Quantum diffusion in biased washboard potentials: strong friction limit

Kinetic Energy of a Free Quantum Brownian Particle

We consider a paradigmatic model of a quantum Brownian particle coupled to a thermostat consisting of harmonic oscillators. In the framework of a generalized Langevin equation, the memory (damping) kernel is assumed to be in the form of exponentially-decaying oscillations. We discuss a quantum counterpart of the equipartition energy theorem for a free Brownian particle in a thermal equilibrium state. We conclude that the average kinetic energy of the Brownian particle is equal to thermally-averaged kinetic energy per one degree of freedom of oscillators of the environment, additionally averaged over all possible oscillators’ frequencies distributed according to some probability density in which details of the particle-environment interaction are present via the parameters of the damping kernel.

Diffusion and mobility of anisotropic particles in tilted periodic structures

We numerically investigated the transport of anisotropic particles in tilted periodic structures. The diffusion and mobility of the particles demonstrate distinct behaviors dependence on the shape of the particles. In two-dimensional (2D) periodic potentials, we find that the mobility is influenced a little by the anisotropy of the particle, while the diffusion increases monotonically with the increasing of the particle anisotropy for large enough biased force. However, due to the sensitivity of the channels for the particle anisotropy, the transport in smooth channels is obviously different from that in energy potentials. The mobility decreases monotonically with the increasing of the particle anisotropy, while the diffusion can be a non-monotonic function of the particle anisotropy with a peak under appropriate biased force.

Synchronization in a semiclassical Kuramoto model

Synchronization is a ubiquitous phenomenon occurring in social, biological, and technological systems when the internal rythms of their constituents are adapted to be in unison as a result of their coupling. This natural tendency towards dynamical consensus has spurred a large body of theoretical and experimental research in recent decades. The Kuramoto model constitutes the most studied and paradigmatic framework in which to study synchronization. In particular, it shows how synchronization appears as a phase transition from a dynamically disordered state at some critical value for the coupling strength between the interacting units. The critical properties of the synchronization transition of this model have been widely studied and many variants of its formulations have been considered to address different physical realizations. However, the Kuramoto model has been studied only within the domain of classical dynamics, thus neglecting its applications for the study of quantum synchronization phenomena. Based on a system-bath approach and within the Feynman path-integral formalism, we derive equations for the Kuramoto model by taking into account the first quantum fluctuations. We also analyze its critical properties, the main result being the derivation of the value for the synchronization onset. This critical coupling increases its value as quantumness increases, as a consequence of the possibility of tunneling that quantum fluctuations provide.

Quantum stochastic dynamics in the presence of a time-periodic rapidly oscillating potential: nonadiabatic escape rate

Escape from a metastable state in the presence of a high-frequency field (where the driving becomes nonadiabatic) underlies a broad range of phenomena of physics and chemistry, and thus its understanding is of paramount importance. We study the problem of intermediate-to-high-damping escape from a metastable state of a dissipative system driven by a rapidly oscillating field, one of the most important classes of nonequilibrium systems, in a broad range of field driving frequencies (ω) and amplitudes (a). We construct a Langevin equation using quantum gauge transformation in the light of Floquet theorem and exploiting a systematic perturbative expansion in powers of 1/ω using "Kapitza-Landau time window". The quantum dynamics in a high-frequency field are found to be described by an effective time-independent potential. The temperature dependence of escape rate and the change of its form with varying parameters of the field have been analyzed. It may decrease upon increasing the temperature which is contingent on the effects of intricate interplay between external modulation and dissipation. The crossover temperature between tunnelling and thermal hopping increases with an increase in external modulation so that quantum effects in the escape are relevant at higher temperatures. These observations are uncommon and counterintuitive and, therefore, of considerable interest. Our results might be valuable for the exploration of the dynamics of cold atoms in electromagnetic fields.

Electric field driven torque in ATP synthase

F_O-ATP synthase (F_O) is a rotary motor that converts potential energy from ions, usually protons, moving from high- to low-potential sides of a membrane into torque and rotary motion. Here we propose a mechanism whereby electric fields emanating from the proton entry and exit channels act on asymmetric charge distributions in the c-ring, due to protonated and deprotonated sites, and drive it to rotate. The model predicts a scaling between time-averaged torque and proton motive force, which can be hindered by mutations that adversely affect the channels. The torque created by the c-ring of F_O drives the γ-subunit to rotate within the ATP-producing complex (F₁) overcoming, with the aid of thermal fluctuations, an opposing torque that rises and falls with angular position. Using the analogy with thermal Brownian motion of a particle in a tilted washboard potential, we compute ATP production rates vs. proton motive force. The latter shows a minimum, needed to drive ATP production, which scales inversely with the number of proton binding sites on the c-ring.

Escape rate for a quantum particle moving in a time-periodic rapidly oscillating potential: a time-independent approach

We explore, in the quantum regime, the stochastic dynamics of a time-periodic, rapidly oscillating potential (having a characteristic frequency of ω) within the framework of a time-dependent system-reservoir Hamiltonian. We invoke the idea of a quantum gauge transformation in light of the standard Floquet theorem in an attempt to construct a Langevin equation (bearing a time-independent effective potential) by employing a systematic perturbative expansion in powers of ω^{-1} using the natural time-scale separation. The time-independent effective potential (corrected to ω^{-2} in leading order) that acts on the slow motion of the driven particle can be employed for trapping. We proceed further to evaluate the rate of escape of the driven particle from the metastable state in the high-temperature limit. We also envisage a resonance phenomena, a true hallmark of the system-reservoir quantization. This development would thus serve as a model template to investigate the trapping mechanism, as well as an appropriate analog to understand the dynamics of a fluctuation-induced escape process from the trap.

Quantum phase diffusion in a small underdamped Josephson junction

Quantum phase diffusion in a small underdamped Nb/AlO(x)/Nb junction (∼0.4 μm(2)) is demonstrated in a wide temperature range of 25-140 mK where macroscopic quantum tunneling (MQT) is the dominant escape mechanism. We propose a two-step transition model to describe the switching process in which the escape rate out of the potential well and the transition rate from phase diffusion to the running state are considered. The transition rate extracted from the experimental switching current distribution follows the predicted Arrhenius law in the thermal regime but is greatly enhanced when MQT becomes dominant.

Phase induced transport of a Brownian particle in a periodic potential in the presence of an external noise: A semiclassical treatment

We develop, invoking a suitable system-reservoir model, the Langevin equation with a state-dependent dissipation associated with a quantum Brownian particle submerged in a heat bath that offers a state-dependent friction to study the directed motion (by studying the phase-induced current) in the presence of an external noise. We study the phase induced current when both system and bath are subjected to external modulation by the noise and thereby expose the system to two cross-correlated noises. We also demonstrate the well-known fact that two noises remain mutually correlated if they share a common origin. We study the effects of correlation on the current in a periodic potential and envisage that the steady state current increases with increase in the extent of correlation, implying that exercising control on the degree of correlation can enhance the current in a properly designed experiment. To establish our model, we analyze numerically the effect of the external noise on system and bath separately as well as on composition of both.

Development of a semiclassical method to compute mobility and diffusion coefficient of a Brownian particle in a nonequilibrium environment

We explore the issue of a quantum-noise-induced directed transport of an overdamped Brownian particle that is allowed to move in a spatially periodic potential. The established system-reservoir model has been employed here to study the quantum-noise-induced transport of a Brownian particle in a periodic potential, where the reservoir is being modulated externally by a Gaussian-colored noise. The mobility of the Brownian particle in the linear response regime has been calculated. Then, using Einstein's relation, the analytical expression for the diffusion rate is evaluated for any arbitrary periodic potential for the high-temperature quantum regime.

Low-temperature quantum fluctuations in overdamped ratchets

At low temperatures and strong friction the time evolution of the density distribution in position follows a quantum Smoluchowski equation. Recently, also higher-order contributions of quantum fluctuations to drift and diffusion coefficients have been systematically derived. As a nontrivial situation to reveal the impact of subleading quantum corrections and to demonstrate convergence properties of the perturbation series, directed transport in ratchets is studied. It is shown that the perturbation series typically has a nonmonotonous behavior. Depending on symmetry properties higher-order contributions may even compensate current reversals induced by leading quantum fluctuations. This analysis demonstrates how to consistently treat the dynamics of overdamped quantum systems at low temperatures also in numerical applications.

Realization of a Brownian engine to study transport phenomena: a semiclassical approach

Brownian particles moving in a periodic potential with or without external load are often used as good theoretical models for the phenomenological studies of microscopic heat engines. The model that we propose here, assumes the particle to be moving in a nonequilibrium medium and we have obtained the exact expression for the stationary current density. We have restricted our consideration to the overdamped motion of the Brownian particle. We present here a self-consistent theory based on the system-reservoir coupling model, within a microscopic approach, of fluctuation induced transport in the semiclassical limit for a general system coupled with two heat baths kept at different temperatures. This essentially puts forth an approach to semiclassical state-dependent diffusion. We also explore the possibility of observing a current when the temperature of the two baths are different, and also envisage that our system may act as a Carnot engine even when the bath temperatures are the same. The condition for such a construction has been elucidated.

Quantum Smoluchowski equation: a systematic study

The strong-friction regime at low temperatures is analyzed systematically starting from the formally exact path-integral expression for the reduced dynamics. This quantum Smoluchowski regime allows for a type of semiclassical treatment in the inverse friction strength so that higher-order quantum corrections to the original quantum Smoluchowski equation [J. Ankerhold, P. Pechukas, and H. Grabert, Phys. Rev. Lett. 87, 086802 (2001); J. Ankerhold and H. Grabert, Phys. Rev. Lett. 101, 119903 (2008)] can be derived. Drift and diffusion coefficients are determined by the equilibrium distribution in position and are directly related to the corresponding action of extremal paths and fluctuations around them. It is shown that the inclusion of higher-order corrections reproduces the quantum enhancement above crossover for the decay rate out of a metastable well exactly.

Semiclassical treatment of a Brownian ratchet using the quantum Smoluchowski equation

Quantum effects in the noninertial Brownian motion of a particle in a one-dimensional ratchet 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 Wigner function in configuration space. In particular, an analytical expression for the stationary average drift velocity for constant driving forces is presented including quantum corrections to any order in Planck's constant. The corresponding frequency response is determined using continued fractions in both the linear approximation holding for small ac driving amplitude and in the nonlinear regime for arbitrary driving amplitude exhibiting pronounced ac induced frequency dependence of the dc component of the average drift velocity. Moreover, Shapiro steps are apparent in the dc characteristics for strong ac driving just as in the dc current-voltage characteristics of a point Josephson junction.

Quantum Smoluchowski equation for driven systems

We consider a driven quantum harmonic oscillator strongly coupled to a heat bath. Starting from the exact quantum Langevin equation, we use a Green's function approach to determine the corresponding semiclassical equation for the Wigner phase space distribution. In the limit of high friction and high temperature, we apply Brinkman's method to derive the quantum Smoluchowski equation for the probability distribution in position space. We further determine the range of validity of the equation and discuss the special case of a Brownian parametric oscillator.

Quantum effects in the Brownian motion of a particle in a double well potential in the overdamped limit

Quantum effects in the noninertial Brownian motion of a particle in a double well potential are treated via a semiclassical Smoluchowski equation for the time evolution of the reduced Wigner distribution function in configuration space allowing one to evaluate the position correlation function, its characteristic relaxation times, and dynamic susceptibility using matrix continued fractions and finite integral representations in the manner of the classical Smoluchowski equation treatment. Reliable approximate analytic solutions based on the exponential separation of the time scales of the fast intrawell and slow overbarrier relaxation processes are given. Moreover, the effective and the longest relaxation times of the position correlation function yield accurate predictions of both the low and high frequency relaxation behavior. The low frequency part of the dynamic susceptibility associated with the Kramers escape rate behaves as a single Lorentzian with characteristic frequency given by the quantum-mechanical reaction rate solution of the Kramers problem. As a particular example, quantum effects in the stochastic resonance are estimated.

Phase induced current in presence of nonequilibrium bath: A quantum approach

Based on a system-reservoir nonlinear coupling model, where the associated bath is externally driven by a fluctuating force, we present a microscopic approach to quantum state-dependent diffusion and multiplicative noises in terms of a quantum (Markovian) Langevin equation in overdamped limit when the associated bath is in nonequilibrium state. We then explore the possibility of observing a quantum current when the bath is modulated by white noise, the phenomena which is absent in the classical regime.

Smoluchowski equation approach for quantum Brownian motion in a tilted periodic potential

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.

Directed motion in a periodic potential of a quantum system coupled to a heat bath driven by a colored noise

A system-reservoir nonlinear coupling model is proposed for a quantum system when the associated bath is not in thermal equilibrium but is modulated by an external colored noise, to present a microscopic approach to quantum state-dependent diffusion and multiplicative noise in terms of a quantum Langevin description. Consequently, the Fokker-Planck equation in position space, valid in the overdamped limit, for multiplicative colored noise is constructed to explore the possibility of observing a quantum current and dependence of the current on various parameters of external noise is examined.