Numerical investigation of collective effects in quantum optical media via integral operator electric fields

The study of quantum optics is principally concerned with investigating light-matter interactions. Within the discipline, computational simulation is a burgeoning field that can lend new insights into optical phenomena previously uncovered by theory or experiment. Collective emission effects such as superradiance serve as one prominent example. In contrast to ordinary emissions, superradiance involves dipolar coupling within optical ensembles and produces a coherent burst of radiation whose intensity scales with the square of the number of emitters. Whereas theoretical results involving superradiance are often shoehorned into small, ideal systems, numerical simulations permit the examination of much larger realistic systems, and can further aid in verifying experimental results. Studies of other phenomena, such as polarization enhancement, inhomogenenous broadening, and subradiance, benefit similarly.To design new systems that exploit quantum optical effects, we devise in this thesis a new numerical approach that can faithfully simulate dynamics of optical active media. Such material are characterized by their ability to modify and re-emit radiation. Nanoscale semiconductor particles known as quantum dots serve as a prime example. Their larger dipole moments--compared to atoms--enable them to experience strong interactions with radiation fields, and permit the observation of a variety of optical phenomena, including superradiance. Despite this merit, numerical simulation of large ensembles of quantum dots--and for long time periods--is challenging. In contrast to previous counterparts, our computational model, which involves the solution to the Maxwell-Bloch equations via integral operator electric fields, is massively scalable in both time and space. This is facilitated by the Adaptive Integral Method (AIM), which effects FFT-based convolutions to evaluate the field. This allows us to perform large scale simulations that reproduce optical effects such as superradiance.To demonstrate the fidelity of our approach, we evaluate the rate of photon emission from our ensemble and show that it reproduces the quadratic scaling of superradiance. In simulations of medium-sized (N = 50 - 300$) ensembles of quantum dots in a Gaussian cloud, we confirm this quadratic scaling by subtracting independent emissions from total emissions. We also observe anisotropy of emission--another hallmark of superradiance--in the field radiated by the Gaussian cloud. Subradiance is revealed in steady state plots of the population excitation, which display diminished emissions. This effect is amplified by inhomogeneous broadening, which induces greater disorder and thus interference within the ensemble, but diminished by the presence of collective Lamb shifts.Additionally, we compare the results of this calculation to those using another formalism, the Master equation. By applying zero-averaging random initial conditions to the polarization, we achieve strong numerical agreement between the two approaches. We observe both superradiant scaling, and destructive interference among dots separated by half-wavelengths. We remark, however, that the Maxwell-Bloch model is superior to the Master equation in resolving time delays and capturing propagation and memory effects. Hence, simulations involving ensembles of emitters separated far apart in space should opt for the Maxwell-Bloch approach to accurately account for delay effects.