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Entropy

A Numerical Study of Quantum Entropy and Information in the Wigner–Fokker–Planck Equation for Open Quantum Systems

Arash Edrisi, Hamza Patwa, José A. Morales Escalante

Published Mar 14, 2024• 4 citations
Publisher

Abstract

Kinetic theory provides modeling of open quantum systems subject to Markovian noise via the Wigner-Fokker-Planck equation, which is an alternate of the Lindblad master equation setting, having the advantage of great physical intuition as it is the quantum equivalent of the classical phase space description. We perform a numerical inspection of the Wehrl entropy for the benchmark problem of a harmonic potential, since the existence of a steady state and its analytical formula have been proven theoretically in this case. When there is friction in the noise terms, no theoretical results on the monotonicity of absolute entropy are available. We provide numerical results of the time evolution of the entropy in the case with friction using a stochastic (Euler-Maruyama-based Monte Carlo) numerical solver. For all the chosen initial conditions studied (all of them Gaussian states), up to the inherent numerical error of the method, one cannot disregard the possibility of monotonic behavior even in the case under study, where the noise includes friction terms.

DOI
10.3390/e26030263

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Kinetic theory provides modeling of open quantum systems subject to Markovian noise via the Wigner-Fokker-Planck equation, which is an alternate of the Lindblad master equation setting, having the advantage of great physical intuition as it is the quantum equivalent of the classical phase space description. We perform a numerical inspection of the Wehrl entropy for the benchmark problem of a harmonic potential, since the existence of a steady state and its analytical formula have been proven theoretically in this case. When there is friction in the noise terms, no theoretical results on the monotonicity of absolute entropy are available. We provide numerical results of the time evolution of the entropy in the case with friction using a stochastic (Euler-Maruyama-based Monte Carlo) numerical solver. For all the chosen initial conditions studied (all of them Gaussian states), up to the inherent numerical error of the method, one cannot disregard the possibility of monotonic behavior even in the case under study, where the noise includes friction terms.
The harmonic oscillator with dissipation is studied within the framework of the Lindblad theory for open quantum systems. By using the Wang-Uhlenbeck method, the Fokker-Planck equation, obtained from the master equation for the density operator, is solved for the Wigner distribution function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The obtained Wigner functions are two-dimensional Gaussians with different widths. Then a closed expression for the density operator is extracted. The entropy of the system is subsequently calculated and its temporal behaviour shows that this quantity relaxes to its equilibrium value.
The quantum Fokker-Planck equation for a Gaussian-Markovian bath is deduced by applying a method proposed by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1989)]. The results are expressed in the form of simultaneous differential equations in terms of density operators and can treat strong system-bath interactions where the correlated effects of the noise play an important role. The classical Fokker-Planck equation for a Gaussian-Markovian noise is obtained by performing the Wigner transformation, and its equilibrium state is shown to be the Maxwell-Boltzmann distribution. The method is convenient for numerical studies. Calculations for quantum-system harmonic oscillators and the double-well potential problems are demonstrated for cases of Gaussian-white noise and Gaussian-Markovian noise.
Abstract Quantum and classical systems evolving under the same formal Hamiltonian H may exhibit dramatically different behavior after the Ehrenfest timescale tE∼log⁡(ℏ−1)t_E \sim \log (\hbar ^{-1})tE​∼log(ℏ−1) t E ∼ log ( ħ - 1 ) , even as ℏ→0\hbar \rightarrow 0ℏ→0 ħ → 0 . Coupling the system to a Markovian environment results in a Lindblad equation for the quantum evolution. Its classical counterpart is given by the Fokker–Planck equation on phase space, which describes Hamiltonian flow with friction and diffusive noise. The quantum and classical evolutions may be compared via the Wigner-Weyl representation. Due to decoherence, they are conjectured to match closely for times far beyond the Ehrenfest timescale as ℏ→0\hbar \rightarrow 0ℏ→0 ħ → 0 . We prove a version of this correspondence, bounding the error between the quantum and classical evolutions for any sufficiently regular Hamiltonian H(x, p) and Lindblad functions Lk(x,p)L_{k}(x,p)Lk​(x,p) L k ( x , p ) . The error is small when the strength of the diffusion D associated to the Lindblad functions satisfies D≫ℏ4/3D \gg \hbar ^{4/3}D≫ℏ4/3 D ≫ ħ 4 / 3 , in particular allowing vanishing noise in the classical limit. Our method uses a time-dependent semiclassical mixture of variably squeezed Gaussian states. The states evolve according to a local harmonic approximation to the Lindblad dynamics constructed from a second-order Taylor expansion of the Lindbladian. Both the exact quantum trajectory and its classical counterpart can be expressed as perturbations of this semiclassical mixture, with the errors bounded using Duhamel’s principle. We present heuristic arguments suggesting the 4/3 exponent is optimal and defines a boundary in the sense that asymptotically weaker diffusion permits a breakdown of quantum-classical correspondence at the Ehrenfest timescale. Our presentation aims to be comprehensive and accessible to both mathematicians and physicists. In a shorter companion paper, we treat the special case of Hamiltonians that decompose into kinetic and potential energy with linear Lindblad operators, with explicit bounds that can be applied directly to physical systems.
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.
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.