This thesis studies realization-dependent transformations in the generalized rate-operator quantum-jump formalism for open quantum systems. The aim is to examine whether these transformations can be used to steer the deterministic part of individual quantum trajectories toward a prescribed fixed target state, while preserving the physicality of the stochastic trajectories. This question is motivated by state-preparation tasks, where a quantum system must be driven to a known state quickly and with high probability, as in qubit resetting, qubit initialization, and the preparation of resourceful states. The theoretical part of the thesis introduces the basic framework of quantum mechanics, open quantum system dynamics, master equations, and stochastic quantum trajectories. The numerical part focuses on qubit dynamics and compares two examples: pure dephasing dynamics and Pauli master equation dynamics. For each fixed transformation, the simulations check the positivity condition, the attraction toward the target state, and the survival probability of the deterministic no-jump evolution. The results show that the simple fixed transformation considered for pure dephasing does not provide simultaneous attraction and positivity. In contrast, for Pauli master equation dynamics, fixed transformations are found that steer the deterministic trajectory toward the target while keeping the generalized rate operator positive. Because the isotropic Pauli master equation treats all directions on the Bloch sphere equally, the specific choice of the target state is just a reference direction, and the result carries over to any pure target state by symmetry. The survival probability analysis shows that steering can be achieved with quite high no-jump probability for the suitable parameter choices. The results suggest that realization-dependent transformations can provide useful trajectory level control in suitable open system dynamics.
This thesis studies realization-dependent transformations in the generalized rate-operator quantum-jump formalism for open quantum systems. The aim is to examine whether these transformations can be used to steer the deterministic part of individual quantum trajectories toward a prescribed fixed target state, while preserving the physicality of the stochastic trajectories. This question is motivated by state-preparation tasks, where a quantum system must be driven to a known state quickly and with high probability, as in qubit resetting, qubit initialization, and the preparation of resourceful states. The theoretical part of the thesis introduces the basic framework of quantum mechanics, open quantum system dynamics, master equations, and stochastic quantum trajectories. The numerical part focuses on qubit dynamics and compares two examples: pure dephasing dynamics and Pauli master equation dynamics. For each fixed transformation, the simulations check the positivity condition, the attraction toward the target state, and the survival probability of the deterministic no-jump evolution. The results show that the simple fixed transformation considered for pure dephasing does not provide simultaneous attraction and positivity. In contrast, for Pauli master equation dynamics, fixed transformations are found that steer the deterministic trajectory toward the target while keeping the generalized rate operator positive. Because the isotropic Pauli master equation treats all directions on the Bloch sphere equally, the specific choice of the target state is just a reference direction, and the result carries over to any pure target state by symmetry. The survival probability analysis shows that steering can be achieved with quite high no-jump probability for the suitable parameter choices. The results suggest that realization-dependent transformations can provide useful trajectory level control in suitable open system dynamics.
Every open-system dynamics can be associated to infinitely many stochastic pictures, called unravelings, which have proved to be extremely useful in several contexts, both from the conceptual and the practical point of view. Here, focusing on quantum-jump unravelings, we demonstrate that there exists inherent freedom in how to assign the terms of the underlying master equation to the deterministic and jump parts of the stochastic description, which leads to a number of qualitatively different unravelings. As relevant examples, we show that a fixed basis of post-jump states can be selected under some definite conditions, or that the deterministic evolution can be set by a chosen time-independent non-Hermitian Hamiltonian, even in the presence of external driving. Our approach relies on the definition of rate operators, whose positivity equips each unraveling with a continuous-measurement scheme and is related to a long known but so far not widely used property to classify quantum dynamics, known as dissipativity. Starting from formal mathematical concepts, our results allow us to get fundamental insights into open quantum system dynamics and to enrich their numerical simulations.
The dynamics of open quantum systems is often solved by stochastic unravelings where the average over the state-vector realizations reproduces the density matrix evolution. We focus on quantum-jump descriptions based on the rate-operator formalism. In addition to displaying and exploiting different equivalent ways of writing the master equation, we introduce state-dependent rate-operator transformations within the framework of stochastic pure state realizations, allowing us to extend and generalize the previously developed formalism. As a consequence, this improves the controllability of the stochastic realizations and subsequently greatly benefits when searching for optimal simulation schemes to solve open system dynamics. At a fundamental level, intriguingly, our results show that it is possible to have positive unravelings, without reverse quantum jumps and avoiding the use of auxiliary degrees freedom, in a number of example cases even when the corresponding dynamical map breaks the property of P divisibility, thus being in the strongly non-Markovian regime.
Non-Hermitian evolution can be realized through post-selection on stochastic pure-state trajectories arising in continuously monitored open quantum systems. The rate operator formalism provides a versatile and systematic framework for unraveling a master equation into stochastic pure-state evolutions, offering enhanced control over the resulting non-Hermitian dynamics. In the present work, we explore the applicability of the rate operator formalism as a tool for engineering non-Hermitian dynamics. Specifically, we apply this approach to the Bose-Hubbard model subject to environmental dephasing, examining its consequences for controlled state manipulation. Our analysis is framed within the broader contexts of quantum state engineering and measurement-induced phase transitions. We demonstrate that the rate operator formalism enables the construction of effective non-Hermitian Hamiltonians exhibiting a unique steady state-even in regimes where the standard Monte Carlo wavefunction method fails to produce one. Furthermore, we show that this framework facilitates transitions between distinct steady-state phases, governed by tunable parameters such as the interaction strength and a non-Hermiticity control parameter introduced via the rate operator formalism.
Stochastic unravelings are a widely used tool to solve open quantum system dynamics, in which the exact solution is obtained via an average over a stochastic process on the set of pure quantum states. Recently, the generalized rate operator unraveling formalism was derived, allowing not only for the engineering of stochastic realizations, but also to unravel without reverse jumps even for some dynamics in which P-divisibility is violated, thus greatly improving the simulation efficiency. This is possible because the unraveling depends on an arbitrary nonlinear transformation which can incorporate the memory effects. In this work, a stochastic Schrödinger equation for this formalism is derived, both for the cases with and without reverse jumps. It is also shown that a failure of this method can be used to witness master equations leading to unphysical time evolutions, independently of the particular nonlinear transformation considered.
Stochastic quantum trajectories, such as pure state evolutions under unitary dynamics and random measurements, offer a crucial ensemble description of many-body open system dynamics. Recent studies have highlighted that individual quantum trajectories also encode essential physical information. Prominent examples include measurement-induced phase transitions, where a pure quantum state corresponding to fixed measurement outcomes (trajectories) exhibits distinct entanglement phases, depending on the measurement rate. However, direct observation of this effect is hindered by an exponential postselection barrier, whereby the probability of realizing a specific trajectory is exponentially small. We propose a deterministic method to efficiently prepare quantum trajectories in polynomial time using imaginary time evolution and, thus, overcome this fundamental challenge. We demonstrate that our method applies to a certain class of quantum states, and argue that universal approaches do not exist for any quantum trajectories. Our result paves the way for experimentally exploring the physics of individual quantum trajectories at scale and enables direct observation of certain postselection-dependent phenomena.
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.