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arXiv:2004.12072 · Physical Review Letters

Diffusive Limit of Non-Markovian Quantum Jumps

Kimmo Luoma, Walter T. Strunz, Jyrki Piilo

Published Oct 9, 2020• 19 citations
arXivHTMLPDF

Abstract

We solve two long-standing problems for stochastic descriptions of open quantum system dynamics. First, we find the classical stochastic processes corresponding to non-Markovian quantum state diffusion and non-Markovian quantum jumps in projective Hilbert space. Second, we show that the diffusive limit of non-Markovian quantum jumps can be taken on the projective Hilbert space in such a way that it coincides with non-Markovian quantum state diffusion. However, the very same limit taken on the Hilbert space leads to a completely new diffusive unraveling, which we call non-Markovian quantum diffusion. Further, we expand the applicability of non-Markovian quantum jumps and non-Markovian quantum diffusion by using a kernel smoothing technique allowing a significant simplification in their use. Lastly, we demonstrate the applicability of our results by studying a driven dissipative two level atom in a non-Markovian regime using all of the three methods.

DOI
10.1103/physrevlett.125.150403

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We solve two long-standing problems for stochastic descriptions of open quantum system dynamics. First, we find the classical stochastic processes corresponding to non-Markovian quantum state diffusion and non-Markovian quantum jumps in projective Hilbert space. Second, we show that the diffusive limit of non-Markovian quantum jumps can be taken on the projective Hilbert space in such a way that it coincides with non-Markovian quantum state diffusion. However, the very same limit taken on the Hilbert space leads to a completely new diffusive unraveling, which we call non-Markovian quantum diffusion. Further, we expand the applicability of non-Markovian quantum jumps and non-Markovian quantum diffusion by using a kernel smoothing technique allowing a significant simplification in their use. Lastly, we demonstrate the applicability of our results by studying a driven dissipative two level atom in a non-Markovian regime using all of the three 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.
Abstract We consider quantum stochastic processes and discuss a level 2.5 large deviation formalism providing an explicit and complete characterisation of fluctuations of time-averaged quantities, in the large-time limit. We analyse two classes of quantum stochastic dynamics, within this framework. The first class consists of the quantum jump trajectories related to photon detection; the second is quantum state diffusion related to homodyne detection. For both processes, we present the level 2.5 functional starting from the corresponding quantum stochastic Schrödinger equation and we discuss connections of these functionals to optimal control theory.
Several key properties of quantum evolutions are characterized by divisibility of the corresponding dynamical maps. In particular, a Markovian evolution respects CP-divisibility, whereas breaking of P-divisibility provides a clear sign of non-Markovian effects. We analyze a class of evolutions which interpolates between CP- and P-divisible classes and is characterized by dissipativity -- a long known but so far not widely used formal concept to classify open system dynamics. By making a connection to stochastic jump unravellings of master equations, we demonstrate that there exists inherent freedom in how to divide the terms of the underlying master equation into the deterministic and jump parts for the stochastic description. This leads to a number of different unravelings, each one with a measurement scheme interpretation and highlighting different properties of the considered open system dynamics. Starting from formal mathematical concepts, our results allow us to get fundamental insights in open system dynamics and to ease their numerical simulations.
Markovian approximation is a widely-employed idea in the descriptions of the dynamics of open quantum systems. Although it is usually claimed to be a concept inspired by classical Markovianity, the physical meaning of this concept, however, is vague. In this thesis, I compare the descriptions of classical stochastic processes and open quantum systems, and show that those inherent differences lead to the non-trivial problem of characterizing quantum non-Markovianity. With various examples, I argue that the current most often used definitions of quantumMarkovianity in the literature do not fully capture the memoryless property of OQSs. To address this issue, I take a different approach by studying a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, while others are first proposed in this thesis. I define all these concepts under a unified framework, which allows one to rigorously build hierarchy relations among them. In this sense, quantum non-Markovianity is highly context-dependent. The results in this thesis, summarized as a hierarchy figure, are expected to bring clarity to the nature of quantumnon-Markovianity.
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.