Phase space master equations for quantum Brownian motion in a periodic potential: comparison of various kinetic models

Abstract

The dynamics of quantum Brownian particles in a cosine periodic potential are studied using the phase space formalism associated with the Wigner representation of quantum mechanics. Various kinetic phase space master equation models describing quantum Brownian motion in a potential are compared by evaluating the dynamic structure factor and escape rate from the differential recurrence relations generated by the models. The numerical solution is accomplished via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The results of numerical calculations of the escape rate from a well of the cosine potential are compared with those given analytically by the quantum-mechanical reaction rate theory solution of the Kramers turnover problem for a periodic potential, given by Georgievskii and Pollak (1994 Phys. Rev. E 49 5098), enabling one to appraise each model.

Wigner–Lindblad Equations for Quantum Friction

“…The fundamental reason for the Ehrenfest equation violation is the ubiquitous usage of A (eq ), which is taken to be linear with respect to the coordinate and momentum (see the comment after eq ). (ii) Non-Lindblad models obeying the Ehrenfest relations that preserve state’s positivity for sufficiently high temperatures are discussed in refs and − . Contrary to the claims, the master equations in refs − belong to the same category.…”

Quantum tunnelling and thermally driven transitions in a double-well potential at finite temperature

“…It has been shown in [19] that it is possible to derive master equations for the Wigner function evolution by incorporating the constraint that the thermal state is a fixed point of the dynamics. We will now present a master equation derivation for the Harmonic oscillator that parallels this approach; however, in our analysis, we will take a step further by explicitly connecting with the Lindblad equation.…”

“…Our approach shares some similarities with the work of Lampo et al [18], in extending the CL equation for low-temperature applications, but with a crucial distinction: we treat the conservation of the quantum Gibbs state as a constraint, leading to a distinct model. Similarly, Cleary et al [19], developed a thermal state-preserving master equation in the Wigner function framework, akin to our model. Yet, our work advances by converting this equation into Lindblad form, enabling exploration of its stochastic unravelling dynamics.…”

Spin Relaxation in Phase Space

“…Moreover, the main advantage of the phase space approach now becomes apparent, namely it provides a master equation that may be solved using the methods [6,71] associated with the classical theory of the Brownian motion in a potential, allowing one to study the quantumclassical correspondence for dissipative systems (see, e.g., [131,132,[135][136][137][138][139][140][141]). Many other examples of the use of the Wigner function representation of the density matrix in various applications in physics and chemistry may be found in the books [42,46,47,77] and references cited therein.…”