DIFFUSION FOR A DISCRETE LINDBLAD MASTER EQUATION WITH PERIODIC HAMILTONIAN By Jacob Gloe A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics – Doctor of Philosophy 2023 ABSTRACT A quantum particle restricted to a lattice of points has been well studied in many different contexts. In the absence of considering the interaction with its environment, the particle simply undergoes ballistic transport for many suitable Hamiltonian operators. The evolution becomes much more complicated when considering environmental interaction, which leads to the so-called Lindblad master equation. When considering this master equation, the Lindbladian term dominates the dynamics of the particle, leading to diffusive propagation. In this document, we prove diffusion is indeed present in the context of a periodic Hamiltonian. Additionally, we show that the diffusion constant is inversely proportional to the particles’ coupling strength with its environment. This dissertation is dedicated to my wife, Lilianne who has been an amazing motivator throughout the writing process iii ACKNOWLEDGEMENTS This document is based on a paper written with my research advisor Jeffrey Schenker, whose guidance has been an incredible help throughout my graduate career. iv TABLE OF CONTENTS GLOSSARY OF TERMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER 2: STATEMENTS OF MAIN RESULTS . . . . . . . . . . . . . . . . . 4 CHAPTER 3: BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1: The Open Quantum System . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2: Derivation of the Master Equation . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.1: The Schrödinger Picture . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.2: The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.3: Lindblad’s Representation Theorem . . . . . . . . . . . . . . . . . . 10 3.3: The Translation-Covariance Assumption . . . . . . . . . . . . . . . . . . . . 11 3.4: Generators of Markov Jump Processes . . . . . . . . . . . . . . . . . . . . . 12 CHAPTER 4: LINDBLADIAN STRUCTURE AND ASSUMPTIONS . . . . . . . 15 4.1: Quasi-Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2: The Jump Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3: Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4: The Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5: A Generalized Dissipation Condition . . . . . . . . . . . . . . . . . . . . . . 27 CHAPTER 5: DIFFUSIVE PROPAGATION FOR MASTER EQUATION WITH PERIODIC HAMILTONIAN . . . . . . . . . . . . . . . . . . . 29 5.1: Proof of Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2: The Small g Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 CHAPTER 6: CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . 41 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 APPENDIX A: A SPECTRUM RESULT FOR COMMUTING OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 APPENDIX B: NEGATIVITY OF A JUMP PROCESS . . . . . . . . . . . . . . . 46 APPENDIX C: A GENERALIZED LIMIT FOR RESOLVENTS . . . . . . . . . . 48 v GLOSSARY OF TERMS • BpHq “ tA : H Ñ H : ||A||H ă 8u, the set of bounded operators in a Hilbert space H • B1 pHq “ tA : H Ñ H : trA ă 8u, the set of operators in H with finite trace • Md pAq, the set of d ˆ d matrices with entries in A • SApHq, the set of self-adjoint operators on a Hilbert space H • C0 pXq, the set of continuous functions on a metric space X which vanish at infinity • pτy f qpxq “ f px ` yq, the translation operator by y • pXj f qpxq “ xj f pxq, the position operators vi CHAPTER 1: INTRODUCTION Consider a single quantum particle living in a closed quantum system given by the Hilbert space H. The pure states of this particle are given by wave functions |ψt y P H which evolve in time via the Schrödinger Equation Bt |ψt y “ ´iH|ψt y for some self-adjoint Hamiltonian operator H P SApHq which represents the energy of the system. When analyzing entangled quantum systems, the associated density matrices ρt :“ |ψt yxψt | P B1 pHq are typically used. A density matrix is any bounded positive operator ρ satisfying trρ “ 1. For convenience, we will denote ρt px, yq :“ xx|ψt yxψt |yy as the kernel of ρt . The equation governing the evolution of density matrices in a closed quantum system is simply given by the related von Neumann equation Bt ρt “ ´irH, ρt s, (1) where here rA, Bs “ AB ´ BA represents the usual commutator. In this document, we consider a quantum particle restricted to a discrete lattice of points Zd , which amounts to letting our Hilbert space be H “ ℓ2 pZd q. This could simulate the particle being in a rigid crystalline structure and such models are widely used in modern day literature (See for example [1], [2], [3]). On its own, the solution ρt of equation (1) will behave ballistically in the limit t Ñ 8 for many standard Hamiltonian operators, i.e. translation invariant or periodic operators. That is, for the 2nd position moments given by ÿ xXt2 y :“ |x|2 ρt px, xq, xPZd we have the following asymptotic relation: xXt2 y „ t2 for large times t. Many recent works have shown that the particle’s dynamics will drastically 1 change to exhibit diffusion, i.e. xXt2 y „ t when subject to some form of random disorder. For instance, [3], [4] showed diffusive prop- agation for the tight binding Markov-Schrödinger model consisting of a random potential which fluctuates stochastically in time. In [5] and [6], diffusion was proven for a quantum particle coupled to a field of bosons having random thermal state in dimensions d ě 3, as well as a quantum particle coupled to an array of heat baths, respectively. Additionally, it is conjectured that the Anderson model consisting of a random static potential will similarly produce diffusion for dimension d ě 3 provided the disorder strength is sufficiently small. Heuristically, the random potential produces this diffusive effect due to the wave scatter- ing off of the random background and producing random phases. These phases eventually build up over time and lead to an overall decoherence of the wave. Most of the recent works in this area do not consider how the particle couples with its environment. For example, in the consideration of the particle being trapped in a crystal, the wave function could interact with free boson gasses in the crystal caused by quantized vibrations [1]. This situation is a lot more nuanced, though the equations of motion for such a particle have been well-established in both open quantum theory [7] as well as quantum information theory [8]. In the thermodynamic limit, the evolution of a one-particle density matrix taking into account environmental coupling may be approximated by the Lindblad equation ˆ ˙ 1 ˚ Bt ρt “ ´irH, ρt s ` g Ψpρt q ´ tΨ pIq, ρt u , (2) 2 where Ψ is some completely positive operator, g ą 0, and tA, Bu “ AB ` BA is the anti-commutator (See Section 4 for a derivation of this equation). The new term 1 Lpρt q :“ Ψpρt q ´ tΨ˚ pIq, ρt u (3) 2 2 in this expression is called the Lindbladian operator, which describes the coupling of the particle with its environment. A parameter g is introduced in (2) to allow us to control the strength of this coupling. In [2] and [9], it was shown that Lindbladian interaction induces diffusive behavior for quantum particles with translation-invariant Hamiltonian operators similar to the effect of adding a disordered potential. In [1], diffusion was shown for a model involving the Anderson Hamiltonian and an environmental interaction term similar to but distinct from a Lindbladian. It is thus natural to wonder whether the Lindbladian will be sufficient to contribute to diffusive behavior in other contexts as well. The present document continues this work by proving diffusion for a single quantum particle in an open quantum system coupled with an environment in the case of a periodic Hamiltonian. 3 CHAPTER 2: STATEMENTS OF MAIN RESULTS For the remainder of this document, we will assume H : B1 pℓ2 pZd qq Ñ B1 pℓ2 pZd qq is Q-periodic. That is, let Q P Md pZq be an invertible matrix so that tQx : x P Zd u defines a sublattice of points in Zd . Then, assume the Hamiltonian operator H satisfies rH, τQx s “ 0 for all x P Zd where τx denotes the translation operator by x. The various assumptions required for the Lindbladian are given in Section 5.3. However, I will outline in general what we need here. First, after diagonalizing the Lindbladian via a Fourier transform, we may fiber along the momentum variable k, which yields Lbk “ Tk ´ Dk for Tk an integral operator and Dk a multiplication operator. This is described by some authors as the gain-loss framework (See [10], [1]) where Tk is the gain term and Dk is the loss term. At the zero fiber, we must guarantee that the kernel of the Lindbladian is nondegenerate, which will help in calculations involving the spectrum. For the gain term, we must assume a local uniform lower bound, and for the loss term, we must assume a uniform upper and lower bound. Finally, the Lindbladian must abide by reflection invariance, in order to reflect certain symmetries present in the environment. Some authors [1], [9] may utilize certain physically realizable assumptions such as detailed balance or a gapping in the spectrum of the Lindbladian at the zero fiber. However, as we will see later, our assumptions are sufficient to prove a gapping in the spectrum of the Lindbladian and Hamiltonian together at the zero fiber. The main result proved in this document is the following central limit theorem: Theorem 1. Let H be a Q-periodic Hamiltonian and L a Lindbladian satisfying Assump- tions 1-4 in Section 5.3. Then, there exists a drift constant v P Rd and a positive definite 4 diffusion matrix D “ pDi,j qi,j P Md pCq such that for all initial conditions ρ0 P B1 pℓ2 pZd qq, i ?1τ px´τ tvq¨Q´1 k ÿ ř Di,j ki kj lim e ρτ t px, xq “ rtrρ0 se´t i,j , (4) τ Ñ8 xPZd where ρt is a solution of (2). In addition, if the initial condition ρ0 satisfies the regularity assumption ÿ |x|2 ρ0 px, xq ă 8, xPZd then the drift and diffusion constants are equivalent to: 1 ÿ v “ lim xρt px, xq, (5) tÑ8 rtrρ0 st xPZd 1 ÿ Di,j “ lim ppQT q´1 px ´ tvqqi ppQT q´1 px ´ tvqqj ρt px, xq. (6) tÑ8 2rtrρ0 st xPZd For Q “ I, this yields the known result of diffusion for translation-invariant Hamiltonian operators given in [2], [9]. Some authors proving similar central limit theorems assume the initial condition ρ0 “ δ0 or a zero-drift condition v “ 0 to simplify the calculations. Our method of proof allows for a generalization of this, as given above. In order to derive the Lindblad equation of motion (2), one must assume that the coupling strength with the environment is small. Therefore, of particular interest is the case 0 ă g ! 1. The method of proof used in this document allows for the diffusion matrix to be expressed as a function of this parameter, which is an improvement over even the translation-invariant case in [2], [9]. Assuming a uniform upper bound on the gain term, as well as an ergodicity assumption, we are able to prove the following result concerning the asymptotics of the diffusion for small g: Theorem 2. Let Dpgq be the diffusion matrix in Theorem 1 and suppose L additionally satisfies Assumptions 5 and 6 in Section 5.3. Then 0 ă lim` gDpgq ă 8. (7) gÑ0 1 That is, Dpgq „ g for small g. This is consistent with previous results since turning off 5 the coupling with the environment will simply lead to ballistic motion. The rest of this document is organized as follows. In Chapter 4, we provide some background into the Lindblad equation (2), including a derivation found in various quantum information theory sources, i.e. [8]. We also discuss Markov jump processes, which are necessary to fully describe the structure of the Lindbladian. In this chapter, we also provide an assumption from [11], and show how it is utilized to express the Lindbladian in a simpler form as is done in [9]. In Chapter 5, we introduce some structure and properties of the Lindbladian. First, we partially diagonalize the Hamiltonian and Lindbladian operators using a generalized Fourier transform. This allows us to state the various assumptions necessary to prove Theorems 1 and 2. We also compute the spectrum of our operators using results from [12]. Theorems 1 and 2 are proved in Chapter 6, and we discuss in Chapter 7 some additional research questions related to this work. 6 CHAPTER 3: BACKGROUND In this chapter, we provide some history into the Lindbladian as well as continuous-time Markov processes, which will be necessary to analyze certain properties of the Lindbladian. We also state the translation-covariance assumption, and show how this is used to decom- pose the Lindbladian in a nicer way. 3.1 The Open Quantum System In an arbitrary open quantum system, the total state space HT is given by a composite system comprised of a system of interest HS and the system corresponding to the environ- ment HE . That is, HT “ HS b HE . The goal is to derive the equations of motion for the system of interest (2). There are two main approaches for deriving equation (2), one of which comes from open quantum theory, and can be found in various sources such as [13], [7]. The central idea is to assume the total system HT is a closed quantum system, and thus its density matrices ρT satisfy (1). In order to determine the evolution equation for the density matrices ρ in the system of interest, we must trace out the extraneous degrees of freedom in the environment, i.e. ρ “ trE ρT . After using various physical assumptions such as the system of interest and the environment are noncorrelated for all times, the state of the environment is thermal for all times, and the so-called rotating wave approximation, we arrive at the master equation (2). What I present in Section 4.2 is a different approach from quantum information theory using generators of dynamical semigroups. 3.2 Derivation of the Master Equation 3.2.1 The Schrödinger Picture In quantum information theory, the evolution equation for density matrices ρt in the 7 system of interest HS should be of the form Bt ρt “ Gρt for some time-independent operator G. For our purposes, we shall assume that G is bounded; however, this assumption was relaxed in [14]. This differential equation is easily solved as ρt “ etG ρ0 , (8) and we assume that Φt :“ etG is a dynamical semigroup with generator given by G. That is, Φt : B1 pHq Ñ B1 pHq is a bounded one-parameter family of operators satisfying Φt`s “ Φt ˝ Φs for all t, s ě 0 and lim` tr|Φt ρ ´ ρ| “ 0 for all ρ P B1 pHq. As evidenced by tÑ0 equation (8) and the fact that density matrices have unit trace, it is reasonable to assume trpΦt ρq “ trρ for all ρ P B1 pHq, i.e. Φt is trace-preserving. Furthermore, due to the composite nature of the problem, we must guarantee positivity not only for the semigroups acting on the system of interest HS , but for semigroups acting on larger systems containing HS as a subsystem. In mathematical terms, we must guarantee positivity for the extended operators Φt b 1n : Mn pB1 pHqq Ñ Mn pB1 pHqq given by Φt b 1n pρ b Eij q “ Φt ρ b Eij for all n P N where Eij , 1 ď i, j ď n are matrix units spanning Mn pCq. That is, we require Φt to be completely positive. Thus, this approach amounts to finding an explicit form for the generator of a completely positive trace-preserving (CPTP) dynamical semigroup known as a quantum Markov semigroup. 3.2.2 The Heisenberg Picture Thus far, we have considered everything in the Schrödinger picture; that is, the states depend on time whereas operators/observables are time-independent. However, most au- 8 thors (See for instance [8]) categorize the generators of quantum Markov semigroups in the Heisenberg picture instead. In this picture, states are time-independent whereas observ- ables Xt P BpHq are dependent on time. Operators and density matrices are related due to the following. For any map A : B1 pHq Ñ B1 pHq in the Schrödinger picture, there is a corresponding map AT : BpHq Ñ BpHq uniquely defined in the Heisenberg picture by the relation trpXApρqq “ trpAT pXqρq and vice-versa. In the Heisenberg picture, we wish to categorize the generators of the T semigroups ΦTt :“ etG . In this space, the condition lim` tr|Φt ρ ´ ρ| is replaced with norm- tÑ0 continuity; that is, lim` ||ΦTt ´ 1|| “ 0. The trace-preserving condition is replaced by tÑ0 ΦTt pIq “ I, i.e. ΦTt must be unital. So in the Heisenberg picture, we wish to completely categorize the generators of completely positive unital dynamical semigroups ΦTt : BpHq Ñ BpHq. Related to this categorization arises the concept of a completely dissipative operator. For a bounded operator L, Lindblad [8] defines the dissipation function DpLq : BpHq ˆ BpHq Ñ BpHq by DpL; X, Y q :“ LpX : Y q ´ LpX : qY ´ X : LpY q. A bounded operator L is then said to be completely dissipative if the following conditions hold: (i) Lp1q “ 0, (ii) LpX : q “ LpXq: for all X P BpHq, and (iii) DpL b 1n , X, Xq ě 0 for all X P Mn pBpHqq and all n P N. 9 3.2.3 Lindblad’s Representation Theorem In [8], Lindblad produces the following representation theorem for the generators of completely positive, unital, and norm-continuous dynamical semigroups in the Heisenberg picture: Proposition 1 (Lindblad, 1976). Let L : BpHq Ñ BpHq be a bounded map and define Φt :“ etL . The following are equivalent: (i) Φt is completely positive, unital, and norm-continuous (ii) L is completely dissipative (iii) There exists a completely positive map Ψ and a self-adjoint operator H such that for all X P BpHq, 1 LpXq “ irH, Xs ` ΨpXq ´ tΨpIq, Xu. (9) 2 Transforming (9) back into the Schrödinger picture will yield the arbitrary form for a Lindbladian acting on density matrices. Assume G T : BpHq Ñ BpHq is the bounded generator given in (9) with corresponding completely positive map Ψ˚ . Then trpXGpρqq “ trpG T pXqρq 1 “ trpirH, Xsρ ` Ψ˚ pXqρ ´ tΨ˚ pIq, Xuρq 2 1 ˚ “ trXp´irH, ρs ` Ψpρq ´ tΨ pIq, ρuq. 2 Thus, Gpρq “ ´irH, ρs ` Ψpρq ´ 12 tΨ˚ pIq, ρu and the evolution equation for density matrices ρt in an open quantum system is given by the master equation (2), after introducing the scaling parameter g. 10 3.3 The Translation-Covariance Assumption As the evolution equation (2) may be decomposed into a purely Hamiltonian part and a purely Lindbladian part, let us begin by considering these terms separately. For the Lindbladian operator (3), we may yield a further decomposition using the Choi-Kraus Theorem given in [15], [16]: Proposition 2 (Choi/Kraus, 1975). A linear map A : B1 pHq Ñ B1 pHq is completely positive if and only if ÿ Aρ “ Aj ρA:j j for Aj P B1 pHq. Applying this to the completely positive maps Ψ, we may express (3) as ÿ 1 Lρ “ pVj ρVj: ´ tVj: Vj , ρuq j 2 for Vj P B1 pHq. Due to Proposition 1, there exists a corresponding quantum Markov semigroup Φt : B1 pHq Ñ B1 pHq given by Φt “ etL . Let us assume that this Markov semigroup is translation-covariant. That is, for τx the translation operator by x, assume Φt pτx˚ ρτx q “ τx˚ Φt pρqτx (10) for every density matrix ρ P B1 pHq and every x P Zd . This assumption reflects certain symmetries present in the environment and is thus a very physically reasonable assumption to make. Assuming translation-covariance of the semigroup allows us to categorize a certain class of Lindbladians by utilizing a very helpful theorem from Holevo [11], which I state here in the Schrödinger picture: Proposition 3 (Holevo, 1993). Let L be the generator of a translation-covariant trace- 11 preserving dynamical semigroup in B1 pℓ2 pZd qq. Then ż ÿ Lρ “ ´irH, ρs ` pVθ Lj pθqρLj pθq˚ Vθ˚ ´ tLj pθq˚ Lj pθq, ρuqdθ, Td j where H P SApℓ2 pZd qq satisfies rH, τx s “ 0 for all x P Zd , Vθ is a unitary representation of Td , Lj pθq are weak-* measureable functions satisfying rLj pθq, τx s “ 0 for all x P Zd , and the integral ż ÿ Lj pθqLj pθq˚ Td j weak-* converges. Utilizing Proposition 3, our Lindbladian operator may further be decomposed as ż ÿ Lpρq “ peiθX Lj pθqρLj pθq˚ e´iθX ´ tLj pθq˚ Lj pθq, ρuqdθ Td j where X represents the position operator. That is, we may express the completely positive map Ψ in (3) as ż Ψpρq “ dθeiθX Mθ pρqe´iθX (11) Td where the operators ÿ Mθ pρq :“ Lj pθqρLj pθq˚ j commute with translations. An important result of this decomposition is that the oper- ators Ψ are also translation-covariant. This yields a very nice structure for the Lindbladian. 3.4 Generators of Markov Jump Processes In this section, we give some background into continuous-time Markov processes, which may be found in various sources such as [17], [18]. In particular, we will focus on the specific Markov process known as a jump process. Jump processes are related to the structure of the Lindbladian, as will be shown in more detail in Section 5.2. In Chapter 6, we will 12 utilize this structure to prove the small g asymptotic result given by Theorem 2. Let Xt be a continuous-time Markov process on a locally compact metric space with homogeneous transition functions given by Tt . That is, Erf pXt q|Fs s “ Tt´s f pXs q, where Ft is the natural filtration pFt q “ pσpXu , u ď tqq. According to the Chapman- Kolmogorov equation, the family of transition functions tTt : t ě 0u form a dynamical semigroup (Also see Hille-Yosida theory for more details). Hence we may define the in- finitesimal generator A of the Markov process in the usual way by 1 pAf qpxq :“ lim` pTt f ´ f q. tÑ0 t The generator of a Markov process is a way to describe how the process moves from point to point in infinitesimally small increments, and thus it is important to be able to categorize the process in a meaningful way. In general, generators of Markov processes may be very complex. However, we may restrict to a specific class of Markov process to be able to categorize them quite nicely. For instance, if we assume the transition functions act on the space C0 pXq and are contractive (||Tt || ď 1 @t) and norm-continuous ( lim` ||Tt f ´ f || “ tÑ0 0 @f P C0 pXq), the Markov process is called a Feller process. For a Feller process, the infinitesimal generator may be categorized via the following Proposition by Revuz and Yor [17]: Proposition 4. Let Xt be a real-valued Feller process on a locally compact smooth manifold X. Then the infinitesimal generator A is given by 1 pAf qpxq “ cpxqf pxq ` bpxq ¨ ∇f pxq ` div apxq∇f pxq ż „ 2 ȷ y´x ` f pyq ´ f pxq ´ ¨ ∇f pxq Rpdy|xq, Xztxu 1 ` |y ´ x|2 where Rpdy|xq is a positive conditional Radon measure on Xztxu, apxq is symmetric and nonnegative, and cpxq ď 0. 13 Heuristically, this generator describes a Markov process which moves from a position x via translation by bpxq, diffuses via a gaussian with covariance apxq, and jumps via Rp¨|xq. The term cpxq describes the killing probability, allowing for the process to be terminated at some future time. While this is the most general form for the generator of a Feller process, we are only interested in the special case where a particle’s movement is governed solely by jumps, i.e. a pure jump process. In this process, the particle waits an exponential time at a position x, jumps to a position y instantaneously, then repeats this process, jumping to a new position. For a pure jump process, the infinitesimal generator will simply be given by ż pAf qpxq “ rf pyq ´ f pxqsRpdy|xq, (12) Xztxu where the rate at which the particle jumps from y to x is given by Rpdy|xq. 14 CHAPTER 4: LINDBLADIAN STRUCTURE AND ASSUMPTIONS 4.1 Quasi-Momentum Space Let us now take a Fourier transform and consider our operators in the momentum representation. We define our Fourier transform on the square-integrable kernel ρt px, x1 q in the following way. First, we may split Zd into a finite set of equivalence classes Σ “ Zd { „ such that for x, y P Zd , x „ y if and only if x ´ y P tQn : n P Zd u. Then for σ, σ 1 P Σ, ÿ ´1 p´x1 ¨Q´1 p1 ρbt pp, p1 qσ,σ1 “ e´ipx¨Q q ρ px, x1 q t (13) xPσ,x1 Pσ 1 where ρbt pp, p1 q : T2d Ñ B1 pC|Σ| q is now matrix-valued. Applying this transform to the maps (11) yields ż ÿ 1 ´1 pp´Qθq´x1 ¨Q´1 pp1 ´Qθqq Ψpbρt qpp, p qσ,σ1 “ dθ e´ipx¨Q Mθ pρt qpx, x1 q Td xPσ,x1 Pσ 1 ż \ 1 “ dθM θ pρt qpp ´ Qθ, p ´ Qθqσ,σ 1 . Td Using the specific form for Mθ , we decompose this operator further. We note that since Lj pθq are translation-invariant, we may define Lj pθ; x ´ yq :“ xx|Lj pθq|yy and using this notation, \ 1 M θ pρt qpp, p qσ,σ 1 ÿ ÿ|Σ| ÿ ´1 p´x1 ¨Q´1 p1 q “ e´ipx¨Q Lj pθ; x ´ yqρt py, y 1 qLj pθ; x1 ´ y 1 q˚ xPσ,x1 Pσ 1 n,m“1 yPσn ,y 1 Pσm 1 ÿ|Σ| ÿ ÿ ´1 p 1 ´1 p1 “ e´ix¨Q Lj pθ; xqb ρt pp, p1 qσn ,σm1 pe´ix ¨Q Lj pθ; x1 qq˚ n,m“1 xPσ´σn x1 Pσ 1 ´σm 1 ÿ “ pLbj pθ; pqb ρt pp, p1 qL bj pθ; p1 q: qσ,σ1 , j 15 where ÿ ´1 p L bj pθ; pqσ,σ1 :“ e´ix¨Q Lj pθ; xq. xPσ´σ 1 Therefore, ż Ψpb 1 ρt qpp, p q “ dθM cθ pp ´ Qθ, p1 ´ Qθqrb ρt pp ´ Qθ, p1 ´ Qθqs Td for the operators M cθ given by ÿ cθ pp, p1 qrAs :“ M L bj pθ; pqAL bj pθ; p1 q: . j We then define ˆ ˙ k k ρbt;k ppq :“ ρbt p ´ ,p ` (14) 2 2 where p, k P Td and we think about ρbt;k as fibers over ρbt , indexed by k P Td . We note that for the density matrix ρt :“ |ψt yxψt |, we have ÿ ´1 k 1 ´1 k p|ψ\t yxψt |qk ppqσ,σ 1 “ e´ipx¨Q pp´ 2 q´x ¨Q pp` 2 qq ψt˚ pxqψt px1 q xPσ,x1 Pσ 1 ˆ ˙ ˆ ˙ k k “ ψbt˚ p´ ψbt p` 2 σ 2 σ1 where ÿ ´1 p ψbt ppqσ :“ eix¨Q ψt pxq. xPσ Since |ψt y P ℓ2 pZd q, |ψbt y P L2 pTd ; C|Σ| q. Cauchy-Schwarz then yields 1 d p|ψ\ |Σ| t yxψt |qk P L pT ; B1 pC qq. By extension, we have ρ bt;k P L1 pTd ; B1 pC|Σ| qq given any density matrix ρt P B1 pℓ2 pZd qq. It will be useful later to define the pairing ż xA, By :“ tr dpAppqBppq (15) Td whenever AppqBppq P L1 pTd ; B1 pC|Σ| qq. 16 Using this fibering, we may write equation (11) as ż ˆ ˙ k k Ψpb ρt;k qppq “ dθMθ p ´ ´ Qθ, p ` ´ Qθ rb c ρt;k pp ´ Qθqs “: pTk ρbt;k qppq. (16) Td 2 2 Remark: At the k “ 0 fiber, this integral takes on the simple form ż pT0 Aqppq “ dp1 M cQ´1 pp´p1 q pp1 , p1 qrApp1 qs. Td To compute Ψ˚ pIq, we observe: trpΨ˚ pIqbρt pp, p1 qq “ trpΨpb ρt qpp, p1 qq ż ż ÿ “ dp dθ trpL bj pθ; p ´ Qθq: L bj pθ; p ´ Qθqb ρt pp ´ Qθ, p ´ Qθqq Td Td j ż ÿ “ tr dθ bj pθ; pq|2 ρbt pp, p1 q, |L Td j and so pΨ˚ pIqb ρt qpp, p1 q “ Dppqb ρt pp, p1 q where ż ÿ Dppq :“ dθ |L bj pθ; pq|2 . (17) Td j Similarly, pbρt Ψ˚ pIqqpp, p1 q “ ρbt pp, p1 qDpp1 q. This allows us to write the simplified form of our Lindblad operator (3) in the momentum representation as Lbk “ Tk ´ Dk , where ˆ ˆ ˙ ˆ ˙˙ 1 k k pDk Aqppq “ D p´ Appq ` AppqD p ` 2 2 2 and Tk is given in (16). Let us now focus on the Hamiltonian term. In the momentum representation, we note 17 that due to the periodicity of the Hamiltonian, pHρt qpx, x1 q becomes ÿ ÿ|Σ| ÿ ´ipx¨Q´1 p´x1 ¨Q´1 p1 q dt qpp, p1 qσ,σ1 “ pHρ e Hpx, yqρt py, x1 q xPσ,x1 Pσ 1 n“1 yPσn ÿ|Σ| “ b σ,σn ρbt pp, p1 qσn ,σ1 , Hppq n“1 where ÿ ´1 p Hppq b σ,γ :“ e´ix¨Q Hpx ` γ, γq. (18) xPσ´γ Similarly, ÿ|Σ| pρd 1 t Hqpp, p qσ,σ 1 “ ρbt pp, p1 qσ,σn Hpp b 1 qσn ,σ1 n“1 and therefore applying the Fourier transform to rH, ρt spx, x1 q yields ˆ ˙ ˆ ˙ k k pJk ρbt;k qppq :“ H p ´ b ρbt;k ppq ´ ρbt;k ppqH p ` b . 2 2 Combining this with the Lindblad operator, we may write the evolution equation (2) as Bt ρbt;k “ ´Gk ρbt;k (19) where Gk “ iJk ´ gpTk ´ Dk q. (20) 4.2 The Jump Process As our method of proof involves a perturbation argument similar to approaches taken in [1], [3], and [2], it is natural to consider the k “ 0 fiber. At k “ 0, we have some additional structure for the Lindbladian that will be very useful in the proof of positivity for the density matrix D. First, for the translation-invariant case Q “ I, the operators L bj 18 and ρbt are no longer matrix-valued, and hence commute. This yields ż ÿ ż ÿ 2 bj pθ; pq|2 ρbt ppq. pL0 ρbt qppq “ b dθ |Lj pθ; p ´ θq| ρbt pp ´ θq ´ b dθ |L Td j Td j ř b This is of the form (12) where the rate of jumping from θ to p is given by |Lj pp ´ θ; θq|2 dθ j and thus Lb0 is the generator for a pure jump process in the translation-invariant case. However, this is not true for an arbitrary Q. In order to yield this structure for the Lindbladian, we first must project onto the subspace given by kerJ0 . Lemma 1. Let Π be the projection onto kerJ0 . Then ΠLb0 Π is the generator for a jump process on C|Σ| ˆ Td . Proof. Since Hppqb is a |Σ| ˆ |Σ| matrix, we may list the eigenvalues as λ1 ppq, ¨ ¨ ¨ , λ|Σ| ppq with corresponding eigenvectors ψ1 ppq, ¨ ¨ ¨ , ψ|Σ| ppq. Denote Eij ppq :“ |ψi ppqyxψj ppq| as the corresponding matrix element in this basis. As the kernel of J0 is the commutant of H, b in the above framework, we may write that Π is the projection onto diagonal matrices in the |Σ| basis tψi ppqui“1 , i.e. |Σ| ÿ pΠAqppq “ Aii ppqEii ppq. i“1 Furthermore, for any matrix Appq, ÿ ż pΠT0 ΠAqppq “ Ejj ppq dp1 M cQ´1 pp´p1 q pp1 , p1 qrΠApp1 qsEjj ppq j Td ÿż “ dp1 Aii pp1 qxψj ppq|M cQ´1 pp´p1 q pp1 , p1 qrEii pp1 qs|ψj ppqyEjj ppq i,j Td and similarly, ÿ ÿ pΠD0 ΠAqppq “ Ejj ppq Aii ppqpDppqEii ppq ` Eii ppqDppqqEjj ppq j i ÿ “ Ajj ppqxψj ppq|Dppq|ψj ppqyEjj ppq. j 19 The operator ΠLb0 Π will thus be given by ÿ "ż ż * 1 1 1 1 1 pΠL0 ΠAqppq “ b dp rppj, pq, pi, p qqApi, p q ´ dp rppi, p q, pj, pqqApj, pq Ejj ppq i,j Td Td (21) where rppj, pq, pi, p1 qq :“ xψj ppq|M cQ´1 pp´p1 q pp1 , p1 qrEii pp1 qs|ψj ppqy ÿ “ |xψj ppq|Lbk pQ´1 pp ´ p1 q; p1 q|ψi pp1 qy|2 ě 0 k and Apj, pq “ Ajj ppq. This is again of the form (12) and hence ΠLb0 Π is the genera- tor for a jump process on C|Σ| ˆ Td with rate of jumping from pi, p1 q to pj, pq given by rppj, pq, pi, p1 qqdp1 . 4.3 Assumptions At this point, we make some additional assumptions, which are slightly stronger con- ditions than are often taken for Lindblad operators of this form (See for instance [1]). We assume the following: Assumptions: 1. (Nondegeneracy of the kernel) kerLbT0 “ xIy, 1 2. (Uniform Dissipation at all Momenta) C ď Dppq ď C for some C ą 0, 3. (Reflection invariance) rL, Rs “ 0 for the reflection operator pRψqpxq “ ψp´xq, 4. (Local Uniform Positivity of the Integral Kernel) There exist constants δ ą 0 and χ ą 0 such that cQ´1 pp´p1 q pp1 , p1 qrApp1 qs ě 1 I M χ for all operators A P L1 pTd ; B1 pC|Σ| qq satisfying App1 q ě 0 and trApp1 q “ 1 whenever |p ´ p1 | ă δ, 20 5. (Uniform Boundedness of the Integral Kernel) There exists a constant χ ą 0 such that cQ´1 pp´p1 q pp1 , p1 qrApp1 qs ď χI M for all operators A P L1 pTd ; B1 pC|Σ| qq satisfying App1 q ě 0 and trApp1 q “ 1 and all p, p1 P Td , ş 6. (Ergodicity of ΠLb0 Π) For a.e. p P Td and every function ϕ ě 0 with Td dpϕppq “ 1, ´ ¯n there exists n P N such that ΠD10 Π ΠT0 Π ϕppq ą 0. Assumptions 1-4 are needed to prove Theorem 1, and Theorem 2 additionally requires Assumptions 5 and 6. I now describe these assumptions in detail, as well as some useful implications of each. We note that clearly, LbT0 I “ 0. The first assumption guarantees that I is in fact the only equilibrium eigenvector for LbT0 . The second assumption is utilized in Lemma 2 to guarantee a gapping in the spectrum of G0 . Looking closer at the third assumption, we see that the reflection operator R is actually the operator pRA b k qppqσ,σ1 “ A´k p´pq´σ,´σ1 in momentum space. Hence the condition rR, b Lbk s “ 0 yields ´p∇k Lbk |k“0 Aqppqσ,σ1 “ pR∇ b k Lbk |k“0 Aqp´pq´σ,´σ1 “ p∇k Lbk |k“0 RAqp´pq b ´σ,´σ 1 “ p∇k Lbk |k“0 Aqppqσ,σ1 . So the third assumption guarantees ∇k Lbk |k“0 “ 0. (22) That is, we have zero-drift for a particle governed solely by the Lindbladian. Let us define the operator On :“ e´rn piJ0 `gD0 q T0 e´rn´1 piJ0 `gD0 q T0 ¨ ¨ ¨ T0 e´r0 piJ0 `gD0 q 21 for some real numbers r0 , ¨ ¨ ¨ , rn ą 0. We remark that ppiJ0 ` gD0 qAqppq “ KppqAppq ` AppqK : ppq (23) for g Kppq :“ iHppq b ` Dppq 2 is simply a sum of multiplication operators. This implies : ppq pe´tpiJ0 `gD0 q Aqppq “ e´tKppq Appqe´tK . Let A P L1 pTd ; B1 pC|Σ| qq satisfy Appq ě 0 for all p P Td and xI, Ay “ 1. Assumption 4 then yields the following: pOn Aqppq “ pe´rn piJ0 `gD0 q T0 On´1 Aqppq : “ e´rn Kppq pT0 On´1 Aqppqe´rn K ppq "ż * “e ´rn Kppq cQ´1 pp´p q pp1 , p1 qrpOn´1 Aqpp1 qs e´rn K : ppq dp1 M 1 Td ż 1 ´rn Kppq 2 ě |e | dp1 trppOn´1 Aqpp1 qq. χ |p´p1 |ăδ Due to Gronwall’s Inequality and Assumption 2, |e´rn Kppq |2 ě e´Cgrn and hence, ż 1 pOn Aqppq ě e´Cgrn dp1 trppOn´1 Aqpp1 qq. χ |p´p1 |ăδ Repeating this argument n times, we have |Σ|n´1 ´Cgpr0 `¨¨¨`rn q ż ż pOn Aqppq ě e dp1 ¨ ¨ ¨ dpn trpAppn qq. χn |p´p1 |ăδ |pn´1 ´pn |ăδ If n is sufficiently large (nδ ą 2π), we will have ż ż ż dp1 ¨ ¨ ¨ dpn trpAppn qq ě Cn,δ trpAppqq “ Cn,δ |p´p1 |ăδ |pn´1 ´pn |ăδ Td 22 for some constant Cn,δ ą 0 and so pOn Aqppq ą 0 for all p P Td . Let us now define the family of operators xptq :“ etpiJ0 `gD0 q e´tG0 for t ą 0. For this family, we have d xptq “ getpiJ0 `gD0 q T0 e´tpiJ0 `gD0 q xptq. dt Hence, ÿ8 żt ż s1 ż sn´1 xptq “ g n ds1 ds2 ¨ ¨ ¨ dsn es1 piJ0 `gD0 q T0 eps2 ´s1 qpiJ0 `gD0 q T0 ¨ ¨ ¨ T0 e´sn piJ0 `gD0 q n“0 0 0 0 for some n P N and some constants s1 , ¨ ¨ ¨ , sn ą 0. This finally implies ÿ8 żt ż s1 ż sn´1 ´tG0 n pe Aqppq “ g ds1 ds2 ¨ ¨ ¨ dsn pOn Aqppq n“0 0 0 0 where the real numbers r0 , ¨ ¨ ¨ , rn in On are defined by r0 “ sn , r1 “ sn´1 ´ sn , ¨ ¨ ¨ , rn´1 “ s1 ´ s2 , rn “ t ´ s1 . So Assumption 4 guarantees that pe´tG0 Aqppq ą 0 @ A P L1 pTd ; B1 pC|Σ| qq satisfying Appq ě 0 @ p P Td and xI, Ay “ 1. (24) The fifth assumption will be utilized in Section 6.2 to bound the invariant state of G. Finally, the sixth assumption is utilized to guarantee ergodicity of the underlying jump for the jump process ΠLb0 Π. 4.4 The Spectrum In order to compute the spectrum of G0 , we must utilize a result from Deimling [12] and Schaefer [19] on the eigenvalues on the boundary of the spectral radius. For a Banach space X, we define a total cone K Ă X to be a closed convex set such that λK Ă K for all λ ě 0, K X p´Kq “ t0u, and K ´ K “ X. The dual cone of K is then defined as K ˚ :“ tx˚ P X ˚ : Rex˚ pxq ě 0 on Ku. 23 An operator T P BpXq is considered to be quasicompact if T n “ T1 ` T2 for some n P N, T1 is bounded with rpT1 q ă prpT qqn and T2 is compact. Here, rpT q “ sup |λ| denotes the λPσpT q spectral radius of T . Proposition 5 (Theorem 19.5 in Deimling, 1985 and Proposition 5.1 in Schaefer, 1974). Let X be a Banach space, K Ă X a total cone, and T P BpXq a positive quasicompact operator satisfying that for each x P Kzt0u, there exists n P N such that x˚ pT n xq ą 0 for all x˚ P K ˚ zt0u. Then for rpT q the spectral radius of T , we have the following: (a) rpT q ą 0 and rpT q is a simple eigenvalue with a positive eigenvector v such that x˚ pvq ą 0 for all x˚ P K ˚ zt0u. (b) |λ| ă rpT q for all λ P σpT qztrpT qu. We can then utilize Proposition 5 to prove the following Lemma regarding the spectrum of our operator at the zero fiber: Lemma 2. The operator G0 defined in (20) has spectrum given by σpG0 q “ t0u Y Σ0 , where Σ0 Ď tRez ě δg u for some δg ą 0. Furthermore, 0 is a nondegenerate eigenvalue of G0 for which the corresponding eigenvector Feq satisfies Feq ppq ą 0 for a.e. p P Td . Proof. Due to (20) we have σess pG0 q “ σess piJ0 ` gD0 q since T0 is compact and essential spectrum is invariant under compact perturbations. Due to (23) and Lemma 4, we have ď σess piJ0 ` gD0 q “ σpKppq ¨ ` ¨ K : ppqq p ď` ˘ Ď σpKppq¨q ` σp¨K : ppqq . p Assumption 2 yields ď ď ! g ) σpKppq¨q, σp¨K : ppqq Ď z P C : Rez ě p p 2C 24 Im z Im z Re z Re z δg (a) σpG0 q (b) σpe´G0 q Figure 1: A visualization of the spectrum of G0 . In (a), σpG0 q is shown with isolated eigenvalue at 0 and the remaining spectrum included in the right half plane after a gap δg . In (b), σpe´G0 q is shown with isolated eigenvalue at 1 and the remaining spectrum included in the unit circle. ␣ g ( and so σess pG0 q Ď z P C : Rez ě C . The remaining spectrum will be discrete spectrum. As suggested in Figure 1, we will compute the discrete spectrum of G0 by consider- ing the related operator e´G0 and utilizing Proposition 5. This operator will act on ele- ments of L1 pTd ; SApC|Σ| qq. Therefore, we will consider the total cone given by K “ tA P L1 pTd ; SApC|Σ| qq : Appq ě 0 for a.e. p P Td u. We note that e´G0 is certainly a positive operator. To see that it is quasicompact, we first realize that σpG0 q Ď tz P C : Rez ě 0u implies σpe´G0 q Ď tz P C : |z| ď 1u. Then due to our above observations about the essential spectrum, we additionally have σess pe´G0 q Ď tz P C : |z| ď cu for some c ă 1. Define the counter-clockwise contour Γ such that |z| ă 1 for all z P Γ, σess pe´G0 q Ă int Γ, and Γ does not intersect any eigenvalues of e´G0 . Then for 1 ş 1 T1 :“ 2πi Γ e´z z´G 0 dz and T2 :“ e´G0 ´ T1 , T1 will be bounded and T2 will be compact since it is finite rank. Additionally, rpT1 q ă 1 “ rpe´G0 q due to the fact that G0: I “ 0. Thus, e´G0 is quasicompact. Due to (24), we may apply Proposition 5 to the operator e´G0 . This yields 1 is a simple eigenvalue, e´G0 Feq “ Feq for some strictly positive equilibrium eigenvector Feq , and |λ| ă 1 for all eigenvalues λ ‰ 1. This is equivalent to G0: having a one-dimensional kernel given by xIy. This then implies 25 that G0 has a one-dimensional kernel as well. Furthermore, the rest of the eigenvalues of : e´G0 will lie strictly inside B1 p0q. Hence the rest of the discrete spectrum of G0 will lie in ` ˘ tRez ě cg u for some constant cg ą 0. So Lemma 2 holds with δg :“ min Cg , cg . This Lemma states that there is a unique density matrix in the kernel of G0 . We shall label this equilibrium eigenvector Feq . Since 0 is an isolated point of the spectrum, we may define the Riesz projection onto this eigenvector in the normal way as ż 1 1 P0 “ dz “ Feq xI, ¨y, (25) 2πi Γ z ´ G0 where Γ is a counterclockwise contour in ρpG0 q whose interior contains the eigenvalue 0 and no other point of σpG0 q, and the pairing x¨, ¨y is given in (15). Im z δg {4 Epkq Re z 3δg {4 Figure 2: A visualization of the spectrum of Gk for sufficiently small k, which consists of a simple isolated eigenvalue Epkq P tz P C : |z| ď δg {4u and the remaining spectrum in some set Σk Ď tRez ě 3δg {4u. We also note that the spectrum of Gk moves continuously with k and hence for k sufficiently small, σpGk q “ tEpkqu Y Σk where Epkq is some isolated nondegenerate eigenvalue in tz P C : |z| ď δg {4u and Σk Ď tRez ě 3δg {4u (See Figure 2). Since Epkq is isolated and nondegenerate, we may con- sequently define the one-dimensional Riesz projection onto the corresponding eigenvector 26 as ż 1 1 Pk “ dz, (26) 2πi Γ z ´ Gk where Γ is a counterclockwise contour in ρpGk q whose interior contains the eigenvalue Epkq and no other point of σpGk q. 4.5 A Generalized Dissipation Condition Let us now return to the operator G0T acting on observables X P BpL2 pTd qq in the Heisenberg picture. Due to Proposition 1, G0T will be completely dissipative. In this section, we prove a generalized version of dissipation using the results from Lemma 2. First, note that G0T “ iJ0 ´ g LbT0 where 1 LbT0 X “ ΨpXq ´ tΨpIq, Xu 2 for every X P L2 pTd ; BpC|Σ| qq and ż ż ÿ T bj pθ; pq: Xpp ` QθqL ΨpXqppq “ dθMθ pp, pqrXpp ` Qθqs “ c dθ L bj pθ; pq. Td Td j Due to Proposition 2 in Lindblad [8], we may further decompose this operator as ż ΨpXqppq “ dθWθ: ppqpXpp ` Qθq b 1qWθ ppq Td for some functions W : Td ˆ Td Ñ BpC|Σ| b Kq and some auxiliary Hilbert space K. Recall the dissipation function DpG0T q : L2 pTd ; BpC|Σ| qq ˆ L2 pTd ; BpC|Σ| qq Ñ L2 pTd ; BpC|Σ| qq given by DpG0T ; X, Y q :“ G0T pX : Y q ´ G0T pX : qY ´ X : G0T pY q. Using our decomposition of Ψ, we observe that ż T DpG0 ; Xppq, Xppqq “ ´ dθ|pXpp ` Qθq b 1qWθ ppq ´ Wθ ppqXppq|2 ď 0. Td 27 For a function F P L2 pTd ; BpC|Σ| qq, define the following inner product on L2 pTd ; BpC|Σ| qq: ż 1 xX, Y yF :“ tr F ppqtX : ppq, Y ppqu. 2 Td We then have for Feq the equilibrium eigenfunction of G0 , ż 1 T RexX, G0 XyFeq “ ´ tr dpFeq ppqrDpG0T ; Xppq, Xppqq ` DpG0T ; X : ppq, X : ppqqs 4 d ż T ż 1 “ “ tr dpFeq ppq dθ |pXpp ` Qθq b 1qWθ ppq ´ Wθ ppqXppq|2 4 Td Td ‰ ` |pX : pp ` Qθq b 1qWθ ppq ´ Wθ ppqX : ppq|2 . Hence, since Feq ppq ą 0 for a.e. p P Td by Lemma 2, RexX, G0T XyFeq “ 0 if and only if pXpp ` Qθq b 1qWθ ppq “ Wθ ppqXppq and pX : pp ` Qθq b 1qWθ ppq “ Wθ ppqX : ppq for a.e. p, θ P Td . Multiplying by Wθ ppq: and integrating, these imply ΨpXqppq “ ΨpIqppqXppq and ΨpXqppq “ XppqΨpIqppq for a.e. p P Td . Hence pLbT0 Xqppq “ 0 and so by Assumption 1, X “ I and pG0T Xqppq “ 0. This string of implications yields the observation that pG0T Xqppq ‰ 0 for a.e. p P Td ñ RexX, G0T XyFeq ą 0. (27) 28 CHAPTER 5: DIFFUSIVE PROPAGATION FOR MASTER EQUATION WITH PERIODIC HAMILTONIAN 5.1 Proof of Main Result We now have everything we need to prove Theorem 1. Proof. To prove the central limit theorem (4), we note that ż ż ÿ ÿ ´1 k 1 ´1 k ÿ ´1 tr dpb ρt;k ppq “ dp e´ipx¨Q pp´ 2 q´x ¨Q pp` 2 qq ρt px, x1 q “ eix¨Q k ρt px, xq Td Td σPΣ x,x1 Pσ x and hence, ż i ?1τ px´τ tvq¨Q´1 k ´i ?1τ τ tv¨Q´1 k ÿ e ρτ t px, xq “ e tr dpbρτ t;k{?τ ppq. x Td where the drift constant v is to be chosen later. Since ρbt;k “ e´tGk ρb0;k , this then gives that this term is equivalent to ż ´i ?1τ τ tv¨Q´1 k ` ? ˘ e tr dp e´τ tGk{ τ ρb0;k{?τ ppq. Td Consider the Riesz projection Pk{?τ as defined in (26). Introducing the projections Pk{?τ and 1 ´ Pk{?τ after the semigroup yields i ?1τ px´τ tvq¨Q´1 k ÿ e ρτ t px, xq x „ ż ȷ ? ´i ?1τ τ tv¨Q´1 k ´τ tEpk{ τ q ` ´τ tGk{?τ ˘ “e xI, ρb0;k{ τ ye ? ` tr dp e p1 ´ Pk{?τ qbρ0;k{?τ ppq . Td Let us first deal with the second term in this expression. Define a contour Γ surrounding ? σpGk{?τ qztEpk{ τ qu such that ReΓ ě δg {2 (which is possible for sufficiently large τ , say 1 τě ε for some ε ą 0). As Gk{?τ is bounded, we may additionally choose Γ to be bounded, and ˇˇ ˇˇ ˇˇ 1 ˇˇ sup ˇˇˇˇ ˇˇ “: M ă 8. 1 τ ě ε ,zPΓ z´G ? k{ τ ˇˇ 29 We then have ˇˇ ż ˇˇ ˇˇ ´i ?1 τ tv¨Q´1 k ` ´τ tG ? ˘ ˇˇ ˇˇe τ tr dp e k{ τ p1 ´ Pk{ τ qb ? ρ0;k{ τ ppqˇˇˇˇ ? ˇˇ d ˇTˇ ż ż ˇˇ ˇˇ ´τ tz 1 ˇˇ ď ˇˇtr ˇ ˇ dp dze ρb0;k{ τ ppqˇˇˇˇ ? Td z ´ Gk{ τ? ˇˇ Γ ˇˇ ˇˇ 1 ˇˇ ´τ t inf Rez ď sup ˇˇˇˇ ρ0;k{?τ || |Γ|e zPΓ ˇˇ ||b τ ě 1 ,zPΓ z ´ Gk{ τ ? ˇ ˇ ε ď M |Γ|xI, ρb0;k{?τ ye´τ tδg {2 . We note that i ?1τ x¨Q´1 k ÿ xI, ρb0;k{?τ y “ e ρ0 px, xq Ñ trρ0 x as τ Ñ 8 and hence this term will vanish in the large τ limit. For the first term, we use the Taylor expansion ˆ ˙ ? 1 ÿ 1 ÿ 1 Epk{ τ q “ Ep0q ` ? Bi Ep0qki ` Bi Bj Ep0qki kj ` o . τ i 2τ i,j τ Since Ep0q is the isolated eigenvalue of G0 , Lemma 2 gives Ep0q “ 0. Using Feynman- Hellman, Bi Ep0q “ xI, Bi Gk |k“0 Feq y. Hence, if we choose the drift constant to be v “ iQT xI, ∇k Gk |k“0 Feq y, (28) ´i ?1τ τ tv¨Q´1 k 1 ř then e will cancel with e´τ t τ i Bi Ep0qki . In this case, ˜ ¸ ´ 2t ř ÿ i ?1 px´τ tvq¨Q´1 k ÿ i ?1 x¨Q´1 k Bi Bj Ep0qki kj e τ ρτ t px, xq “ e τ ρ0 px, xq e i,j ` op1q. x x Again using the fact that xI, ρb0;k{?τ y Ñ trρ0 and choosing the diffusion matrix D to be defined as 1 Di,j “ Bi Bj Ep0q, (29) 2 30 this yields i ?1τ px´tvq¨Q´1 k ÿ ř Di,j ki kj lim e ρτ t px, xq “ rtrρ0 se´t i,j τ Ñ8 x as desired. ř Now let us assume the initial condition ρ0 satisfies |x|2 ρ0 px, xq ă 8. The solution xPZd to the evolution equation (19) is given by ρbt;k “ e´tGk ρb0;k and so we have the following for the right-hand side of (5): ż 1 ÿ i T xρt px, xq “ ´ Q tr dp∇k ρbt;k ppq|k“0 t d t Td xPZ ż i T “ ´ Q tr dp∇k pe´tGk ρb0;k qppq|k“0 . t Td : After using the formula for the derivative of a semigroup and using the fact that e´tGk I “ I due to conservation of quantum probabilities, we have ż żt ż 1 ÿ i T ´sG0 i T xρt px, xq “ Q tr dp ds∇k Gk |k“0 e ρb0;0 ppq ´ Q tr p∇k ρb0;0 qppq. t d t Td 0 t Td xPZ The second term in this expression trivially vanishes in the large t limit. For the first term, we insert the Riesz projections P0 and 1 ´ P0 after the semigroup to yield the two terms ż żt ż żt i T i T Q tr dp ds∇k Gk |k“0 Feq ppqxI, ρb0;0 y ` Q tr dp ds∇k Gk |k“0 e´sG0 p1 ´ P0 qb ρ0;0 ppq. t Td 0 t Td 0 For the projection off of P0 , we may draw a bounded contour Γ around σpG0 qzt0u such that δg ReΓ ě 2 . Hence ż żt i T Q tr dp ds∇k Gk |k“0 e´sG0 p1 ´ P0 qb ρ0;0 ppq t Td 0 ż żt ż i T 1 1 “ Q tr dp ds∇k Gk |k“0 dze´sz ρb0;0 ppq t Td 0 2πi Γ z ´ G0 ż ż T 1 1 1 “ iQ tr dp∇k Gk |k“0 p1 ´ e´tz q ρb0;0 ppq. Td 2πi Γ tz z ´ G0 This will vanish in the large t limit since tz1 p1 ´ e´tz q Ñ 0 as t Ñ 8 for Rez ą 0. Therefore, ż 1 ÿ T lim xρt px, xq “ iQ tr dp∇k Gk |k“0 Feq ppqxI, ρb0;0 y “ iQT rtrρ0 sxI, ∇k Gk |k“0 Feq y. tÑ8 t Td xPZ d 31 1 ř So v “ lim xρt px, xq due to our definition of the drift constant (28). tÑ8 rtrρ0 st xPZd We perform a similar analysis in calculating the expression (6) for the diffusion matrix D. We observe: 1 ÿ ppQT q´1 px ´ tvqqi ppQT q´1 px ´ tvqqj ρt px, xq 2t d xPZ ż 1 “ ´ tr dppBi ´ itppQT q´1 vqi qpBj ´ itppQT q´1 vqj qb ρt;k ppq|k“0 2t Td ż 1 “ ´ tr dprBi Bj ´ itppQT q´1 vqi Bj ´ itppQT q´1 vqj Bi 2t Td ´ t2 ppQT q´1 vqi ppQT q´1 vqj spe´tGk ρb0;k qppq|k“0 . : After using the formula for the derivative of a semigroup and using the fact that e´tGk I “ I due to conservation of quantum probabilities, we have 7 1 ÿ ÿ ppQT q´1 xqi ppQT q´1 xqj ρt px, xq “ Nn ptq 2t d n“1 xPZ where ż 1 N1 ptq “ ´ tr dppBi Bj ρb0;0 qppq, 2t Td ż żt 1 ␣ ( N2 ptq “ ´ tr dp ds Bi Gk |k“0 e´sG0 pBj ρb0;0 qppq ` Bj Gk |k“0 e´sG0 pBi ρb0;0 qppq , 2t d żT ż0t ż s 1 ␣ N3 ptq “ ´ tr dp ds dr Bi Gk |k“0 e´ps´rqG0 Bj Gk |k“0 e´rG0 ρb0;0 ppq 2t Td 0 0 ( ` Bj Gk |k“0 e´ps´rqG0 Bi Gk |k“0 e´rG0 ρb0;0 ppq , ż żt 1 N4 ptq “ tr dp dsBi Bj Gk |k“0 e´sG0 ρb0;0 ppq, 2t d 0 żT 1 ␣ ( N5 ptq “ tr dp ippQT q´1 vqi pBj ρb0;0 qppq ` ippQT q´1 vqj pBi ρb0;0 qppq , 2 d żT żt 1 ␣ N6 ptq “ tr dp ds ippQT q´1 vqi Bj Gk |k“0 e´sG0 ρb0;0 ppq 2 Td 0 ( `ippQT q´1 vqj Bi Gk |k“0 e´sG0 ρb0;0 ppq , and t N7 ptq “ ppQT q´1 vqi ppQT q´1 vqj xI, ρb0;0 y. 2 We first note that lim N1 ptq “ 0 so this first term is negligible in the large time limit. Let tÑ8 32 us now consider projecting onto and off of the eigenspace xFeq y using the Riesz projection P0 given in (25). Introducing P0 ` p1 ´ P0 q after every semigroup in the above expression yields the following. For the second term, we have ż żt 1 ␣ N2 ptq “ ´ tr dp ds Bi Gk |k“0 e´sG0 p1 ´ P0 qpBj ρb0;0 qppq 2t Td 0 ( `Bj Gk |k“0 e´sG0 p1 ´ P0 qpBi ρb0;0 qppq 1␣ ( ´ ippQT q´1 vqi xI, Bj ρb0;0 y ` ippQT q´1 vqj xI, Bi ρb0;0 y , 2 the second part of which simply cancels with N5 ptq. For the first part of this term, we use the fact that off of the equilibrium eigenvalue, we may draw a contour Γ enclosing the rest of the spectrum of G0 such that ReΓ ě C ą 0 as per Lemma 2. This yields ż żt " ż 1 1 1 N2 ptq ` N5 ptq “ ´ tr dp ds Bi Gk |k“0 e´sz pBj ρb0;0 qppq 2t Td 0 2πi Γ z ´ G0 ż * 1 ´sz 1 `Bj Gk |k“0 e pBi ρb0;0 qppq 2πi Γ z ´ G0 ż " ż 1 1 1 1 “ ´ tr dp Bi Gk |k“0 p1 ´ e´tz q pBj ρb0;0 qppq 2t Td 2πi Γ z z ´ G0 ż * 1 1 ´tz 1 `Bj Gk |k“0 p1 ´ e q pBi ρb0;0 qppq . 2πi Γ z z ´ G0 1 Since ReΓ ě C ą 0, lim p1 ´ e´tz q “ 0 for z P Γ. Hence this term vanishes as well for tÑ8 tz t Ñ 8. For the fourth term, we note that ż żt 1 N4 ptq “ tr dp dsBi Bj Gk |k“0 e´sG0 p1 ´ P0 qb ρ0;0 ppq 2t T d 0 ż 1 ` tr dpBi Bj Gk |k“0 Feq ppqxI, ρb0;0 y. 2 Td In a very similar manner to the previous calculation, the first part of this term vanishes as 33 t Ñ 8. For N6 ptq, another similar calculation yields ż żt 1 ␣ N6 ptq “ tr dp ds ippQT q´1 vqi Bj Gk |k“0 e´sG0 p1 ´ P0 qb ρ0;0 ppq 2 Td 0 ( `ippQT q´1 vqj Bi Gk |k“0 e´sG0 p1 ´ P0 qb ρ0;0 ppq ´ tppQT q´1 vqi ppQT q´1 vqj xI, ρb0;0 y. Due to the lack of a t in the denominator, the first part of this term will not vanish as t Ñ 8. In fact, the first part of this term will tend to ż 1 tr dptippQT q´1 vqi Bj Gk |k“0 G0´1 p1 ´ P0 qbρ0;0 ppq 2 Td ` ippQT q´1 vqj Bi Gk |k“0 G0´1 p1 ´ P0 qb ρ0;0 ppqu. Finally, for N3 ptq, applying P0 to both semigroups yields ż żt żs 1 ␣ ´ tr dp ds dr Bi Gk |k“0 e´ps´rqG0 P0 Bj Gk |k“0 e´rG0 P0 ρb0;0 ppq 2t Td 0 0 ( `Bj Gk |k“0 e´ps´rqG0 P0 Bi Gk |k“0 e´rG0 P0 ρb0;0 ppq ż żt żs 1 “ ´ tr dp ds dr tBi Gk |k“0 Feq ppqxI, Bj Gk |k“0 Feq y 2t Td 0 0 `Bj Gk |k“0 Feq ppqxI, Bi Gk |k“0 Feq yu xI, ρb0;0 y 1 ␣ (1 “´ ippQT q´1 vqi ¨ ippQT q´1 vqj ` ippQT q´1 vqj ¨ ippQT q´1 vqi t2 xI, ρb0;0 y 2t 2 t “ ppQT q´1 vqi ppQT q´1 vqj xI, ρb0;0 y. 2 This will fully cancel with N7 ptq and the second part of N6 ptq. For the remainder of the 34 terms in N3 ptq, we again use the contour Γ. We observe that ż żt żs 1 ␣ ´ tr dp ds dr Bi Gk |k“0 e´ps´rqG0 P0 Bj Gk |k“0 e´rG0 p1 ´ P0 qb ρ0;0 ppq 2t Td 0 0 ( `Bj Gk |k“0 e´ps´rqG0 P0 Bi Gk |k“0 e´rG0 p1 ´ P0 qb ρ0;0 ppq ż żt żs 1 ␣ “ ´ tr dp ds dr Bi Gk |k“0 Feq ppqxI, Bj Gk |k“0 e´rG0 p1 ´ P0 qb ρ0;0 y 2t Td 0 0 ( `Bj Gk |k“0 Feq ppqxI, Bi Gk |k“0 e´rG0 p1 ´ P0 qb ρ0;0 y ż żt żs 1 ␣ “ ´ tr dp ds dr ippQT q´1 vqi Bj Gk |k“0 e´rG0 p1 ´ P0 qb ρ0;0 ppq 2t Td 0 0 ( `ippQT q´1 vqj Bi Gk |k“0 e´rG0 p1 ´ P0 qb ρ0;0 ppq ż 1 ␣ Ñ ´ tr dp ippQT q´1 vqi Bj Gk |k“0 G0´1 p1 ´ P0 qbρ0;0 ppq 2 Td ( `ippQT q´1 vqj Bi Gk |k“0 G0´1 p1 ´ P0 qbρ0;0 ppq . This precisely cancels with the first part of N6 ptq as shown above. Similarly, placing the projections in the reverse order for N3 ptq gives ż żt żs 1 ␣ ´ tr dp ds dr Bi Gk |k“0 e´ps´rqG0 p1 ´ P0 qBj Gk |k“0 e´rG0 P0 ρb0;0 ppq 2t Td 0 0 ( `Bj Gk |k“0 e´ps´rqG0 p1 ´ P0 qBi Gk |k“0 e´rG0 P0 ρb0;0 ppq ż 1 ␣ Ñ ´ tr dp Bi Gk |k“0 G0´1 p1 ´ P0 qBj Gk |k“0 Feq ppq 2 Td ( `Bj Gk |k“0 G0´1 p1 ´ P0 qBi Gk |k“0 Feq ppq xI, ρb0;0 y. Finally, performing the projection 1´P0 on both semigroups in N3 ptq requires two contours 35 Γ and Γ1 , each with strictly positive real part and such that Γ X Γ1 ‰ H. We then have ż żt żs 1 ␣ ´ tr dp ds dr Bi Gk |k“0 e´ps´rqG0 p1 ´ P0 qBj Gk |k“0 e´rG0 p1 ´ P0 qb ρ0;0 ppq 2t Td 0 0 ( `Bj Gk |k“0 e´ps´rqG0 p1 ´ P0 qBi Gk |k“0 e´rG0 p1 ´ P0 qb ρ0;0 ppq ż " ż ż 1 1 1 “ ´ tr dp Bi Gk |k“0 dz dz 1 2t Td 2πi Γ 2πi Γ 1 ˆ ˙ 1 1 ´tz 1 ´tz 1 1 1 p1 ´ e q ´ p1 ´ e q B j Gk |k“0 ρb0;0 ppq z 1 ´ z tz tz 1 z ´ G0 z 1 ´ G0 ż ż 1 1 ` Bj Gk |k“0 dz dz 1 2πi Γ 2πi Γ1 ˆ ˙ * 1 1 ´tz 1 ´tz 1 1 1 p1 ´ e q ´ 1 p1 ´ e q Bi Gk |k“0 1 ρb0;0 ppq z 1 ´ z tz tz z ´ G0 z ´ G0 Ñ 0. Putting all terms Nn ptq together and taking the limit as t Ñ 8 then gives the following expression for the right-hand side of (6): 1 ÿ lim ppQT q´1 px ´ tvqqi ppQT q´1 px ´ tvqqj ρt px, xq tÑ8 2rtrρ0 st xPZd ż 1 “ ´ tr dpBi Gk |k“0 G0´1 p1 ´ P0 qBj Gk |k“0 Feq ppq 2 Td ż 1 ´ tr dpBj Gk |k“0 G0´1 p1 ´ P0 qBi Gk |k“0 Feq ppq (30) 2 d żT 1 ` tr dpBi Bj Gk |k“0 Feq ppq. 2 Td Using second-order perturbation theory, we have Bi Bj Ep0q “ xI, Bi Bj Gk |k“0 Feq y ` xI, Bi Gk |k“0 G0´1 p1 ´ P0 qBj Gk |k“0 Feq y ` xI, Bj Gk |k“0 G0´1 p1 ´ P0 qBi Gk |k“0 Feq y and hence due to our definition of the diffusion matrix (29), 1 1 ÿ Di,j “ Bi Bj Ep0q “ lim ppQT q´1 px ´ tvqqi ppQT q´1 px ´ tvqqj ρt px, xq. 2 tÑ8 2rtrρ0 st d xPZ It is clear that the diffusion matrix D is symmetric. We wish to further show that this 36 matrix is positive definite. To show positivity for the final term in (30), let us consider a solution ρbt;k of the evolution equation (19) with initial condition ρb0;k “ Feq . That is, ρbt;k “ e´tGk Feq . Similar to the above analysis, we have for z P Cd , ˇ ˇ2 1 ÿ ˇÿ ˇ 0ď ρt px, xq ˇ ppQT q´1 xqi zi ˇ ˇ ˇ 2t d ˇ i ˇ xPZ 1 ÿ ÿ “ ppQT q´1 xqi ppQT q´1 xqj ρt px, xqzi˚ zj 2t i,j xPZd ż 1 ÿ “´ tr dpBi Bj pe´tGk Feq ´ Feq qppq|k“0 zi˚ zj . 2t i,j Td ´tGk F ´F As e´tGk is a dynamical semigroup, lim e t eq eq “ ´Gk . Hence taking a limit as t Ñ 0 tÑ0 of the above expression yields ż 1ÿ tr dpBi Bj Gk |k“0 Feq ppqzi˚ zj ě 0. 2 i,j Td To show positivity for the first two terms in (30), we note that due to (22), the expression may be simplified using the modified inner product introduced in Section 5.5. We observe: ż 1 tr dpBi Jk |k“0 G0´1 p1 ´ P0 qBj Jk |k“0 Feq ppq 2 T d ż 1 ` tr dpBj Jk |k“0 G0´1 p1 ´ P0 qBi Jk |k“0 Feq ppq 2 Td ż 1 ´1 ´1 “ tr dppBi HppqG b 0 p1 ´ P0 qpBj HFeq qppq ` Bi HppqG0 p1 ´ P0 qpFeq Bj Hqppqq b b b 2 Td ż 1 ´1 ´1 ` tr dppBj HppqG b 0 p1 ´ P0 qpBi HFeq qppq ` Bj HppqG0 p1 ´ P0 qpFeq Bi Hqppqq b b b 2 Td “ RexBj H, b pG T q´1 p1 ´ P0 qBi Hy b Feq 0 ` RexBi H,b pG T q´1 p1 ´ P0 qBj Hy b Feq . 0 Therefore, for z P Cd zt0u, ÿ ÿ Rexz, Dzy ě 2Rex zi Bi H, b pG T q´1 p1 ´ P0 q zi Bi Hy b Feq 0 i i “ 2RexΦ, G0T p1 ´ P0 qΦyFeq 37 ř where Φ :“ zi pG0T q´1 Bi H. b Due to (27), this expression will be strictly positive and so D i is positive definite. 5.2 The Small g Limit ´ ¯ 1 We wish to show the diffusion is O g in the small g limit as per Theorem 2. This requires us to first analyze the limit of the equilibrium eigenvector Feq . Lemma 3. The equilibrium eigenvector Feq for G0 converges weakly as g Ñ 0 to the equilibrium eigenvector for ΠLb0 Π, where Π is the projection onto ker J0 . Furthermore, this eigenvector is strictly positive. Proof. First, we must guarantee the existence of w-lim Feq . To do so requires a uniform gÑ0 bound on ||Feq ||2 , which guarantees we may pass to a weakly convergent subsequence. Since Feq P ker G0 , (20) yields ˆ ˙´1 i Feq “ D0 ` J0 T0 Feq . (31) g Since ˆ ˙ ˆ ˙ ˆ ˙ i 1 i b 1 i b D0 ` J0 Appq “ Dppq ` Hppq Appq ` Appq Dppq ´ Hppq g 2 g 2 g is a sum of multiplication operators, (31) becomes ż8 1 i b 1 i b Feq “ dte´tp 2 Dppq` g Hppqq pT0 Feq qppqe´tp 2 Dppq´ g Hppqq . 0 1 i 1 Due to Gronwall’s inequality and Assumption 2, we have ||e´tp 2 Dppq˘ g Hppqq || ď e´ 2C t , and b hence ż8 1 i 1 i dt||e´tp 2 Dppq` g Hppqq || ||pT0 Feq qppq||2 ||e´tp 2 Dppq´ g Hppqq || b b ||Feq ||2 ď ż08 1 ď dte´ C t ||pT0 Feq qppq||2 0 ż ďC dp1 ||M cQ´1 pp´p1 q pp1 , p1 qrFeq pp1 qs||2 . Td 38 Due to Lemma 2, we may apply Assumption 5 to bound this kernel to yield ż ||Feq ||2 ď Cχ dp1 ||I||2 trFeq pp1 q “ Cχ|Σ|1{2 . Td Hence ||Feq ||2 is uniformly bounded and a certain subsequence of Feq converges weakly as 0 g Ñ 0 to some matrix Feq . This implies Feq converges weakly as well and it must also 0 converge to Feq . Due to Lemma 6, taking the weak limit of (31) as g Ñ 0 yields 0 0 Feq “ ΠpΠD0 Πq´1 ΠT0 Feq (32) 0 where Π is the projection onto ker J0 . Hence Feq is an equilibrium eigenvector for ΠLb0 Π. 0 Due to Lemma 1, Feq will be nonnegative, as it is the equilibrium eigenvector for the 0 generator of a jump process. In addition, Feq ‰ 0 since xI, Feq y “ 1 for all g implies 0 xI, Feq y “ 1 as well. Therefore, due to Assumption 4 (using a suitable normalization of 0 0 Feq ), Feq will in fact be strictly positive. |Σ| ř 0 0 Lemma 3 shows that Feq converges weakly to Feq satisfying Feq ppq “ wi ppqEii ppq for i“1 |Σ| some wi ppq and the matrix elements Eij ppq “ |ψi ppqyxψj ppq| for the basis tψi ppqui“1 of Hppq b given in the proof of Lemma 1. Utilizing Corollary 1 in Appendix C, we can now prove Theorem 2. Proof. To begin, consider expression (30). To leading order in g, the diffusion will be ż 1 Di,j “ tr dpBi Jk |k“0 piJ0 ´ g Lb0 q´1 Bj Jk |k“0 Feq ppq 2 Td ż 1 ` tr dpBj Jk |k“0 piJ0 ´ g Lb0 q´1 Bi Jk |k“0 Feq ppq ` Op1q. 2 Td Multiplying by g and taking the limit g Ñ 0` using Corollary 1 yields ż 1 0 gDi,j Ñ ´ tr dpΠBi J0 |k“0 pΠLb0 Πq´1 ΠBj Jk |k“0 Feq 2 T d ż 1 0 ´ tr dpΠBj J0 |k“0 pΠLb0 Πq´1 ΠBi Jk |k“0 Feq , 2 T d 39 0 where Π is the projection onto kerJ0 and Feq “ lim Feq as in the proof of Lemma 3. For gÑ0 an arbitrary function F ppq, we calculate |Σ| 1ÿ pΠBj Jk |k“0 F qppq “ ´ Eii ppqpBj HppqF b ppq ` F ppqBj HppqqEb ii ppq. 2 i“1 In particular, this yields |Σ| ÿ 0 pΠBj Jk |k“0 Feq qppq “ ´ wi ppqxψi ppq|Bj Hppq|ψb i ppqyEii ppq. i“1 |Σ| ř ř d Let z P C zt0u. If we denote Φppq :“ xψi ppq| Bj Hppqz b j |ψi ppqyEii ppq, we will have i“1 j 0 lim` Rexz, gDzy “ ´RexΦ, pΠLb0 Πq´1 ΦFeq y gÑ0 0 0 ´1 0 “ ´RexΦFeq , pΠLb0 ΠFeq q ΦFeq y 0 0 : ´1 0 0 ´1 0 “ ´xΦFeq , ppΠLb0 ΠFeq q q RepΠLb0 ΠFeq qpΠLb0 ΠFeq q ΦFeq y 0 ´1 0 0 0 ´1 0 “ ´RexpΠLb0 ΠFeq q ΦFeq , ΠLb0 ΠFeq pΠLb0 ΠFeq q ΦFeq y. Due to the proof of Lemma 3, ΠLb0 Π is the generator for a jump process with a unique 0 positive invariant state given by Feq . Hence Lemma 5 applies and this term will be strictly ´ ¯ positive. Hence Dpgq “ O g1 . 40 CHAPTER 6: CONCLUSIONS AND FUTURE WORK A quantum particle’s dynamics are seemingly governed solely by its interaction with the environment in the case of a Lindblad master equation. Indeed, [9] and [2] showed diffusion was present in this context for a translation-invariant Hamiltonian. This document showed that in the more general Q-periodic Hamiltonian context, diffusive propagation also occurred. In [1], diffusion was shown for an Anderson Hamiltonian (though the Lindbladian used in that paper was not the generator of a completely positive semigroup). In each case, the presence of a Lindbladian caused the dynamics of the particle to exhibit diffusion. It is then natural to wonder whether this behavior occurs for other Hamiltonians as well. For instance, consider the Anderson model (2) whose Hamiltonian operator is given by Hω “ ´∆ ` λVω (33) where the potentials Vω are diagonal operators with ω given by i.i.d. random variables and the parameter λ measures the strength of the disorder. We should also suspect diffusion to be present for this Hamiltonian in the context of Lindbladian environmental interaction. However, one would expect the disorder to affect the asymptotics of the diffusion for small g, as the g “ 0 case should yield localization for large enough disorder. Thus we make the following conjecture. Conjecture 1. Let ρt be a solution of (2) with initial condition ρ0 P B1 pℓ2 pZd qq and g ą 0. If Hω satisfies (33) with λ sufficiently large, then the quantum particle whose density matrix is given by ρt exhibits diffusive propagation with diffusion matrix Dpgq satisfying Dpgq “ Opgq for small g. A particularly interesting subcase of the one-dimensional Anderson model is the random dimer model. In this model, the random variables ωpxq for x P Z are chosen from the set t´1, 1u with the additional requirement that ωp2xq “ ωp2x ` 1q for every x P Z. That is, the random variables are chosen in dimer pairs. This particular case of the Anderson model 41 is interesting, since it was shown that (without Lindbladian interaction) the dynamics of the quantum particle change depending on the value of λ. For instance, the particle’s dynamics will be localized as with the usual Anderson model for λ ą 1, yet diffusive for λ “ 1 and superdiffusive for 0 ă λ ă 1 [20], [21]. This leads us to the following conjecture. Conjecture 2. Let ρt be a solution of (2) with initial condition ρ0 P B1 pℓ2 pZd qq and g ą 0. If Hω is the random dimer Hamiltonian, then the quantum particle whose density matrix is given by ρt exhibits diffusive propagation with diffusion matrix Dpgq. For small g, we have the following asymptotics: • If 0 ď λ ď 1, then Dpgq “ Opg ´1`2λ q • If λ ą 1, then Dpgq “ Opgq. 42 BIBLIOGRAPHY [1] Jürg Fröhlich and Jeffrey Schenker. Quantum brownian motion induced by thermal noise in the presence of disorder. Journal of Mathematical Physics, 57(2):023305, 2016. [2] Jeremy Thane Clark. Diffusive limit for a quantum linear boltzmann dynamics. An- nales Henri Poincaré, 14(5):1203–1262, 2012. [3] Yang Kang and Jeffrey Schenker. Diffusion of wave packets in a markov random potential. Journal of Statistical Physics, 134(5-6):1005–1022, 2009. [4] Jeffrey Schenker, F. Zak Tilocco, and Shiwen Zhang. Diffusion in the mean for a periodic schrödinger equation perturbed by a fluctuating potential. Communications in Mathematical Physics, 377(2):1597–1635, 2020. [5] W. De Roeck and A. Kupiainen. Diffusion for a Quantum Particle Coupled to Phonons in d ě 3. Communications in Mathematical Physics, 323(3):889–973, November 2013. [6] W De Roeck, J Fröhlich, and A Pizzo. Quantum Brownian Motion in a Simple Model System. Communications in Mathematical Physics, 293(2):361–398, September 2009. [7] D. Manzano and P.I. Hurtado. Harnessing symmetry to control quantum transport. Advances in Physics, 67(1):1–67, 2018. [8] G. Lindblad. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48(2):119–130, 1976. [9] Jeremy Clark, W. De Roeck, and Christian Maes. Diffusive behavior from a quantum master equation. Journal of Mathematical Physics, 52(8):083303, 2011. [10] R. Alicki. Invitation to quantum dynamical semigroups. Dynamics of Dissipation, page 239–264, 2002. [11] A.S. Holevo. A note on covariant dynamical semigroups. Reports on Mathematical Physics, 32(2):211–216, 1993. [12] Klaus Deimling. Nonlinear functional analysis. Springer-Verlag, 1985. [13] Daniel Manzano. A short introduction to the lindblad master equation. AIP Advances, 10(2):025106, 2020. [14] I. Siemon, A. S. Holevo, and R. F. Werner. Unbounded generators of dynamical semigroups. Open Systems amp; Information Dynamics, 24(04):1740015, 2017. [15] Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10(3):285–290, 1975. [16] K Kraus. General state changes in quantum theory. Annals of Physics, 64(2):311–335, 1971. 43 [17] Daniel Revuz and Marc Yor. Continuous Martingales and Brownian motion. Springer, 2005. [18] Yacine Aı̈t-Sahalia, Lars Peter Hansen, and José A. Scheinkman. Operator methods for continuous-time markov processes. Handbook of Financial Econometrics: Tools and Techniques, page 1–66, 2010. [19] Helmut H. Schaefer. Banach lattices and positive operators. Springer-Verlag, 1974. [20] David H. Dunlap, H-L. Wu, and Philip W. Phillips. Absence of localization in a random-dimer model. Physical Review Letters, 65(1):88–91, 1990. [21] A Bovier. Perturbation theory for the random dimer model. Journal of Physics A: Mathematical and General, 25(5):1021–1029, 1992. [22] B. Szőkefalvi-Nagy. Sur les contractions de l’espace de hilbert. Acta Sci. Math. Szeged, 15:87–92, 1956. 44 APPENDIX A: A SPECTRUM RESULT FOR COMMUTING OPERATORS Lemma 4. Let A and B be bounded operators on a Banach space with rA, Bs “ 0. Then σpA ` Bq Ď σpAq ` σpBq. Proof. We first note that rA, Bs “ 0 implies rpA ´ zq´1 , pB ´ wq´1 s “ 0 for all z P ρpAq, w P ρpBq since rpA ´ zq´1 , pB ´ wq´1 s “ pA ´ zq´1 pB ´ wq´1 rA ´ z, B ´ wspB ´ wq´1 pA ´ zq´1 “ 0. Let z R σpAq ` σpBq so that σpAq and z ´ σpBq are two disjoint compact sets. We may thus define a bounded counterclockwise contour Γ enclosing σpAq such that intΓ contains no part of z ´ σpBq. Using this contour, we define an operator ż 1 Φ :“ dwpw ` B ´ zq´1 pw ´ Aq´1 2πi Γ which will be bounded since distpΓ, σpAqq, distpΓ, z ´ σpBqq ą 0. Since the resolvents of A and B commute, ż 1 pA ` B ´ zqΦ “ dwpA ´ wqpw ` B ´ zq´1 pw ´ Aq´1 2πi Γ ż 1 ` dwpB ` w ´ zqpw ` B ´ zq´1 pw ´ Aq´1 2πi Γ ż ż 1 ´1 1 “´ dwpw ` B ´ zq ` dwpw ´ Aq´1 2πi Γ 2πi Γ “ 0 ` I. So Φ “ pA ` B ´ zq´1 and since Φ is bounded, z P ρpA ` Bq. 45 APPENDIX B: NEGATIVITY OF A JUMP PROCESS Lemma 5. Suppose L is the generator for a jump process on a compact metric space X with nonnegative jump rate rpx, yqdy, i.e. ż ż pLf qpxq “ dy rpx, yqf pyq ´ dy rpy, xqf pxq “: pKf qpxq ´ Dpxqf pxq. X X 1 Also, suppose for a.e. x P X, C ď Dpxq ď C for some C ą 0 and let L have a unique nonnegative invariant state w. Assume that for a.e. x P X and every nonnegative function ş ϕ with X ϕpxqdx “ 1, there exists n P N such that pK n ϕqpxq ą 0. Then for all f ‰ constant a.e., ´Rexf, Lpwf qy ą 0. Proof. We first observe that the adjoint operator will be given by ż : pL f qpxq “ rpy, xqpf pyq ´ f pxqqdy. X We then have since Lw “ 0, 1 1 ´Rexf, Lpwf qy “ ´ Rexf, Lpwf qy ´ RexL: f, pwf qy 2 ż ż 2 1 ˚ “ ´ Re dxdy f pxqrrpx, yqwpyqf pyq ´ rpy, xqwpxqf pxqs 2 X X ż ż 1 ´ Re dxdy rpy, xqrf ˚ pyq ´ f ˚ pxqswpxqf pxq 2 ż ż X X 1 “ Re dxdy rpx, yqwpyqr|f pxq|2 ´ 2f ˚ pxqf pyq ` |f pyq|2 s 2 ż żX X 1 “ dxdy rpx, yqwpyq|f pxq ´ f pyq|2 . 2 X X This integral is clearly nonngative. Since w is an invariant state for L, we will have ˆ ˙ ˆ ˙n 1 1 1 wpxq “ K wpxq “ K wpxq ě n pK n wqpxq D D C for each n P N and a.e. x P X. Using a suitable normalization of w, this implies w ą 0 a.e. By way of contradiction, suppose ´Rexf, Lpwf qy “ 0. Then for a.e. x, y P X with 46 rpx, yq ą 0, we have f pxq “ f pyq. However, our assumption yields ż ż ż dy1 ¨ ¨ ¨ dyn´1 dyrpx, y1 qrpy1 , y2 q ¨ ¨ ¨ rpyn´2 , yn´1 qrpyn´1 , yqϕpyq ą 0 X X X ş for a.e. x P X and every function ϕ ě 0 with X ϕpxqdx “ 1. Hence there exists a positive measure set E Ď X n´1 such that rpx, y1 q, rpy1 , y2 q, ¨ ¨ ¨ , rpyn´2 , yn´1 q, rpyn´1 , yq ą 0 for a.e. x, y P X and a.e. py1 , ¨ ¨ ¨ , yn q P E. Therefore, for a.e. x, y P X and a.e. py1 , ¨ ¨ ¨ , yn q P E, f pxq “ f py1 q “ f py2 q “ ¨ ¨ ¨ “ f pyn´2 q “ f pyn´1 q “ f pyq and f “ constant a.e., a contradiction. 47 APPENDIX C: A GENERALIZED LIMIT FOR RESOLVENTS In [1], the following limit result was proved for resolvents of a particular form: Proposition 6 (Fröhlich/Schenker, 2016). Let H be a Hilbert space. Suppose A is a normal operator on H with ReA ě 0 and B is a bounded operator on H with ReB ě c ą 0. Then if Π denotes the projection onto the kernel of A, lim xϕ, pλA ` Bq´1 ψyH “ xΠϕ, pΠBΠq´1 ΠψyranΠ λÑ8 for all ϕ, ψ P H. This Proposition, while elegant, has a couple of drawbacks. For one, it requires that A be normal. Also, it does not hold up to compact perturbations of B. What we prove below is a slightly stronger generalization of this Proposition using the concept of dilation spaces given in [22]. Lemma 6. Let H be a Hilbert space. Let A and B be operators on H with ReA ě 0, ReB ě c ą 0 and B bounded. Then if w-limλÑ8 ψλ “ ψ P H, lim xϕ, pλA ` Bq´1 ψλ yH “ xΠϕ, pΠBΠq´1 ΠψyranΠ λÑ8 for any ϕ P H, where Π denotes the projection onto the kernel of A. Proof. Let z P C be such that Rez ą 0 and consider the operator F pzq :“ pλA ` zq´1 for some fixed λ ą 0. As this is an operator-valued Herglotz function with Rexϕ, F pzqϕy ě 0, there exists a positive operator-valued measure M such that ż 1 F pzq “ dM ptq. iλt ` z Based on results in [22], there exists a dilation space K Ą H and a minimal dilation N of M onto K where N is a projection-valued measure on the real line. Furthermore, if P is 1 the projection of K onto H, then since iλt`z is bounded, ż ż 1 1 dM ptq “ P dN ptqP. iλt ` z iλt ` z 48 Since N is a projection-valued measure on R, the spectral theorem dictates that H :“ ş tdN ptq defines a self-adjoint operator. Hence by spectral mapping, F pzq “ P piλH ` zq´1 P. Now consider the operator pλA ` B ` zq´1 . Since B is bounded, a Taylor expansion yields ÿ8 pλA ` B ` zq ´1 “ F pzq p´BF pzqqn n“0 ÿ8 “ P piλH ` zq´1 p´P BP piλH ` zq´1 qn P n“0 “ P piλH ` P BP ` zq´1 P “ P piλH ` P pB ´ cqP ` z ` cq´1 P. Since ReA ą 0 and ReB ě c ą 0, the limit z Ñ 0 may be taken on the left-hand side. Similarly, since c ą 0, RepiHq “ 0, and ReP pB ´ cqP ě 0, we may take a limit as z Ñ 0 on the right-hand side as well to give pλA ` Bq´1 “ P piλH ` P pB ´ cqP ` cq´1 P. Denote hλ :“ piλH`P pB´cqP `cq´1 P ψλ . We note that RepiHq “ 0 and ReP pB´cqP `c ě c ą 0. Hence hλ is bounded since c||hλ ||2 ď Re xhλ , P ψλ y ď ||hλ || ||P ψλ || implies ||hλ || ď c´1 ||P ψλ ||. Then since P ψλ “ iλHhλ ` rP pB ´ cqP ` cshλ , this gives |λ| ||iHhλ || ď p1 ` c´1 ||P pB ´ cqP ` c||q||P ψλ ||. Now since ψλ á ψ, Banach-Steinhaus yields P ψλ is uniformly bounded. Therefore, pI ´ ΠH qhλ á 0 where ΠH is the projection onto the kernel of H. Since iH is normal, ΠH commutes with iH. Hence ΠH rP pB ´ cqP ` cshλ “ ΠH P ψλ . 49 Furthermore, since ΠH rP pB ´ cqP ` csΠH is boundedly invertible, we have pΠH rP pB ´ cqP ` csΠH q´1 ΠH P ψλ á pΠH rP pB ´ cqP ` csΠH q´1 ΠH P ψ. These together imply that ΠH hλ á pΠH rP pB ´ cqP ` csΠH q´1 ΠH P ψ. That is, lim xϕ, P pλiH`P pB ´ cqP ` cq´1 P ψλ yH λÑ8 “ xΠH P ϕ, pΠH rP pB ´ cqP ` csΠH q´1 ΠH P ψyranΠH P for all ϕ P H. Now the kernel of A corresponds with the atoms of the operator-valued measure M . These clearly correspond to the atoms of the projection-valued measure N , which then further correspond with the kernel of H. Hence the kernel of H coincides with the kernel of A and we may replace ΠH by Π. Finally, since P is a projection from K onto H, P may be removed from the final expression. Corollary 1. Let H be a Hilbert space. Let A, B P BpHq with ReA ě 0 and ReB ě c ą 0, and let K be compact in H. Then if w-limλÑ8 ψλ “ ψ P H and kerΠpB ` KqΠ “ kerΠ for Π the projection onto the kernel of A, lim xϕ, pλA ` B ` Kq´1 ψλ yH “ xΠϕ, pΠpB ` KqΠq´1 ΠψyranΠ λÑ8 for any ϕ P H. Proof. For an arbitrary compact K, we may write K “ K 1 ` F where ReK 1 ą ´c and F has finite rank. As RepB ` K 1 q ě c1 ą 0 for some constant c1 , we may thus assume without loss of generality that K “ F has finite rank. Let us write pλA ` B ` Kq´1 “ pλA ` Bq´1 ´ pλA ` Bq´1 KpλA ` B ` Kq´1 or Sλ pλA ` B ` Kq´1 “ pλA ` Bq´1 50 where Sλ :“ I ` pλA ` Bq´1 K. Consider the decomposition H “ H1 ‘ H2 where H1 “ ranP and H2 “ ranp1 ´ P q for P the projection onto pkerKqK . This decomposition yields ¨ ˛ ´1 ˚1 ´ P p1 ´ P qpλA ` Bq K ‹ Sλ “ ˝ ‚. 0 P ` P pλA ` Bq´1 K Assuming for the moment that P ` P pλA ` Bq´1 K is invertible, Schur complement gives ¨ ˛ ´1 ´1 ´1 ˚1 ´ P ´p1 ´ P qpλA ` Bq KpP ` P pλA ` Bq Kq ‹ Sλ´1 “ ˝ ‚. (34) ´1 ´1 0 pP ` P pλA ` Bq Kq We observe that P ` P pλA ` Bq´1 K will be invertible if and only if P pλA ` Bq´1 K does not have a nontrivial eigenvector corresponding to the eigenvalue -1. By Lemma 6, pλA ` Bq´1 K converges weakly to ΠpΠBΠq´1 ΠK as λ Ñ 8. Then since K is compact and P is the projection onto ranK, P pλA ` Bq´1 K actually converges to P ΠpΠBΠq´1 ΠK in norm. Hence taking λ Ñ 8 in the eigenvalue equation P pλA ` Bq´1 Kψ “ ´ψ yields ΠpΠBΠq´1 ΠKψ “ ´ψ for ψ P ranP . In fact, this equation reveals that ψ P ranΠ as well and so pΠBΠq´1 ΠKΠψ “ ´ψ for ψ P ranP X ranΠ. That is, ΠpB ` KqΠψ “ 0. Since we have assumed kerΠpB ` KqΠ “ kerΠ, ψ “ 0 and therefore, P ` P ΠpΠBΠq´1 ΠK is invertible. Using perturbation theory and norm convergence, we may conclude that P ` P pλA ` Bq´1 K is invertible for sufficiently large λ and so Sλ is also invertible with inverse given in block form by (34). Then since pλA ` B ` Kq´1 “ 51 Sλ´1 pλA ` Bq´1 , we have for any ϕ, ψλ P H with ψλ á ψ, xϕ, pλA ` B ` Kq´1 ψλ y “ xp1 ´ P qϕ, p1 ´ P qpλA ` Bq´1 ψλ y ´ xp1 ´ P qϕ, p1 ´ P qpλA ` Bq´1 KpP ` P pλA ` Bq´1 Kq´1 P pλA ` Bq´1 ψλ y ` xP ϕ, pP ` P pλA ` Bq´1 Kq´1 P pλA ` Bq´1 ψλ y. Since P is the projection onto a finite rank operator, we may decompose each projection P in the above expression into a finite sum. Then utilizing Lemma 6, the limit of each term should exist as λ Ñ 8. In particular, we should have lim xϕ, pλA ` B ` Kq´1 ψλ y “ λÑ8 xp1 ´ P qϕ, p1 ´ P qΠpΠBΠq´1 Πψy ´ xp1 ´ P qϕ, p1 ´ P qΠpΠBΠq´1 ΠKpP ` P ΠpΠBΠq´1 ΠKq´1 P ΠpΠBΠq´1 Πψy ` xP ϕ, pP ` P ΠpΠBΠq´1 ΠKq´1 P ΠpΠBΠq´1 Πψy ´1 “ xϕ, S8 ΠpΠBΠq´1 Πψy where ¨ ˛ ´1 ´1 ˚1 ´ P p1 ´ P qΠpΠBΠq ΠK ‹ S8 :“ ˝ ‚. ´1 0 P ` P ΠpΠBΠq ΠK Working backwards, this yields S8 “ I ` ΠpΠBΠq´1 ΠK and therefore, we have lim xϕ, pλA ` B ` Kq´1 ψλ y “ xϕ, pI ` ΠpΠBΠq´1 ΠKq´1 ΠpΠBΠq´1 Πψy. λÑ8 By decomposing the space H into H “ ranp1 ´ Πq ‘ ranΠ, this may be simplified to lim xϕ, pλA ` B ` Kq´1 ψλ y “ xΠϕ, pΠpB ` KqΠq´1 Πψy λÑ8 as desired. 52