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arXiv:quant-ph/0106151

Lindblad Evolution and Stochasticity: the Case of a Two-Level System

Salgado, D., J. L. Sánchez‐Gómez

Published Jun 26, 2001• 0 citations
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Abstract

We consider the simple hypothesis of letting quantum systems have an inherent random nature. Using well-known stochastic methods we thus derive a stochastic evolution operator which let us define a stochastic density operator whose expectation value under certain conditions satisfies a Lindblad equation. As natural consequences of the former assumption decoherence and spontaneous emission processes are obtained under the same conceptual scheme. A temptative solution for the preferred basis problem is suggested. All this is illustrated with a comprehensive study of a two-level quantum system evolution.

DOI
10.48550/arxiv.quant-ph/0106151

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We consider the simple hypothesis of letting quantum systems have an inherent random nature. Using well-known stochastic methods we thus derive a stochastic evolution operator which let us define a stochastic density operator whose expectation value under certain conditions satisfies a Lindblad equation. As natural consequences of the former assumption decoherence and spontaneous emission processes are obtained under the same conceptual scheme. A temptative solution for the preferred basis problem is suggested. All this is illustrated with a comprehensive study of a two-level quantum system evolution.
It is shown how any Lindbladian evolution with selfadjoint Lindblad operators, either Markovian or nonMarkovian, can be understood as an averaged random unitary evolution. Both mathematical and physical consequences are analyzed. First a simple and fast method to solve this kind of master equations is suggested and particularly illustrated with the phase-damped master equation for the multiphoton resonant Jaynes-Cummings model in the rotating-wave approximation. A generalization to some intrinsic decoherence models present in the literature is included. Under the same philosophy a proposal to generalize the Jaynes-Cummings model is suggested whose predictions are in accordance with experimental results in cavity QED and in ion traps. A comparison with stochastic dynamical collapse models is also included.
Recently a generalized master equation was derived that extends the Lindblad theory to highly non-Markovian quantum processes [H.-P. Breuer, Phys. Rev. A 75, 022103 (2007)]. We perform a stochastic unraveling of this master equation by considering nnn random state vectors that satisfy the corresponding stochastic differential equation for a piecewise deterministic process. As an application we consider a two-state system randomly coupled to an environment consisting of two energy bands with finite number of levels. Our numerical results are compared to results obtained from the time-convolutionless projection operator method using correlated projection superoperators and the exact solution of the Schr\"odinger equation for this system.
The structure of the initial system-environment state is fundamental to determining the nature and characteristics of the evolution of such an open quantum system. The usual assumption is to consider that the initial system-environment state is separable. Here, we go beyond this simple case and derive the evolution equations, up to second order in a weak-coupling expansion, that describe the evolution of the reduced density matrix of the system for any arbitrary system-environment initial state. The structure of these equations allows us to determine the initial conditions for which the evolution of the reduced density matrix can be written in terms of a set of Lindblad-like equations, once considering the Markov and secular approximations. Moreover, we show that for initial states belonging to a subset of separable states such Lindblad-like equations become actual Lindblad equations and that the evolution of ${\ensuremath{\rho}}_{s}$ thereby obtained is also trace and positive preserving.
This chapter presents the theoretical framework that allows us to describe evolutions in the general case using Kraus operators as the main tool. It considers in detail the phenomenon of decoherence and gives examples of such maps. It shows that any evolution can be considered as unitary by going in a larger Hilbert space. The Lindblad equation for the evolution of the density matrix appears as a particular case of evolution in the short memory or Markov approximation. Up-jumps and down-jumps are also described within this framework using cavity damping, spontaneous emission, and the shelving technique as examples.
We consider a model of environment-induced dissipationless decoherence of a quantum system where the system is coupled to the bath degrees of freedom via the system Hamiltonian itself. We solve exactly for the reduced density operator of the system for an arbitrary spectral density of the thermal bath and also write down an exact master equation in the Lindblad form. We compare and contrast the above results with those obtained by considering the system frequencies to be randomly modulated as in stochastic models. We observe that a coupling to the bath as above necessarily induces a Kerr-like coherent contribution in the reduced dynamics of the system. This Kerr-like term is a reflection of the quantum nature of the bath and cannot be obtained from stochastic models. For the special case of a harmonic oscillator we consider the influence of decoherence and its relation with phase diffusion. Our numerical results exhibit oscillations in the evolution of system variables which overall is a signature of the quantum nature of the environment.
Starting from the standard Lindblad form of a dissipative master equation, we identify a stochastic parameter in the quantum time evolution. This lacks dynamic evolution but its value is affected by the dissipative part of the master equation. An unravelling of the master equation in terms of quantum trajectories then shows this parameter performing the erratic variation characteristic of a stochastic variable. When the two evolutions separate, the stochastic Markov process can be extracted and solved independently and imposed on the parameter of the quantum evolution. This is the traditional form of treatment of stochastic variables in quantum mechanics. The relations to earlier works are established, and the approach is illustrated by simple model examples.