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arXiv:1205.6952 · Physical Review A

Non-Markovian waiting-time distribution for quantum jumps in open systems

Kimmo Luoma, Kari Härkönen, Sabrina Maniscalco, Kalle‐Antti Suominen, Jyrki Piilo

Published Aug 1, 2012• 8 citations
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Abstract

Simulation methods based on stochastic realizations of state vector evolutions are commonly used tools to solve open quantum system dynamics, both in the Markovian and non-Markovian regimes. Here, we address the question of the waiting time distribution (WTD) of quantum jumps for non-Markovian systems. We generalize Markovian quantum trajectory methods in the sense of deriving an exact analytical WTD for non-Markovian quantum dynamics and show explicitly how to construct this distribution for certain commonly used quantum optical systems.

DOI
10.1103/physreva.86.022102

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Simulation methods based on stochastic realizations of state vector evolutions are commonly used tools to solve open quantum system dynamics, both in the Markovian and non-Markovian regimes. Here, we address the question of the waiting time distribution (WTD) of quantum jumps for non-Markovian systems. We generalize Markovian quantum trajectory methods in the sense of deriving an exact analytical WTD for non-Markovian quantum dynamics and show explicitly how to construct this distribution for certain commonly used quantum optical systems.
The simulation methods based on stochastic realizations of state vector evolutions derived from the time convolutionless master equation are investigated. The quantum jumps for non-Markovian systems and their waiting time distribution is mainly focused on. With the help of an exact analytical waiting time distribution, the non-Markovian quantum trajectory for a stochastic process of a driven two-level system is generated. The effectiveness and superiority of the algorithm is verified by the performance comparing to our previous work in the sense of Markovian prosperities.
Recently, a large class of quantum non-Markovian piecewise dynamics for an open quantum system obeying closed evolution equations has been introduced [1]. These dynamics have been defined in terms of a waiting-time distribution between quantum jumps, along with quantum maps describing the effect of jumps and the system evolution between them. Here, we present a quantum collision model with memory, whose reduced dynamics in the continuous-time limit reproduces the above class of non-Markovian piecewise dynamics, thus providing an explicit microscopic realization.
We discuss in detail how non-Markovian open system dynamics can be described in terms of quantum jumps [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. Our results demonstrate that it is possible to have a jump description contained in the physical Hilbert space of the reduced system. The developed non-Markovian quantum jump approach is a generalization of the Markovian Monte Carlo wave function (MCWF) method into the non-Markovian regime. The method conserves both the probabilities in the density matrix and the norms of the state vectors exactly and sheds new light on non-Markovian dynamics. The dynamics of the pure state ensemble illustrates how local-in-time master equation can describe memory effects and how the current state of the system carries information on its earlier state. Our approach solves the problem of negative jump probabilities of the Markovian MCWF method in the non-Markovian regime by defining the corresponding jump process with positive probability. The results demonstrate that in the theoretical description of non-Markovian open systems, there occurs quantum jumps which recreate seemingly lost superpositions due to the memory.
A long-standing open problem in the non-Markovian quantum state diffusion (QSD) approach to open quantum systems is to establish the non-Markovian QSD equations for multiple-qubit systems. In this paper, we settle this important question by explicitly constructing a set of exact time-local QSD equations for NNN-qubit systems. Our exact time-local (convolutionless) QSD equations have paved the way towards simulating quantum dynamics of many-body open systems interacting with a common bosonic environment. The applicability of this multiple-qubit stochastic equation is exemplified by numerically solving several quantum open many-body systems concerning quantum coherence dynamics and dynamical control.
Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of ``typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of ``exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...
Time-evolution of open, dissipative quantum system is a stochastic process that consists of a series of quantum jumps that take place at random times. Between quantum jumps quantum system idles for some time in a particular quantum state. Based on superfermion/superboson formalism and general Kossakowski- Lindblad master equation for an open quantum system we develop a method to compute distribution of waiting times between quantum jumps. We illustrate the use of the theory by computing waiting time distribution for model Fermi-system in thermodynamic equilibrium.