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arXiv:1412.1892 · The Journal of Physical Chemistry Letters

Wigner–Lindblad Equations for Quantum Friction

Denys I. Bondar, Renán Cabrera, Andre G. Campos, Shaul Mukamel, HERSCHEL A. RABITZ

Published Apr 14, 2016• 25 citations
arXivHTMLPDF

Abstract

Dissipative forces are ubiquitous and thus constitute an essential part of realistic physical theories. However, quantization of dissipation has remained an open challenge for nearly a century. We construct a quantum counterpart of classical friction, a velocity-dependent force acting against the direction of motion. In particular, a translationary invariant Lindblad equation is derived satisfying the appropriate dynamical relations for the coordinate and momentum (i.e., the Ehrenfest equations). Numerical simulations establish that the model approximately equilibrates. These findings significantly advance a long search for a universally valid Lindblad model of quantum friction and open opportunities for exploring novel dissipation phenomena.

DOI
10.1021/acs.jpclett.6b00498

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Dissipative forces are ubiquitous and thus constitute an essential part of realistic physical theories. However, quantization of dissipation has remained an open challenge for nearly a century. We construct a quantum counterpart of classical friction, a velocity-dependent force acting against the direction of motion. In particular, a translationary invariant Lindblad equation is derived satisfying the appropriate dynamical relations for the coordinate and momentum (i.e., the Ehrenfest equations). Numerical simulations establish that the model approximately equilibrates. These findings significantly advance a long search for a universally valid Lindblad model of quantum friction and open opportunities for exploring novel dissipation phenomena.
By means of studying the evolution equation for the Wigner distributions of quantum dissipative systems we derive the quantum corrections to the classical Liouville dynamics, taking into account the standard quantum friction model. The resulting evolution turns out to be the classical one plus fluctuations that depend not only on the ℏ size but also on the momentum and the dissipation parameter (i.e., the coupling with the environment). On the other hand, we extend our studies of a paradigmatic system based on the kicked rotator, and we confirm that by adding fluctuations only depending on the size of the Planck constant we essentially recover the quantum behavior. This is systematically measured in the parameter space with the overlaps and differences in the dispersion of the marginal distributions corresponding to the Wigner functions. Taking into account these results and analyzing the Wigner evolution equation we conjecture that the chaotic nature of our system is responsible for the independence on the momentum, while the dependence on the dissipation is provided implicitly by the dynamics.
We introduce a type of quantum dissipation -- local quantum friction -- by adding to the Hamiltonian a local potential that breaks time-reversal invariance so as to cool the system. Unlike the Kossakowski-Lindblad master equation, local quantum friction directly effects unitary evolution of the wavefunctions rather than the density matrix: it may thus be used to cool fermionic many-body systems with thousands of wavefunctions that must remain orthogonal. In addition to providing an efficient way to simulate quantum dissipation and non-equilibrium dynamics, local quantum friction coupled with adiabatic state preparation significantly speeds up many-body simulations, making the solution of the time-dependent Schrödinger equation significantly simpler than the solution of its stationary counterpart.
Open quantum systems play a central role in contemporary nanoscale technologies, including molecular electronics, quantum heat engines, quantum computation and information processing. A major theoretical challenge is to construct dynamical models that are simultaneously consistent with classical thermodynamics and complete positivity. In this work, we develop a framework that addresses this issue by extending classical stochastic dynamics to the quantum domain. We begin by formulating a generalized Langevin equation in which both friction and noise act symmetrically on the two Hamiltonian equations. From this, we derive a generalized Klein-Kramers equation expressed in terms of Poisson brackets, and we show that it admits the Boltzmann distribution as its stationary solution while satisfying the first and second laws of thermodynamics along individual trajectories. Applying canonical quantization to this classical framework yields two distinct quantum master equations, depending on whether the friction operators are taken to be Hermitian or non-Hermitian. By analyzing the dynamics of a harmonic oscillator, we determine the conditions under which these equations reduce to a Lindblad-type generator. Our results demonstrate that complete positivity is ensured only when friction and noise are included in both Hamiltonian equations, thus fully justifying the classical construction. Moreover, we find that the friction coefficients must satisfy the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a form of universality that transcends the specific operator representation. The formalism offers a versatile tool for deriving quantum versions of the thermodynamic laws and is directly applicable to a wide class of nonequilibrium nanoscale systems.
Using the classical trajectory we quantize the motion of a pointlike particle of mass m and energy 1/2${\mathit{mx}\mathit{\ifmmode \dot{}\else \.{}\fi{}}}_{0}^{2}$ under the influence of the force Fn{\mathit{F}}_{\mathit{n}}Fn​=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}}$, \ensuremath{\alpha}\ensuremath{\gtrsim}0, n=1,2,.... Such forces are frictional only for x\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\gtrsim}0 and antifrictional for x\ifmmode \dot{}\else \.{}\fi{}0 and can be simulated through the gradient of a one-parameter class of energy-dependent potentials Vn{\mathit{V}}_{\mathit{n}}Vn​(x;x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{0}$), which increase monotonically in x. This permits one to construct a class of Lagrange and Hamilton functions that do not contain any integrating functions. We then proceed to quantization and study the quantum scattering problem assuming that Fn{\mathit{F}}_{\mathit{n}}Fn​ acts on a finite interval of the x axis only. The stationary problem can be solved analytically for all n, and quantitative examples are shown for n=2. The time-dependent scattering problem is discussed with emphasis on Ehrenfest's theorems and on linear damping n=1. When the quantum-mechanical probability for reflection is small the method provides an approximation to the motion of a wave packet through a dissipative medium, with velocity-power forces Fdn{\mathit{F}}_{\mathit{dn}}Fdn​=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$x\ifmmode \dot{}\else \.{}\fi{}. \textcopyright{} 1996 The American Physical Society.
In a former publication [D. Schuch and M. Moshinsky, Phys. Rev. A 73, 062111 (2006)] we have shown that the time evolution of a quantum system with at most quadratic Hamiltonian that can be described by different methods, namely, the time-dependent Schrödinger equation, the time propagator or Feynman kernel, and the Wigner function is connected via a dynamical invariant, the so-called Ermakov invariant. Since exact invariants of this type also exist for certain effective descriptions of dissipative quantum systems, we will show the relations between these descriptions and the corresponding invariants and will discuss how the results obtained for the conservative systems must be modified in the presence of a linear velocity dependent frictional force.