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Limits in the quantum characteristic function description of open quantum systems

Sabrina Maniscalco

Published Mar 14, 2005• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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 analyze non-Markovian evolution of open quantum systems. It is shown that any dynamical map representing the evolution of such a system may be described either by a nonlocal master equation with a memory kernel or equivalently by an equation which is local in time. These two descriptions are complementary: if one is simple, the other is quite involved, or even singular, and vice versa. The price one pays for the local approach is that the corresponding generator keeps the memory about the starting point "t{0}." This is the very essence of non-Markovianity. Interestingly, this generator might be highly singular; nevertheless, the corresponding dynamics is perfectly regular. Remarkably, the singularities of the generator may lead to interesting physical phenomena such as the revival of coherence or sudden death and revival of entanglement.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. We approach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non-Markovian characteristics are demonstrated.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. Weapproach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non- Markovian characteristics are demonstrated.
Do phenomenological master equations with a memory kernel always describe a non-Markovian quantum dynamics characterized by reverse flow of information? Is the integration over the past states of the system an unmistakable signature of non-Markovianity? We show by a counterexample that this is not always the case. We consider two commonly used phenomenological integro-differential master equations describing the dynamics of a spin 1/21/21/2 in a thermal bath. By using a recently introduced measure to quantify non-Markovianity [Breuer et al., Phys. Rev. Lett. 103, 210401 (2009)] we demonstrate that as far as the equations retain their physical sense, the key feature of non-Markovian behavior does not appear in the considered memory kernel master equations. Namely, there is no reverse flow of information from the environment to the open system. Therefore, the assumption that the integration over a memory kernel always leads to a non-Markovian dynamics turns out to be vulnerable to phenomenological approximations. Instead, the considered phenomenological equations are able to describe time-dependent and unidirectional information flow from the system to the reservoir associated with time-dependent Markovian processes.