structure and reactions
of loosely bound nuclei far from stability requires the correct unified treatment of the bound
states and continuum. Cold atoms in traps and optical lattices can give rise to new effects of
coupling, entanglement and transfer of information. Solid-state micro- and nano-devices, including Josephson junctions and spintronics, are probably the most developed arrangements
of this type. Fig. 5.1 depicts two examples of the aforementioned open quantum systems,
an unstable nucleus and a quantum wire attached to leads.

97

From a general point of view, in all cases we have to deal with a mesoscopic system 1
of interacting constituents that serves as a guide for the transmission of a quantum signal.
The system can have intrinsic degrees of freedom which can be excited and deexcited by the
signal. The coupling to the external world is realized through a certain amount of channels
characterized by the asymptotic quantum numbers of emitted particles or quanta and the
final state of the system. Each channel has an energy threshold where it becomes open and
connects to the environment. In the absence of decoherence through external noise or a heat
bath, the transmission at given energy is described by the unitary scattering matrix in the
space of channels open at this energy. All these features are common for numerous loosely
bound or marginally stable mesoscopic systems and play a key role in determining their main
observable properties.
A convenient mathematical formalism for the description of such systems is given by the
effective non-Hermitian Hamiltonian; this method, based on the Feshbach projection formalism, is formally exact but very flexible and can be adjusted to many specific situations. This
approach provides an alternative to popular methods such as the Feynman diagrammatic
techniques and the master equation that are commonly used for studying open quantum
systems. The suggested method, being practically simpler, reveals new physics, including
the sharp redistribution of decay widths and the emergence of short-lived superradiant states
and long-lived trapped states for sufficiently strong coupling to the continuum.
Using the non-Hermitian Hamiltonian framework, three different problems are considered
in this thesis. Even though the systems are physically distinct, the emergence of superradiant
states as a result of interaction with the external world is the common thread between all
1 Mesoscopic physics is concerned with matter in intermediate length scales, between elementary particles
and macroscopic objects. Mesoscopic systems can contain a relatively large number of particles, however,
in contrast with macroscopic systems, material properties are defined by individual quantum states and not
the average over their ensemble.

98

three cases. An overview of these three problems and the thesis organization is given in the
next section.

decay

(a)

Lead

Quantum
Wire

Leads are considered as
environment

Insulator

Gate

(b)

Figure 5.1: Examples of open quantum systems: (a) an unstable nucleus breaking apart
and undergoing nuclear decay and (b) a quantum wire attached to leads acting as electron
reservoirs.

5.2

Organization

The derivation of the main mathematical framework used throughout this part of the thesis,
the effective non-Hermitian Hamiltonian, is presented in Chapter 6. Dicke superradiance

99

in quantum optics is described briefly in this chapter. It is shown that the phenomenon of
superradiance is a generic feature of all open quantum systems. The conditions under which
superradiance occurs are discussed in detail. In this chapter we closely follow the approach
used in [93, 99].
Chapters 7, 8 and 9 consider various physical systems which are all studied within the
effective Hamiltonian framework. The problem of quantum transport through two different
topologies is considered in Chapter 7. In the first section, signal transmission is studied
through a one-dimensional chain with a single qubit attached to its center. In the second
case, a two-channel quantum wire is considered. For both cases, transition to superradiance
and its effect on transport properties are discussed in detail.
Chapter 8 proposes a novel solid state charge qubit suitable for performing quantum
operations. The system consists of a one-dimensional wire with a pair of qubits embedded at
its center. It is shown that the system supports collective states localized in the left and right
sides of the wire and therefore, as a whole, performs as a single qubit. Here, the superradiant
states effectively protect the remaining internal states from decaying into the continuum and
hence increase the lifetime of the qubit. Furthermore, environmental noise is introduced by
considering random Gaussian fluctuations of electronic energies. The interplay between
decoherence and superradiance is studied by solving the stochastic Liouville equation. It is
shown that, in addition to increasing the lifetime, the emergence of the superradiant states
increases the qubit coherence.
In Chapter 9 we consider one-dimensional structures of plasmonic metallic nanospheres.
For a single sphere, solving the Maxwell equations results in quasi-stationary eigenmodes.
It is shown that systems of two or more coupled spheres can be described by the effective
non-Hermitian Hamiltonian matrix, even though our treatment of this problem is entirely
100

classical. This allows us to predict the occurrence of superradiance phenomena in systems
of plasmonic spheres. In particular, when two spheres couple through a single continuum
channel and the effect of coherent interaction between the spheres becomes noticeable, the
eigenmodes of the system fall into two distinct categories; superradiant states with enhanced
radiation and dark states with extremely suppressed radiation properties. The transmission
through plasmonic waveguides, one-dimensional chains with an arbitrary number of spheres,
is also considered. Similar to the quantum transport problem, superradiance enhances energy
transport through the plasmonic waveguides. Part 4 includes the conclusion and future
directions.

101

For want of a better term, a gas which is radiating strongly
because of coherence will be called ‘superradiant’.
Robert Henry Dicke

Chapter 6
The Effective non-Hermitian
Hamiltonian

6.1

Derivation-Feshbach Projection Formalism

Consider a quantum system described by the Hamiltonian H0 and its discrete set of eigenvectors |i that interacts with its surrounding environment. The environment is thermodynamically large and is characterized by infinitely many channels with continuous energy
spectrum |c; E . The total Hamiltonian H is then the sum of the Hamiltonians of the system,
the environment and their interaction.
Let us divide the Hilbert space of this problem into two subspaces with the help of
projection operators Q and P. The operator Q only acts on the subspace of the closed
system and the operator P acts on the environment only. Without loss of generality, one can
further assume that the projections are orthogonal, therefore P + Q = 1 and PQ = QP = 0.

102

The total Hamiltonian H is then decomposed into H = HQQ + HQP + HPQ + HPP ,
where HQQ = QHQ, HQP = QHP, HPQ = PHQ and HPP = PHP. The effective
Hamiltonian is achieved by projecting the stationary wave function in the Sschr¨odinger
equation, HΨ = EΨ, into the subspace of the closed system

Heff (E)QΨ = EQΨ,

(6.1)

where
Heff (E) = HQQ + HQP

1
H .
E − HPP PQ

(6.2)

The first term in the right-hand side of (6.2) is the Hamiltonian of the closed system QHQ =
H0 . The second term in (6.2), that incorporates the interaction with the environment, has
a simple interpretation. The operator HPQ takes the particle out of the system to the
environment, where it propagates through the environment via (E − HPP )−1 , and finally
returns back to the system through HQP . This term can be further simplified by calculating
the matrix element between two intrinsic states of the closed system |i and |j

1
H
|j =
i|HQP
E − HPP PQ
c

Aci (E )Acj ∗ (E )
dE
,
E − E + i0

(6.3)

where Aci (E) is the transition amplitude from continuum channel |c; E to internal state |i ;
Aci (E) = i|HQP |c; E . The shift in the energy, E → E + i0, is required in order to avoid
the singularity on the real energy axis. The transition amplitudes are in general energy
dependent; the channel c is open only if the running energy is greater than the channel
energy threshold and closed otherwise. Using the Sokhotski-Plemelj theorem, the integral in

103

(6.3) can be decomposed into Hermitian and anti-Hermitian parts

dE

Aci (E )Acj ∗ (E )

c

P.V.

dE

E − E + i0

Aci (E )Acj ∗ (E )

c

E−E

=

(6.4)

Aci (E)Acj ∗ (E),

− iπ
copen

where P.V. denotes the Cauchy principal value. Accordingly, two operators are defined;
∆(E), corresponding to the Hermitian component in (6.4) with matrix elements

P.V.

∆ij (E) =

dE

Aci (E )Acj ∗ (E )

c

E−E

,

(6.5)

and W (E) corresponding to the anti-Hermitian component in (6.4), with matrix elements

Aci (E)Acj ∗ (E).

Wij (E) = 2π

(6.6)

copen

Thus, in operator form, the energy-dependent effective Hamiltonian is

Heff (E) = H0 + ∆(E) −

i
W (E).
2

(6.7)

The two terms, ∆(E) and W (E), also known as the self energy, completely take the effect of
the interaction with the environment into account. The Hermitian part, ∆(E), renormalizes
the energies of the closed system. The summation in (6.5) runs over all continuum channels,
open and closed, and therefore it takes into account virtual transitions to the environment.
On the other hand W (E), which is responsible for the decay width of the energy states
of the effective Hamiltonian (6.7), only includes contributions from real transitions to the

104

continuum channels and hence it only runs over the channels open at a given energy.
In many situations, including the cases we are concerned with in this thesis, the energy
interval of interest is relatively small and the transition amplitudes, Aci (E), can be considered
to be smooth functions of energy. Consequently, the energy dependence of the amplitudes can
be neglected. Then the principal value integral in (6.5) vanishes and the effective Hamiltonian
reduces to
Heff = H0 −

i
W.
2

(6.8)

For the remainder of the thesis, the effective Hamiltonian (6.8) is used as the starting
point for investigating various open systems.

6.2

The Superradiance Phase Transition

We now discuss the superradiance transition starting with the original idea predicted by
Dicke more than half a century ago [97]. Dicke superradiance in quantum optics is described
qualitatively. It is then argued that the phenomenon of superradiance is a general feature
of open quantum systems. The existence of intrinsic states coupled to a few continuum
channels can strongly affect the dynamics of the system. In this situation a phase transition
can occur and broad superradiant states that acquire almost the entire available width can
form. Consequently the remaining states become long-lived and trapped within the system.

6.2.1

Dicke Superradiance in Quantum Optics

It is well known that when a system of dilute gas is excited, for instance by a laser, the
individual atoms or molecules of the gas release their energy due to the so-called spontaneous radiation phenomenon. This radiation is isotropic and incoherent and the intensity of
105

radiation, I, which exponentially decays with characteristic time τr , is proportional to the
number of atoms N , I ∝ N . This situation is graphically shown in Fig. 6.1.

Classical dilute gas

Intensity

Spontaneous emission

Time

Figure 6.1: Spontanous emission of a dilute classical gas initially prepared in the excited
state. The radiation is isotropic and the intensity is proportional to the number of atoms.

In 1954, Dicke predicted the phenomenon of coherent spontaneous radiation. It was
shown that a system with a relatively large number of two-level atoms that interact through
the background radiation which has a much larger wavelength than the dimensions of the
system, supports broad resonances. In his paper [97] Dicke coined the term superradiant for
such states:

“For want of a better term, a gas which is radiating strongly because of coherence
will be called superradiant.”
When the system is initially prepared in the many-body superradiant state, the emission is
anisotropic and it occurs as a burst of radiation, indicating the small characteristic decay
time, τs.r. , of the state (See Fig. 6.2). Moreover, Dicke showed that the intensity of the
radiation is proportional to the squared of the number of atoms: I ∝ N 2 . Dicke superradiant
emission is graphically shown in Fig. 6.2.
106

Quantum gas : identical two-level
atoms coupled via background
common radiation.
Intensity

Dicke superradiant radiation
Coherent spontaneous emission

Time
Figure 6.2: Dicke coherent spontanous radiation. The radiation is directional and the
intensity is proportional to the squared of the number of atoms.

6.2.2

Superradiance in Open Systems

As mentioned, the phenomenon of superradiance is a generic feature of open quantum system.
In order to see this, it helps to consider the statistics of the quasi-stationary eigenenergies
of the effective Hamiltonian (6.8)
i
Er = Er − Γr ,
2

(6.9)

where Γr is the width of the state and related to the lifetime by τr = /Γr . The positiveness
of Γr is guaranteed by the Cholesky factorized form of W in (6.6) which makes W a positive
definite matrix. We define the quantity Ξ that parameterizes the strength of interaction with
the external world, thus
i
Heff = H0 − ξ W.
2

(6.10)

Because the framework is exact and no approximation was used, Ξ can take arbitrarily
small values representing weak interactions or extremely large values representing strong
interactions with the external world. For weak interactions, when Ξ is small, the anti-

107

Hermitian component, W is a perturbation to H0 . The complex eigenenergies are then
narrow resonances with an almost uniform width distribution [89, 99]. In the opposite limit
of strong interaction, the anti-Hermitian component becomes the dominant term and H0
is the perturbation. It is clear from (6.6) that the rank of W is equal to the number of
open channels which is normally much smaller than the number of intrinsic states of the
closed system. Therefore a few states become giant resonances that steal the entire width
and the remaining states become long-lived and effectively decoupled from the environment.
Due to the resemblance of this phenomenon to the Dicke superradiance in quantum optics,
we term the broad, short-lived resonances as superradiant states and the narrow, shortlived resonances as subradiant states. It was shown in [93, 99] that the superradiance phase
transition occurs when Γ /D ≈ 1, where Γ is the average widths of the energies in (6.9),
and D is the mean level spacing of the closed system.

6.3

Scattering Matrix and Transmission

The non-Hermitian Hamiltonian formalism provides a suitable platform for studying scattering properties of open quantum systems. Consider process a → b, where a particle enters
from channel a, which is coupled to the intrinsic state of the closed system |i with the
coupling amplitude Aai , propagates through the system with possible virtual excursions to
the continuum and back, and finally exits from the intrinsic state |j to the decay channel b
with the amplitude Abj . The amplitude of this process is given by [99]:

Z a→b (E) =

Ab∗
i
i, j

108

1
E − Hef f

Aaj .
ij

(6.11)

It was shown in [93] that the corresponding scattering matrix, S ba = δ ba −iZ ba , is unitary [93]
and the transmission coefficient is

2

T ba (E) = Z ba (E) .

(6.12)

It might be convenient to perform the transformation to the (biorthogonal) basis |r
of eigenfunctions of the effective Hamiltonian. The complex energies Er = Er − (i/2)Γr
correspond to the poles of the scattering matrix, while the process amplitude (6.11) still has
factorized residues transformed to the eigenbasis,

Z ba (E)

A˜br A˜ar
.
=
E − Er
r

(6.13)

We can note parenthetically that this description can be treated as a simple superposition
of interfering resonances only approximately, namely if the energy dependence of continuum
amplitudes Aan is neglected as it is done in our consideration. In this approximation the time
decay curve of a single isolated resonance |r would be pure exponential with the width Γr .

109

It has today occurred to me that an amplifier using
semiconductors rather than vacuum is in principle possible.
William Shockley’s Laboratory notebook, 29 Dec 1939

Chapter 7
Quantum Transport

7.1

Introduction

Thanks to the development of nanotechnology, the preparation of low-dimensional assemblies
of nanoparticles becomes a routine experimental task [109]. This renews the attention to
the study of quantum properties of relatively simple low-dimensional mesoscopic systems
which reveal a rich physical behavior. In this context, the exactly solvable models as tightbinding one- or two-dimensional chains coupled to adatoms or impurities are of special
interest [106, 110–112]. In light of this, quantum transport in two different nanostructure
topologies are considered in this section.

110

7.2

Quantum Signal Transmission Through a SingleQubit Chain

In what follows we consider a system with a two-level atom of an impurity (qubit) inserted
into an open periodic chain. The model is simple enough to allow for the exact solution; at
the same time the model turns out to be rich enough to demonstrate interesting physics of the
signal transmission. An analytical consideration is supplemented by the detailed numerical
study for the finite chains
We first consider a closed chain of 2N identical cells with the nearest neighbor hopping
interaction. Two arms of the chain are connected through the central cell occupied by
a two-level atom (qubit). We find the energy spectrum of the system that consists of a
normal band of delocalized Bloch standing waves and two additional states outside the band
corresponding to the excited and the ground states of the qubit. We study the evolution of
the energy spectrum as a function of the chain-qubit coupling strength.
Next, we describe the energy spectrum of the open system coupled to the continuum
through its edge states. The former stationary states acquire decay widths that change as a
function of the coupling constants. The decay widths of the qubit states are small compared
to those of the Bloch waves. The qubit states remain essentially localized at the qubit even
when the coupling of the chain to the continuum becomes sufficiently strong to allow for the
formation of super-radiant states.
In addition, we study the transmission through the chain as a function of the coupling
parameters. In both limits of the edge continuum coupling being weak and strong compared
to the coupling between the qubit levels, the resonances are well isolated all having narrow
decay widths and perfect transmission at resonance energies. When both couplings are of
111

comparable strength, the resonances are overlapped and the transmission is below the perfect
level. The results are summarized in the Discussion.

7.2.1

Closed Chain

7.2.1.1

Symmetric and Antisymmetric Modes

We consider a linear chain of 2N identical cells numbered as n = −N, −(N − 1), ..., −1 and
n = 1, ..., N − 1, N , while the central cell, n = 0, is occupied by a qubit, a two-level atom
with states |0 and |e , excitation energy e − 0 = ∆, and matrix element λ of the qubit
excitation. The energy 0 is the level position in all cells; for simplicity we put the lower
level of the isolated qubit at the same position. Introducing the hopping (tunneling) matrix
element v between the neighboring cells, we come to the Hamiltonian of the closed chain:

Hnn = 0 ,

Hn,n+1 = Hn+1,n = v,

Hee = ∆,

n = −N, ..., 0, ..., N ;

H0e = H0e = Îť,

(7.1)

(7.2)

where all matrix elements can be considered as real.
Before introducing the coupling to the outside world, we briefly characterize the solution
for the closed chain. A general stationary state |E with energy E can be presented as a
superposition
N

|E =

cn (E)|n + b(E)|e .

(7.3)

n=−N

The boundary conditions for the chain closed at the edges are c−N −1 = cN +1 = 0. The

112

coefficients of the superposition (7.3) satisfy the obvious equations:

(E − 0 )cn − v(cn−1 + cn+1 ) = 0,

n = 0,

(7.4)

(E − 0 )c0 − v(c−1 + c+1 ) = λb,

(7.5)

(E − ∆)b = λc0 .

(7.6)

Eqs. (7.5) and (7.6) can be treated as boundary conditions for the two chains (left and right)
implying that
c0 (E) =

v(E − ∆)
(c−1 + c1 ).
(E − 0 )(E − ∆) − λ2

(7.7)

Following the same procedure as for a single chain [99], see also Appendix in Ref. [103],
we find that the solutions on both sides of the qubit have the form

cn =



n + BΞ n , n < 0;
 AΞ+
−

(7.8)


 A Ξ n + B Ξ n , n > 0.
−
+
The amplitudes Ξ¹ are given by

Ξ¹ =

1
E− 0±
2v

(E − 0 )2 − 4v 2 ,

(7.9)

and ξ+ ξ− = 1. The edge conditions determine

B = −A

ξ−
Ξ+

N +1

,

B = −A

Ξ+
ξ−

N +1

.

(7.10)

The central point determines two classes of solutions. The antisymmetric states, cn +

113

c−n = 0, have c0 = 0, so that these wave functions are decoupled from the excited qubit
that lives on the excited level E = ∆ (in this situation the qubit can be excited or deexcited
only by an additional external coupling). Because of c0 = 0 and interaction only between
the neighboring cells, the two sides of the chain are also decoupled from each other so that
such modes do not take part in the transport through the whole chain. The spectrum Eq of
2N +2 = 1 that can be parameterized
the antisymmetric states is determined by the roots of ξ−

by the even-number quantized quasimomentum q, or by the phase ϕq ,

Eq = 2v cos ϕq ,

ϕq =

πq
,
2N + 2

q even,

(7.11)

and the local amplitudes (7.8) of the wave functions are

cn (q) = iq

1
sin(nϕq ).
N +1

(7.12)

The energies (7.11) are inside the Bloch band (−2v, +2v).
The symmetric solutions, c1 = c−1 , involve the qubit into dynamics and the corresponding roots are determined by

2N +2 =
Ξ+

λ2 + (E − ∆) (E − 0 )2 − 4v 2
.
λ2 − (E − ∆) (E − 0 )2 − 4v 2

(7.13)

To derive this relation, it is convenient to use eq. (7.4) and connect the amplitude c1 , or
c−1 , to the central amplitude c0 ,
N − ξN
Ξ+
−
c1 = c0 N +1
.
N +1
ξ+ − ξ−

114

(7.14)

With the qubit disconnected, λ → 0, we have the missing in eq. (7.11) q-odd part of the band
spectrum with the amplitudes (7.12) where sin(nϕq ) is changed to cos(nϕq ). Depending on
the relative position of ∆ with respect to the band width 2v, there are two cases; below
throughout the section we set 0 = 0.

7.2.1.2

Case ∆ > 2v

In this case, the upper level is always outside the band, E > 2v; the lower level also leaves
the band at a finite value of λ when E < −2v. The full spectrum of twelve energy levels for
N = 5 is shown as a function of λ in Fig.7.1. The upper state starts at E = ∆ and grows
approximately linearly with Îť. At small Îť, this is the state with the excited qubit and only
weak admixtures of intrinsic sites.
With increasing Îť, the symmetric and antisymmetric states inside the band become
degenerate, while the “excited” (upper) wave function is spread almost equally over two
states of the qubit. The orthogonal combination of the excited and ground state of the qubit
gives the lower state (at sufficiently large values of Îť). In this limit the picture effectively is
of Rabi oscillations between the qubit levels with only a small probability of hopping along
the chain. The chain then is almost decoupled from the qubit. This evolution of the two
wave functions is illustrated by Fig. 7.2.

7.2.1.3

Case ∆ < 2v

Here we consider the spectrum of the energy levels as a function of Îť when the qubit excitation
energy is inside the band (∆ < 2v). The full spectrum of twelve energy levels for N = 5,
∆ = 0.5 is shown as a function of λ in Fig. 7.3. When λ = 0, all the energies are inside
the band; with increasing Îť, both the upper and lower energies move out of the band (as

115

Energy

6
5
4
3
2
1
0
−1
−2
−3
−4
0

Upper state
Lower state
Antisymmetric states
Symmetric states

0.5

1

1.5

2

2.5

Îť

3

3.5

4

3

4

5

Îť=1

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

3

4

5

Îť=3

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

3

4

5

Îť=10
0
−5 −4 −3 −2 −1 0 1 2
Cell number

3

4

5

Probability

0.5

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

0.5

Probability

Probability

0.5

Îť=0.1

0.5

Probability

Probability

0.5

1

0.5

Probability

Probability

0.5

Probability

Figure 7.1: Energy levels for a system of the closed chain of eleven sites and the excited
qubit state in the middle as a function of the qubit coupling strength Îť; the hopping
amplitude is set to v = 1, and the excitation energy of the qubit ∆ = 2.5.

0.5

1
2N identical cells
Excited state

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

0
−5 −4 −3 −2 −1 0 1 2
Cell number

3

4

5

3

4

5

3

4

5

3

4

5

Figure 7.2: Squared components of the lower (left column) and upper (right column)
eigenstates as a function of the qubit excitation strength Îť, for a chain N = 5 and v = 1.

116

Energy

5
4
3
2
1
0
−1
−2
−3
−4
−5
0

Upper state
Lower state
Antisymmetric states
Symmetric states

0.5

1

1.5

2

Îť

2.5

3

3.5

4

Figure 7.3: Energy levels for a system of the closed chain of eleven sites and the excited
qubit state as a function of the qubit coupling strength Îť; the hopping amplitude is set to
v = 1, and the excitation energy of the qubit ∆ = 0.5.
before). As Îť increases the symmetric states and the qubit level inside the band merge with
corresponding antisymmetric states.
Fig. 7.4 shows the eigenstates corresponding to the lowest and highest eigenenergies,
upper and lower wave functions. Unlike in the case B, when Îť = 0.1, in the upper wave
function there is a small admixture of the excited qubit state.
The special situation emerges if ∆ is degenerate with the energy of one of the antisymmetric states. In this case, c0 = 0, however b is not necessarily vanishing. Eq. (5) results
in c1 + c−1 = −(λ/v)b. As a consequence we obtain here two eigenstates which do not fall
into the symmetric or anti-symmetric category. In Fig. 7.5 we show the two eigenstates for
√
N = 5, v = 1, ∆ = 3. As λ increases the two state become localized either in the left
chain or in the right chain.

117

Probability

Probability

0.5

0.5

5

Probability

Probability

0.5

5

0.5

5

Probability

Probability

0.5

0.5

0.5

5

Probability

Probability

Îť=0.1
0.5

1

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

3

4

Îť=1

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

3

4

Îť=3

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

3

4

Îť=10

0
−5 −4 −3 −2 −1 0 1 2
Cell number

3

4

1

2N identical cells
Excited state

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

0
−5 −4 −3 −2 −1 0 1 2
Cell number
1

0
−5 −4 −3 −2 −1 0 1 2
Cell number

3

4

5

3

4

5

3

4

5

3

4

5

Figure 7.4: Squared components of the lower (left column) and upper (right column) of the
wave functions as a function of the qubit excitation strength λ, for a chain N = 5, ∆ = 0.5
and v = 1.
7.2.1.4

Large N Chain

For two qubit levels outside of the band we can derive in case of large N a simple expression.
For the upper qubit level, E > 2v, we always have ξ+ > ξ− . Hence, for large N we may
N in eq. (7.14) to obtain c = c /Ξ = c Ξ . Accordingly, for the lower qubit level,
neglect ξ−
1
0 +
0 −
N in eq. (7.14). This allows us to find from expressions (7.5)
E < −2v, we may neglect ξ+

and (7.6) equations for the two levels genetically related to the qubit:

E 2 − 4v 2 = ±

Îť2
,
E−∆

where plus (minus) corresponds to the upper (lower) qubit level.
The wave functions of these states are given by

118

(7.15)

|Ψ∓ = B∓


N


|n|

ξ∓ |n +

−N




Îť
|e ,
E−∆ 

(7.16)

where the normalization constant B∓ is defined by

2 1+
B∓

ε2∓
−
4
= 1,
ε2∓ − 1
(E − ∆)2
Îť2

(7.17)

with the same identification of the signs.
In fact, the asymptotic expression (7.15) is good even for N = 5. For example, for
E/2v = 1.1 we have the ratio (ξ− /ξ+ )5 = 0.01, while at N = 10 the ratio (ξ− /ξ+ )10 is of
the order of 10−4 . Therefore, for the qubit levels positioned out of the band, the expression
(7.15) is good for any N ≥ 5 and for any value of ∆. As before, these two states move out
of the band as Îť increases. For the states inside the band, there are no simple expressions
for large N . However, their behavior is similar to that shown in Fig.7.1: as Îť increases all
symmetric states merge with antisymmetric states.

7.2.2

Open Chain

Now we assume that the edge states are coupled to the outside world by ideal leads and
the situation becomes identical with what can be described by the effective non-Hermitian
Hamiltonian (6.8),
Heff = H −

i
W,
2

(7.18)

acting only in the intrinsic space of the closed system. The anti-Hermitian part W is given
√
by equation (6.6). In our case, the only non-vanishing amplitudes are AL
ÎłL and
−N =
√
AR
N = ÎłR describing the coupling of the left (right) edge to the left (right) decay channel.
119

Probability

5

Probability

5

Probability

5

Probability

Probability
Probability
Probability
Probability

5

1

Îť=0.1
0.5
0
−5

0
Cell number

1

0.5 Îť=1
0
−5

0
Cell number

1

Îť=3
0.5
0
−5

0
Cell number

1

Îť=10
0.5
0
−5

0
Cell number

1
0.5
0
−5

2N identical cells
Excited state

0
Cell number

5

0
Cell number

5

0
Cell number

5

0
Cell number

5

1
0.5
0
−5
1
0.5
0
−5
1
0.5
0
−5

Figure 7.5: Squared components of two wave functions for degenerate level as a function of
√
the qubit excitation strength λ, for a chain N = 5, ∆ = 3 and v = 1.
In the case of a linear chain we allow only left and right decays through the edge states (it
would be also interesting to study the decoherence through the coupling with many “random”
weak channels connected to the intrinsic sites). The factorized nature of the operator W
(dictated essentially by requirements of unitarity of the scattering matrix in the channel
space, see for example [113]), shows that this operator has only few non-zero eigenvalues
their number being equal to the number of open channels. The corresponding eigenstates
of W are obviously just the edge states directly coupled to the continuum. Therefore it is
sufficient in our scheme to introduce two complex energies, R and L for the states at the
edges, where L,R = 0 − (i/2)γL,R . Such an open system without a qubit was analyzed in
Refs. [99, 101, 103, 105]. In this approach the states directly coupled to the continuum play
the role of doorways [114], and the remaining states can get their widths (finite lifetimes)
only through their coupling to the doorways.
120

The typical situation for the chain without a qubit is evolving as a function of parameters
ÎłL,R compared to the level spacing D in the closed system. At weak coupling, every intrinsic
state becomes a resonance with a small decay width determined by the overlap of the Bloch
state with the edges; the final width distribution has, for ÎłL = ÎłR , a maximum in the
center of the band. In the limit of strong continuum coupling, we have a super-radiant
situation when the central state in the spectrum accumulates almost the entire width while
the remaining states become very long-lived (trapped). In the case of ÎłL = ÎłR there occur
two super-radiant transitions [103] with the maximum of the signal transmission in between.
In the site representation, the super-radiant states with energies in the center of the band are
concentrated at the edges [99]. This picture survives also the possible presence of disorder
in the intrinsic wells [101].
The results for the chain with the qubit are illustrated by the series of graphs, where the
chain consists of 5+1+5=11 cells with the qubit in the middle, altogether 12 intrinsic states.
Here we diagonalize the effective Hamiltonian in the doorway representation: the continuum
coupling occurs only at the edges which serve as doorways, and the matrix elements of the
anti-Hermitian part of the effective Hamiltonian (7.18), which couples the states |q with the
outside world, are given in the band representation by

Wqq = γL c−N (q)c−N (q ) + γR cN (q)cN (q ).

(7.19)

Fig. 7.6 shows the evolution (as a function of Îť) of the resonance energies for the case of
weak continuum coupling, ÎłL = ÎłR = 0.1 (the scale is fixed by the band width, v = 1). At
small λ and ∆ > 2, we have the parabolic distribution of widths with the maximum at the
center of the band, as known from previous studies [99,101], and the decoupled excited qubit

121

Decay Width Γ

Decay Width Γ

0.03 Îť=0.1
0.02
0.01
0
−4

−2

0

2

4

6

0.03 Îť=1
0.02
0.01
0
−4

−2

0.03 Îť=3
0.02
0.01
0
−4

−2

0

2

0

2

4

6

4

6

Energy
Decay Width Γ

Decay Width Γ

Energy

4

6

Energy

0.03 Îť=5
0.02
0.01
0
−4

−2

0

2

Energy

Figure 7.6: Resonance complex energies (eigenvalues of the effective Hamiltonian for weak
continuum coupling, Îł = 0.1) evolve as a function of the qubit excitation strength Îť. Two
qubit states are effectively decoupled from the chain and move along the real energy axis.
state above the band with zero width. As Îť increases, the symmetric and antisymmetric
states merge becoming effectively decoupled from the qubit. The upper qubit state and
emerging outside the band the lower qubit state are still almost stationary (Rabi regime).
They are moving along the real energy axis being repelled by the band.
The situation changes when we come to the strong continuum coupling, Fig. 7.7. Here
we again follow the evolution as a function of Îť but at Îł = 20. At small Îť = 0.1, the two
coinciding super-radiant states are formed in the center of the band (here we keep ÎłL = ÎłR ),
and the remaining ten states are trapped, effectively returning to the non-overlap regime
(the left lower plot shows, at a much smaller width scale, the parabolic width distribution).
With increase of Îť, the super-radiant states survive, while the qubit states again are repelled
by the band along the real energy axis having still very small widths.
It is instructive to take a look of the width evolution as a function of the continuum
coupling strength Îł. The typical process is presented by Fig. 7.8, where the trajectories of
122

Îť=0.1

Decay Width Γ

Decay Width Γ

20

10

0
−2

−1

0

1

2

20

Îť=5

10

0
−4

−2

0.1 Îť=0.1

0
−2

−1

0

0

2

4

6

4

6

Energy
Decay Width Γ

Decay Width Γ

Energy

1

2

Energy

0.1 Îť=5

0
−4

−2

0

2

Energy

Figure 7.7: Resonance complex energies (eigenvalues of the effective Hamiltonian for strong
continuum coupling, Îł = 20) show segregation of two super-radiant states in the middle of
the band from ten trapped states which include the strongly localized states of the qubit.
all twelve complex poles in the lower half of the complex plane are shown. In the limit of
very weak continuum coupling, part (a) of this figure, the width distribution is parabolic,
being proportional, as a function of real energy, to the group velocity of the band states. It is
transformed with increase of Îł. All widths, except for the super-radiant states in the center
of the energy band, turn back after reaching their maximum values. Their corresponding
trajectories are almost symmetric with respect to their maxima which is typical for the
phenomenon of super-radiance in a space of fixed dimension. Indeed, after the segregation of
the super-radiant state(s), the remaining trapped states are essentially in the same situation
as they were in the beginning of the process; this symmetry for the finite dimension of intrinsic
space is a characteristic feature [103] that appears also in the statistical distribution of
neutron widths for thermal-energy neutron resonances [115]. Part (b) of the figure selects, on
a detailed energy scale, the complex-plane evolution of the excited qubit state that becomes
extremely long-lived. The details of interference between neighboring resonances were also
123

1

10

(a)
0

Γ

10

−1

10

−2

10
−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

E

−4

Γ

10

−5

(b)

10

−6

10

2.8215

2.82152

2.82154

2.82156

E
Figure 7.8: Complex-plane trajectories of eigenstates of the effective Hamiltonian; the
parameter values are v = 1, λ = 0.8, ∆ = 2.5. The arrows show the direction of the
evolution as Îł changes from 0 to 10. Panel (a) shows the behavior of 12 states in the chain
while panel (b) singles out the state genetically connected to the excited qubit state
located outside the band.
discussed repeatedly in the context of the electron conduction in nano-scale systems, see for
example [116].
It was noticed long ago [117] that, in the description of an open system with the aid of the
effective Hamiltonian, the Hermitian and non-Hermitian parts of the interaction act in the
opposite way. The real (Hermitian) perturbation repels the levels but, through the mixing
mechanism, attracts the widths of unstable states. Contrary to that, the imaginary (nonHermitian) interaction through the continuum repels the widths (the road to superradiance
and trapping) but attracts real energies of resonances. This attraction is seen in Fig. 7.9 for
a larger value of Îť: as the continuum coupling Îł increases along the road to super-radiance,
the real energies of the poles move to the middle of the band, where the super-radiant states
are located.

124

1

10

(a)
0

Γ

10

−1

10

−2

−1.5

−1

−0.5

0.5

1

1.5

2

(c)

(b)

Γ

−6

Γ

10

0

E

−2

−2.204

−2.202

−2.2

10

3.843586

E

3.84359

E

Figure 7.9: Complex-plane trajectories of eigenstates of the effective Hamiltonian; the
parameters values are v = 1, λ = 2.1, ∆ = 2.5. Ten states, including the super-radiant at
the edges of the chain, are moved closer inside the band, panel (a), two states mainly
localized at the qubit have energies outside the band and essentially interact only with each
other having a large lifetime with respect to tunneling through the chain, panels (b) and (c)
.

125

1

10

(a)
0

Γ

10

−1

10

−2

10
−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

E
(c)

(b)

Γ

Γ

−8

10

−10

10

−10

10
−7.016353

−7.016352

9.438383

E

E

Figure 7.10: Complex-plane trajectories of eigenstates of the effective Hamiltonian; the
parameters values are v = 1, λ = 8, ∆ = 2.5. Ten states, including the super-radiant at
the edges of the chain, are moved closer inside the band, panel (a). The qubit states, the
ground state, (b), and the excited state, (c), are essentially decoupled.
Finally, the limit of very strong coupling between the qubit levels is shown in Fig. 7.10
where Îť = 8. With increasing continuum coupling Îł, as antisymmetric and symmetric
roots merge, we see, panel (a), the evolution of pairs of states (ten of them inside the band
including the super-radiant ones). The time arrow of this evolution is again in the direction
of attraction for the real energies of resonances. The qubit states, the ground state, (b),
and the excited state, (c), are essentially decoupled. They do not shift and only gradually
increase their (still small) decay widths as Îł increases.
Next we briefly consider the case when qubit energy ∆ is inside the band. In Fig. 7.11
we show the resonance complex energies for ∆ = 0.5 and for weak continuum coupling,
Îł = 0.1, as a function of the qubit excitation strength Îť. We have already seen that at
small λ there are two eigenstates with eigenenergies close to ∆ = 0.5. Their decay width
is smaller since there is a strong contribution from the excited qubit state to these two.
126

Decay Width Γ

Decay Width Γ

0.03 Îť=0.1
0.02
0.01
0

−5

0

5

0.03 Îť=1
0.02
0.01
0

−5

0.03 Îť=3
0.02
0.01
0

−5

0

0

5

Energy
Decay Width Γ

Decay Width Γ

Energy

5

0.03 Îť=5
0.02
0.01
0

−5

Energy

0

5

Energy

Figure 7.11: Resonance complex energies (eigenvalues of the effective Hamiltonian) for
∆ = 0.5 and for weak continuum coupling, γ = 0.1 as a function of the qubit excitation
strength Îť.
As Îť increases, the eigenenergies inside the band merge,so that the two qubit states are
effectively decoupled from the chain moving along the real energy axis. Resonance complex
energies for strong continuum coupling γ = 20 are shown in Fig. 7.12 for ∆ = 0.5. Here the
situation is qualitatively identical with that in Fig. 7.7. At small Îť = 0.1, the two coinciding
super-radiant states are formed in the center of the band, and the remaining ten states are
trapped. With the increase of Îť, the superradiant states survive, while the qubit states again
are repelled by the band along the real energy axis having still very small widths.

7.2.3

Transmission Through The Chain

Here we briefly consider the transmission of an external signal through the chain with the
inserted qubit, similarly to the consideration made earlier for the uniform chain or multidimensional lattices [103] and for the star graph [108]. As discussed earlier in Chapter 6, the

127

Decay Width Γ

Decay Width Γ

20

Îť=0.1

10
0
−2

−1

0

1

2

20

Îť=5

10
0
−5

0.1 Îť=0.1

0
−2

−1

0

0

5

Energy
Decay Width Γ

Decay Width Γ

Energy

1

2

0.1 Îť=5

0
−5

Energy

0

5

Energy

Figure 7.12: Resonance complex energies (eigenvalues of the effective Hamiltonian) for
∆ = 0.5 and for strong continuum coupling, γ = 20 as a function of the qubit excitation
strength Îť.
internal propagation of the signal of given energy E in the open system is described by the
propagator
G(E) =

1
,
E − Heff

(7.20)

where Heff is the effective Hamiltonian (7.18) with the imaginary part (6.6) that describes
multiple excursions of the signal into continuum and back. Our simple geometry has two
open channels, left and right (labeled in eq. (7.19) L and R). The full amplitude Z LR and
the transmission coefficient T LR of the process between channels were given earlier by (6.11)
and (6.12), respectively.
For the calculation of transmission we adopt the approach of Ref. [103]. For the open
chain of Sec. 3, the transmission is determined by the edge couplings which we again assume
here to be equal, ÎłL = ÎłR = Îł. Similarly to Ref. [103], the transmission coefficient (6.12)

128

can be written as
T RL (E)

=

T LR (E)

(γ/v 2 )(E − ∆)

=

2N +2
r=1 [(E

− Er )/v]

2

.

(7.21)

Below we show the transmission results for the chain of N = 5 (twelve intrinsic states)
and various combinations of the parameters. The resulting picture is determined by the
counterplay of the trend to super-radiation and decoupling of the qubit.
Starting with the weak qubit excitation amplitude, Fig. 7.13 for Îť = 0.1, we follow the
evolution of the transmission as a function of the continuum coupling Îł. At small Îł = 0.1,
panel (a), we see twelve isolated resonances all having narrow decay widths; one of them
is outside the energy band as we discussed earlier. When Îł is growing, panels (b) and (c),
the resonances start overlapping. When two super-radiant states merge at Îł = 2.4, panel
(c), they disappear from the transmission spectrum, so that panel (d) for Îł = 4 shows ten
separated resonances. At each resonance the transmission is perfect, T = 1.
Next four panels, Fig. 7.14, correspond to the intermediate value Îť = 2. Again the case
of weak continuum coupling, panel (a), Îł = 0.1, reveals twelve resonances; now the two
states associated with the qubit are outside the energy band, while all resonances still show
perfect transmission. At Îł = Îť = 2, panel (b), the competition between the transmission
through the chain and Rabi dynamics of the qubit leads to an almost random pattern of
overlapping resonances with transmission below perfect, similarly to Ericson fluctuations
[107] or universal conductance fluctuations [118]. The relation between those well known
pictures and necessary changes due to the effects of unitarity in exact theory were discussed
in Ref. [102]. At large Îł

Îť, panels (c) and (d), the continuum coupling prevails leading

to the narrow resonances coming from trapped intrinsic states.
Finally, the large value of the qubit excitation strength, Îť = 5, changes the transmission

129

Transmission

Îł=0.1

0.5
0

−2

0
Energy

2

1 (c)

Transmission

Transmission
Transmission

1 (a)

Îł=2.4

0.5
0

−2

0
Energy

2

1 (b)
Îł=1.8

0.5
0

−2

1

0
Energy

2

0
Energy

2

(d)
Îł=4

0.5
0

−2

Transmission

1 (a)
Îł=0.1

0.5
0
−4

−2

0
Energy

2

4

1 (c)

Transmission

Transmission

Transmission

Figure 7.13: Evolution of the transmission for small Îť = 0.1 as a function of the continuum
coupling parameter ÎłL = ÎłR = Îł, from 12 isolated resonances (one outside the energy
band) through overlap and super-radiance to ten resonances corresponding to trapped
states. As earlier, ∆ = 2.5, v = 1.

Îł=8

0.5
0
−4

−2

0
Energy

2

4

1 (b)
Îł=2

0.5
0
−4

−2

0
Energy

2

4

−2

0
Energy

2

4

1 (d)
Îł=20

0.5
0
−4

Figure 7.14: The same as Fig. 7.13, with Îť = 2.

130

Transmission

Îł=0.1

0.5
0

−5

0
Energy

5

1 (c)

Transmission

Transmission
Transmission

1 (a)

Îł=10

0.5
0

−5

0
Energy

5

1

(b)
Îł=2

0.5
0
1

−5

0
Energy

5

0
Energy

5

(d)
Îł=40

0.5
0

−5

Figure 7.15: The same as Fig. 7.13, with Îť = 5.
picture at not very large Îł, Fig. 7.15. At small continuum coupling, Îł

Îť, panel (a), the

two states associated with the qubit produce two resonances outside the band, with transmission equal to 1. At Îł = 2, panel (b), we observe some kind of an intrinsic resonance
between propagation and internal oscillations which, along with the emergence of superradiance, almost kills the transmission at other energies within the band. With further growth
of Îł, panels (c) and (d), the perfect transmission through trapped states, including those
associated with the qubit, is gradually restored. This abundance of possible regimes opens
the way to various applications.
At the conclusion of this section we show the transmission across the chain for the case
when ∆ is inside the band, Fig. 7.16, and for the case of large N , Fig. 7.17.

131

Transmission

Îł=0.1

0.5
0

−2

0
Energy

2

1 (c)

Transmission

Transmission
Transmission

1 (a)

Îł=2.4

0.5
0

−2

0
Energy

2

1

(b)
Îł=1.8

0.5
0

−2

0
Energy

2

0
Energy

2

1 (d)
Îł=4

0.5
0

−2

Figure 7.16: Transmission through the chain for N = 5, λ = 0.1, ∆ = 0.5 as function of
continuum coupling Îł

(a)
Îł=0.1

Transmission

1
0.5
0
1

−2

0
Energy

2
Transmission

Transmission

Transmission

.

(c)
Îł=2.4

0.5
0

−2

0
Energy

2

1

(b)
Îł=1.8

0.5
0
1

−2

0
Energy

2

0
Energy

2

(d)
Îł=4

0.5
0

−2

Figure 7.17: Transmission through the chain for N = 20, λ = 0.1, ∆ = 2.5 as function of
continuum coupling Îł
.

132

7.2.4

Discussion

Using the formalism of the effective non-Hermitian Hamiltonian, we studied a model of
quantum signal transmission through a linear periodic chain with a qubit placed at the
center. For the closed chain, the intrinsic eigenstates form two classes, antisymmetric that
includes standing waves with the excluded center site and two independent subchains, and
symmetric that reveals the qubit dynamics with two additional eigenstates. When the system
is coupled to the environment at its entrance and exit points, we have found the spectrum of
quasistationary states characterized by complex energies and finite lifetimes. Two of those
states are genetically related to the ground and excited state of the qubit with real energy
outside the Bloch band.
The most interesting feature is the stability of states related to the qubit, small decay
widths and correspondingly long lifetimes. Further, these states are only weakly perturbed
when the coupling of the chain to the continuum becomes strong. In the limit of strong
coupling, the qubit states and remaining trapped Bloch states are practically shielded from
the external world by the two superradiant states localized at the edges of the chain. The
stability of the qubit states and the possibility to switch on and off the access to them
suggests that the simple configuration considered above may serve as a building block of a
quantum computer.
We discussed also the transmission through the chain as a function of the coupling
strength to the continuum. In the limits of weak and strong continuum coupling (as compared to the excitation amplitude of the qubit) the chain reveals well separated narrow
resonances with perfect transmission at corresponding energy. In the intermediate regime,
when the continuum coupling and the excitation strength of the qubit are comparable, the

133

resonances overlap with the transmission below the perfect level.
There are many possibilities to enrich this prototypical model. For a realistic situation,
for example a chain of quantum dots, one should carefully determine the lifetimes of the
qubit states. The geometry of the system can be made more complicated in various ways
including the transition to more-dimensional schemes. More qubits and more branches can be
added approaching a complicated network. Another direction is to include electron-phonon
interactions in order to consider vibrational effects [122]. It would be also interesting to
extend the ideology of an open quantum system to the study of coherent photon transport
in continuous waveguides [119] and circuit quantum electrodynamics [95, 120, 121].

7.3

Two-Channel Quantum Wire

In this section, we study superradiance and its effect on transport properties in a quantum
wire with a different geometry: a one-dimensional chain of two level systems where the edges
are coupled to the outside world presented by a continuum of channels. The two-channel
wire has served as a model in [123] where superconductivity in ladder structures was studied
and in [110] where a two-dimensional chain was coupled to an adatom. This specific system
was not earlier analyzed from the viewpoint of superradiance.
The steps taken to attack this problem are in general similar to the previous section. First,
we consider the isolated wire, a closed chain of N identical two-level cells with the nearest
neighbor hopping interaction between the ground and excited states of neighboring cells.
A closed form solution describes the two energy bands of the system. Several symmetries
associated with the eigenvectors of the system are explained as well. Similar to the single
qubit chain case, the system is then opened through the coupling of the left and right

134

edges of the chain to the continuum. As a result of the coupling the eigenenergies acquire
an imaginary part. The evolution of the complex energies as a function of the coupling
strength is studied in detail. The superradiance transition emerges as in the previous case.
It is shown that the width is uniformly distributed at weak coupling. However, the picture
changes at strong couplings: the states are not uniformly broadened. The superradiant
states, their number being equal to the number of intrinsic doorway states directly coupled to
the continuum, are greatly broadened while the remaining states acquire only small widths.
Next, the transport properties of the system are studied. It is shown that the transition
between the weak and the strong coupling regimes i.e. the superradiance transition, greatly
influences the scattering properties of the system, particularly the transmission coefficient.
Energy-dependent coupling to the continuum due to the presence of the decay threshold is
considered as well. It is shown that superradiance plays a significant role, even when the
energy levels are in proximity to their cutoff thresholds.

7.3.1

The Isolated Two-Channel Wire

The wire of the model consists of a periodic chain of N identical two-level cells. The energy
levels of the ground state |n, g and the excited state |n, e of each cell are

and

+ ∆,

respectively. The matrix element for excitation between the ground and excited state of
each cell is denoted Îť, and the hopping amplitudes between the neighboring ground and
excited states are v1 and v2 , respectively. Fig. 7.18 shows a schematic of the system under
study.
The Hamiltonian of the closed chain is

135

𝒗𝟐 = ⟨𝒏, 𝒆𝒔|𝑯𝒐 𝒏 + 𝟏, 𝒆𝒔

𝑛 = 1, 𝑒𝑠

Δ+

𝑛 = 𝑁, 𝑒𝑠

𝜆 = ⟨𝒏, 𝒈𝒔|𝑯𝒐 𝒏, 𝒆𝒔

𝑛 = 1, 𝑔𝑠



𝒗𝟏 = ⟨𝒏, 𝒈𝒔|𝑯𝒐 𝒏 + 𝟏, 𝒈𝒔

𝑛 = 𝑁, 𝑔𝑠

Figure 7.18: Schematic of the two-channel wire.

N

H0 =

2

n=1

|n, g n, g| +

+∆
|n, e n, e|
2

+ λ |n, g n, e| + v1 |n, g n − 1, g|
+ v2 |n, e n − 1, e| + h.c. .

(7.22)

In order to diagonalize the Hamiltonian a stationary state with energy E is considered
as a superposition of localized states,

N

an (E) |n, g + cn (E) |n, e

|q =

.

(7.23)

n=1

The Schr¨odinger equation with the Hamiltonian given in eq. (7.22) determines a set of
two coupled linear recurrence equations for the amplitudes an and cn :

( + ∆)cn + λan + v2 (cn−1 + cn+1 ) = Ecn ,

an + λcn + v1 (an−1 + an+1 ) = Ean ,

136

n = 1, 2, .., N,

n = 1, 2, .., N.

(7.24)

(7.25)

Ordinary decoupling of the equations would lead to a fifth order recurrence relation.
However, the decoupled equations can be simplified with the realization of a symmetry
resulting from the parallel ladder structure of the two-channel wire:

cn = m(E)an .

(7.26)

Rewriting eq. (7.24) using eq. (7.26) results in a second order difference equation:

( + ∆) +

Îť
m

cn + v2 (cn−1 + cn+1 ) = Ecn .

(7.27)

The solution to this equation, after applying the boundary condition cn=0 = cn=N +1 = 0,
is a standing wave with the quasimomentum q,

cn = A(−1)n sin(nφq ),

φq =

πq
.
N +1

(7.28)

One can find the energy levels of the system using eqs. (7.26), (7.28), and the original set
of the two recurrence relations. The eigenenergies are solutions to the following quadratic
equation:
( + ∆) +

Îť2
− 2v2 cos φq = E.
E − + 2v1 cos φq

137

(7.29)

Hence the energy levels are

Eq± = 12 − αq ±

αq2 − 4βq ,

αq = 2v1 cos φq − + 2v2 cos φq − ( + ∆),
βq = −2 v2 cos φq + 4v1 v2 cos2 φq
−λ2 + ( + ∆)( − 2v1 cos φq ).

(7.30)

The coefficient m(E) is found simultaneously:

mÂą
q (E)

Eq± − + 2v1 cos φq
.
=
Îť

(7.31)

Finally, the normalization of the eigenvectors results in

q,Âą

cn (E) = (−1)n

2
N +1

mÂą
q
2
1+(mÂą
q)

sin(nφq ),

q,Âą

c
(E)
q,Âą
an (E) = n Âą ,
m

q = 1, 2, ..., N.

(7.32)

q

Similar to the single-qubit wire case, the eigenvectors can be classified into two categories:
symmetric and anti-symmetric states with respect to the center of the wire. From the
equation for the energy levels, eq. (7.29), it is easy to establish an extra symmetry associated
with the eigenvectors,

Eq+ Eq− − ( + 2v1 cos φq )(Eq+ + Eq− ) + ( + 2v1 cos φq )2 = −λ2 ,

138

(7.33)

and use eq. (7.33) to show that
|m−
q|
2
1 + (m−
q)

1

=

2
1 + (m+
q)

.

(7.34)

As a result, for a fixed state q,

q,Âą

q,∓

|cn | = |an |,

n = 1, 2, ..., N.

(7.35)

Fig. 7.19 shows the energy levels of the two bands as functions of the coupling between the
ground and the excited states (Îť) for a system with N = 5. As expected, the energy plot
is similar to a typical semiconductor band structure; the bands repel each other when the
excitation amplitude Îť grows.
15

10

Energy

5

0

-5

-10
0

1

2

3

4

5

6

7

8

9

10

𝜆
Figure 7.19: Energy levels of a two-band wire as a function of the coupling between the
ground and excited states of each cell, the parameters of the wire are:
N = 5, = 0, ∆ = 5, v1 = 1 and v2 = 2.

139

7.3.2

Coupling to the Continuum and Superradiance

Now we open the system coupling the excited states at the edges to the ideal leads which
are characterized by a continuum of states. This situation can be described by the effective
Hamiltonian (6.8).
We assume that the left, |n = 1, e , and the right, |n = N, e , edge state are each coupled
to one channel in the continuum. Correspondingly, the squares of the coupling amplitudes
determine the partial widths Îł L and Îł R of the left and right edge states decoupled from the
chain,

Ac=left
n=1,e =

Îł L,

c=right

ÎłR.

An=N,e =

(7.36)

In the basis of unperturbed chain states, the operator W has only two matrix elements,

W|n=1,e ,|n=1,e = Îł L ,

W|n=N,e ,|n=N,e = Îł R .

(7.37)

In order to diagonalize the effective Hamiltonian, W is transformed into the eigenvector
basis of the closed system, |q, Âą .

q,Âą q ,Âą

q,Âą

q ,Âą

W|q,Âą ,|q ,Âą = Îł L cn=1 cn=1 + Îł R cn=N cn=N ,

q,Âą

where cn

(7.38)

are given in eq. (7.32). The effective Hamiltonian can now be diagonalized which

140

leads to the following secular equation for the complex energies Îľ:

2 (ε) − P 2 (ε)) = 0,
Ω(ε) ≡ 1 + iP+ (ε)(γ L + γ R ) + γ L γ R (P−
+

(7.39)

where P+ (ε) and P− (ε) are defined as

P+ (Îľ) =

1
N +1

(1)q
q, Âą

sin2 φq m±
q
2
(ε − Eq± )(1 + (m±
q) )

and

P− (ε) =

1
N +1

(−1)q
q, Âą

sin2 φq m±
q
2
(ε − Eq± )(1 + (m±
q) )

.

Fig. 7.20 shows the resonance complex energies E − (i/2)Γ of our system with N =5,
when the two ends are symmetrically coupled to the continuum, Îł L = Îł R = Îł. At weak
coupling, when γ=0.1, all eigenenergies gain small decay width Γ due to interaction with
the environment. Further increase of Îł initially results in the increase of the decay widths
for all resonances as illustrated in the upper right panel (Îł = 1). The complex pole dynamics changes at larger values of Îł (see [93] and references therein). At a critical point,
when Îł/D

1, where D is the mean level spacing of the energy levels of H0 , the system

undergoes a superradiance transition. Beyond this point the decay widths do not monotonically increase with the increase of Îł. In fact, only some states (superradiant) continue to
increase their decay widths while the remaining states become long-lived (trapped) states.
This phenomenon is illustrated in the lower left panel of Fig. 7.20, where Îł=10 and the
two super-radiant states, in accordance with the number of open channels, have obtained
larger decay widths compared to the other states. The picture remains similar at larger Îł.
141

At Îł=20, the lower right panel, the two overlapping super-radiant states are shown in the
smaller window. Note that the number of superradiant states is always equal to the number
of continuum channels coupled to the wire; it is also important to notice the different vertical

𝛾 = 0.1

Decay Width Γ

0.06
0.04
0.02
0
-5
10

0

Energy

5

10

𝛾 = 10

5

0
-5

0

Energy

1

𝛾=1

0.5

0 0.06
-5 0.04

1

0

0.02

Decay Width Γ

Decay Width Γ

Decay Width Γ

scales on these panels.

5

10

20

𝛾 = 020

0
-5

Energy
5

5

10

0.5

0
-5

10

0

5

10

10

10

17.8
5

0
-5

0
-5

0

5

0

10

Energy

17.6

5

5.41

5.42

10

Figure 7.20: Complex eigenenergies for different values of the coupling strength to the
continuum, γ. The parameters of the wire are: N = 5, = 0, ∆ = 5, λ = 3, v1 = 1 and
v2 = 2.

It is instructive to consider the evolution of the resonances in the complex plane as Îł
increases. This is depicted in Fig. 7.21.

7.3.3

Transmission Through The Wire

Scattering properties of the two-channel quantum wire, in particular transmission of a signal
through the wire at weak, intermediate and strong coupling regimes, are considered in this
section. This concerns the features which could be the most significant for any practical

142

10

10

2 Super-radiant
states

1

0

Γ

10

2

10

10

-1

-2

-4

-2

0

2

E

4

6

8

10

Figure 7.21: Evolution of the energies in the complex plane. Up to the critical point all the
decay widths increase as Îł increases. A superradiant transition occurs as Îł exceeds the
critical value. After passing the critical point, the decay width of the two superradiant
states increases with the increase of Îł, whereas the decay width of the other eight states
decreases. The parameters of the wire are similar to the one in Fig . 7.20. The arrows
show the direction of the evolution as Îł varies from 0 to 100.
application.
The transmission coefficient is calculated according to eqn. (6.12). Of course, one can
also look at the reflection channels; the reflectivity coefficient is R = 1 − T , since the
form of the anti-Hermitian part (6.6) preserves the unitarity of the scattering matrix. the
transmission coefficient T L↔R (E) is depicted for different values of γ, Fig. 7.22. At small γ,
the perfect transmission, T = 1, occurs at narrow resonances. With increasing the coupling,
the resonances broaden and start to overlap. At the super-radiance transition, the maximum
transmission is achieved as a result of the broadening of the states, the case of Îł=4. With
further increase of Îł there are only three resonances left in the upper band, the two superradiant states are out of the picture since they are extremely broaden (Îł=100). Note that
the resonances become narrow as the trapped states lose their widths.
143

1

𝛾 = 0.1

0.5
0
-5
1

0

5

10

0

5

10

0

5

10

0

5

10

0

5

10

5

10

𝛾=1

𝑇𝑙𝑒𝑓𝑡↔𝑟𝑖𝑔ℎ𝑡 𝐸

0.5
0
-5
1
0.5
0
-5
1

𝛾=4

𝛾=6

0.5
0
-5
1
0.5

𝛾 = 10

0
-5
1
𝛾 = 100
0.5
0
-5

0

Energy

Figure 7.22: Transmission through the two-channel wire for different values of Îł. The
parameters of the wire are similar to those in Fig . 7.20.

7.3.4

Energy-Dependent Coupling

In previous sections it was assumed that the matrix elements of the effective Hamiltonian
do not depend on the running energy E. The energy dependence appeared only due to
the resonance denominators. Here we briefly address a more realistic situation with energydependent coupling to the continuum. This dependence arises inevitably when the energy
of an experiment is in the proximity of thresholds of the continuum channels (separation
energy in nuclei or the work function in solids). As a result, one should consider the energy
dependency of the coupling amplitudes. When a simple case of the s-wave form dependence
144

is considered, the coupling amplitudes can be taken as

L
L
AL
n=1,e (E) = α Θ(E − Ec )
R
R
AR
n=N,e (E) = α Θ(E − Ec )

E − EcL
E − EcR ,

(7.40)

where ÎąL and ÎąR are the strengths of the coupling to the left and to the right edge, respectively. In general these coefficients could be different; similar to the previous section
we assume a symmetric coupling to the continuum, hence ÎąL = ÎąR = Îą. EcL and EcR are
the cutoff energies of the continuum coupling also assumed to be the same, Ec , for both
channels. Θ is the step function, which guarantees that the system is decoupled from the
environment at energy below threshold.
Fig. 7.23 shows the complex energies of the same system when the cutoff threshold
Ec is approximately located in between the two bands of the isolated wire. Note that the
superradiant transition still occurs as the coupling strength Îą increases. The four states with
energy below threshold do not acquire any width being decoupled from the environment. The
four trapped states above threshold initially gain width as Îą increases, however, beyond the
superradiant transition, when Îą=2, they become long-lived states. On the lower right panel
the two overlapping super-radiant states with the large decay width are shown in a smaller
window.
In Fig. 7.24 we show the transmission coefficients for different values of the strength
coupling Îą. It is clear that the states below threshold do not participate in the transport of
the signal. Similar to the energy-independent case, Fig. 7.22, the transmission increases in
the super-radiant transition when the resonances start overlapping.

145

𝐸𝑐 = 0

0
-5

0

Energy

5

𝛼=1

0

Energy 5

𝛼 = 0.5

0.5

10

2

0
-5

1

Decay Width Γ

4

Decay Width Γ

𝛼 = 0.1

0.02

Decay Width Γ

Decay Width Γ

0.04

10

0.04

1

0
-50.02
20

0
-5

0
0

Energy
5

𝛼4 = 2

5

10

10

0.5
0
-5

0

5

10

19.5
19.4

2

19.3

10
0
-5

0
-5

19.2
0

5

0

10

Energy 5

5.36

5.365

5.37

10

Figure 7.23: Complex energies in the case of energy-dependent couplings for different
values of the coupling strength, Îą. The cutoff threshold is shown with a vertical line,
Ec =0. The wire parameters are the same as in Fig . 7.20.

7.3.5

Discussion

The effective non-Hermitian Hamiltonian was used to study an open quantum wire of twolevel atoms. The dynamics of the states when the wire is coupled to the continuum undergo
dramatic change as the strength of the coupling to the continuum varies crossing the superradiant segregation point separating the broad (short-lived) states from long-lived (trapped)
states. It was demonstrated that the width distribution of the states significantly alters
transport properties of the wire. Maximum transmission is achieved at the super-radiance
transition. The future studies should include the practically important current-voltage characteristic of a quantum wire in different coupling regimes. This can be done considering the
wire connected with reservoirs of electrons with different changeable chemical potentials.

146

1

𝛼 = 0.1

0.5
0
-5
1

𝐸𝑐 = 0
0

5

10

0

5

10

0

5

10

0

5

10

0

5

10

5

10

𝛼 = 0.5

𝑇𝑙𝑒𝑓𝑡↔𝑟𝑖𝑔ℎ𝑡 𝐸

0.5
0
-5
1

𝛼=1

0.5
0
-5
1
0.5
0
-5
1

𝛼 = 1.5

𝛼=2

0.5
0
-5
1

𝛼=3

0.5
0
-5

0

Energy

Figure 7.24: Transmission through the two-channel wire with energy-dependent couplings
for different values of the coupling strength, Îą. The cutoff threshold located between the
two bands is shown with a red vertical line, Ec =0. The wire parameters are the same as in
Fig. 7.20.

147

Nanotechnology will let us build computers that are incredibly
powerful. We’ll have more power in the volume of a sugar cube
than exists in the entire world today.
Ralph Merkle

Chapter 8
Solid State Quantum Computing

8.1

Introduction

Despite great theoretical advances in the field of quantum information, the best system to
realize the underlying physical layer of quantum computers is still a matter of debate. The
problem arises with the most fundamental element of the quantum computer, a single qubit.
Ideally, error-free qubits are desired due to the fact that unknown quantum states cannot
be replicated without loss of information, a consequence of the so-called no-cloning theorem.
Even with advances in error correcting codes and fault-tolerant quantum computing, we still
require qubits which are inherently resistive to noise and decoherence and therefore have
long life and coherence times capable of outlasting gate operation timespans.
Over the past two decades, many systems have been proposed for the physical implementation of a qubit. Some examples are photonic qubits in optical quantum computers [124,125], collective spin states in nuclear magnetic resonance (NMR) systems [126–128],

148

electronic states in trapped ions [129–131], charge, flux or phase qubits in superconducting circuits [132–135], electronic states on the surface of superfluid helium [136, 137] and
electronic charge or spin in solid state systems [138–140]. For a comprehensive review and
discussion of advantages and disadvantages of various qubit implementations see [142].
Recently, solid-state based quantum components have sparked great interest, mainly due
to their scalability. In addition, the accumulated knowledge in nano-circuit fabrication, along
with existing infrastructure, can be combined to more easily realize nano-scale integrated
quantum computers [141].
In any physical implementation, it is critical to consider not only the system of interest
but also the constituents that the system interacts with once placed in a greater final design.
In the case of solid-state based quantum computing systems, the qubits interact with devices
for writing and reading information in and out, such as leads and charge detectors. These
devices can be modeled as a continuum due to the large density of states they possess.
Consequently, one has to deal with an open mesoscopic quantum system where the intrinsic
states are coupled to the external world through a number of channels. Similarly to a nuclear
or molecular reaction, each channel is characterized by the energy of the final states and their
quantum numbers. The channels have energy thresholds when the coupling opens connecting
the system to the environment. Such situations require a correct unified treatment of the
discrete bound states and the continuum.
As demonstrated earlier, a convenient framework for such problems is given by the effective non-Hermitian Hamiltonian approach introduced by Feshbach [91]. The description is
non-perturbative and formally exact, treating the dynamics of systems with weak, intermediate and strong couplings to the environment on equal footing. This framework is quite flexible
and can be adjusted to various types of problems ranging from nuclear reactions [99, 145] to
149

electronic transport in mesoscopic physics [108, 146] and quantum optics [147]. An overview
of the approach and some of its results can be found in [93]. To stress the large spectrum
of applications using this approach we note its utilization in plasmonic antenna arrays [95]
and biological light harvesting complexes [96].
Earlier in Chapter 7, a simple system of a quantum wire with a two-level atom (qubit)
inserted in the middle was considered using the effective Hamiltonian [148]. The qubit was
used to regulate transport in the wire. In fact, insertion of qubits in various systems was
found to be useful in different applications. For instance, in [149], a charge detector is
proposed consisting of a qubit attached to a quantum wire. In this section we consider a
similar system; a quantum wire with two embedded qubits which, as a whole, behave like a
single qubit.
We first introduce a closed one-dimensional chain of 2N identical sites with hoping between adjacent cells. The center of the chain is occupied by an asynchronized pair of twolevel atoms: the couplings between the ground states and excited states of the two qubits
are inversely proportional. The band structure of the system consists of delocalized states
extended over the chain and additional states outside the Bloch band confined to the central
qubits. Depending on the coupling strengths within the qubits, the states inside the band
are localized in the left arm, the right arm, or evenly distributed across the entire chain.
Consequently the system acts as a distributed charge qubit with collective right and left
states.
Next the open system with the edge sites coupled to the continuum is considered. The
continuum represents ideal leads or charge detectors attached to the edges. Due to the
interaction with the leads the energy states acquire decay widths influenced by the continuum
coupling in a non-trivial manner. At sufficiently strong coupling, protective superradiant
150

edge states are formed. These states steal the entire width and hence prevent the remaining
states from decaying into the environment.
Finally, we perform a numerical study on the effect of noise and investigate the phenomenon of decoherence. The noise in the environment is modeled by a random Gaussian
process. The situation is described by the stochastic Liouville equation that determines the
evolution of the density matrix. It is shown that the formation of superradiant states can
maximize the coherence time of the distributed qubit system.

8.2

Closed System

The proposed system consists of a nano-wire with a pair of two-level atoms embedded at its
center. The wire is considered to be a one-dimensional chain of 2N identical sites numbered
as n = −N, −(N −1), ..., −1 and n = 1, ..., N −1, N . The chain is modeled by a tight-binding
Hamiltonian with coupling between adjacent neighbors:
N

HW =

†
0 cn cn

n=−N
n=0

†

+

†

ν cn cn + c cn ,
n

(8.1)

n,n

†

where 0 is the on-site energy, ν is the hopping integral, and cn and cn are creation and
annihilation operators at site |n , respectively. The typical value for ν is within the range of
1-100 ÂľeV in quantum dot systems [140, 143, 144]. The second sum in the Hamiltonian (8.1)
runs over the nearest neighboring cells only.
Two asynchronized qubits are symmetrically connected to the center of the wire. The left

151

qubit with excited state |eL and energy level δL is connected to site |−1 with Hamiltonian

†

†

†

L =δ c c +Ν c c
HQ
L L L
L −1 + c−1 cL ,

(8.2)

†

where Îť is the matrix element of the qubit excitation and cL and cL are creation and
annihilation operators for the left qubit excited state, respectively. Similarly, the right qubit
with excited state |eR and energy level δR is connected to site |1
R = δ c† c + κ c† c + c† c .
HQ
R R R
1 R
Îť R 1

(8.3)

†

Here, cR and cR create and annihilate an excitation in the qubit upper level, respectively.
Îş/Îť is the coupling strength between the ground and excited states of the two-level atom.
The left and right qubits are fully asynchronized, i.e. at strong coupling between the excited
and ground states of the left qubit, the two states of the right qubit are weakly coupled and
vice versa; Îş is the asynchronization parameter. In practice this parameter can be tuned by
introducing a local electric field and adjusting the field strength [150, 151].
The total Hamiltonian of the closed system is the sum

Q

Q

H0 = HW + HL + HR .

(8.4)

A generic stationary wave function of the system with energy E is represented as
N

|ψ(E) =

an (E) |n + bL (E) |eL + bR (E) |eR .

(8.5)

n=−N
n=0

In order to fulfill the Schr¨odinger equation, the coefficients of the superposition in (8.5)
152

satisfy the linear three-term recurrence relation

(E − 0 )an − ν(an−1 + an+1 ) = 0,

n = 0, Âą1.

(8.6)

The discontinuities at Âą1 are due to the inserted qubits in the center. This naturally divides
the chain into two regions, left and right. The solutions in the two sides of the chain are
given by
an =



n
n
 aL
n = AL ζ+ + BL ζ− ,

−N ≤ n < −1,


aR = A Μ n + B Μ n ,
R −
R +
n

(8.7)
1 < n ≤ N,

where Μ¹ are the roots of the characteristic polynomial of the recurrence relation,

Μ¹ =

1
E− 0±
2ν

(E − 0 )2 − 4ν 2 ,

(8.8)

with the obvious property ζ+ ζ− = 1. The remaining equations resulting from solving the
Schr¨odinger equation can be used as a boundary condition to connect the two regions in
(8.7) in order to find the constants AL , AR , BL and BR . At n = Âą1 we have

(E − 0 )a−1 − ν(a−2 + a1 ) = λbL ,

(8.9)

Îş
(E − 0 )a1 − ν(a2 + a−1 ) = bR ,
Îť

(8.10)

(E − δL )bL = λa−1 ,

(8.11)

Îş
a .
Îť 1

(8.12)

(E − δR )bR =

For two important cases, the solution can be analytically obtained assuming, for simplic-

153

ity, that δL = δR = δ. In the first case, when λ → 0, the system supports states that are
either fully localized in the right side or in the left side of the chain. In the second case we
consider Îť2 = Îş, where the excitation is equally distributed between the left and the right
sides of the chain. The system is analogous to a qubit where the two states are extended over
the entire left or right side of the wire, with Îť serving as the coupling parameter, regulating
the population in each side and enabling us to perform gate operations. In accordance, we
adopt a special notation throughout the section. The stationary state (8.5) is denoted as

|ψ(E) = |LΝ (E) + |RΝ (E) ,

(8.13)

where |LÎť (E) contains the components of the wave function in the left side of the chain,
−1

aL
n (E) |n + bL (E) |eL ,

|LÎť (E) =

(8.14)

n=−N

while |RÎť (E) contains the components in the right side,
N

aR
n (E) |n + bR (E) |eR .

|RÎť (E) =

(8.15)

n=1

The subscript Îť indicates that the left and right states are changed as Îť takes different
values.

8.2.1

Case λ → 0

In the limit of very weak coupling between the two states in the left qubit, λ → 0, the
eigenstates of the system fall into two categories: states that are confined in the two central
qubits and Bloch waves fully localized in the right or the left side of the chain. Since the
154

left qubit is decoupled from the chain, there exists a state with the only non-vanishing wave
function component bL = 1 and energy E = δ. The right qubit states can be found using
eqs. (8.10) and (8.12). The two states have energies

E=−

1
Îş
1
Îş
+ ( 0 + δ) and E =
+ ( 0 + δ),
Îť
2
Îť
2

(8.16)

√
√
corresponding to eigenstates a1 = −bR = 1/ 2 and a1 = bR = 1/ 2, respectively.
The states of the second type are distributed over the wire, localized either in the left
side between the sites |−N and |−1 , or in the right side between |1 and |N . According
to (8.12), since κ/λc → ∞, we have aR
1 = 0. This breaks the symmetrical structure of the
wire, creating a longer chain on the left and leaving the right chain shorter, which is the key
point in realizing a qubit structure using the proposed system. For the states in the left side
R
L
(only aL
n = 0), applying the boundary conditions a1 = 0 and a−N −1 = 0 to (8.7) provides
2N +2 = 1. Thus the energies can be parameterized by a positive
an equation for energies, ζ−

even number, the quantized quasi-momentum k,

EkL = 0 + 2ν cos ϕL
k,

ϕL
k =

πk
,
2N + 2

k even,

(8.17)

and the corresponding amplitudes (8.7) are of the Bloch-wave type,

k
aL
n (k) = i

2
sin(nϕL
k ).
N +1

(8.18)

In what follows, according to (8.13), these states are denoted as |L0 (E) .
Similarly, for the states in the right side of the wire (only aR
n = 0) with boundary

155

R
2N
conditions aR
1 = 0 and aN +1 = 0 we have Îś+ = 1. Thus

EkR = 0 + 2ν cos ϕR
k,

ϕR
k =

πk
,
2N

k even,

(8.19)

with eigenfunctions
k
aR
n (k) = i

2
sin(nϕR
k ).
N

(8.20)

These states are denoted as |R0 (E) .
As k varies, the energy states (8.17) and (8.19) consecutively alternate and come in
pairs. In each pair, the higher energy state is extended in the left side and the lower level
in the right side, being associated with wave functions |L0 and |R0 , respectively. Clearly,
in the extreme limit of λ → ∞ we have the reverse situation. Since aL
−1 = 0, the right
chain becomes longer and the upper energy state in each pair is localized in the right side
of the chain and therefore can be denoted as |R∞ . Consequently, the lower energy state is
localized on the left which is indicated by |L∞ .

8.2.2

Case Îť2 = Îş

For finite values of Îť the two sides of the chain are not decoupled and states are extended
over the entire chain. The parameter Îť therefore acts as a knob to control the population in
the left and right regions. We now consider a special case of Îť2 = Îş where the population
is equally divided between the two sides of the wire. Due to symmetry, there exist two
R
types of eigenfunctions: fully symmetric states with aL
n = an and anti-symmetric states
R
with aL
n = −an . Using eqs. (8.9) and (8.11) or eqs. (8.10) and (8.12) we obtain an equation

156

for the energies,
sin (N + 1)φ
Îş
= Âą1 +
,
sin(N φ)
ν(E − δ)
where sin φ = (1/2ν)

(8.21)

(E − 0 )2 − 4ν 2 , with the positive (negative) sign corresponding to

symmetric (anti-symmetric) states. Analogous to an equal superposition of the two states
√
in a qubit, we denote the symmetric and anti-symmetric states as (1/ 2)(|L√κ + |R√κ )
√
and (1/ 2)(|L√κ − |R√κ ), respectively. The evolution with varying λ from zero to infinity
and controlling the state of the qubit is graphically shown in Fig. 8.1 using the Bloch sphere.
Assuming that at λ = 0 the state is localized on the left, |L0 , a 90◦ rotation (π/2 pulse)
is performed by adiabatically moving Îť to Îť =

√

Îş. Further increasing Îť to extreme values

localizes the particle in the right side of the chain, |R∞ .

Figure 8.1: Graphical presentation of qubit rotation on the Bloch sphere. Gate operations
can be performed by varying
the value of Îť. The initial state at Îť = 0 is localized in the
√
left side, |L0 . At λ = κ the state is 90◦ rotated becoming an equal superposition of the
left and right states. When λ → ∞, the state becomes localized on the right.

The complete band structure as a function of Îť is shown in Fig. 8.2(a). The parameters
of the system are N = 10, 0 = 0, ν = 1, δ = 2.5 and κ = 4 (for the remainder of the section,
0

is set to zero and the scale is fixed by setting ν = 1). The black curves correspond to

states localized in the central qubits and the red curves are pairs of extended states over

157

the wire. Fig. 8.2(b) shows pair I in a smaller energy scale. It is clear that any rotation on
the Bloch sphere can be performed by varying Îť. The arrow indicates the avoided crossing
point, when Îť =

√

Îş. The difference between two consecutive energy solutions of eq. (8.21)

is the Rabi frequency, Ω, at which the population oscillates between the left and right sides
of the chain.

5
4
3
2
1
1.95

0
1.94-6
1.93
1.95
1.92
1.94
1.91
1.93
1.9-6
1.92

-4

-2

0

2

4

6

2

4

6

(a)

-4

-2

0
(b)

1.91

Figure 8.2: (a) Upper half of the band structure as a function of Îť for a system with
N = 10, 0 = 0, t = 1, δ = 2.5 and κ = 4. The black curves correspond to states localized
1.9-6 red curves
in the central qubits and the
are pairs
-4 -2
0 of Bloch
2 waves.
4
6(b) A closer view of pair
I as a function of Îť.

The squared components of the wave function (8.5) of the upper and lower states in pair
I as a function of Îť are shown in Fig. 8.3. The figure only considers the components in the

158

chain and not those of the excited states of the central qubits (an ’s in the wave function
(8.5)). At Îť = 0 the upper and lower states are fully localized in the left and right sides,
respectively. At Îť =

√
Îş = 2, both upper and lower states are in equal superpositions. As

Îť increases, the upper state quickly becomes localized in the right side. Similarly, the lower
state develops into a localized state in the left.

(a)

(b)

Figure 8.3: Evolution of the squared components of the (a) upper state and (b) lower state
wave functions in pair one indicated in Fig. 8.2. Only the components in the chain are
considered in the figure: amplitudes a(n) in the wave function (8.5).

All pairs within the Bloch band in Fig. 8.2 possess a qubit structure and therefore can
be considered for implementing a qubit. The wave function profile, however, varies for each
pair. Fig. 8.4 shows the upper state wave function in pair III as Îť takes different values.
159

The behavior is qualitatively the same as in pair I, when Îť increases, the state evolves from
being fully localized in the left to being fully localized in the right.

Figure 8.4: Evolution of the squared components of the upper state wave function in pair
three indicated in Fig. 8.2. Only the components inside the chain are considered in this
figure.

8.3

Open System - Superradiance and Emergence of
Protecting Edge States

In this section we consider the open system with the two edge sites of the chain coupled to
the external world. This represents the interaction of the system with devices such as charge
detectors and leads connected to the two ends of the wire. We apply the non-Hermitian
Hamiltonian technique discussed in Chapter 6 to the distributed qubit system.
We open our system by coupling the left-most and right-most sites (|−N and |N )
c

c

L and A R , respectively. We further assume that
to ideal leads by the amplitudes A−N
N

the energy dependence of the amplitudes can be ignored and the couplings are symmetric,
R
AL
−N = AN =

√
Îł. Here Îł is the parameter representing the interaction strength with the

decay channels. The effective Hamiltonian (6.8), with H0 being the Hamiltonian of the closed
160

system (8.4), thus fully describes the situation. Since the chain is only coupled through the
edge sites, the operator W takes on a simple form: according to (6.6) the only non-zero
matrix elements of this operator are

W−N,−N = WN,N = γ.

(8.22)

The behavior of the system is strongly influenced by the dimensionless parameter Îł/D,
where D is the mean energy level spacing of the closed system. A parametric study for
a chain without qubits and a chain with a single qubit was performed in [105] and [148],
respectively. In both cases, the typical picture was found to be as follows. At weak coupling
to the environment, all intrinsic states acquire a small decay width. The width distribution
among the states is almost uniform with the maximum at the center of the Bloch band. When
Îł grows, the distribution abruptly changes at Îł

D. Beyond this point, further increasing

the coupling results in the segregation of states into long-lived narrow and short-lived broad
resonances. As previously, in analogy to Dicke superradiance in quantum optics [97], we
term these emergent giant resonances as superradiant states.
Adopting a similar approach to our system we consider a chain with 20 intrinsic cells
(N = 10), a pair of asynchronized qubits attached at the center, and the two end sites
coupled to the continuum. The results of this study are presented with a series of figures (in
all figures the scale is fixed by setting ν = 1).
In the first step we diagonalize the effective Hamiltonian,

H |q = Eq |q ,

161

(8.23)

The trajectories of the energies Eq in the complex plane as a function of Îł are shown in
Fig. 8.5. The x and y axes represent the real and imaginary parts of the energies, respectively
(Eq = Eq − (i/2)Γq ). The parameters of the system are δ = 2.5, λ = 0.01, κ = 4. States
within the band are shown in the top panel. Because Îť is small, the states are fully localized
in both sides of the chain. The five pairs indicated in the figure correspond to the pairs of the
closed system shown in Fig. 8.2. The emergence of superradiant states is clear in the figure.
At small values of Îł all states are narrow resonances. The decay widths grow as Îł increases.
At the critical value of Îł

2.5 the sharp superradiant transition occurs. Beyond this point,

the superradiant states (pair V in the figure) become broad resonances, essentially protecting
the remaining states from decaying into the continuum. The lower panel in the figure shows
the states outside of the band. These states are localized in the central qubits and have
extremely small decay widths. For these states, the shift in the real energy is negligible and
therefore after the superradiance transition they trace back the same trajectories.
It was noticed in [152] that the real part of the effective Hamiltonian repels the levels
while attracting the widths. On the contrary, the anti-Hermitian part attracts the real
energies but repels the widths. This phenomenon can be seen in Fig. 8.5, where on the road
to superradiance the energy levels are attracted to the center of the band.
The evolution of complex energies of the same system when Îť = 2 is shown in Fig. 8.6.
All pairs are now evenly distributed over the entire chain. The picture is qualitatively the
same as in the previous case. The two superradiant states are again placed in the center of
the band. At large values of coupling to the continuum, the superradiant states acquire the
entire width, leaving the remaining states to be long-lived.
In order to better envision the protecting role of the superradiant states we compare the
lifetime of a particle inside the chain for different initial conditions. In the first case, the
162

101
10-1
10-3
-2

-1

0

1

2

(a)

10-10
10-14

10-14

10-15

10-15

-11

10

10-16

-16

10

-398.7544

2.5000

401.2544

(b)

Figure 8.5: Evolution of complex eigenenergies as a function of coupling to the continuum
for (a) states within the Bloch band and (b) states outside the band. The arrows indicate
the direction in which Îł evolves from 0.05 to 20 with 0.01 increments. The parameter
values are δ = 2.5, Ν = 0.01 and κ = 4.
particle is initialized in the upper energy state of pair I shown in Fig. 8.6. In the second
case, the initial state is that of the upper energy of pair V (superradiant pair) shown in the
same figure. We calculate the survival probability P (t) according to

m |ψ(t)

P (t) =

2

,

(8.24)

m

where |m ’s are the intrinsic states of the closed system ({|n }, |eL and |eR in the wave
163

101
10-1
10-3
-2

-1

0

1

2

(a)

10-5

10-10

10-11

10-6

10-12
3.4646

-2.3704

3.9773

(b)

Figure 8.6: Evolution of the complex eigenenergies as a function of coupling to the
continuum for (a) states within the Bloch band and (b) states outside the band. Îł varies
from 0.05 to 20 with 0.01 increments. The parameter values are δ = 2.5, Ν = 2 and κ = 4.
function (8.5)) and ψ(t) is the result of quantum evolution,

ψ(t) = e−iHeff t ψ0 ,

(8.25)

where ψ0 is the initial state. The initial state can be expanded in the biorthogonal space of

164

the eigenstates of the effective Hamiltonian (8.23), ψ0 =

cq e−iEq t m |q

P (t) =

q cq |q

2

.

. Consequently we have

(8.26)

m,q

The results for different values of the coupling constant Îł are shown in Fig. 8.7. In
panel (a) the particle was initialized in the upper state of the first energy pair in the band
structure. Compared to the weak coupling case (Îł = 0.25), at the superradiance transition
(Îł = 2.5) the lifetime is by an order of magnitude smaller, which makes the situation convenient for measurement and fast readout of the qubit system. At strong coupling and beyond
the superradiance transition, when Îł = 25, the lifetime increases and allows for storing information or performing operations on the qubit. Contrary to this, it is shown in panel (b)
that the lifetime of a particle initialized in the superradiant state monotonically decreases as
the coupling strength is increased. Consequently, other pairs in the band structure become
protected from decaying into the leads.
Next we consider the band structure of the open system. Fig. 8.8 shows the upper half
of the band structure when Îł = 3. Here the y-axis gives the real part of energy. The picture
is similar to the band structure of the closed system (Fig. 8.2) with all pairs slightly pushed
towards the center of the band. In addition, pair V has now become a superradiant pair
where the two states have short lifetimes and are insensitive to the parameter Îť. Therefore
the pair is no longer a suitable candidate for our qubit system and its role is solely protecting
other pairs from decaying into continuum.
It is interesting to monitor the superradiant wave function profile in the chain as a
function of Îł. We again consider two cases here: Îť = 0.01 and Îť = 2. The two superradiant
wave functions for the first case are shown in Fig. 8.9. For small values of Îł the upper and

165

1
0.8
0.6
0.4
0.2
0
10-1

100

101

102

103

(a)

1
0.8
0.6
0.4
0.2
0
10-4

10-2

100

102

(b)

Figure 8.7: Lifetime for different values of coupling to the continuum Îł for a particle
initialized in (a) the upper state of pair I and (b) the upper state of pair V (superradiant
pair) in Fig. 8.6. The parameter values are δ = 2.5, Ν = 2 and κ = 4.
lower states are Bloch waves confined to the left and the right sides of the chain, respectively.
As Îł increases, the states quickly become localized at the edges of the wire.
The wave functions for the second case, Îť = 2, are shown in Fig. 8.10. The picture is
similar to the previous case with the difference that, at weak coupling, the states are extended
over the entire chain. It is apparent that at strong coupling the superradiant states become
localized in the two edges [153].

166

5
4
3
2
1
0
-6

-4

-2

0

2

4

6

Figure 8.8: Upper half of the band structure as a function of Îť for the open system with
δ = 2.5, κ = 4 and γ = 3. The black curves correspond to states localized in the central
qubits and the red curves are pairs of Bloch waves.

8.4

Noise and Decoherence

So far, we have studied the system under ideal conditions i.e. assuming that all processes are
fully coherent. In reality, due to the interaction with the surrounding the system is subject
to perturbations that might destroy the phase relations of the components of a wave function
(decoherence). Because superradiance is a direct consequence of quantum superposition and
therefore a coherent phenomenon, we expect such fluctuations to have a great impact on the
dynamics of the system.
In this section we study the interplay between superradiance and decoherence by considering a model originally introduced by Haken and Strobl [154]. The model has been
extensively used for studying the role of dephasing in various physical situations such as radiative decay in molecular aggregates [155], quantum teleportation and the implementation
of a quantum controlled-NOT gate [156]. The dephasing process is introduced by adding a

167

(a)

(b)

Figure 8.9: Evolution of the squared components of the wave functions for (a) upper and
(b) lower superradiant states. The parameters are δ = 2.5, κ = 4 and Ν = 0.01. Only the
components inside the chain are considered in the figure.
time-dependent Hamiltonian to the effective non-Hermitian Hamiltonian,
N

Hφ (t) =

†

†

†

δ n (t)cn cn + δ L (t)cL cL + δ R (t)cR cR ,

(8.27)

n=−N
n=0

where δ (t) describe stochastic Gaussian processes representing rapid fluctuations of on-site

168

(a)

(b)

Figure 8.10: Evolution of the squared components of the wave functions for (a) upper and
(b) lower superradiant states. The parameters are δ = 2.5, κ = 4 and Ν = 2. Only the
components inside the chain are considered in the figure.
electronic energies with zero mean and delta-function correlations in time

δ i (t) = 0,

(8.28)

δ i (t)δ j (t ) = αφ δi,j δ(t − t ).

(8.29)

Here αφ is the parameter representing the dephasing strength; subscripts i and j run over

169

cell numbers, n = −N, ..., −1, 1, ..., N , as well as the excited states of the central qubits, L
and R. The symbol

denotes averaging over the statistical ensemble.

The evolution of the density matrix ρ(t) is governed by the stochastic Liouville equation.
In the site representation we have (see the appendix for derivation)

∂
ρ(t) i,j = −i Heff , ρ(t) i,j − 2αφ (1 − δi,j ) ρ(t) i,j .
∂t

(8.30)

Both superradiance and dephasing result in the decay of off-diagonal elements of the density
matrix. As a measure of coherence, we define a new quantity, R(t), according to

R(t) =

ρ(t) i,j .

(8.31)

i=j

Furthermore, we define the coherence time, τcoh , as the time duration for which R(τcoh ) =
−αφ t

R(0)/e. Without coupling to the continuum, Îł = 0, we have R(t) = R(0)e

and

τcoh = αφ−1 . For an open system, γ = 0, the Liouville equation (8.30) was numerically solved
using the ordinary differential equations package in Matlab. Here we consider three cases,
αφ = 10−3 , αφ = 10−2 and αφ = 10−1 . For each case, the coherence time is calculated
for systems with different numbers of sites. The initial density matrix in all calculations
corresponds to the upper state in the highest energy pair inside the Bloch band (pair I in
Fig. 8.8) when Îť2 = Îş.
The results for the case of the weak dephasing strength, αφ = 10−3 , are shown in Fig. 8.11.
As expected, regardless of the site number, the coherence time is equal to τcoh = 103 when
Îł = 0. As Îł increases, coherence time is governed by superradiance dynamics. At the
transition to superradiance, the initial state achieves its maximum decay width and hence

170

τcoh reaches its minimum due to fast decay into the continuum. However, the reduction in
τcoh is quite different for chains with varying numbers of sites. For larger chains, states have
a smaller share of the total width and therefore longer lifetimes compared to shorter chains.
Consequently, at the superradiance transition, longer chains have larger decoherence time.
A similar type of resistance and robustness to noise with an increase in the number of sites
was observed in nanoscale rings within light-harvesting systems [157]. At larger values of Îł,
the lifetime of the initial state increases which in turn increases τcoh .

1000
750
N=10
N=20
N=30
N=40
N=50

500
250
00

10

20

30

Figure 8.11: Coherence time as defined in (8.31) for systems with different numbers of cells
when αφ = 10−3 . The initial state is the upper state
√ of the highest pair inside the Bloch
band. Other parameters are Ν = κ = 2 and δ = 2.5.
Fig. 8.12 shows the results for αφ = 10−2 . The effect of superradiance is apparent only
for systems with a smaller number of sites. For systems with N = 10 and N = 20, τcoh is
decreased due to the decay into the continuum (for smaller systems the lifetime of the initial
state is shorter since there are less states that share the entire decay width).
The results associated with the case of αφ = 10−1 , are presented in Fig. 8.13. The strong
dephasing quickly dissipates the off-diagonal elements of the density matrix. This happens
before the particle gets a chance to escape the wire and therefore the superradiance effect
is suppressed by the dephasing phenomenon. As expected and demonstrated by the figures,
171

100
75
N=10
N=20
N=30
N=40
N=50

50
25
00

10

20

30

Figure 8.12: Coherence time of systems with different numbers of cells when αφ = 10−2 .
The initial state is the upper state of the highest pair inside the Bloch band.
superradiance survives only in the presence of relatively weak dephasing.

10
7.5
N=10
N=20
N=30
N=40
N=50

5
2.5
00

10

20

30

Figure 8.13: Coherence time of systems with different numbers of cells when αφ = 10−1 .
The initial state is the upper state of the highest pair inside the Bloch band.

8.5

Discussion

We proposed a novel solid-state based system for implementing a qubit. The system consists
of a one-dimensional chain and a pair of two-level atoms inserted at its center where the
couplings between their ground and excited states are inversely proportional. The energy
eigenstates within the Bloch band exhibit a qubit-like behavior. The coupling between the
172

ground and excited states of the two central qubits can be used to perform gate operations
and initialize the system in a given state. The effect of connecting the wire to ideal leads and
charge detectors (coupling to the continuum) was discussed in detail by exploiting the nonHermitian effective Hamiltonian approach. In the case of strong continuum coupling, two
broad resonances localized at the edges of the wire are formed. These emerging superradiant
edge states increase the lifetime of the remaining states making them suitable candidates for
qubit implementation. The effect of ambient noise was included by considering the stochastic
Liouville equation. The interplay between superradiance and decoherence was discussed
for various dephasing strengths. Superradiant effects are prominent for weak dephasing
strengths and fade away as the noise increases.

173

For the rest of my life I will reflect on what light is.
Albert Einstein

Chapter 9
Non-Hermitian Plasmonics Antennas
and Waveguides

9.1

Introduction

Manipulation of light in nanometer scales via surface plasmonic resonances of metallic structures has attracted a great deal of attention over the past two decades [159]. Optical antennas
capable of localizing light in sub-wavelength regions have resulted in a new generation of photonic devices with applications ranging from imaging [160–162] to biosensing [163, 164] and
emission enhancement of photon sources [165–167]. In addition, plasmonics is ought to play
an important role in the efficient reception and transport of optical energy in light harvesting devices [174]. Therefore, various waveguide structures are being investigated in order to
control and further improve the propagation of light in micrometer length scales [172, 173].
More recently, it has been shown that surface plasmons can exhibit quantum interfer-

174

ence [168, 169]. This has sparked a great interest in studying the quantum properties of
surface plasmons and exploiting plasmonic devices as potential building blocks of quantum
computers and quantum circuits [170, 171].
It was previously suggested that plasmonic structures could be mapped to quantum
systems governed by non-Hermitian Hamiltonians [95, 175]. In [95], the radiation properties of an array of optical dipole antennas are manipulated by altering the anti-Hermitian
coupling strength between the elements of the array. However, due to the complexity of
nano-dipole antennas and the lack of closed form expressions for the fields, the mapping to
a non-Hermitian Hamiltonian was achieved via numerical simulation and curve fitting. In
this section we consider systems of plasmonic nano-spheres using the effective Hamiltonian
framework introduced in Chapter 6. This framework provides a general platform for studying different physical systems; it has been previously utilized in various problems ranging
from quantum signal transmission in nano-structures [105, 148] to solid state quantum computing [90] and nuclear reactions [101, 114]. The description of plasmonic structures via the
effective Hamiltonian is achieved due to the correspondence between the Feshbach formalism
and the coupled mode theory of optical resonators. The effective non-Hermitian Hamiltonian
approach allows us to translate phenomena already known in condensed matter and nuclear
systems to the plasmonic system under study. One such example is the existence of superradiant and subradiant, or dark, states. In addition, the effective Hamiltonian framework
can be readily used in order to calculate observables, such as the transmission coefficient
through a plasmonic waveguide.
We first discuss the coupled mode formalism used for studying optical and plasmonic
systems. The connection between the effective Hamiltonian and coupled mode theory is
illustrated through simple two-level examples. We then consider a single plasmonic metallic
175

nanosphere. The wave equation is solved in order to find the natural resonant frequencies
of the single sphere, and discuss the intrinsic radiative nature of nanospheres. This provides
a basis for the consideration of two spheres via coupled mode theory. The signature of
superradiance emerges when the interaction between adjacent optical nano antennas occurs
through a single continuum channel, resulting in states with enhanced radiation and confined
dark modes. The effect of these states on energy transmission through a one-dimensional
chain of spheres is also considered, with applications to optical frequency nanoscale antennas
and waveguide-like structures.

9.2

Coupled Mode Theory and The Non-Hermitian
Hamiltonian

In this thesis, we employ the coupled mode theory in order to study systems of interacting
plasmonic spheres. This method, is reminiscent of the time-dependent perturbation theory
in quantum mechanics; it has been used for the investigation of coupled resonators in optical systems [176–178], wireless energy transfer loop antennas [179] and plasmonic structures
including antennas [95, 179] and waveguides [180–182]. The coupled mode approach significantly simplifies the complexity of the problem: instead of solving the wave equation one
needs to solve a system of linear algebraic equations. In addition, it provides a clear and
intuitive picture of how interaction between the constituents of the system can dramatically
change the dynamics.
The formulation provided in this thesis, similar to [183], is rather general. No specific
boundary conditions are assumed and hence it is applicable to the system of coupled plasmonic nanoantennas discussed in the future sections.
176

Consider two non-magnetic dielectric resonators with relative dielectric constants 1 (r)
and 2 (r). The resonators occupy a volume in space, V1 and V2 , respectively. In addition,
consider that the relative dielectric constants 1 (r) and 2 (r), are equal to unity for points
outside of the first and second resonators, respectively. In a time harmonic scenario each
resonator, when isolated, satisfies the wave equation

∇ × ∇ × Enα (r) −

ωα,n 2
Îą
Îą (r)En (r) = 0,
c

(9.1)

where Îą = 1, 2 denotes the resonator number and c is the speed of light in the background
medium which is assumed to be the free space for simplicity. Due to the sharp discontinuity between the resonator and the background at the resonator boundaries, the modes are
quantized and characterized by the integer number n = 1, 2, 3, ... and their eigenfrequency
ωα,n . The modes of isolated resonators are normalized according to

V

α∗
Îą
3
Îą (r)Em (r).En (r)d r

= δmn ,

(9.2)

where V is the total volume and m and n are the mode indices and δmn is the Kronecker
delta. The normalization expression is of crucial importance and its form is dictated by the
boundary conditions of the problem. For instance in [183] the normalization is similar to
(9.2), however with no complex conjugation. As we will see, the normalization expression
has to be modified when discussing spherical plasmonic particles in the following sections.
For now, without loss of generality, we assume that the normalization rule is given by the
general dot product definition provided in (9.2) with the integration volume being all space,
as this is usually the case in electromagnetic textbooks.

177

Next we assume that, for a system of two coupled resonators, the total electric field,
E(r), can be written as a superposition of a finite number of individual modes of the two
resonators:
N

a1 (n)En1 (r) + a2 (n)En2 (r) ,

E(r) =

(9.3)

n=1

where N is the total number of modes. The total electric field satisfies the wave equation

∇ × ∇ × En (r) −

ωn 2
(r) En (r) = 0,
c

(9.4)

where ωn are the eigenfrequencies of the coupled system and (r) is the dielectric constant
at a given point, r, when both resonators are simultaneously present. The function (r) is
equal to 1 (r) and 2 (r) for points inside the first and second resonator, respectively, and is
equal to unity otherwise. Plugging the ansatz (9.3) into the wave equation (9.4) and using
its linearity and (9.1) we arrive at
N

2

a1 (n) ω1,n

1
1 (r)En (r) + a2 (n)

ω2,n

2

2
2 (r)En (r)

n=1
N

=

ωn2

a1 (n)En1 (r) + a2 (n)En2 (r) .

(r)

(9.5)

n=1
1 (r) and E 2 (r) for all values of m:
Using the normalization rule (9.2) to project (9.5) onto Em
m

m = 1, 2, .., N , we obtain a system of 2N linear equations. In the matrix form




T 11

T 12



T 21

T 22



2
 Ω1

0


 
 
11
12
L
L  A1 
0  A1 
2
 ,
  = ω 
21
22
2
A2
A2
L
L
Ω2

(9.6)

where A1 and A2 are N × 1 vectors of the coefficients a1 (n) and a2 (n) in the ansatz (9.3),

178

respectively. Ω1 and Ω2 are N × N diagonal matrices containing the eigenfrequencies of the
isolated resonators with matrix elements

Ωα mn = ωα,n δmn

(9.7)

where as previously α = 1, 2. The matrix elements of the four square N × N matrices T αβ ,
where ι, β = 1, 2, are given by

β
α∗
3
β (r)Em (r).En (r)d r.

ιβ

Tmn =

V

(9.8)

According to (9.2), T 11 and T 22 are equal to the identity matrix 1. Finally the elements
of the matrices Lι,β are given by

∗

ιβ

Lmn =

β

Îą (r).E (r)d3 r.
(r)Em
n

V

(9.9)

Because the dielectric function (r) is the sum of the two dielectric function, the matrix
12
elements L12
m,n and Tm,n are related via

12
L12
mn = Tmn +

V1

1 (r) − 1

∗

1 (r).E 2 (r)d3 r,
Em
n

(9.10)

where the integration is carried out over the volume of the first resonator, V1 only. Accordingly, we define the matrix K 12 with matrix elements

12 =
Kmn

V1

1 (r) − 1

179

∗

1 (r).E 2 (r)d3 r.
Em
n

(9.11)

Therefore
L12 = T 12 + K 12 .

(9.12)

21
Similarly L21
mn is related to Tmn as

21
L21
mn = Tmn +

V2

2 (r) − 1

∗

2 (r).E 1 (r)d3 r.
Em
n

(9.13)

Correspondingly K 21 is defined with matrix elements

21 =
Kmn

V2

2 (r) − 1

∗

2 (r).E 1 (r)d3 r.
Em
n

(9.14)

Hence
L21 = T 21 + K 21 .

(9.15)

In order to simplify (9.6), we accept a number of approximations that are commonly used
in studying systems of weakly coupled resonators [184–186]. We assume that the diagonal
22
matrix elements in (9.9) are approximately equal to unity, i.e. L11
m,n = Lm,n ≈ 1 and

therefore L11 = L22 ≈ 1. This is justified due to the strong field confinement within
the dielectric regions. Furthermore, we assume that the coupling is weak and therefore the
12 T 21
coupling elements in (9.10) satisfy the condition Tmn

mn

1. Using these approximations

along with (9.12) and (9.15), the coupled mode equation (9.6) reduces to

 1

−K 21

−K 12
1



2
 Ω11



0

 
 
A
0  A1 
2  1
  = ω  .
Ω222
A2
A2

(9.16)

It is also helpful to linearize the system of equations (9.16). This can be done by noting that

180

the eigenmodes of the isolated resonators are not far apart and are clustered around their
mean value [187], i.e. ω ≈ ωα,n . Under this approximation ω and ωα,n satisfy the following

ω 2 − ωα,n

2

≈ 2ωα,n (ω − ωα,n ).

(9.17)

This brings us to the final form of the coupled mode equations



1



− 21 K 21

− 21 K 12  Ω11
1



0

 
 
0  A1 
A1 
  = ω .
Ω22
A2
A2

(9.18)

In the simplest situation when the two resonators are identical and only one mode of an
isolated resonator is considered, the electric field of the coupled system can be expressed as
E(r) = a1 E 1 (r) + a2 E 1 (r). According to (9.18) the coupled mode equations are then given
by

ω0 a1 + κ a2 = ωa1 ,
ω0 a2 + κ∗ a1 = ωa2 ,

(9.19)

where ω0 is the eigenfrequency of the isolated resonators. Using (9.11) and (9.18), κ is given
by
1
κ = − ω0
2
V1

1 (r) − 1

∗

E 1 (r).E 2 (r)d3 r.

(9.20)

The coupling coefficient Îş has a simple interpretation: it is the interaction energy between
the field generated by the second resonator and the dipole moment of the first resonator
averaged over one period. The complex conjugation of the coupling coefficient in (9.19) is
dictated by the energy conservation, assuming there is no loss or gain in the system [187].
181

The eigenfrequencies of the coupled system, which are guaranteed to be real due to the
Hermitian form of the equations in (9.19), are

ω± = ω0 ± |κ|,

(9.21)

where the frequencies of the coupled system, ω+ and ω− , correspond to the symmetric
√
√
eigenstate with a1 = a2 = 1/ 2 and the anti-symmetric eigenstate with a1 = −a2 = 1/ 2,
respectively.
One can readily see the similarity between the coupled mode theory and the quantum
theory as both are a theory of waves. Equation (9.19) is the Schr¨odinger equation for a twolevel system (a qubit). In quantum mechanical language, ω0 is the energy of the unperturbed
states and the off-diagonal matrix element Îş represents the interaction strength between the
two states which is responsible for the level repulsion and avoided crossing of the final mixed
states.
An interesting dynamic of the two-level system is the so-called Rabi oscillation. Let
the system at time t = 0 be prepared in the unperturbed state with energy ω1 . Then the
probability P (t) to find the system in the same state at time t is [188]
ω t
|Îş|2
P (t) = 1 − 4 2 sin2 R ,
2
ωR

(9.22)

where ωR = ω+ − ω− = 2|κ| is the Rabi frequency of the excitation oscillating back and
forth between the two levels. Because the unperturbed energies of the two states are equal,
the probability goes through the minimum, P = 0, which indicates that the excitation can
be completely transferred, leaving no residue in the initial state. The Rabi oscillation was

182

predicted in systems of optical waveguides [189] and in coupled ring resonators [190].
We now focus on a more realistic case: two coupled identical dielectric resonators where
in general, due to damping and leakage of the resonators, the energy is no longer conserved.
Therefore the governing equations need not be Hermitian. In the case of open systems one
has to modify the normalization expression (9.2) which leads to an altered expression for Îş.
The coupling coefficient in this case is similar to (9.20) but with no complex conjugation
(see [183] for detail). The coupled equations (9.19) are modified to a more general form

ω0 a1 + κa2 = ωa1
ω0 a2 + κa1 = ωa2 ,

(9.23)

where ω0 can now be complex: ω0 = η0 − iγ0 /2 representing loss and radiation. Similarly, κ
is in general complex as well: κ = κ − iκ where κ and κ are real numbers. The right hand
side of the coupled equations (9.23) can then be written as the summation of two matrices,
a Hermitian matrix, H0 , and an anti-Hermitian matrix W :




i
ω0 κ 

 = H0 − W ,
2
κ ω0

(9.24)

where




Ρ 0 κ 
H0 = 

κ Ρ0

(9.25)

and




 γ0
W =
2Îş

183

2κ 
.
Îł0

(9.26)

The similarity between the coupled mode theory and the effective non-Hermitian Hamiltonian formalism becomes apparent by comparing (6.8) and (9.24). The problem of two
coupled dielectric resonators is mapped to a two level quantum system where in general
each level is coupled to an independent continuum channel. This is because in general the
rank of W is 2, therefore, according to (6.6), one requires two independent open channels to
construct the anti-Hermitian matrix W . In the particular case when Îł0 = Âą2Îş , the rank
of W is equal to unity, therefore the superradiance condition is fulfilled and only one open
channel is required to construct the matrix W . In this case, the effective Hamiltonian has
two distinct eigenvalues, a purely real eigenvalue or the subradiant state , reminiscent of dark
modes in open quantum systems, with eigenfrequency η0 −κ , and a complex eigenvalue with
enhanced radiation properties and eigenfrequency η0 + κ − iγ0 , which is the superradiant
state.

9.3

A Single Metallic Sphere

In this section we consider a single isolated metallic sphere embedded in a homogeneous
background dielectric material. The problem is treated classically by solving Maxwell equations. In the absence of external sources and assuming a harmonic time dependence of the
form
E(r, t) = eiωt E(r),

(9.27)

the governing equation is the well known Helmholtz equation

∇2 E(r) − k 2 E(r) = 0.

184

(9.28)

The wave number k is defined for the interior and exterior regions of the plasmonic sphere
according to



√

kin = ωc in
k=


√

kout = ωc out

r ≤ a,
(9.29)
r > a,

where c is the speed of light in vacuum and a is the radius of the sphere which is located at the
origin of the coordinate system. The relative dielectric constants of the metallic sphere and
the background medium are denoted by in and out , respectively. Plasmonic structures are
usually made of noble metals, such as gold and silver, with face-centered cubic lattice type
or alkali metals, such as sodium and potassium, with body-centered cubic crystal structures.
Due to their symmetric crystal lattice types, they are isotropic to light and their relative
permittivity is characterized by a scalar. This dielectric constant is well described by the
Drude-Sommerfeld model

in (ω)

ωp2

= ∞− 2
,
ω − iωγs

(9.30)

where ωp is the plasma frequency which is defined by the electron effective mass m∗ , the
vacuum permittivity 0 , electron charge e, and electron density n; ωp2 = ne2 / 0 m∗ . The loss
within the dielectric material due to various processes, such as electron-phonon interaction,
impurities and scattering, is incorporated into the relaxation rate Îłs . The negative sign of
this term in the denominator is dictated by the phase convention adopted in (9.27). The
phenomenological parameter ∞ accounts for the contribution of the bound electrons to the
polarization of the dielectric material. For typical metals the plasma frequency, ωp , ranges
from 3 to 15 eV (700-3600 THz) which mainly falls into the ultraviolet spectrum [191–195].
185

The damping rate Îłs is much smaller than the plasma frequency, Îłs

ωp , being of the

order 10−2 − 10−1 eV (2.4-24 THz). Finally, the correction term ∞ typically ranges from
1 to 10 [196]. In an ideal electron gas, ∞ = 1 and γs = 0, therefore the dielectric function
(9.30) reduces to in (ω) = 1 − ωp2 /ω 2 . Below the plasma frequency the dielectric function
is negative and the field can not penetrate inside the metal. For frequencies larger than
the plasma frequency however, the dielectric constant become positive and the fields can
penetrate the metal.
Using the spherical coordinate system, the solutions of the Helmholtz equation (9.28)
for the plasmonic sphere can be divided into two categories: transverse magnetic (TM)
modes with vanishing radial magnetic field and transverse electric (TE) modes with no radial
electric field component. In this work we consider the TM modes only. The components of
the electric field are given by [197]:

Er = Îś C(r; a) ( + 1)

f (kr) m
Y (θ, φ),
(r)kr

Eθ = Μ C(r; a)

∂
∂ m
1
krf (kr)
Y (θ, φ),
(r)kr ∂(kr)
∂θ

Eφ = ζ C(r; a)

1
∂
1 ∂ m
krf (kr)
Y (θ, φ),
(r)kr ∂(kr)
sin θ ∂φ

(9.31)

where Îś is the normalization constant discussed in detail in the next section. The dielectric
constant is equal to in and out for r ≤ a and r > a, respectively. Y m (θ, φ) is a spherical
harmonics with

= 0, 1, 2, ... and m = 0, Âą1, Âą2, ..., Âą . The case of

= 0 results in the

trivial solution. The first non-trivial solution corresponds to the dipole mode,

= 1. The

function f (kr) is equal to the spherical Bessel function of the first kind and the spherical

186

Hankel function of the second kind, for r ≤ a and r > a, respectively.

f (kr) =





j (kin r)

r ≤ a,
(9.32)




h(2) (kout r) r > a.
The Bessel function j (kin r) represent standing waves within the plasmonic sphere while, not(2)

ing the phase convention (9.27), the Hankel function h

(kout r) describes radially outward

traveling waves which satisfy the Sommerfeld radiation boundary condition. The coefficient
C(r; a) guarantees that the boundary conditions are satisfied at the boundary of the sphere
(see [198] for details)

C(r; a) =





 j (kin a) −1

r ≤ a,




 h(2) (kout a) −1

r > a.

(9.33)

Matching the interior and the exterior fields leads to the characteristic equation of the discrete
eigenfrequencies of the system:
(2)

in

1 + kout a

h

(2)

h

(kout a)

= out 1 + kin a

(kout a)

j (kin a)
.
j (kin a)

(9.34)

Here, the prime denotes differentiation with respect to the argument of the function, i.e.
j (kin r) = ∂j (kin r)/∂(kin r). For a given radius, different modes can be labeled by : ω0, ,
where the subscript 0 denotes isolated single spheres. In case of small spherical particles,
when ka

1, considering the dipole mode

= 1, the spherical Bessel and Hankel functions
(2)

can be approximated by their leading order terms: j1 (kin a) ∟ kin a/3 and h1 (kout a) ∟
i(kout a)−2 . Therefore the characteristic equation (9.34) reduces to in = −2 out which leads
187

√
to the well known resonance frequency of ω = ωp / 3 for an ideal electron gas with ∞ = 1
and Îłs = 0.
Next, we numerically solve eq. (9.34) for a silver sphere with a free space background.
The parameters of the Drude-Sommerfeld model for silver are [196]: the plasma frequency
ωp = 8.9 eV, the damping rate γs = 0.1 eV, and ∞ = 5. The eigenfrequencies are always complex, which indicates the radiative nature of the nanospheres [199]. The real and
imaginary components of the eigenfrequencies as a function of radius and for various values
of

= 1, 2, 3, 4, are shown in Fig. 9.1. The real part is the frequency required to excite

a mode, for instance with a laser, and the imaginary component is the associated width
of the mode. For all the modes, as expected, the real component decreases monotonically
as the radius increases. To see the capability of the plasmonic sphere to manipulate light
in sub-wavelength dimensions consider the dipole resonance ( = 1) for a 50 nm sphere.
The resonance frequency is about 3 eV corresponding to a free space wavelength of approximately 413 nm which is an order of magnitude larger than the radius of the sphere. The
imaginary part of the eigenfrequencies consists of both non-radiative and radiative components, Im(ω0, ) = γ nrad + γ rad . The non-radiative damping is associated with the loss
within the plasmonic sphere. It was discussed in [199] that the non-radiative component can
approximately be considered size-independent and is of equal value for all different modes,
Îł nrad = (1/2)Îłs . It is therefore clear from Fig. 9.1(b) that the dipole is the most radiative
mode. For

= 1, initially the sphere becomes more radiative as the radius increases. How-

ever, larger spheres have less pronounced radiation properties. For all higher order modes,
the imaginary part grows as the radius increases.
The electric field patterns of a silver sphere with a radius of 40 nm and various values
of

are shown in Fig. 9.2. The field values are in logarithmic scale being normalized to
188

4
real( 7
h ! ) [eV]

real( 7
h ! ) [eV]

4
3
2
1

3
2
1100 50150 100

50

Radius [nm]

150

Radius [nm]
(a)

imag( 7h ! ) [eV]

real( 7h ! ) [eV]

4
3
2
1

50

0.6
0.4
0.2
0100 50 150 100

Radius [nm]

150

Radius [nm]
(b)

Figure 9.1: Eigenfrequencies for a metallic silver sphere with free space background as a
function of radius for various values (a) real part and (b) imaginary part. The left and
right axes represent values in units of eV and THz, respectively.
the maximum of the electric field. As the

value increases the fields become more tightly

bound to the surface of the sphere. This becomes important when we consider the coupling
between two spheres in the next section.

189

log10|Etotal|

log10|Etotal|

0

0

-400

-400

-200

-1 -200

0

0
-2

200

-1

-2

200

400

400
-400 -200

0

200

400

-3

-400 -200

0

(a)

200

400

-3

(b)
log10|Etotal|

0
-400
-200

-1

0
-2

200
400
-400 -200

0

200

400

-3

(c)

Figure 9.2: Electric field patterns of a silver sphere with a radius of 40 nm for (a) dipole
mode, = 1, (b) quadrupole mode, = 2, and (c) octupole mode, = 3. The values are
normalized to the maximum of the electric field. The projection m is equal to zero for all
three plots.
It is also instructive to look at the electric field pattern for the dipole mode. Fig. 9.3
represents the electric field pattern in polar coordinates for various radial distances from the
center of the sphere. The black arrow represents the dipole orientation. In all six figures, ρ is
the radial distance, perpendicular to the dipole axis and z is the direction along the dipole.
190

The blue line represents the relative field strength at a given polar angle. Furthermore, the
field strength is normalized to the maximum value of the electric field. Fig. 9.3(a) shows
the pattern right at the surface of the plasmonic sphere, r/a = 1. Closer to the surface of
the sphere the fields reach the maximum along the direction of the dipole. At r/a = 3 and
r/a = 5 the patterns are almost omnidirectional (see Figs. 9.3(b) and (c)). The well known
doughnut shape radiation pattern of the dipole only emerges in the far field. This is shown
in Figs. 9.3(d), (e) and (f).
The eigenmodes and plasmonic resonances of single metallic nanospheres of various sizes
and the effect of different background dielectrics have been considered in detail in [201] for
gold nanoparticles and in [200] for alkali metals such as sodium, lithium and Cesium; see [199]
for a more detail description of the size dependency properties of nanospheres. Below we
consider the case of two identical plasmonic spheres and discuss the coupling between them
and the eigenmodes of the system.

9.4

Superradiant and Dark States in System of Two
Coupled Spheres

We now consider the case of two coupled metallic spheres within the coupled mode theory
framework discussed earlier. The interaction between the spheres can greatly alter the radiation properties of the system and result in resonance frequencies profoundly different from
those for the isolated spheres. Here, we limit our consideration to the dipole modes only.
This is justified by the earlier discussion that the dipole mode is the most radiative mode
and also has the longest range compared to higher multipolarities. The isolated frequencies discussed in the previous section serve as the diagonal elements in the coupled mode
191

|Etotal| for r/Rsphere = 3, Rsphere = 40 nm

1

1

0.5

0.5

0

0

z

z

|Etotal| for r/Rsphere = 1.001, Rsphere = 40 nm

-0.5

-0.5

-1

-1
-1

-0.5

0


0.5

1

-1

-0.5

(a)

0.5

0.5

0

0

z

z

1

-0.5

-0.5

-1

-1
0


0.5

1

-1

-0.5

(c)

0.5

0.5

0

0

z

z

1

-0.5

-0.5

-1

-1
0


0


0.5

1

|Etotal| for r/Rsphere = 30, Rsphere = 40 nm

1

-0.5

1

(d)

|Etotal| for r/Rsphere = 20, Rsphere = 40 nm

-1

0.5

|Etotal| for r/Rsphere = 10, Rsphere = 40 nm

1

-0.5



(b)

|Etotal| for r/Rsphere = 5, Rsphere = 40 nm

-1

0

0.5

1

-1

(e)

-0.5

0


0.5

1

(f)

Figure 9.3: Electric field pattern for (a) r/a = 1, (b) r/a = 3, (c) r/a = 5, (d) r/a = 10, (e)
r/a = 20, and (f) r/a = 30. ρ is the radial distance and z is the direction along the dipole.
The values are normalized with respect to the maximum of the electric field.

192

matrix (9.24). The real and imaginary components of the dipole eigen mode are the diagonal elements of the Hermitian (9.25) and the anti-Hermitian (9.26) matrices, respectively.
The coupling between two modes, Îş, makes up the off-diagonal matrix elements of the final
matrix (9.24).
However, a difficulty arises due to the normalization of the single sphere modes. In the
far field region, kout r

1, the spherical Hankel functions behave asymptotically as

(2)

h

−ikout r

e
(kout r) ≈ (i) +1

kout r

.

(9.35)

Because of the complex nature of the eigenfrequencies, the asymptotic form of the fields given
by (9.35) grows exponentially in space as r → ∞. This growth is however compensated by
the exponential decay in time in the complete expression of the field (9.27) when the time
dependency is considered. As a result the amplitude of the wave front of the total field
(9.27) reaching any point in the asymptotic region is proportional to 1/r, as it is expected.
Nevertheless, the modes (9.31) should be properly normalized. The correct normalization of
such modes was discussed in [202–204] for one-dimensional problems. The generalization to
three dimensions by three different methods is discussed in [205], [206], and [207]. However,
in [208], it was shown that all three expressions are compatible.
The normalization condition is given by [208]:

σ(r, ω)E(r).E(r)d3 r +
V

i out
E(r).E(r)d2 r = 1
2k ∂V

(9.36)

where V is the integration volume and ∂V is its surface. The integration volume is assumed to
be sufficiently large, so the fields at its surface are accurately approximated by asymptotic

193

expressions of the spherical Hankel function provided in (9.35). The modified dielectric
function σ(r, ω) which incorporates the dispersiveness of the medium is given, according
to [209] as
σ(r, ω) =

1 ∂
ω 2 (r, ω) .
2ω ∂ω

(9.37)

Contrary to the normalization discussed earlier, the dot-product in (9.36) does not require
any complex conjugation of the fields. It is therefore easier to use the so-called tesseral
harmonics instead of the conventional spherical harmonics in the field expression (9.31).
The tesseral harmonics (sometimes also called real spherical harmonics) are nothing but
even and odd superpositions of the traditional spherical harmonics, see Appendix C. The
normalization condition (9.36) defines the constant Îś in the field expressions (9.31) up to a
phase. It is shown in Appendix B that assuming the volume of integration as a sphere, the
volume and surface terms in the normalization expression can be evaluated explicitly. It is
furthermore proved in the same appendix that the condition (9.36) reduces to:

(2)

I j (kin a) − I h

(kout a) = 1,

(9.38)

where the functional I f (kr) is given by (B.5)

( + 1)
I f (kr) = σ(r, ω) ξ 2 C 2 (r; a)
rf 2 (kr) + kr2 f (kr)f (kr)
2
k
2
k r3 2
+
f (kr) − f −1 (kr)f +1 (kr) .
2

(9.39)
(9.40)

Once the normalization constant Îś is found the coupling between two modes can be

194

calculated according to

1
κ = − ω0
2
V1

1 (r) − 1

E 1 (r).E 2 (r)d3 r,

(9.41)

where V1 is again the volume of sphere 1. Contrary to (9.20), the fields from both spheres
are treated on equal footing and no complex conjugation is required due to the normalization
definition (9.36).
In what follows we show the eigenfrequencies of a system of two coupled spheres with
different sizes and for different separation distances. The coupling Îş is calculated numerically
and the coupled mode matrix is constructed and diagonalized. In the first case two identical
silver spheres with radii of 10 nm are considered and the eigenfrequencies of the system are
calculated for two different dipole orientations and as a function of the separation distance
between the spheres. It is important to note that due to the symmetry of the dipole modes,
dipoles with perpendicular orientations do not couple. Therefore we consider two orthogonal
orientations only.
Fig. 9.4(a) shows the case where the two dipoles are parallel (vertical orientation). Both
the real and imaginary components of the eigenfrequencies are plotted as a function of the
separation d between the two spheres normalized by the sphere radius a. The maximum
coupling occurs when the two spheres are in contact with one another which in turn results
in the largest deviation from the unperturbed frequency of a single sphere (dotted black
lines). As the separation increases, the coupling decreases and the two eigenfrequencies
approach the unperturbed eigenmode. A similar phenomenon is seen for the case of two
dipoles in Fig. 9.4(b) (horizontal orientation). However, for a specific separation distance
between the spheres, the splitting of the two eigen modes is larger compared to the previous

195

case. This is because, as shown in the previous section, the electric field is maximum along
the dipole direction in the near field (see Fig. 9.3). Consequently, this orientation results in
larger coupling coefficients between the spheres.
According to our findings in the previous section, silver spheres with larger diameters are
more radiative. It is therefore desirable to look at the case of coupling between larger spheres
since the coupling is stronger. Fig. 9.5 shows the eigenfrequencies of a system of two silver
spheres with radii of 40 nm. We only consider the case of the horizontal dipole orientation
when that the coupling coefficient is larger. The difference between the superradiant and
subradiant states is more pronounced in this case. At d/a ≈ 3 there is a distinct difference
between the imaginary components of the two eigen modes. Similar to the previous case,
the two eigen frequencies approach the unperturbed resonance as the separation distance
increases.
According to our calculations, an exact dark mode does not exist for a system of two
silver plasmonics dipoles. Therefore a numerical search over the parameters of the DrudeSommerfeld dielectric function (9.30) was performed in order to find material properties for
which two plasmonic spheres can support a dark mode. Fig. 9.6 shows the eigenfrequencies
of a system of two spheres with Drude-Sommerfeld parameters of ∞ = 1, ωp = 10.918 eV
and γs = 0. At d/a ≈ 3.5, the rank of the anti-Hermitian part of the coupled mode matrix
is almost unity indicating that the interaction between the two spheres occurs through a
single continuum channel. Consequently, the imaginary component of the subradiant mode
is extremely small, in the order of 10−3 eV. This is indicated with a black circle in the figure.

196

0.056
imag( 7h ! ) [eV]

real( 7h ! ) [eV]

3.5
3.4
3.3
3.22

4

6

8

0.054
0.052
0.05
0.0482

10 12

4

6

4

6

8

10 12

(a)

0.056
imag( 7h ! ) [eV]

real( 7h ! ) [eV]

3.8
3.6
3.4
3.2
32

4

6

8

0.054
0.052
0.05
0.0482

10 12

8 10 12
d/a

(b)

Figure 9.4: Real and imaginary components of the eigenfrequencies of a system of two
coupled identical silver spheres with (a) vertical and (b) horizontal dipole orientations. The
radii of the spheres are 10 nm. d is the center-to-center separation and a is the radius of
the spheres. The black dotted lines represent the unperturbed eigenfrequency of a single
sphere.

197

imag( 7h ! ) [eV]

real( 7h ! ) [eV]

3.4
3.2
3
2.82 4 6 8 10 12

0.25
0.2
0.15
2 4 6 8 10 12

Figure 9.5: Real and imaginary components of the eigenfrequencies of a system of two
coupled identical silver spheres with horizontal dipole orientation. The radii of the spheres
are 40 nm. d is the center-to-center separation and a is the radius of the spheres. The
black dotted lines represent the unperturbed eigenfrequency of a single sphere.

9.5

Plasmonic Waveguide

We now consider the signal transmission through a plasmonic waveguide, namely a onedimensional chain of identical spheres. It is assumed that the two edges of the waveguide
are connected to an instrument capable of exciting the system of spheres with frequency ωe
and measuring the electric field intensity. Fig. 9.7 depicts the schematic of the plasmonic
waveguide and the two probes symmetrically coupled to the edges of the chain with coupling
constants Îłe . Similar to the tight binding model of crystals in condensed matter physics,
it is further assumed that each sphere in the chain only interacts with its nearest neighbor.

198

imag( 7h ! ) [eV]

real( 7h ! ) [eV]

7
6
5
4
2 4 6 8 10 12

0.6
0.4
0.2
02 4 6 8 10 12

Figure 9.6: Real and imaginary components of the eigenfrequencies of a system of two
coupled identical spheres with horizontal dipole orientation. The material properties of the
sphere are: ∞ = 1, ωp = 10.918 eV and γs = 0. The radii of the spheres are 20 nm. d is
the center to center separation and a is the radii of the spheres. The black dotted lines
represent the unperturbed eigenfrequency of a single sphere.
This system can be modeled with the effective non-Hermitian Hamiltonian (6.8)




i
 2 γe







Heff = 







+ ω0

Îş

0

...

0

0

Îş

ω0

Îş

...

0

0

0

Îş

ω0

Îş

...

0

..
.

..
.

..
.

..

.

..
.

..
.

0

0

0

...

Îş

i
2 Îłe

+ ω0








,







(9.42)

where ω0 is the unperturbed dipole frequency of the spheres, and κ is the coupling coefficient
(9.41) between adjacent dipoles. It is important to mention that the addition of the antiHermition matrix elements, 2i Îłe , with a positive sign is due to the phase convention adopted

199

earlier in (9.27).

ℏ𝝎𝒆

Figure 9.7: Schematics of a plasmonic waveguide; a one-dimensional chain of five silver
spheres with nearest neighbor coupling Îş. The two edges are symmetrically coupled to
continuum, the excitation source with frequency ωe , with coupling coefficient γe .

Through its coupling to the two probes, the system can undergo an additional superradiance phase transition, other than that discussed in the previous section. This is illustrated by
considering two different plasmonic waveguides. In both cases, according to our findings in
the last section, in order to maximize the coupling between the neighboring sites the spheres
are in contact with one another and the dipole orientation of the spheres is considered to be
along the waveguide (horizontal orientation). In the first case, a chain of five silver spheres
with radii of 10 nm is considered (Fig. 9.7). The resulting effective Hamiltonian (9.42)
describing the system is a 5x5 square matrix with diagonal elements, ω0 = 3.3468 + i0.0519
eV, and off-diagonal matrix elements κ = −0.2459 + i0.0029 eV. The continuum coupling
coefficient Îłe is treated as a variable that changes from small, Îłe = 0.01 eV, to extreme
values Îłe = 10 eV. The evolution of the complex eigenvalues of the effective Hamiltonian as
the coupling to the continuum varies is shown in Fig. 9.8(a). At small values of Îłe all the
eigenvalues acquire a small width through the coupling to the continuum. The widths of
the complex eigenmodes almost uniformly increase as the system is more strongly coupled
200

to the continuum. However, the situation changes dramatically at γe ≈ 1. The eigenvalues
have reached their maximum width and, with further increasing Îłe , the system undergoes
a phase transition (superradiance transition) when the eigenmodes become segregated into
two distinct categories: superradiant and subradiant states. At strong coupling, the two
superradiant states, their number being equal to the number of continuum channels (two
probes), steal the entire available width of the system and leave the remaining states as
narrow resonances.
The second waveguide differs only in the size of the spheres which have radii of 40 nm.
In this case, the diagonal unperturbed frequencies are ω0 = 3.1172 + i0.1910 eV and the
off-diagonal coupling coefficients are κ = −0.2606 + i0.0475 eV. The continuum coupling γe
is again varied from Îłe = 0.01 eV to Îł = 10 eV, and the complex eigenvalues are plotted
in Fig. 9.8(b). In general the picture is similar to the previous case. The superradiant
transition can be clearly seen as the coupling Îłe increases to extreme values.
We now study the propagation of a signal through the two waveguides using the transmission coefficient (6.12). Using (6.11) and (6.12) we arrive at the following expression for
the transmission coefficient

T ( ωe ) =

Îłe /Îş
N
r=1

( ωe − ωr )/κ

2

,

(9.43)

where ωr are the complex frequencies of the effective Hamiltonian (9.42) and N is its dimension.
Transmission as a function of the excitation frequency, ωe , is shown in Fig. 9.9 for
the waveguide with 10 nm spheres. At weak coupling to the continuum, Îłe = 0.03 eV,
Fig. 9.9(a), the five resonances are distinguishable. However, the resonances are not well

201

101
100
10-1
10-2

3

3.2

3.4

3.6

3.8

(a)

101

100

2.6

2.8

3

3.2

3.4

3.6

(b)

Figure 9.8: Complex-plane trajetories of the effective Hamiltonian for a one-dimensional
chain of five identical silver spheres with radii of (a) 10 nm and (b) 40 nm. The spheres are
in contact with one another and the dipoles are oriented along the waveguide. The arrows
show the direction of the evolution as Îłe changes from 0.01 eV to 10 eV.

202

separated due to the complex coupling coefficient between spheres, Îş, which provide the
eigenvalues of the effective Hamiltonian an initial width even for the closed system (Îłe = 0).
The case of intermediate coupling when Îłe = 0.55 eV is shown in Fig. 9.9(b). This is
when the system is on the road to superradiance transition and all the eigenvalues of the
Hamiltonian have large widths. Consequently, the resonances overlap and the transmission
is dramatically enhanced. The case of strong couplings, Fig. 9.9(c), has a picture similar
to that of the weak coupling case. However, only three resonances are remained. The two
giant superradiant states do not participate in signal transmission and transmission is greatly
suppressed due to the small width of the remaining subradiant states.
We follow the same steps of weak, intermediate and strong coupling to continuum in order
to study transmission through the waveguide with 40 nm spheres. Due to larger coupling, Îş,
between adjacent spheres the eigenvalues of the effective Hamiltonian poses a relatively large
initial width even for small coupling to the continuum. Therefore, contrary to the previous
case, the resonances overlap and are not separated even at weak coupling, Fig. 9.10(a).
Similar to before, the transmission is greatly enhanced at the superradiance transition, and
at extreme couplings, we are back to suppressed transmission.

9.6

Discussion

We studied the resonant frequencies of plasmonic spherical nanoantennas by solving the full
wave equation. These eigenfrequencies are always complex due to radiation and damping.
Utilizing the non-Hermitian effective Hamiltonian framework, it was shown that a system
of coupled two spheres can have modes with distinct properties; a superradiant mode with
enhanced radiation and a dark mode with extremely damped radiation.

203

0.025

0.4

0.02

0.3

0.015

0.2
0.01

0.1

0.005
02.6

2.8

3

3.2

3.4

3.6

3.8

02.6

4

2.8

3

3.2

(a)

3.4

3.6

3.8

4

(b)

0.03
0.02
0.01
02.6

2.8

3

3.2

3.4

3.6

3.8

4

(c)

Figure 9.9: Transmission through a one-dimensional chain of five identical spheres with
radii of 10 nm as a function of excitation source frequency ωe for (a) γe =0.03 eV, (b)
Îłe =0.55 eV, and (c) Îłe =10 eV.
Signal transmission through one dimensional chains was also considered. The coupling of
the edge spheres to the continuum can drastically change transport properties of the system.
A different superradiant transition arises through this interaction. Transmission is greatly
enhanced at this transition.
A possible direction to improve the accuracy of the results is to modify the DrudeSommerfeld model by including terms that take into account surface scattering effects. It
204

-3

2.5 x 10

0.04
2

0.03

1.5
1

0.02

0.5

0.01

02.4

2.6

2.8

3

3.2

3.4

3.6

3.8

02.4

4

2.6

2.8

3

(a)

3.2

3.4

3.6

3.8

4

(b)
-3

3 x 10

2

1

02.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

(c)

Figure 9.10: Transmission through a one-dimensional chain of five identical spheres with
radii of 40 nm as a function of excitation source frequency ωe for (a) γe =0.05 eV, (b)
Îłe =0.7 eV, and (c) Îłe =10 eV.
would be interesting to study the contribution of surface scattering to the total damping
and radiation of the nanospheres. Another possibility is to consider higher order modes and
their effect on the eigenfrequencies of the coupled system. These are left for the future.

205

Part 4
Conclusion and Future Work

206

To achieve great things, two things are needed; a plan, and not
quite enough time.
Leonard Bernstein

Chapter 10
Conclusion

10.1

Summary

10.1.1

The Novel Microwave Imaging System

In the first part of this thesis, a novel microwave imaging system was proposed. The primary component of the system is a tunable reflectarray antenna with high beam steering
capabilities. The unit cell of the array operates at two frequency bands and requires only
one varactor diode to dynamically alter the phase of the scattered field. This significantly
reduces the manufacturing cost of the array compared to more complicated designs that
require multiple varactors. The unit cell was evaluated using both full-wave simulations and
equivalent circuit modeling. The equivalent circuit provides a simple description of the unit
cell in terms of passive circuit components and requires far less computation time compared
to the full wave simulations. The 10x10 reflectarray built using the proposed unit cell shows
excellent steering capabilities. The beam can be deflected up to ±60◦ in both frequency
207

bands. This steering range has not yet been reported in literature.
The inclusion of the beam scanning array in the proposed microwave imaging system
significantly reduces cost and operational complexities compared to traditional setups. In
conventional systems, the region of interest is enclosed by a costly array of transceiver antennas which additionally requires a complicated switching circuitry. Our design eliminates
the need for multiple antennas and the involved circuitry. In addition, there is no constraint
on the size of the object under test. Consequently, the inspection of large objects such as
airplane wings and wind turbine blades would be achievable.
A prototype of the imaging system was built and experimentally verified. Using the time
reversal signal processing technique, the system was able to detect and locate cylindrical
targets of different sizes and positions. The system was also used for NDE applications aimed
at detecting different types of anomalies in dielectric materials. Specifically, two types of
anomalies were considered in the thesis: defects and metallic impurities. It was demonstrated
that the system was capable of accurately detecting both defects and impurities.

10.1.2

Superradiance in Open Quantum Systems

In the second part of the thesis the following three problems,
1. quantum transport through one-dimensional nanostructures,
2. quantum computation with a novel solid state charge qubit and,
3. plasmonic nanoantennas and waveguides,
were investigated within the framework of the non-Hermitian Hamiltonian. This approach
provides an alternative to popular methods such as the Feynman diagrammatic techniques

208

and the master equation approach that are commonly used for studying open quantum systems. Not only is the suggested method less complex, it also reveals new physics, namely
the sharp redistribution of decay widths and the emergence of short-lived “superradiant”
states for sufficiently strong coupling to the continuum. The effects of the emergence of
superradiant states on the properties of the system and various observables were discussed
in detail for all three problems. It was shown that quantum signal transmission is greatly
enhanced at the superradiance transition. The superradiance phenomenon is also responsible for the longer lifetime and protection of the internal states of the proposed qubit against
noise and decoherence. Finally, the superradiance transition provided a tool to manipulate
light via plasmonic structures and control the properties of nanoantennas and waveguide.
These findings can have a significant impact on the future of the semiconductor, quantum
computation, and imaging industries. Transistors could be engineered to be more power
efficient. Noise-resistant solid state qubits could maintain coherence much longer than gate
operation times and plasmonic field enhancement via superradiance in sub-wavelength dimensions could improve imaging resolution and sensing capabilities.

10.2

Future Work

10.2.1

The Novel Microwave Imaging System

One possible direction to improve the capability of the proposed system is to leverage the
3D beam steering ability of the reflectarray. This can be achieved by controlling individual
unit cells of the array as opposed to entire columns in the two dimensional case. Therefore,
while the prototype was used in a 2D setup, it can be readily extended to 3D applications.
One can also increase the beam steering range and go beyond ±60◦ and achieve larger gain
209

values by extending the surface area of the reflectarray and increasing the number of unit
cells.
Another possible improvement to the existing system would be to enhance the image
reconstruction technique. Though time reversal signal processing is fast, robust, and simple,
one drawback is the need for prior knowledge of a healthy sample, which is unavailable in
many cases. A potential solution would be to utilize more advanced reconstruction algorithms such as limited angle tomographic techniques [87, 88, 211, 212]. This would allow for
a reconstruction of target spatial profiles, moving beyond mere defect detection. Moreover,
the system can be easily scaled to operate in other frequency bands to detect targets and
defects with smaller or larger dimensions. It therefore has a range of potential applications
such as medical imaging, defect detection in extended composite materials, et cetera. These
are left for future studies.

10.2.2

Superradiance in Open Quantum Systems

With regard to the quantum transport problem and the models discussed in this thesis,
there are many possibilities to enrich the prototypical models. For a realistic situation, for
example a chain of quantum dots, one should carefully determine the lifetimes of the qubit
states. The geometry of the system can be made more complicated in various ways including
the transition to more-dimensional schemes. More qubits and more branches can be added
approaching a complicated network. It would be also interesting to extend the ideology of an
open quantum system to the study of coherent photon transport in continuous waveguides
and circuit quantum electrodynamics.
Another possibility is the inclusion of vibrational modes [122] and their coupling to electrons. In many physical systems such as bio and DNA molecules electron-phonon interaction
210

plays a significant role in the transport properties. The same can be considered for the proposed qubit. More accurate calculations of lifetime and decoherence can be made by explicitly
considering the interaction between electrons and phonons instead of the assumptian made
in this thesis, i.e. random gaussian noise.
The results found in the plasmonic problem need to be verified against exact numerical
solutions. Even though the coupled mode theory is intensively used within the optic and
plasmonic community it is necessary to verify the results against numerics since the coupled
mode framework is essentially a perturbation theory. It is also important to consider higher
order modes in the perturbation expansion. These are left for the future.

211

APPENDICES

212

Appendix A
The Stochastic Liouville Equation
Here we provide a simple derivation of the stochastic Liouville equation using perturbation
theory. The case considered here is a special case of the more general model where the
presence of phonons was accounted for by a heat bath affecting the electrons in a stochastic
fashion. Similarly, we include the vibrational effects, which lead to dephasing, by the addition
of the time-dependent Hamiltonian (8.27). Fluctuations of electronic energies are modeled
using Gaussian processes with the standard stochastic properties (8.28) and (8.29). The
total Hamiltonian is then H = H + Hφ and the evolution of the density operator is governed
by the von Neumann equation,
ρ˙ = −i[H, ρ].

(A.1)

Next we define superoperators Leff and Lφ according to

Leff ρ = [H, ρ],
Lφ ρ = [Hφ , ρ].

(A.2)
(A.3)

The von Neumann equation (A.1) in terms of the superoperators reads

ρ˙ = −iLρ = −i(Leff + Lφ )ρ.

213

(A.4)

Using

ρI (t) = eiLeff t ρ(t),
LIφ = eiLeff t Lφ e−iLeff t ,

(A.5)
(A.6)

the time evolution equation, (A.4), transforms to the interaction picture,

∂ I
ρ (t) = −iLIφ ρI .
∂t

(A.7)

The solution of eq. (A.7) up to the second order in the perturbation expansion is
t

ρI (t) =ρI (0) − i
0

dt LIφ (t )ρI (0)
t

t

+ i2

dt LIφ (t )LIφ (t )ρI (0).

dt
0

0

(A.8)

This formal solution only makes sense after averaging over the ensemble. Using the
properties of the bath given in eqs. (8.28) and (8.29) we have
t

ρI (t) = ρI (0) −

t

dt
0

0

dt LIφ (t )LIφ (t ) ρI (0).

(A.9)

Differentiating (A.9) we arrive at the equation of motion for ρI (t) ,
t
∂ I
ρ (t) = −
dt LIφ (t)LIφ (t ) ρI (t) .
∂t
0

(A.10)

In obtaining (A.10) it was assumed that the bath memory is short compared to the time
scales of the density operator. Therefore ρI (0) was replaced by ρI (t) . Evaluating the

214

integral with the help of (8.29) and using the definition provided in (A.2) we arrive at

∂ I
ρ (t) = −αφ
∂t

(cIj )† cIj , (cIj )† cIj , ρI (t)

,

(A.11)

j

where j runs over cells, n = −N, ..., −1, 1, ..., N , as well as the excited states of the central
qubits, L and R. Going back to the Schr¨odinger picture we have

∂
ρ(t) = −iLeff ρ(t) − αφ
∂t

†

†

cj cj , cj cj , ρ(t)

.

(A.12)

j

Finally, by calculating the matrix element of the density operator in the site representation, we arrive at the expression given in (8.30),

∂
ρ(t) i,j = −i Heff , ρ(t) i,j − 2αφ (1 − δi,j ) ρ(t) i,j .
∂t

(A.13)

Even though we have used perturbation expansion to derive the above stochastic Liouville
equation, the final result is exact [158] due to the Markovian character of the random process,
eqs. (8.28) and (8.29).

215

Appendix B
Quasi Normal Modes Normalization
In this appendix we explicitly normalize the fields of a plasmonic sphere by evaluating the
normalization expression (9.36)

σ(r, ω)E(r).E(r)d3 r +
V

i out
E(r).E(r)d2 r = 1.
2k ∂V

(B.1)

Due to the homogeneity of the sphere dielectric function in , and the surrounding background
out ,

the modified dielectric function σ(ω) given in (9.37) is only a function of frequency and

can be taken out of the integral. In what fallows we first evaluate the volume term in (B.1)
assuming the normalization volume itself is a sphere with a radius R where a

R. We first

consider the volume term. Using the field expressions given in (9.31) the volume term of the
normalization (9.36) is expressed as

V

+

R

E(r).E(r)d3 r = σ(ω) ζ 2

σ(ω)

1 ∂
krf (kr)
kr ∂(kr)

drr2 C 2 (r; a)

0
2

dΩ

( + 1)

2

f (kr) 2
kr

2
2
∂ m
1
∂ m
(θ,
φ)
Y (θ, φ) +
Y
∂θ
sin2 θ ∂φ

dΩ Y m (θ, φ)
,

(B.2)

where dΩ = sin θdθdφ is the solid angle differential in spherical coordinates. The integrals
involving spherical harmonics can be evaluated by using the orthogonality relation of the

216

2

tesseral harmonics and identity (C.2) in Appendix C. Thus, (B.2) reduces to

E(r).E(r)d3 r =

σ(ω)
V

R

σ(ω) ζ 2 ( + 1)

drr2 C 2 (r; a)

( + 1)

0

f (kr) 2
+
kr

1 ∂
krf (kr)
kr ∂(kr)

2

.
(B.3)

Due to the disconitiouty of the function f (kr) and the coefficients C(r; a) at the surface of
the plasmonic sphere [see eqn. (9.33)], the radial integral in (B.3) has to be divided into two
terms: 0R = 0a + aR . Each term can be evaluated with the help of (C.3) and (C.5). This
brings us to the final expression for the volume term of the normalization

(2)

σ(ω)
V

E(r).E(r)d3 r = I j (kin a) − I h

(2)

(kout a) + I h

(kout R) ,

(B.4)

where the functional I f (kr) is defined as

I f (kr) =

(B.5)

2 3
( + 1)
2 (kr) + kr 2 f (kr)f (kr) + k r f 2 (kr) − f
σ(ω) ζ 2 C 2 (r; a)
rf
−1 (kr)f +1 (kr)
2
k2

.

∂ f (kr). Note
As before, f (kr) implies differentiation with respect to the argument i.e. ∂(kr)

that in evaluating the first term of the right hand side of (B.4), I j (kin a) , one has to use all
the parameters corresponding to the region interior to the plasmonic sphere. i.e. in (ω), kin
and C(r; a) for r ≤ a as it is defined in (9.33). Accordingly, the same applies to the second
(2)

and third terms, I h

(2)

(kout a) and I h

(kout R) , for which one has to use the parameters

corresponding to the background material.

217

(2)

Next we show that the last term in the right hand side of (B.4), I h

(kout R) , exactly

cancels out with the surface term in (9.36). Therefore, as expected, the normalization
condition becomes independent of the integration volume. Because the integration sphere
was assumed to be sufficiently large, asymptotic expressions can be used to evaluate both
these terms. The spherical Hankel functions at large radial distances have the following
asymptotic form
(2)

h

e
(kr) ∟ i( +1)

−ikr

1−i

kr

( + 1)
.
2kr

(B.6)

The last term in the right hand side of (B.5) can therefoe be approximated as

(2)

I h

(kout R) ≈

−i out 2
C (r; a)(−) +1 e−2ikout R .
2k

(B.7)

We now consider the surface term of the normalization condition (9.36). Considering that
in the far field the dominant terms are Eθ and Eφ and using the asymptotic form of the
spherical Hankel function (B.6), the leading order of the surface term is

i out
i out 2
E(r).E(r)d2 r ≈
C (r; a)(−) +1 e−2ikout R
3
2k ∂V
2k

(B.8)

It is clear that the surface term (B.8) and (B.7) of the volume term cancel out. With
this, the normalization condition (B.1) reduces to

(2)

I j (kin a) − I h

which provide us the coefficient Îś in (9.31).

218

(kout a) = 1,

(B.9)

Appendix C
Tesseral Harmonics and Spherical
Functions Identities
This appendix contains the definition of tesseral harmonics and a list of identities that are
used throughout Chapter 9.
The tesseral harmonic are linear superpositions of the complex spherical harmonics with
same

and opposite sign m values. Therefore the azimuthal dependency of the functions

are in the form of sin(mφ) and cos(mφ) instead of the usual exponential form eimφ . They
are defined as [210]:

Y m (θ, φ) =





















2 +1 ( −|m|)! P |m| (cosθ)sin(|m|φ)
2π ( +|m|)!

m<0

2 +1 P m
4π

m=0

2 +1 ( −m)! P m (cosθ)cos(mφ)
2π ( +m)!

m>0

(C.1)

where P m are the associated Legendre polynomials. The tesseral harmonics satisfy the same
orthogonality relation as the complex spherical harmonics.
The tesseral harmonics also satisfy the following identity

219

2
2
∂ m
∂ m
1
Y (θ, φ) +
Y
(θ,
φ)
∂θ
sin2 θ ∂φ

dΩ

=

( + 1).

(C.2)

This can be proven by starting with the fact that the harmonics fulfill the identity
r2 ∇2 Y m (θ, φ) = − ( + 1)Y m (θ, φ). One can then get to (C.2), by calculating the matrix
element of the operator r2 ∇2 and using integration by parts.
A useful identity of the spherical Bessel and Hankel functions is the following

dr

( + 1)f 2 (kr) +

2
∂
krf (kr)
∂(kr)

rf 2 (kr) + kr2 f (kr)f (kr) + k 2

=

drr2 f 2 (kr).

(C.3)

where as previously f (kr) implies differentiation with respect to the argument. The
equality can be proven by using integration by parts and the spherical Bessel differential
equation to get

( + 1)f 2 (kr) = k 2 r2 f (kr)f (kr) + 2krf (kr)f (kr) + k 2 r2 f 2 (kr),

(C.4)

and
drr2 f 2 (kr) =

r3 2
f (kr) − f −1 (kr)f +1 (kr) .
2

220

(C.5)

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