Generalized Quantum Langevin Equation Research Articles

Monte Carlo study of transport in low-dimensional quantum disorder systems at finite temperature

Using the quantum generalized Langevin equation and the path integral Monte Carlo approach, we study the transport dynamics of low-dimensional quantum disorder systems at finite temperature. Motivated by the nature of the classical-to-quantum transformation in fluctuations in the time domain, we extend the treatment to the spatial domain and propose a quantum random-correlated potential, describing specifically quantum disorder. For understanding the Anderson localization from the particle transport perspective, we present an intuitive treatment using a classical analogy in which the particle moves through a flat periodic crystal lattice corrugated by classical or quantum disorder. We emphasize an effective classical disorder potential in studying the quantum effects on the transport dynamics. Compared with the classical case, we find that the quantum escape rate from a disordered metastable potential is larger. Moreover, the diffusion enhancement of a quantum system moving in a weak, biased, periodic disorder potential is more significant compared with the classical case; for an effective rock-ratcheted disorder potential, quantum effects increase the directed current with decreasing temperature. For the classical case, we explore surface diffusion on a two-dimensional biased disorder potential at finite temperature; surprisingly, the optimal angle of the external bias force is found to enhance diffusion in the biased disorder surface. Furthermore, to explain the quantum transport dynamics in a disorder potential, we adopt the barrier-crossing mechanism and the mean first passage time theory to establish the probability distribution function.

Environment-dependent vibrational heat transport in molecular junctions: Rectification, quantum effects, vibrational mismatch.

Vibrational heat transport in molecular junctions is a central issue in different contemporary research areas such as chemistry, materials science, mechanical engineering, thermoelectrics, and power generation. Our model system consists of a chain of molecules which are sandwiched between two solids that are maintained at different temperatures. We employ a quantum self-consistent reservoir model, which is built on a generalized quantum Langevin equation, to investigate quantum effects and far from equilibrium conditions on thermal conduction at nanoscale. The present self-consistent reservoir model can easily mimic the phonon-phonon scattering mechanisms. Different thermal environments are modeled as (i) Ohmic, (ii) sub-Ohmic, and (iii) super-Ohmic environments, and their effects are demonstrated for the thermal rectification properties of the system with spring graded or mass graded features. The behavior of heat current across molecular junctions as a function of chain length, temperature gradient, and phonon scattering rates are studied. Further, our analysis reveals the effects of vibrational mismatch between the solids phonon spectra on heat transfer characteristics in molecular junctions for different thermal environments.

Phononic heat transport in molecular junctions: Quantum effects and vibrational mismatch.

Problems of heat transport are ubiquitous to various technologies such as power generation, cooling, electronics, and thermoelectrics. In this paper, we advocate for the application of the quantum self-consistent reservoir method, which is based on the generalized quantum Langevin equation, to study phononic thermal conduction in molecular junctions. The method emulates phonon-phonon scattering processes while taking into account quantum effects and far-from-equilibrium (large temperature difference) conditions. We test the applicability of the method by simulating the thermal conductance of molecular junctions with one-dimensional molecules sandwiched between solid surfaces. Our results satisfy the expected behavior of the thermal conductance in anharmonic chains as a function of length, phonon scattering rate, and temperature, thus validating the computational scheme. Moreover, we examine the effects of vibrational mismatch between the solids' phonon spectra on the heat transfer characteristics in molecular junctions. Here, we reveal the dual role of vibrational anharmonicity: It raises the resistance of the junction due to multiple scattering processes, yet it promotes energy transport across a vibrational mismatch by enabling phonon recombination and decay processes.

C-number quantum generalized Langevin equation for an open system

We derive a $c-$number Generalised Langevin Equation (GLE) describing the evolution of the expectation values $\left\langle x_{i}\right\rangle_{t}$ of the atomic position operators $x_{i}$ of an open system. The latter is coupled linearly to a harmonic bath kept at a fixed temperature. The equations of motion contain a non-Markovian friction term with the classical kernel {[}L. Kantorovich - PRB 78, 094304 (2008){]} and a zero mean \emph{non-Gaussian} random force with correlation functions that depend on the initial preparation of the open system. We used a density operator formalism without assuming that initially the combined system was decoupled. The only approximation made in deriving quantum GLE consists in assuming that the Hamiltonian of the open system at time $t$ can be expanded up to the second order with respect to operators of atomic displacements $u_{i}=x_{i}-\left\langle x_{i}\right\rangle_{t}$ in the open system around their exact atomic positions $\left\langle x_{i}\right\rangle_{t}$ (the "harmonisation" approximation). The noise is introduced to ensure that sampling many quantum GLE trajectories yields exactly the average one. An explicit expression for the pair correlation function of the noise, consistent with the classical limit, is also proposed. Unlike the usually considered quantum operator GLE, the proposed $c-$number quantum GLE can be used in direct molecular dynamic simulations of open systems under general equilibrium or non-equilibrium conditions.

Modeling the quasistatic energy transport between nanoparticles.

We consider phononic energy transport between nanoparticles mediated by a quantum particle. The nanoparticles are considered as thermal reservoirs described by ensembles of finite numbers of harmonic oscillators within the Drude-Ullersma model having, in general, unequal mode spacings Δ(1) and Δ(2), which amount to different numbers of atoms in the nanoparticles. The quasistatic energy transport between the nanoparticles on the time scale t∼1/Δ(1,2) is investigated using the generalized quantum Langevin equation. We find that double degeneracy of system's eigenfrequencies, which occurs in the case of identical nanoparticles, is removed when the mode spacings become unequal. The equations describing the dynamics of the averaged eigenmode energies are derived and solved, and the resulting expression for the energy current between the nanoparticles is obtained and explored. Unlike the case when the thermodynamic limit is assumed resulting in time-independent energy current, finite-size effects result in temporal behavior of the energy current that evinces reversibility features combined with decay and possesses peculiarities at time moments t=2πn/Δ(1)+2πm/Δ(2) for non-negative integers n and m. When Δ(1,2)→0, an expression for the heat current obtained previously under assumption of the thermodynamic limit is reproduced. The energy current between two platinum nanoparticles mediated by a carbon oxide molecule is considered as an application of the developed model.

Quasi-Static Energy Transport between Nanoparticles

ABSTRACTWe consider energy transfer between non-equal nanoparticles mediated by a quantum system. The nanoparticles are considered as thermal reservoirs described as ensembles of finite numbers of harmonic oscillators within the Drude-Ullersma model having mode spacings Δ1 and Δ2. Our approach is based on the generalized quantum Langevin equation. The quasi-static energy transport between the thermal reservoirs is investigated. As is shown, the double degeneracy of the mode frequencies, which occurred in the previously considered case when Δ1 = Δ2, is removed in the present case of non-equal mode spacings. Equations describing long-time (t ∼1/Δ1,2) relaxation for the mode temperatures (or the ensemble averaged mode energies) are solved and the resulting expression for the total energy current between the nanoparticles is derived and explored.

Gaussian entanglement induced by an extended thermal environment

We study stationary entanglement among three harmonic oscillators which are dipole coupled to a one-dimensional or a three-dimensional bosonic environment. The analysis of the open-system dynamics is performed with generalized quantum Langevin equations which we solve exactly in Fourier representation. The focus lies on Gaussian bipartite and tripartite entanglement induced by the highly non-Markovian interaction mediated by the environment. This environment-induced interaction represents an effective many-parties interaction with a spatial long-range feature: a main finding is that the presence of a passive oscillator is detrimental for the stationary two-mode entanglement. Furthermore, our results strongly indicate that the environment-induced entanglement mechanism corresponds to uncontrolled feedback which is predominantly coherent at low temperatures and for moderate oscillator-environment coupling as compared to the oscillator frequency.

Size effects in long-term quasistatic heat transport

We consider finite-size effects on heat transfer between thermal reservoirs mediated by a quantum system, where the number of modes in each reservoir is finite. Our approach is based on the generalized quantum Langevin equation and the thermal reservoirs are described as ensembles of oscillators within the Drude-Ullersma model. A general expression for the heat current between the thermal reservoirs in the long-time quasistatic regime, when an observation time is of the order of Δ(-1) and Δ is the mode spacing constant of a thermal reservoir, is obtained. The resulting equations that govern the long-time relaxation for the mode temperatures and the average temperatures of the reservoirs are derived and approximate analytical solutions are found. The obtained time dependencies of the temperatures and the resulting heat current reveal peculiarities at t=2πm/Δ with non-negative integers m and the heat current vanishes nonmonotonically when t→∞. The validity of Fourier's law for a chain of finite-size macroscopic subsystems is considered. As is shown, for characteristic times of the order of Δ(-1) the temperatures of subsystems' modes deviate from each other and the validity of Fourier's law cannot be established. In a case when deviations of initial temperatures of the subsystems from their average value are small, t→∞ asymptotic values for the mode temperatures do not depend on a mode's number and are the same as if Fourier's law were valid for all times.

Heat exchange mediated by a quantum system

We consider heat transfer between two thermal reservoirs mediated by a quantum system using the generalized quantum Langevin equation. The thermal reservoirs are treated as ensembles of oscillators within the framework of the Drude-Ullersma model. General expressions for the heat current and thermal conductance are obtained for arbitrary coupling strength between the reservoirs and the mediator and for different temperature regimes. As an application of these results we discuss the origin of Fourier's law in a chain of large but finite subsystems coupled to each other by the quantum mediators. We also address a question of anomalously large heat current between the scanning tunneling microscope (STM) tip and substrate found in a recent experiment. The question of minimum thermal conductivity is revisited in the framework of scaling theory as a potential application of the developed approach.

An approach towards a simple quantum Langevin equation

We offer a simple way to include energy dissipation in small spatial–dimensional quantum dynamics simulations of a primary quantum system coupled to its surroundings. We show how a quantum Langevin equation can be generated by including a friction operator, F ˆ , in the primary quantum system’s Hamiltonian operator. We demonstrate this approach’s computational ease and flexibility through numerical results on both harmonic and anharmonic primary quantum systems. Also, we show how the present approach can be easily generalized to non-Markovian, ‘memory’ friction. Thus, the approach presented in this work can be used to generate a quantum generalized Langevin equation.

Generalized Einstein Relation in Tilted Periodic Potential: A Semiclassical Approach

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.

Investigation of Noise-Induced Escape Rate: A Quantum Mechanical Approach

A quantum system coupled to a heat-bath in non-equilibrium environment is considered to study the problem of noise-induced escape rate from a metastable state in the moderate to strong friction limit (Kramers’ regime). It is known that starting from an initial coherent state representation of bath oscillators, one can derive a c-number generalized quantum Langevin equation where the quantum correction terms appear as a coupled infinite set of hierarchy of equations. For practical purpose, one should truncate these equations after a certain order. In our present development, we calculate the quantum correction terms in a closed analytical form based on a systematic perturbation technique and then derive the lowest order quantum correction factor exactly in the case of an Ohmic dissipative bath. Finally, to demonstrate its applicability, the effective equation of motions has been used to study the barrier crossing dynamics which incorporates the quantum correction factors.

Coupled electron–phonon transport from molecular dynamics with quantum baths

Based on generalized quantum Langevin equations for the tight-binding wavefunctionamplitudes and lattice displacements, electron and phonon quantum transport areobtained exactly using molecular dynamics (MD) in the ballistic regime. Theelectron–phonon interactions can be handled with a quasi-classical approximation. Bothcharge and energy transport and their interplay can be studied. We compare theMD results with those of a fully quantum mechanical nonequilibrium Green’sfunction (NEGF) approach for the electron currents. We find a ballistic to diffusivetransition of the electron conduction in one-dimensional chains as the chain lengthincreases.

The dynamical resonance of a harmonic oscillator coupled to a heat bath with harmonic velocity noise and harmonic noise

For a harmonic oscillator coupled to a heat bath with harmonic velocity noise and harmonic noise, we obtain the exact average energy by resoving the corresponding generalized quantum Langevin equation. Due to the different power spectra of the two kinds of noise, the harmonic oscillator driven by external harmonic force exhibits different resonant features. This is an approach for examing the characterization of internal noise.

Fundamental aspects of quantum Brownian motion

With this work we elaborate on the physics of quantum noise in thermal equilibrium and in stationary nonequilibrium. Starting out from the celebrated quantum fluctuation-dissipation theorem we discuss some important consequences that must hold for open, dissipative quantum systems in thermal equilibrium. The issue of quantum dissipation is exemplified with the fundamental problem of a damped harmonic quantum oscillator. The role of quantum fluctuations is discussed in the context of both, the nonlinear generalized quantum Langevin equation and the path integral approach. We discuss the consequences of the time-reversal symmetry for an open dissipative quantum dynamics and, furthermore, point to a series of subtleties and possible pitfalls. The path integral methodology is applied to the decay of metastable states assisted by quantum Brownian noise.

Solution of quantum Langevin equation: approximations, theoretical and numerical aspects.

Based on a coherent state representation of noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillators, a generalized quantum Langevin equation has been recently developed [Phys. Rev. E 65, 021109 (2002); 66, 051106 (2002)] to derive the equations of motion for probability distribution functions in c-number phase-space. We extend the treatment to explore several systematic approximation schemes for the solutions of the Langevin equation for nonlinear potentials for a wide range of noise correlation, strength and temperature down to the vacuum limit. The method is exemplified by an analytic application to harmonic oscillator for arbitrary memory kernel and with the help of a numerical calculation of barrier crossing, in a cubic potential to demonstrate the quantum Kramers' turnover and the quantum Arrhenius plot.

Decoherence in nanostructures and quantum systems

Decoherence phenomena are pervasive in the arena of nanostructures but perhaps even more so in the study of the fundamentals of quantum mechanics and quantum computation. Since there has been little overlap between the studies in both arenas, this is an attempt to bridge the gap. Topics stressed include (a) wave packet spreading in a dissipative environment, a key element in all arenas, (b) the definition of a quantitative measure of decoherence, (c) the near zero and zero temperature limit, and (d) the key role played by initial conditions: system and environment entangled at all times so that one must use the density matrix (or Wigner distribution) for the complete system or initially decoupled system and environment so that use of a reduced density matrix or reduced Wigner distribution is feasible. Our approach utilizes generalized quantum Langevin equations and Wigner distributions.

Direct experimental access to microscopic dynamics in liquid hydrogen.

We have obtained the double-differential incoherent neutron scattering cross section of liquid and solid parahydrogen in various thermodynamic conditions using TOSCA, a time-of-flight, inverse geometry, crystal analyzer spectrometer, operating at the pulsed neutron source ISIS. The measured cross section provides direct experimental access to the self part of the center-of-mass inelastic structure factor of the parahydrogen molecules in the system. Data have been corrected for the experimental effects and then analyzed in the framework of the Young-Koppel model and the Gaussian approximation. The velocity autocorrelation functions and their energy spectra have been obtained from a fitting procedure, making use of the quantum generalized Langevin equation and of model memory functions, and finally compared to the most recent results of both molecular centroid dynamics and self-consistent quantum mode-coupling theory. Some dynamic quantities were also related to simple equilibrium properties and simulated through a standard path integral Monte Carlo code. Results are very interesting but still urge for further developments of theoretical and dynamic simulation approaches, as well as for more extensive experimental efforts.

Generalized quantum Fokker-Planck, diffusion, and Smoluchowski equations with true probability distribution functions

Traditionally, quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasiprobability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using true probability distribution functions is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their coordinates and momenta, we derive a generalized quantum Langevin equation in c numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion, and Smoluchowski equations are the exact quantum analogs of their classical counterparts. The present work is independent of path integral techniques. The theory as developed here is a natural extension of its classical version and is valid for arbitrary temperature and friction (the Smoluchowski equation being considered in the overdamped limit).

A self-consistent mode-coupling theory for dynamical correlations in quantum liquids: Rigorous formulation

A quantum molecular hydrodynamic formalism is developed for the study of dynamical correlations in dense quantum liquids. The approach is based on augmenting an exact closed, self-consistent quantum generalized Langevin equation for the Kubo transform of the dynamical correlation of interest, with a suitable approximation for the memory kernel obtained within the framework of a quantum mode-coupling theory. The solution to the quantum generalized Langevin equation requires as input static equilibrium information which is generated from a path-integral Monte Carlo method. Examples are given for the intermediate and self-intermediate scattering functions, and for the velocity autocorrelation function. The attractive advantages of the present approach are discussed.