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arXiv:0806.2984

QUANTUM FOKKER-PLANCK MODELS: THE LINDBLAD AND WIGNER APPROACHES

Anton Arnold, Franco Fagnola, Lukas Neumann

Published Oct 1, 2008• 10 citations
arXivHTMLPDF

Abstract

In this article we try to bridge the gap between the quantum dynamical semigroup and Wigner function approaches to quantum open systems. In particular we study stationary states and the long time asymptotics for the quantum Fokker-Planck equation. Our new results apply to open quantum systems in a harmonic confinement potential, perturbed by a (large) sub-quadratic term.

DOI
10.1142/9789812835277_0003

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In this article we try to bridge the gap between the quantum dynamical semigroup and Wigner function approaches to quantum open systems. In particular we study stationary states and the long time asymptotics for the quantum Fokker-Planck equation. Our new results apply to open quantum systems in a harmonic confinement potential, perturbed by a (large) sub-quadratic term.
In this article we study stationary states and the long time asymptotics for the quantum Fokker–Planck equation. We continue the investigation of an earlier \nwork in which we derived convergence to a steady state if the Lindblad condition $D_{pp}D_{qq} −D_{pq} \n\\ge \\gamma^2/4$ is satisfied with strict inequality. Here we extend our results to the limiting case that turns out to be more difficult because irreducibility \nof the quantum Markov semigroup does not follow from triviality of the generalized commutator with position and momentum operators.
The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator. AMS 1991 Subject classification: 35Q40, 35S10, 81Q99, 81V99 Key words: Open quantum system, Wigner equation, large-time behavior, self-similarity. 1. Introduction This paper is concerned with the analysis of a class of dissipative quantum models that arise when considering the motio...
Abstract We consider a frictionless system coupled to an external Markovian environment. The quantum and classical evolution of such systems are described by the Lindblad and the Fokker–Planck equation, respectively. We show that when such a system is given by an at most quadratically growing Hamiltonian and at most linearly growing real jump functions, the quantum and classical evolutions remain close on time scales much longer than Ehrenfest time. In particular, we show that the evolution of a density matrix by the Lindblad equation is close in trace norm to the quantization of the corresponding evolution by the Fokker–Planck equation. Such agreement improves upon recent results (Galokowski and Zworski in Classical quantum correspondence in Lindblad evolution, 2024. arXiv:2403.09345 ; Hernández et al. in Decoherence ensures classicality beyond the Ehrenfest time as ℏ→0\hbar \rightarrow 0ℏ→0 ħ → 0 , 2023. arXiv:2306.13717 , Hernández et al. in The limit of open quantum systems with general Lindbladians: vanishing noise ensures classicality beyond the ehrenfest time, 2023. arXiv:2307.05326 ), which proved long-time agreement in weaker norms.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad's theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function is also solved by using the Wang-Uhlenbeck method of transforming it into a linearized partial differential equation for the Wigner function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The Wigner functions which we obtain are two-dimensional Gaussians with different widths.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function, subject to either the Gaussian type or the δδδ-function type of initial conditions, is also solved by using the Wang-Uhlenbeck method. The obtained Wigner functions are two-dimensional Gaussians with different widths.
In a recent work we have found a contraction semigroup able to correctly approximate a projected and perturbed one-parameter group of isometries in a generic Banach space, in the limit of weak-coupling. Here we study its generator by specializing to W∗W^*W∗-algebras: after defining a Physical Subsystem in terms of a completely positive projecting conditional expectation, we find that it generates a Quantum Dynamical Semigroup. As a consequence of uniqueness and strong generality (well defined dynamics, irrespective of the Physical Subsystem spectral properties or dimensions), its generator deserves to be referred as "the" Quantum Fokker-Planck Equation. We then provide important examples of the limit dynamics, one of which constitutes a new Quantum generalization of the celebrated Fermi Golden Rule.