This article focuses on the general theory of open quantum systems in the Gaussian regime and explores a number of diverse ramifications and consequences of the theory. We shall first introduce the Gaussian framework in its full generality, including a classification of Gaussian (also known as ‘general-dyne’) quantum measurements. In doing so, we will give a compact proof for the parametrisation of the most general Gaussian completely positive map, which we believe to be missing in the existing literature. We will then move on to consider the linear coupling with a white noise bath, and derive the diffusion equations that describe the evolution of Gaussian states under such circumstances. Starting from these equations, we outline a constructive method to derive general master equations that apply outside the Gaussian regime. Next, we include the general-dyne monitoring of the environmental degrees of freedom and recover the Riccati equation for the conditional evolution of Gaussian states. Our derivation relies exclusively on the standard quantum mechanical update of the system state, through the evaluation of Gaussian overlaps. The parametrisation of the conditional dynamics we obtain is novel and, at variance with existing alternatives, directly ties in to physical detection schemes. We conclude our study with two examples of conditional dynamics that can be dealt with conveniently through our formalism, demonstrating how monitoring can suppress the noise in optical parametric processes as well as stabilise systems subject to diffusive scattering.
This article focuses on the general theory of open quantum systems in the Gaussian regime and explores a number of diverse ramifications and consequences of the theory. We shall first introduce the Gaussian framework in its full generality, including a classification of Gaussian (also known as ‘general-dyne’) quantum measurements. In doing so, we will give a compact proof for the parametrisation of the most general Gaussian completely positive map, which we believe to be missing in the existing literature. We will then move on to consider the linear coupling with a white noise bath, and derive the diffusion equations that describe the evolution of Gaussian states under such circumstances. Starting from these equations, we outline a constructive method to derive general master equations that apply outside the Gaussian regime. Next, we include the general-dyne monitoring of the environmental degrees of freedom and recover the Riccati equation for the conditional evolution of Gaussian states. Our derivation relies exclusively on the standard quantum mechanical update of the system state, through the evaluation of Gaussian overlaps. The parametrisation of the conditional dynamics we obtain is novel and, at variance with existing alternatives, directly ties in to physical detection schemes. We conclude our study with two examples of conditional dynamics that can be dealt with conveniently through our formalism, demonstrating how monitoring can suppress the noise in optical parametric processes as well as stabilise systems subject to diffusive scattering.
In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.
This thesis develops a comprehensive framework for the use of continuous variable (CV) Gaussian states under environmental interactions and continuous measurements. Gaussian states provide an experimentally relevant class of quantum states because they are fully described by their first and second statistical moments. This thesis begins by representing Gaussian operations as symplectic transformations on the statistical moments and formulates the evolution of a Gaussian state under both unconditional and conditional open system dynamics. Stochastic methods are introduced in a physicist-accessible manner by sacrificing mathematical rigor over physical intuition. These methods are necessary to model environmental noise and continuous measurements represented by a Wiener process. From stochastic principles, a stochastic version of the Schr\"{o}dinger equation and a master equation are derived that capture the measurement-induced back-action. From these principles a conditional equation of motion for the covariance matrix of a Gaussian state is derived, leading to the Riccati equation. Unconditional evolution always leads to diffusive dynamics when the noise is modeled by a Wiener process. The key difference between unconditional and conditional equations of motion is that monitoring always leads to a reduction of uncertainty because a measurement gains information about the system. At the same time, all correlations to the part of the system being measured are lost, which in many cases can lead to an impure quantum state. At some cases, a sufficient measurement scheme can even be used to stabilize the unconditional equation at all scales. The developed formalism is applied to a system of multiple parametric flux pumps based on spontaneous parametric down-conversion to analyze entanglement generation in a complex network of modes. A simple formula relevant to all symmetric pumping architectures, also relevant for asymmetric cases, is derived with graph theoretical tools. The investigation revealed fundamental challenges in entanglement-generation based on multiple two-mode squeezed modes. Overall, the thesis provides a complete framework of Gaussian state dynamics connecting symplectic geometry, open quantum systems, quantum measurement theory, and stochastic methods. The results highlight the importance of Gaussian states in modern quantum technologies and future improvements in quantum computation and communication.
Gaussian operations, as one usually refers to CP-maps that send Gaussian states to Gaussian states, are apt to model a wide range of situations of practical interest and may often be controlled and applied on demand in the laboratory, in a variety of set-ups (that will be the subject matter of Chapter 9 ). Furthermore, they can be classified in a hierarchy of compact, powerful and far-reaching theoretical descriptions, which will be introduced by degrees and discussed in the current chapter. Note that here we are dealing with discrete quantum operations, rather than dynamical equations, which will be dealt with in the next chapter. We shall start from Gaussian unitary operations, corresponding to the symplectic group already introduced in Chapter 3 , and will also clarify how other basic manipulations such as tensoring and partial tracing carry over to the Gaussian picture. Then, we shall broaden our view to include all deterministic (trace-preserving) maps resulting from Gaussian unitary dilations, which are often known as the set of ‘bosonic Gaussian channels’ and encompass open dynamics subject to noise and ‘decoherence’ (the loss of quantum coherence due to the interaction with an environment). Next, we will move on to consider POVMs corresponding to the well-known homodyne and heterodyne detection schemes which, when acting on a subsystem, preserve the Gaussian character of the initial state both when the system is filtered by recording the measurement outcomes, and when acting unconditionally (that is, when the outcome of the measurement is not recorded). Our description will be further enlarged to the wider class of ‘general-dyne’ detections, and we will also introduce methods to account for imperfect detections. The ability to accommodate noise in the analytical description, often daunting with discrete variable and, more generally, outside the Gaussian regime, is probably one of the greatest strengths of the formalism under discussion. Finally, we shall introduce the Gaussian version of the Choi–Jamiolkowski isomorphism, and attain a unified description of all the Gaussian maps previously defined. Throughout our exposition, we will refer to quantum optical systems to exemplify the practical relevance of the operations encountered.
This chapter describes what could loosely be considered the continuous-time version of such transformations and provides the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, it addresses systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as 'white noise'), and thus gives rise to Markovian dynamics. The chapter the turns its attention to the study of the system's dynamics that result from the measurement of the environment. It uses the Gaussian dynamics as a key to unlock the full stochastic master equations governing generic states. Finally, the chapter builds on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control, and also applies theoretical tools to the enhanced production of squeezing.
A concise and self-contained derivation of hybrid quantum-classical dynamics is given in terms of Markovian master equations. Many previously known results are rederived and revised and some of them are completed or corrected. Using a method as simple as possible, our goal is a brief introduction to the state of the art of hybrid dynamics, with a limited discussion of the implications for foundations and without discussion of further relevance in the measurement problem, quantum gravity, chemistry, numeric methods, etc. Hybrid dynamics is defined as a special case of composite quantum dynamics where the observables of one of the two subsystems are restricted to the commuting set of diagonal operators in a fixed basis. With this restriction, the derivation of hybrid dynamical equations is clear conceptually and simple technically. Jump and diffusive dynamics follow in the form of hybrid master equations. Their stochastic interpretation (called unravelings) is derived. We discuss gauge-type ambiguities, problems of uniqueness, and covariance of the diffusive master equation. Also conditions of minimum noise and of monitoring the quantum trajectory are derived. We conclude that the hybrid formalism is equivalent to the standard Markovian theory of time-continuous quantum measurement (monitoring) on the one hand and is a motivating alternative formalism on the other.
Quantum systems are unavoidably open to their surrounding degrees of freedom. The theory of open quantum systems is thus crucial to understanding the fluctuations, dissipation, and decoherence of a quantum system of interest. Typically, the bath is modeled as an ensemble of harmonic oscillators, which yields Gaussian statistics of the bath influence on the quantum systems. However, there are also phenomena in which the bath consists of two-state systems, spins, or anharmonic oscillators; therefore, the non-Gaussian properties of the bath become important. Nevertheless, a theoretical framework to describe quantum systems under the influence of such non-Gaussian baths is not well established. Here, we develop a theory to describe quantum dissipative systems affected by Poisson noise properties of the bath, because the Lévi-Itô decomposition theorem asserts that Poisson noise is fundamental in describing arbitrary white noise beyond Gaussian properties. We introduce a quantum bath model that allows for the consistent description of dissipative quantum systems. The obtained master equation reveals non-Gaussian bath effects in the white noise regime, and provides an essential step toward describing open quantum dynamics under the influence of generic baths.