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.
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.
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.
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.
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.
Open quantum system dynamics is efficiently described by the quantum master equation formalism. Therein, quantum master equations in Lindblad form constitute an important subclass describing Markovian dynamics. When an open quantum system is in an out-of-equilibrium state, an exchange of particles between the open system and reservoirs takes place yielding to a non-zero average net current and associated current fluctuations, which can be characterised with the quantum jump formalism for quantum master equations expressed in Lindblad form. However, a large class of quantum master equations cannot be described by Lindblad dynamics. Here we assess the validity and the effectiveness of the quantum jump formalism when the dissipators in the quantum master equation describe a non-additive, open quantum system dynamics. We find that an additive unravelling of the non-additive quantum master equation does not generate a completely-positive dynamics in the scenario of perfect jump detection. Nevertheless, allowing for an imperfect jump detection scenario, we find that an additive unravelling is possible that reproduces the current and the fluctuations obtained via the Landauer-Büttiker 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.
Recent advances in quantum technologies and related experiments have created a need for highly accurate, versatile, and computationally efficient simulation techniques for the dynamics of open quantum systems. Long-lived correlation effects (non-Markovianity), system-environment hybridization, and the necessity for accuracy beyond the Born-Markov approximation form particular challenges. Approaches to meet these challenges have been introduced, originating from different fields, such as hierarchical equations of motion, Lindblad-pseudomode formulas, chain-mapping approaches, quantum Brownian motion master equations, stochastic unravelings, and refined quantum master equations. This diversity, while indicative of the field's relevance, has inadvertently led to a fragmentation that hinders cohesive advances and their effective cross-community application to current problems for complex systems. How are different approaches related to each other? What are their strengths and limitations? Here we give a systematic overview and concise discussion addressing these questions. We make use of a unified framework which very conveniently allows to link different schemes and, this way, may also catalyze further progress. In line with the state of the art, this framework is formulated not in a fully reduced space of the system but in an extended state space which in a minimal fashion includes effective reservoir modes. This in turn offers a comprehensive understanding of existing methods, elucidating their physical interpretations, interconnections, and applicability.