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Journal of Physics A Mathematical and General

Decoherence by Lindblad motion

Klaus Dietz

Published May 25, 2004• 15 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/37/23/012

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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.
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 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 Lindblad equation is commonly used to model dissipation and decoherence in open quantum systems. Yet many of the properties of Lindblad dynamics are still little understood. This thesis explores various aspects of the Lindblad equation and its connections with non-Hermitian quantum theories. The evolution speed of Lindblad dynamics is investigated and an explicit expression for the speed is derived. The radial component, connected to the rate at which the purity of the state changes, is shown to be determined by the modified skew information of the Lindblad operators. The semiclassical time evolution of Gaussian states under Lindblad dynamics is then considered and a new type of phase-space dynamics is derived for the centre of a Gaussian Wigner function. By viewing the phase-space Lindblad equation as a Schroedinger equation with a non-Hermitian Hamiltonian, a further set of semiclassical equations are derived. These can describe the interference terms in Wigner functions. Further, a PT-symmetric two-level system driven through two consecutive exceptional points at finite speed is considered and an analytical expression for the probability of a non-adiabatic transition is derived. It is shown that the transition through the exceptional points can be experimentally addressed in a PT-symmetric lattice using Bloch oscillations. Finally, a quantum implementation of this PT-symmetric system using ultracold atoms in an optical lattice with single-particle losses is investigated.
Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity
The Lindblad equation determines the time evolution of the density operator of open quantum systems. While valid for any system size, its use is, in practice, restricted to prototype/surrogate models with the aim of tackling specific aspects of the overall quantum complexity of a multi-atomic system. Path integral molecular dynamics (PIMD) instead provides static and dynamical quantum statistical averages of physical observables for systems in equilibrium composed of up to thousands of atoms over timescales up to nanoseconds, under the condition that short-time quantum coherence is not relevant for the properties of interest. PIMD relies on the well-established technique of molecular dynamics (MD) with its associated classical trajectories. However, it cannot describe a direct time evolution of a system and its convergence to a stationary state in situations out of equilibrium. In this work, we analyze the link between the Lindblad equation and PIMD; specifically, we will discuss how PIMD can actually be used to calculate the time evolution of ensemble-averaged physical observables and their convergence to a stationary state for situations out of equilibrium, bypassing the need of explicitly solving the Lindblad equation. Yet, at the same time, the Lindblad equation and PIMD are linked to one another through a formal relation of equivalence, which provides an argument for the consistency of PIMD results, namely the positivity of the density operator at any time. A numerical study of a prototype system, which is of interest in chemical physics, will be used to showcase the method.