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Physical Review A

Non-Markovian reduced propagator, multiple-time correlation functions, and master equations with general initial conditions in the weak-coupling limit

Inés de Vega, Daniel Alonso

Published Feb 2, 2006• 44 citations
Publisher

Abstract

In this paper we derive the evolution equation for the reduced propagator, an object that evolves vectors of the Hilbert space of a system $\mathcal{S}$ interacting with an environment $\mathcal{B}$ in a non-Markovian way. This evolution is conditioned to certain initial and final states of the environment. Once an average over these environmental states is made, reduced propagators permit the evaluation of multiple-time correlation functions of system observables. When this average is done stochastically the reduced propagator evolves according to a stochastic Schr\"odinger equation. In addition, it is possible to obtain the evolution equations of the multiple-time correlation functions which generalize the well-known quantum regression theorem to the non-Markovian case. Here, both methods, stochastic and evolution equations, are described by assuming a weak coupling between system and environment. Finally, we show that reduced propagators can be used to obtain a master equation with general initial conditions, and not necessarily an initial vacuum state for the environment. We illustrate the theory with several examples.

DOI
10.1103/physreva.73.022102

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In this paper we derive the evolution equation for the reduced propagator, an object that evolves vectors of the Hilbert space of a system S\mathcal{S}S interacting with an environment B\mathcal{B}B in a non-Markovian way. This evolution is conditioned to certain initial and final states of the environment. Once an average over these environmental states is made, reduced propagators permit the evaluation of multiple-time correlation functions of system observables. When this average is done stochastically the reduced propagator evolves according to a stochastic Schr"odinger equation. In addition, it is possible to obtain the evolution equations of the multiple-time correlation functions which generalize the well-known quantum regression theorem to the non-Markovian case. Here, both methods, stochastic and evolution equations, are described by assuming a weak coupling between system and environment. Finally, we show that reduced propagators can be used to obtain a master equation with general initial conditions, and not necessarily an initial vacuum state for the environment. We illustrate the theory with several examples.
We derive the dynamical equation of the reduced propagator, an object that evolves state vectors of the system conditioned to the dynamics of its environment, which is not necessarily in the vacuum state at the initial time. Such a reduced propagator is essential to obtain multiple-time correlation functions (MTCFs). We also study the evolution of MTCFs within the weak-coupling limit and show that the quantum regression theorem is, in general, not satisfied. We illustrate the theory in two different cases: first, solving an exact model, and, second, presenting the results of the numerical integration for a system coupled with a thermal environment through a nondiagonal interaction.
We derive the evolution equations for two-time correlation functions of a generalized non-Markovian open quantum system based on a modified stochastic Schrödinger equation approach. We find that the two-time reduced propagator, an object that used to be characterized by two independent stochastic processes in the Hilbert space of the system, can be simplified and obtained by taking ensemble average over one single noise. This discovery can save the cost of computation, and accelerate the converging process when taking the average over noisy trajectories. As a result, our method can be widely applied to many open quantum models, especially large-scale systems and extend the quantum regression theory to the non-Markovian case. In the short-time simulations, it is observed a significant difference between Markovian and non-Markovian cases, which can be applied to realize the environmental spectrum detection and enhance the measurement sensitivity in varying open quantum systems.
An extremely useful evolution equation that allows systematically calculating the two-time correlation functions (CF's) of system operators for non-Markovian open (dissipative) quantum systems is derived. The derivation is based on perturbative quantum master equation approach, so non-Markovian open quantum system models that are not exactly solvable can use our derived evolution equation to easily obtain their two-time CF's of system operators, valid to second order in the system-environment interaction. Since the form and nature of the Hamiltonian are not specified in our derived evolution equation, our evolution equation is applicable for bosonic and/or fermionic environments and can be applied to a wide range of system-environment models with any factorized (separable) system-environment initial states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equation is valid for both L = L(†) and L ≠ L(†) cases, in contrast to those evolution equations valid only for L = L(†) case in the literature. The derived equation that generalizes the quantum regression theorem (QRT) to the non-Markovian case will have broad applications in many different branches of physics. We then give conditions on which the QRT holds in the weak system-environment coupling case and apply the derived evolution equation to a problem of a two-level system (atom) coupled to the finite-temperature bosonic environment (electromagnetic fields) with L ≠ L(†).
We study the role of correlations with the environment as the source of non-Markovian quantum evolutions. We first focus on the impact that correlations with the environment can have on the dynamical map that evolve the system. We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We discuss the implications and limitations of the Markov approximation necessary to derive the Kossakowski-Lindblad master equation. A generalized non-Markovian master equation is derived from the dynamical map of systems correlated with their environment. The physical meaning of not completely positive maps is studied to obtain a consistent theory of non-Markovian quantum dynamics. These are associated to inverse maps necessary to establish correlations and they give rise to a canonical embedding map that is local in time. This master equation goes beyond the Kossakowski-Lindblad master equation. Non-equilibrium quantum thermodynamics can be be studied within this theory. Through out this discussion, the general dynamics of two interacting qubits is used as an example for illustrations.
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.
The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.