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arXiv:math-ph/0212027 · Journal of Physics A Mathematical and General

Memory effects in long-time Lindblad motion

Klaus Dietz

Published Jan 8, 2003• 8 citations
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Abstract

It is shown that the Lindblad equation accounts for memory effects. That is to say, Lindblad operators can be constructed in a natural manner such that a memory term appears in the asymptotic ( time → ∞ ) region; at the same time the expectation values depend on the initial state. Furthermore, a procedure for extending the Lindblad equation to an equation of motion for an ideal Bose 'gas' of 'particles', i.e. systems with non-trivial internal structure, is described. Initially in some quantum state this collection of 'particles' will asymptotically turn into an equilibrium ensemble whose probability distribution is determined by the Lindblad operators building the dissipative part of the equation of motion.

DOI
10.1088/0305-4470/36/3/101

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It is shown that the Lindblad equation accounts for memory effects. That is to say, Lindblad operators can be constructed in a natural manner such that a memory term appears in the asymptotic ( time → ∞ ) region; at the same time the expectation values depend on the initial state. Furthermore, a procedure for extending the Lindblad equation to an equation of motion for an ideal Bose 'gas' of 'particles', i.e. systems with non-trivial internal structure, is described. Initially in some quantum state this collection of 'particles' will asymptotically turn into an equilibrium ensemble whose probability distribution is determined by the Lindblad operators building the dissipative part of the equation of motion.
The structure of the Lindblad equation of motion of quantum states is discussed. General specifications for this motion to lead asymptotically into equilibrium states are given. Incomplete ‘thermalization’, i.e. the Lindblad motion of only a selected subset of quantum states leads to a reduced quantum system whose observables are explicitly constructed and seen to incorporate memory terms. It is shown that under rather general conditions the absolute value of Lindblad operators is given by the (inverse square root of the) grand-canonical probability distribution.
The predictive power of Lindblad equations for the dynamics of open systems is discussed. In a model with only one Lindblad operator the asymptotic state (density operator) for t → ∞ determines the Lindblad operator up to an isometry. Hence the asymptotic state reached by Lindblad motion appears as an input and has to be obtained from independent physical considerations when setting up the equations of motion. To illustrate this fact we assume an asymptotic Gibbs state (and a Hamiltonian of course) and discuss the temporal behaviour of the statistical entropy. Numerical model calculations are presented which show a non-monotonic behaviour of the statistical entropy during the approach to equilibrium. As a second point in our discussion of the types of predictions derived from Lindblad equations, we show that a certain structure of the Lindblad operators leads to decoherence into superselection sectors. The latter are determined by a spectral decomposition of the Hamiltonian of the open system considered. An explicit construction of such decohering systems is given.
An equation of motion for open quantum systems incorporating memory effects and initial correlations with the environment is presented in terms of an effective Liouville operator that solely acts on states of the system. The environment can induce memory effects via the frequency dependence of the effective Liouville and initial correlations can be mapped to a shifted frequency dependent initial state within the system. The equation of motion generalizes the well known semi-group dynamic equations. In generic systems the effective Liouville has a non-degenerate zero mode. By probability conservation one can demonstrate that a generic open system reaches, in the long time limit, a stationary state, which is independent of any initial condition.
Using an analytical model potential which contains resonant and bound states, we show that the decay of the resonances can be simulated by Lindblad operators. For that purpose, the unitary time evolution of an initial Gaussian wave packet in the model potential is compared with the non-unitary time evolution, obtained by solving the Lindblad equation, of the same wave packet in a potential which coincides with the model potential in the region of interest but does not contain resonances. In the latter case, dissipative effects are accounted for by Lindblad operators which lead to phenomenological friction and diffusion constants in the equations of motion. We suggest how those constants can be determined in a non-heuristic way, being directly connected to the width of the resonance in the model potential which we calculate using the complex rotation method.
The celebrated Lindblad equation governs the nonunitary time evolution of density operators used in the description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently, solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical potential is externally imposed rather than derived from first principles, and hence the corresponding density operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and complementary approach to the current literature that consists in using a generalized system Hamiltonian which includes a term μN. In a previous paper, this empirically well-known term has been formally derived from the von Neumann equation for the specific case of equilibrium. Including μN in the system Hamiltonian leads to a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the quantities involved are obtained from the physics of the system without any external assumptions.
The reduced description of the quantum dynamic processes in the condensed phase environment leads to the equation of motion with a memory kernel. Such a memory effect, termed non-Markovianity, presents more complex dynamics compared to its memoryless or Markovian counterpart, and many chemical systems have been demonstrated through numerical simulations to exhibit non-Markovian quantum dynamics. Explicitly how the memory impacts the dynamic process remains largely unexplored. In this work, we focus on ways to separate the non-Markovian contributions from the dynamics and study the non-Markovian effects. Specifically, we developed a rigorous procedure for mapping the exact non-Markovian quantum propagator to the Lindblad form. Consequently, it allows us to extract the negative decay rate from the Lindbladian that is the signature of the non-Markovianity. By including or excluding the negative rate in the time evolution, we can decisively pinpoint the influence of non-Markovianity on the system's properties such as coherence, entanglement, and equilibrium state distribution. The understanding of such memory effects on the dynamic process suggests the possibility of leveraging non-Markovianity for quantum control.