We present an elementary derivation and generalisation of a recently reported method of simulating feedback in open quantum systems. We use our generalised method to simulate systems with multiple delays, as well as cascaded systems with delayed backscatter. In addition, we derive a generalisation of the quantum regression formula that applies to systems with delayed feedback, and show how to use the formula to compute two-time correlation functions of the system as well as output field properties. Finally, we show that delayed coherent feedback can be simulated as a quantum teleportation protocol that requires only Markovian resources, pre-shared entanglement, and time travel. The requirement for time travel can be avoided by using a probabilistic protocol.
We present an elementary derivation and generalisation of a recently reported method of simulating feedback in open quantum systems. We use our generalised method to simulate systems with multiple delays, as well as cascaded systems with delayed backscatter. In addition, we derive a generalisation of the quantum regression formula that applies to systems with delayed feedback, and show how to use the formula to compute two-time correlation functions of the system as well as output field properties. Finally, we show that delayed coherent feedback can be simulated as a quantum teleportation protocol that requires only Markovian resources, pre-shared entanglement, and time travel. The requirement for time travel can be avoided by using a probabilistic protocol.
The subject of this thesis is the modelling of non-Markovian open quantum systems, in particular those that display memory effects due to time-delayed interactions. I study these systems in the context of a model of an open quantum system that interacts with a bosonic environment by way of particle exchanges. While the model is flexible enough to describe quite general memory effects, this work focuses to a large extent on a single prototypical system. This system emits into a coherent feedback loop with a time delay, and it captures the essential phenomena we aim to model. Within the context of this general model, I develop a general approach to spontaneous emission from singly-excited systems and show how to calculate the corresponding emission spectra. I also evaluate two commonly-used perturbative techniques in the context of spontaneous emission from singly-excited systems, coming to the conclusion that perturbation theory is of limited use in modelling open quantum systems with delayed feedback. I then turn attention to exact equations of motion for more general non-Markovian open quantum systems. I derive Heisenberg–Langevin equations that describe a large class of such systems, and solve these equations analytically for a series of simple bosonic systems. I also use the theory of cascaded open quantum systems to simulate systems with delayed coherent feedback in the limit of long time delay, and compare the results of these simulations with the analytic solutions obtained in the Heisenberg picture. Finally, I show that once we have a solution to a given Heisenberg–Langevin equation we can derive an alternative, time-local equation of motion that describes equivalent dynamics, and that these time-local Heisenberg–Langevin equations can in turn be used to derive corresponding time-local master equations. These master equations provide an exact description of the non-Markovian dynamics, but this treatment is limited to linear, bosonic systems—those systems for which the Heisenberg–Langevin equations can be solved in closed form. I discuss the prospect of generalising these master equations to take into account nonlinear and fermionic systems.
Quantum feedback networks with non-negligible propagation delays in the interconnecting edges have recently gained attention. Grimsmo has analysed the evolution of a Markov quantum system coupled to a single field via a doublepass interaction with a time delay and derived the evolution equation for the state (density operator) of the system (after tracing out the field) using a super-operator notion of partial tracing together with a tensor network representation. However, for the purposes of feedback control (both measurement-based and coherent feedback), the propagating output fields of the system can be used to facilitate feedback interconnections. Thus it is desirable to have an equivalent input-output description of quantum systems that incorporates time delays, without tracing out the field. Here, we first extend the analysis to the setting of a single system coupled to optical fields in multiple channels with a feedback structure in which the fields makes infinite repeated interactions with the system. Then we generalise this to the setting of quantum feedback networks with both instantaneous interconnections and uniform time delays in the interconnection.
The emergence of integrated (or solid state) photonic systems, including quantum dots, waveguides, and cavities, has provided a base to harness quantum optic phenomena for use in current and future quantum technologies. These systems also provide opportunities for exploring fundamentally new regimes in quantum optics. In order to fully realize the potential of these systems, many figures of merit need to be improved, for example by increasing the stability and coherent lifetimes of these systems. Following the notable improvements using measurement-based feedback, time-delayed coherent optical feedback has been proposed as one such method of stabilizing and improving these systems. Furthermore, modelling coherent feedback itself presents an interesting fundamental problem due to its non-Markovian nature. In this thesis, we use quantum trajectory (QT) theory to derive two models for simulating cavity quantum electrodynamic (cavity-QED) systems with time-delayed coherent feedback. First, an explanation of QT theory is given and the related time discretized waveguide (TDW) model is derived. Next we present a model for feedback using the frequency modes of the waveguide and results are presented in the ``one photon in the loop" approximation. We demonstrate how the photon lifetime can be improved in typical cavity-QED with coherent feedback and we explore some nonlinear effects. We then expand the system to allow for two photons in the feedback loop which requires us to use the TDW model. Lastly, this new approach is used to model two two-level systems coupled via a waveguide and connected to a feedback loop. These findings have implications on improving quantum optic systems by creating a new degree of control, as well as for modelling non-Markovian features in quantum optics.
A theory of time-delayed coherent quantum feedback is developed. More specifically, we consider a quantum system coupled to a bosonic reservoir creating a unidirectional feedback loop. It is shown that the dynamics can be mapped onto a fictitious series of cascaded quantum systems, where the system is driven by past versions of itself. The derivation of this model relies on a tensor network representation of the system-reservoir time propagator. For concreteness, this general theory is applied to a driven two-level atom scattering into a coherent feedback loop. We demonstrate how delay effects can qualitatively change the dynamics of the atom and how quantum control can be implemented in the presence of time delays.
We study the nonlinear spectra and multiphoton correlation functions of a waveguide system coupled to a two-level system (including realistic dissipation channels) with a time-delayed coherent feedback. We compute these observables by extending a recent quantum trajectory discretized-waveguide (QTDW) approach which exploits quantum trajectory simulations and a collisional model for the waveguide to efficiently simulate the quantum dynamics. Following a description of the general technique, we show how to calculate the first- and second-order quantum correlation functions, in the presence of a coherent pumping field. With a short delay time, we show how feedback can be used to filter out the central peak of the Mollow triplet or switch the output between bunched and antibunched photons by proper choice of round trip phase. We further show how the loop length and round trip phase affect the zero-time second-order quantum correlation function, an indicator of bunching or antibunching. New resonances introduced through the feedback loop are also shown through their appearance in the incoherent output spectrum from the waveguide. We explain these results in the context of the waiting time distributions of the system output and individual trajectories, uniquely stochastic observables that are easily accessible with the QTDW model.
To continue developing quantum technologies, such as quantum computing, quantum communication and quantum cryptography, we need reliable sources of quantum states of light. These quantum states of light can come in many forms; single photons, entangled pairs, bunched and anti-bunched light, all have uses in current quantum technologies, but for the best performance, this light must maintain its coherence. Time-delayed coherent feedback is a passive feedback method proposed for waveguide quantum electrodynamic systems to improve their performance and unlock new quantum optical phenomena. Additionally, due to its underlying non-Markovian dynamics, feedback is also an interesting theoretical problem to study. In this thesis, we use a quantum trajectory discretized waveguide (QTDW) model to simulate feedback dynamics and investigate its effects in a variety of applications. Originally proposed by Whalen in 2018, we develop the QTDW model beyond its initial implementation to now calculate quantum correlation functions, output spectra, waiting time distributions, and single photon source figures of merit. First, we present an explanation of quantum trajectory theory which is used to model the time dynamics in the QTDW approach, and discuss feedback in the linear regime to highlight the differences between Markovian and non-Markovian dynamics with coherent feedback. We next present nonlinear spectra and multi-photon effects from coherent feedback highlighting how the typical Mollow triplet is dressed by the feedback to create new resonances in the output field. This is followed by a collaborative theory and experiment work which observed our predicted non-Markovian signatures in the nonlinear output spectrum along with new non-Markovian phenomena in the linear regime. We then investigate how pulse-triggered single photon sources can be improved through the inclusion of a time-delayed coherent feedback, presenting the possible improvement to the performance figures of merit. Lastly, the modeling techniques developed for single photon sources with feedback are used to compare the performance of three new off-resonant driving techniques for on-demand single photon sources. These findings show time-delayed coherent feedback is an excellent degree of control to include in future waveguide quantum electrodynamic systems to unlock better performance and new phenomena in the system under investigation.