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Journal of Physics A Mathematical and General

Asymptotic solutions of Lindblad equations

Klaus Dietz

Published Nov 27, 2002• 6 citations
Publisher

Abstract

The temporal motion of observables of a quantum mechanical N -level system is studied. In particular, I investigate the mapping, in its dependence on the matrix V parametrizing the Lindblad generator, of given initial configurations into the resulting configurations at large times t ( t → ∞ ). Explicit solutions are given for a large class of V .

DOI
10.1088/0305-4470/35/49/307

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The temporal motion of observables of a quantum mechanical N -level system is studied. In particular, I investigate the mapping, in its dependence on the matrix V parametrizing the Lindblad generator, of given initial configurations into the resulting configurations at large times t ( t → ∞ ). Explicit solutions are given for a large class of V .
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.
We investigate the long-time behavior of quantum Markovian dynamics generated by time-dependent Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) master equations. We introduce a notion of weak relaxation and derive sufficient conditions guaranteeing asymptotic independence from the initial state. Our results provide a quantitative extension of the Spohn–Frigerio theorem to time-dependent generators, yielding explicit contraction bounds in terms of the instantaneous steady state and the jump operators. For a class of microscopically derived master equations, we further obtain a graph-theoretic characterization of the aforementioned conditions that directly links the structure of the jump operators to the relaxation properties. The general theory is illustrated by applications to driven finite-level systems, including a detailed three-level example, and is extended to a non-Markovian setting by means of time-local master equations that become of GKLS form at long times. These findings pave the way for the development of a more general theory of relaxation beyond the Markovian case.
The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment.