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.
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.
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 unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.
Stochastic methods with quantum jumps are often used to solve open quantum system dynamics. Moreover, they provide insight into fundamental topics, such as the role of measurements in quantum mechanics and the description of non-Markovian memory effects. However, there is no unified framework to use quantum jumps to describe open-system dynamics in any regime. We solve this issue by developing the rate operator quantum jump (ROQJ) approach. The method not only applies to both Markovian and non-Markovian evolutions, but also allows us to unravel master equations for which previous methods do not work. In addition, ROQJ yields a rigorous measurement-scheme interpretation for a wide class of dynamics, including a set of master equations with negative decay rates, and sheds light on different types of memory effects which arise when using stochastic quantum jump methods.
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.
Stochastic unravelings allow to efficiently simulate open system dynamics, yet their application has traditionally been restricted to master equations that preserve both Hermiticity and trace. In this work, we introduce a general framework that extends piecewise-deterministic unravelings to arbitrary trace-nonpreserving master equations, requiring only positivity and Hermiticity of the dynamics. Our approach includes, as special cases, unravelings of arbitrary dynamics in the Heisenberg picture, evolutions interpolating between fully Lindblad and non-Hermitian Hamiltonian generators, and equations employed in the derivation of full counting statistics, for which we show it can be used to obtain the moments of the associated probability distribution. The framework is suitable for both trace-decreasing and trace-increasing processes through stochastic disappearance and replication of the stochastic realizations, and it is compatible with different unraveling schemes and with reverse jumps in the non-Markovian regime. Thereby, our approach provides a powerful and versatile simulation method that significantly broadens the applicability of stochastic techniques for 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.