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Fortschritte der Physik

Steady‐State control of open Quantum Brownian Motion

Ilya Sinayskiy, Francesco Petruccione

Published Oct 5, 2016• 3 citations
Publisher

Abstract

Open Quantum Brownian motion describes a Quantum Brownian particle with an additional internal degree of freedom.The quantum master equation for a free Open Quantum Brownian walker with two‐dimensional internal degree of freedom interacting decoherently with the dissipative environment is derived. Using the analytical solution for the moments of the position distribution of the Brownian walker, one finds that the steady state average position of the walker can be controlled by the strength of an external classical driving field.

DOI
10.1002/prop.201600063

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Open Quantum Brownian motion describes a Quantum Brownian particle with an additional internal degree of freedom.The quantum master equation for a free Open Quantum Brownian walker with two‐dimensional internal degree of freedom interacting decoherently with the dissipative environment is derived. Using the analytical solution for the moments of the position distribution of the Brownian walker, one finds that the steady state average position of the walker can be controlled by the strength of an external classical driving field.
Open Quantum Brownian Motion (OQBM) was introduced as a scaling limit of discrete-time open quantum walks. This limit defines a new class of quantum Brownian motion, which incorporates both the external and internal degrees of freedom of the Brownian particle. We consider a weakly driven Brownian particle confined in a harmonic potential and dissipatively coupled to a thermal bath. Applying the rotating wave approximation (RWA) to the system-bath interaction Hamiltonian, we derive a completely positive Born-Markov master equation for the reduced dynamics. We express the resulting master equation in the coordinate representation and, utilizing the adiabatic elimination of fast variables, derive a completely positive hybrid quantum-classical master equation that defines OQBM. We illustrate the resulting dynamics using examples of initial Gaussian and non-Gaussian distributions of the OQBM walker. Both examples reveal the emergence of Gaussian distributions in the limiting behavior of the OQBM dynamics, which closely matches that of the standard OQBM. With the help of the obtained OQBM master equation, we derive the equations for the nnn-th moments and the cumulants of the position distribution of the open Brownian walker. We subsequently solve these equations numerically for Gaussian initial distributions across various parameter regimes. Notably, we find that the third-order cumulant is nonzero, indicating that the dynamics' intrinsic generator is non-Gaussian.
Using quantum parallelism on random walks as the original seed, we introduce new quantum stochastic processes, the open quantum Brownian motions. They describe the behaviors of quantum walkers—with internal degrees of freedom which serve as random gyroscopes—interacting with a series of probes which serve as quantum coins. These processes may also be viewed as the scaling limit of open quantum random walks and we develop this approach along three different lines: the quantum trajectory, the quantum dynamical map and the quantum stochastic differential equation. We also present a study of the simplest case, with a two level system as an internal gyroscope, illustrating the interplay between the ballistic and diffusive behaviors at work in these processes. Notation : orbital (walker) Hilbert space, in the discrete, in the continuum : internal spin (or gyroscope) Hilbert space : system Hilbert space : probe (or quantum coin) Hilbert space, : density matrix for the total system (walker + internal spin + quantum coins) : reduced density matrix on : : system density matrix in a quantum trajectory: . If diagonal and localized in position: ρ t : internal density matrix in a simple quantum trajectory X t : walker position in a simple quantum trajectory B t : normalized Brownian motion ξ t , : quantum noises
Abstract We investigate open quantum Brownian motions as quantum analogues of classical diffusion processes under interaction with an external environment. Building upon the microscopic derivation by Sinayskiy and Petruccione (2015 Phys. Scr. T165 014017), we revisit the associated master equation and study its formulation as a generalized parabolic system. Employing Fourier transform methods, we derive exact analytical solutions for one-dimensional evolutions of particles with two-level internal degree of freedom.
Open Quantum Brownian Motion (OQBM) is a new class of quantum Brownian motion in which the dynamics of the Brownian particle depend not only on interactions with a thermal environment but also on the state of its internal degrees of freedom. For an Ohmic bath spectral density with a Lorentz--Drude cutoff frequency at a high-temperature limit, we derive the Born--Markov master equation for the reduced density matrix of an open Brownian particle in a harmonic potential. The resulting master equation is written in phase-space representation using the Wigner function, and due to the separation of associated timescales in the high-damping limit, we perform adiabatic elimination of the momentum variable to obtain OQBM. We numerically solve the derived master equation for the reduced density matrix of the OQBM, using both Gaussian and non-Gaussian initial distributions. In each case, the OQBM dynamics converge to several Gaussian distributions. To gain physical insight into the studied system, we also plotted the dynamics of the off-diagonal element of the open quantum Brownian particle and found damped coherent oscillations. Finally, we investigated the time-dependent variance in the position of the OQBM walker and observed initial diffusive spreading, followed by a crossover to a finite steady-state value.