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Quantum Trajectories For, and As, Understanding

Prahlad Warszawski

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

Quantum trajectories provide a fundamental description of the measurement of individual quantum systems. As such, they have wide impact, and application, in the emerging field of quantum technology. Importantly, they also give a mechanism for developing our understanding of the nature of quantum mechanics. \n \nIn Part I of this thesis, we develop and apply quantum trajectory methods, with a focus upon experimentally relevant optomechanical systems. By solving the stochastic master equation for sufficiently simple bosonic systems, and subsequently finding the positive operator-valued measure (POVM), we are able to conduct a detailed study of the use of parametric amplification for quantum state tomography of nonclassical optomechanical states of motion. Homodyne tomography is a cornerstone experimental tool, and an analysis of its convergence is carried out in the presence of realistic imperfections. We complete Part I by conducting a detailed preparatory analysis of superfluid optomechanical systems possessing vorticity. \n \nPart II of this thesis investigates the correspondence between open classical and open quantum systems. We prove a result shows that open quantum systems are, in general, harder to track than open classical systems. We couch this result in terms of physically realisable ensembles (PREs), which can describe the dynamics of a monitored, $D$-dimensional, quantum system obeying a master equation that has reached equilibrium. Associated with the PRE is a measurement scheme that leads to quantum trajectories in which the system evolution consists purely of jumps between the states that are members of the PRE. The occupation of the $K$, generally non-orthogonal, states in the ensemble can be used to track the system. The number of states in the ensemble, $K$, represents the amount of memory that is required to do so. In comparison, a classical $D$-dimensional system requires occupation of the $D$ states to be tracked. After first developing analysis tools that make feasible the discovery of PREs in $D>2$, we prove our main result that there are quantum systems that have a minimal sized PRE with $K>D$.

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Quantum trajectories provide a fundamental description of the measurement of individual quantum systems. As such, they have wide impact, and application, in the emerging field of quantum technology. Importantly, they also give a mechanism for developing our understanding of the nature of quantum mechanics. \n \nIn Part I of this thesis, we develop and apply quantum trajectory methods, with a focus upon experimentally relevant optomechanical systems. By solving the stochastic master equation for sufficiently simple bosonic systems, and subsequently finding the positive operator-valued measure (POVM), we are able to conduct a detailed study of the use of parametric amplification for quantum state tomography of nonclassical optomechanical states of motion. Homodyne tomography is a cornerstone experimental tool, and an analysis of its convergence is carried out in the presence of realistic imperfections. We complete Part I by conducting a detailed preparatory analysis of superfluid optomechanical systems possessing vorticity. \n \nPart II of this thesis investigates the correspondence between open classical and open quantum systems. We prove a result shows that open quantum systems are, in general, harder to track than open classical systems. We couch this result in terms of physically realisable ensembles (PREs), which can describe the dynamics of a monitored, DDD-dimensional, quantum system obeying a master equation that has reached equilibrium. Associated with the PRE is a measurement scheme that leads to quantum trajectories in which the system evolution consists purely of jumps between the states that are members of the PRE. The occupation of the KKK, generally non-orthogonal, states in the ensemble can be used to track the system. The number of states in the ensemble, KKK, represents the amount of memory that is required to do so. In comparison, a classical DDD-dimensional system requires occupation of the DDD states to be tracked. After first developing analysis tools that make feasible the discovery of PREs in D>2D>2D>2, we prove our main result that there are quantum systems that have a minimal sized PRE with K>DK>DK>D.
This thesis considers quantum systems which interact with their surrounding environment. Specifically we consider the underlying stochastic dynamics which arise from the back-action of measurements performed in the environment. These quantum trajectories can be tracked by continuously monitoring the environment, as is now possible in modern experiments. We consider phenomena understood at the average level of the system state density matrix, and explore these at this more information rich level. This is especially relevant for experimental setups and implementation of quantum technologies. Firstly, we consider how systems exhibiting metastability behave at the level of stochastic quantum trajectories. In doing so we make comparisons to classical theories of metastability, where trajectories exhibit fast relaxation into distinct phases and slow transitions between them. We also illustrate when this phenomenology does not hold, with instead the emergence of a slow manifold in the form of a decoherence free subspace. We also highlight the useful role that quantum reset processes can play in the analysis of stochastic quantum trajectories. The dynamics of the stochastic quantum trajectories depends on the choice of environmental measurements. There are many choices which leave the average dynamics unchanged, but give different statistical ensembles of quantum trajectories. This gauge freedom of the average dynamics is well known. We derive the analogous result at the level of quantum trajectory ensembles; giving the measurement protocols which leave the generator of the stochastic trajectory dynamics unchanged. We also extend this to include joint trajectories of conditional system states and their associated measurement records, either full records or coarse-grained. Using these results on gauge freedoms of quantum trajectories, we consider the conditions under which the trajectory dynamics exhibit a weak unitary symmetry. We then consider under what conditions the joint state of the system and environment exhibit symmetric dynamics, including under various levels of dephasing. We find that the joint dynamics always features a separable symmetry directly related to that of the quantum master equation, but for quantum trajectories the corresponding symmetry is present only if the counting measurement satisfies certain conditions.
This thesis is a collection of works focusing on interrelations between the fields of special relativity, open quantum systems and complex networks. As each of the aforementioned fields encompasses a huge variety of topics, the thesis contains a selection of particular connections. The first half of the thesis considers an open quantum systems approach to the description of relativistic phenomena. As the aforementioned phenomena often involve dynamically evolving counterparts, some non-Markovian aspects of open quantum systems are initially explored. Conditions for complete positivity are derived for a type of phase-covariant time-local master equation and then applied to a non-Markovian master equation with heuristically-derived decay rates. This is relevant in cases when the physicality of the master equation cannot be postulated. The first relativistic system considered is a superposition of coherent states in a gravitational gradient. It is shown to exhibit decoherence induced by gravitational time dilation through interaction between inner degrees of freedom and the centre of mass of the superposition. The decoherence is quantified using various, widely-used non-classicality measures. The decoherence rates are then compared to decoherence induced by classical noise to roughly evaluate the experimental precision necessary to detect gravitationally induced decoherence. In the second relativistic system, the Unruh effect is modeled as an open system where an Unruh-deWitt detector is interacting with bosonic fields. A master equation with time-dependent decay rates for the open system is derived by assuming a non-conventional spacetime path profile for the detector. A particular parameter governing the physical evolution is identified. The system is shown to exhibit non-Markovianity within the completely positive domain of the master equation, the latter ensured by the conditions derived before. The second half of the thesis starts with a definition of complex networks, which are yet not a commonly used tool in quantum physics. The concept of pairwise tomography networks is then introduced as a way to represent many-body quantum states. As the number of pairwise connections grows quadratically with the number of nodes, an efficiently scaling measurement scheme to recover the pairwise tomography networks is presented. The scheme and the concept of pairwise tomography networks are demonstrated to be useful in various applications and the results of a proof-of-principle experiment are shown. One application explored is a paradigmatic spin chain model known as the XX model. A pairwise entanglement network representation of the XX model is shown to suggest new phenomena such as gradual establishment of quasi-long range order accompanied by a symmetry regarding single-spin concurrence distributions. The existence of structural classes and a cyclic self-similarity in the state are revealed.
The study of open quantum systems – microscopic systems exhibiting quantum coherence that are coupled to their environment – has become increasingly important in the past years, as the ability to control quantum coherence on a single particle level has been developed in a wide variety of physical systems. In quantum optics, the study of open systems goes well beyond understanding the breakdown of quantum coherence. There, the coupling to the environment is sufficiently well understood that it can be manipulated to drive the system into desired quantum states, or to project the system onto known states via feedback in quantum measurements. Many mathematical frameworks have been developed to describe such systems, which for atomic, molecular, and optical (AMO) systems generally provide a very accurate description of the open quantum system on a microscopic level. In recent years, AMO systems including cold atomic and molecular gases and trapped ions have been applied heavily to the study of many-body physics, and it has become important to extend previous understanding of open system dynamics in single- and few-body systems to this many-body context. A key formalism that has already proven very useful in this context is the quantum trajectories technique. This method was developed in quantum optics as a numerical tool for studying dynamics in open quantum systems, and falls within a broader framework of continuous measurement theory as a way to understand the dynamics of large classes of open quantum systems. In this article, we review the progress that has been made in studying open many-body systems in the AMO context, focussing on the application of ideas from quantum optics, and on the implementation and applications of quantum trajectories methods in these systems. Control over dissipative processes promises many further tools to prepare interesting and important states in strongly interacting systems, including the realisation of parameter regimes in quantum simulators that are inaccessible via current techniques.
We introduce jump-time unraveling as a distinct description of open quantum systems. As our starting point, we consider quantum jump trajectories, which emerge, physically, from continuous quantum measurements, or, formally, from the unraveling of Markovian quantum master equations. If the stochastically evolving quantum trajectories are ensemble-averaged at specific times, the resulting quantum states are solutions to the associated quantum master equation. We demonstrate that quantum trajectories can also be ensemble-averaged at specific jump counts. The resulting jump-time-averaged quantum states are then solutions to a discrete, deterministic evolution equation, with time replaced by the jump count. This jump-time evolution represents a trace-preserving quantum dynamical map if and only if the associated open system does not exhibit dark states. In the presence of dark states, on the other hand, jump-time-averaged states may decay into the dark states and the jump-time evolution may eventually terminate. Jump-time-averaged quantum states and the associated jump-time evolution are operationally accessible in continuous measurement schemes, when quantum jumps are detected and used to trigger the readout measurements. We illustrate the jump-time evolution with the examples of a two-level system undergoing relaxation or dephasing, a damped harmonic oscillator, and a free particle exposed to collisional decoherence.
Measurement and control of quantum systems play an important role in science and technology, from the foundations of quantum mechanics to practical applications. This thesis presents theoretical and experimental work that exploits quantum measurement and measurement-based feedback control to study some new problems in the contexts of solid state physics and optical quantum systems. In the first part, we consider the dynamics of open quantum systems conditioned on continuous measurements. We revisit the fact that dynamical quantum events are detector dependent by framing the problem as a quantum correlation test of the Einstein-Podolsky-Rosen (EPR) steering type. In this regards, a no-go theorem for quantum diusion is derived. In order to study a realistic experimental situation we scrutinise dierent physical systems, Einstein-Podolsky-Rosen steering inequalities and measurement protocol. We show that the no-go result for quantum diusion is not universal and does not apply to quantum jumps by devising a novel adaptive measurement scheme. Since the objectiveness of pure-state dynamical models is ruled out by quantum entanglement, then it is reasonable to say that quantum jumps are more quantum than quantum diusion. In the rst part we also study a stochastic feedback control of quantum transport. We show that using a feedback control strategy it is possible to restrict the dynamics of a double quantum dot system to a non-orthogonal two-state ensemble. The feedback control approach can be less experimentally challenging than an adaptive measurement scheme, as the latter requires implementing local oscillators, which might be dicult to realise in mesoscopic physics.
Quantum mechanics has both deepened our understanding of the physical reality and allowed development of quantum technologies beating their classical counterparts in efficiency, security, and sensitivity. As a major downside, the quantum properties behind these technologies are very sensitive to the effects the system’s interaction with its environment. The theory of open quantum systems describes how such interaction influences the system, and as a consequence, how the quantum features are lost, controlled, and possibly returned by manipulating the environment. This thesis begins with a short description of the most relevant features and results of the open quantum system theory. We continue by presenting some commonly used measures to quantify the information encoded in quantum systems and the correlations between spatially or temporally separated parties. The literature review ends with discussion of non-Markovianity and memory effects which combine the fields of open quantum systems and quantum information in a natural way. The rest of the thesis presents the main results of Publications I – VII. We study the memory effects and physicality conditions of open system dynamics described by the vast family of phase-covariant master equations. We show how the open quantum system picture can be exploited to design probing protocols to extract information without knowing how the probe and system interact. Such protocols are considered also from the point-of view of efficiently obtaining the information required in the probing, both in theory and by experimental implementation. The experiment serves also as evidence for applicability of recently developed geometric tools to design informationally incomplete measurements for solving so-called membership problems. We present and experimentally realize a protocol where control over intial correlations between polarization and frequency gives full freedom to implement any dephasing dynamics in the open system polarization, thus dictating how its information content changes in time. We study the two-qubit memory effects of dephasing dynamics and compare them to the effect of such noise in the transmission efficiency in superdense coding. Optimal noise configurations in different situations are found. Finally, we characterize the families of initial states for which the decay of quantum correlations depends or does not depend on the noise location.