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Fluctuation and Noise Letters

QUANTUM BROWNIAN MOTION IN A PERIODIC POTENTIAL: THE PATH INTEGRAL FOR A SUPER-OHMIC BATH

Manuel Arrayás

Published Mar 1, 2012• 1 citation
Publisher

Abstract

The problem of a Brownian particle in a super-Ohmic bath under the effect of a periodic tilted potential is studied. The quantum dissipative dynamics is formulated in terms of the influence functional and the path integrals are calculated explicitly and analytically.
DOI
10.1142/s0219477512400068

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This paper concerns the investigation of the quantum motion of a system in a dissipative Ohmic heat bath in the presence of an external field using the traditional system-reservoir model. Using physically motivated initial conditions, we then obtain the c-number of the generalized quantum Langevin equation by which we calculate the quantum correction terms using a perturbation technique. As a result of this, one can apply a classical differential equation-based approach to consider quantum diffusion in a tilted periodic potential, and thus our approach is easy to use. We use our expression to calculate the Einstein relation for the quantum Brownian particle in a ratchet-type potential in a very simple closed analytical form using a suitable and convenient approximation. It is found that the diffusion rate is independent of the detailed form of the potential both in quantum and classical regimes, which is the main essence of this work.
We investigate the equilibrium properties of a quantum Brownian particle moving in a periodic potential, specifically addressing the nature of the dissipation-driven Schmid transition in the Ohmic regime. By employing World-Line Monte Carlo in the path-integral formalism and introducing a specific binary order parameter, we demonstrate that the transition belongs to the Berezinskii-Kosterlitz-Thouless universality class. This classification is substantiated through finite-size scaling analysis that reveals the characteristic logarithmic decay of the correlation functions associated with the order parameter at the critical point. Quantum phase transition turns out to be extremely fragile: it disappears in both over- and sub-Ohmic dissipation regimes. Crucially, we find that the presence of the periodic potential does not alter the localization properties in the sub-Ohmic and super-Ohmic regimes, where the system exhibits the same qualitative behavior as the free quantum Brownian particle. These findings highlight that the emergence of critical behavior is strictly governed by the low-frequency form of the environmental spectral function, which determines the long-range temporal decay of the dissipative kernel.
Abstract We introduced the formalism of path integrals to describe the dynamics of a system consisting of two particles. One is located in a periodic potential well (first subsystem), while the other particle is outside the well and subjected to a circular electromagnetic field (second subsystem). Thanks to this formalism, we are able to describe the dynamics of our system by a propagator which is the product of two subsystems of propagators. The propagator of the first subsystem is put in the standard form exp[Action] as a result of the coherent spin states. This form of propagator is possible by replacing the spin with a unit vector aligned along the polar and azimuthal directions. The second subsystem propagator come with the use of the Feynman-Vernon functional on the subsystem. Thanks to its two propagators, we have succeeded in explicitly and analytically evaluating the dynamics of our dissipative quantum system by a single propagator. Using this fundamental result, we evaluated the special case where the second subsystem bathed in a super-ohmic environment.
We study the steady state behaviour of a confined quantum Brownian particle subjected to a space-dependent, rapidly oscillating time-periodic force. To leading order in the period of driving, the result of the oscillating force is an effective static potential which has a quantum dissipative contribution, VQDV_{QD}VQD​, which adds on to the classical result. This is shown using a coherent state representation of bath oscillators. VQDV_{QD}VQD​ is evaluated exactly in the case of an Ohmic dissipation bath. It is strongest for intermediate values of the damping, where it can have pronounced effects.
Quantum corrections to the noninertial Brownian motion of a particle in a one-dimensional tilted cosine periodic potential are treated in the high-temperature and weak bath-particle coupling limit by solving a quantum Smoluchowski equation for the time evolution of the distribution function in configuration space. The theoretical predictions from two different forms of the quantum Smoluchowski equation already proposed-viz., J. Ankerhold [Phys. Rev. Lett. 87, 086802 (2001)] and W. T. Coffey [J. Phys. A 40, F91 (2007)]-are compared in detail in a particular application to the dynamics of a point Josephson junction. Various characteristics (stationary distribution, current-voltage characteristics, mean first passage time, linear ac response) are evaluated via continued fractions and finite integral representations in the manner customarily used for the classical Smoluchowski equation. The deviations from the classical behavior, discernible in the dc current-voltage characteristics as enhanced current for a given voltage and in the resonant peak in the impedance curve as an enhancement of the Q factor, are, respectively, a manifestation of relatively high-temperature nondissipative tunneling (reducing the barrier height) and dissipative tunneling (reducing the damping of the Josephson oscillations) near the top of a barrier.