A Monte Carlo Method for Modeling Thermal Damping: Beyond the Brownian-Motion Master Equation

The “standard” Brownian motion master equation, used to describe thermal damping, is not completely positive, and does not admit a Monte Carlo method, important in numerical simulations. To eliminate both these problems one must add a term that generates additional position diffusion. He we show that one can obtain a completely positive simple quantum Brownian motion, efficiently solvable, without any extra diffusion. This is achieved by using a stochastic Schrödinger equation (SSE), closely analogous to Langevin’s equation, that has no equivalent Markovian master equation. Considering a specific example, we show that this SSE is sensitive to nonlinearities in situations in which the master equation is not, and may therefore be a better model of damping for nonlinear systems. All mechanical oscillators experience frictional damping, in which they lose energy to their environment. This damping is accompanied by thermalization of the oscillator, since the environment is a large system, and thus a thermal bath. One would naturally like to model the effects of this damping on the oscillator, and indirectly on any other devices to which it is coupled, without having to describe the motion of the environment, with its many degrees of freedom. Frictional damping of quantum systems by an environment, often referred to as “quantum Brownian motion”, has many applications. In particular, it is essential in describing the behavior of nanomechanical resonators, and is therefore of current interest in quantum nano-electro-mechanics (QNEMS) [1, 2] and quantum opto-mechanics [3,4]. For classical systems, Langevin’s equation [5], containing only a deterministic frictional force and a Gaussian white noise source, gives a simple and excellent model of damping and thermalization. The situation is much less simple for quantum systems, however. The standard approach to obtaining a quantum equation to model damping from a thermal bath is to derive a master equation for a linear oscillator by coupling it to a continuum of oscillators, and then trace out these oscillators. This was first done by Caldeira and Leggett [6], and was brought to its completion in the tour-de-force by Hu et al. [7], building on the work of Unruh and Zurek [8]. They showed that an exact master equation for the oscillator alone could be derived, and that any non-markovian effects of the bath appear merely as time-dependent coefficients in this equation. This master equation, the exact BME, is [7] ρ̇ = −iΓ(t)[x, {p, ρ}]− ξ(t)[x, [x, ρ]] + ζ(t)[x, [p, ρ]]. (1) Here ρ is the density matrix for the oscillator, {p, ρ} = pρ+ρp is the anti-commutator, and x and p are the dimensionless position and momentum of the oscillator, defined as x = (a+ a)/ √ 2, (2) p = −i(a− a)/ √ 2, (3) where a is the oscillator’s annihilation operator. The three functions of time, Γ, ξ and ζ, are the time-dependent coefficients that encode all non-Markovian effects of the bath, and are determined by the frequency of the resonator, ω, the coupling to the bath (which sets the damping rate, γ), the structure of the bath (ohmic, super-ohmic, etc.), and the initial bath state (which is determined by the temperature, T ). For reference, the physical position and momentum are X = √ h̄/(ωm)x and P = √ h̄ωmp, where m is the mass and ω the frequency of the oscillator. We note