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arXiv:quant-ph/0503125 · Physical Review A

Limits in the characteristic-function description of non-Lindblad-type open quantum systems

Sabrina Maniscalco

Published Aug 30, 2005• 42 citations
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Abstract

In this paper I investigate the usability of the characteristic functions for the description of the dynamics of open quantum systems focussing on non-Lindblad-type master equations. I consider, as an example, a non-Markovian generalized master equation containing a memory kernel which may lead to nonphysical time evolutions characterized by negative values of the density matrix diagonal elements [S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001)]. The main result of the paper is to demonstrate that there exist situations in which the symmetrically ordered characteristic function is perfectly well-defined while the corresponding density matrix loses positivity. Therefore, nonphysical situations may not show up in the characteristic function. As a consequence, the characteristic function cannot be considered an alternative complete description of the non-Lindblad dynamics.

DOI
10.1103/physreva.72.024103

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In this paper I investigate the usability of the characteristic functions for the description of the dynamics of open quantum systems focussing on non-Lindblad-type master equations. I consider, as an example, a non-Markovian generalized master equation containing a memory kernel which may lead to nonphysical time evolutions characterized by negative values of the density matrix diagonal elements [S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001)]. The main result of the paper is to demonstrate that there exist situations in which the symmetrically ordered characteristic function is perfectly well-defined while the corresponding density matrix loses positivity. Therefore, nonphysical situations may not show up in the characteristic function. As a consequence, the characteristic function cannot be considered an alternative complete description of the non-Lindblad dynamics.
In this paper I investigate the usability of the quantum characteristic function for the description of the dynamics of open quantum systems, focussing on the non-Markovian regime. It has been shown that non-Markovian generalized master equations containing memory kernels may lead to non-physical time evolutions characterized by negative values of the density matrix diagonal elements [S.M. Barnett and S. Stenholm, Phys. Rev. A {\\bf 64}, 033808 (2001)]. The main purpose of the paper is to show that there exist situations in which the quantum characteristic function of the non-Markovian open quantum system is perfectly well defined while the corresponding density matrix loses positivity. Therefore unphysical situations may not show up in a description of the dynamics via the quantum characteristic function. I conclude that, in general, the quantum characteristic function cannot be used for describing non-Markovian dynamics.
Semi-Markov processes represent a well-known and widely used class of random processes in classical probability theory. Here, we develop an extension of this type of non-Markovian dynamics to the quantum regime. This extension is demonstrated to yield quantum master equations with memory kernels which allow the formulation of explicit conditions for the complete positivity of the corresponding quantum dynamical maps, thus leading to important insights into the structural characterization of the non-Markovian quantum dynamics of open systems. Explicit examples are analyzed in detail.
Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalization procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)] is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0t>0t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.
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.
We use basic physical motivations to develop sufficient conditions for positive semidefiniteness of the reduced density matrix for generalized non-Markovian integrodifferential Lindblad–Kossakowski master equations with Hermitian generators. We show that it is sufficient for the memory function to be the Fourier transform of a real positive symmetric frequency density function with certain properties. These requirements are physically motivated, and are more general and more easily checked than previously stated sufficient conditions. We also explore the decoherence dynamics numerically for some simple models using the Hadamard representation of the propagator. We show that the sufficient conditions are not necessary conditions. We also show that models exist in which the long time limit is in part determined by non-Markovian effects.
Determining the Markovianity and non-Markovianity of a quantum process is a critical problem in the theory of open quantum systems, as their behaviors differ significantly in terms of complexity. It is well recognized that a quantum process is Markovian if and only if the quantum master equation can be written in the standard Lindblad form with all rates nonnegative for all time. However, here we present a striking result that \textit{any} finite-dimensional open quantum system dynamics can be described by a quantum master equation in the Lindblad form with all rates nonnegative for all time. In fact, it can be shown that one can arbitrarily decide the sign of the rates in any case at any time interval. Note that here we take an unconventional approach where the quantum master equation we construct will in general be state-dependent, which means that the Hamiltonian, jump operators and rates will all depend on the current state of the density matrix ρ(t)ρ(t)ρ(t). Our findings raise serious questions on the current criterion in determining Markovianity and non-Markovianity in open quantum system dynamics.