Caldeira-Leggett Master Equation Research Articles

Transverse quantum decoherence of a fast particle in a gas

The decoherence of a fast quantum particle in a gas is studied by applying the Kramers-Moyal expansion to the quantum master equation for the reduced density matrix of the particle. This expansion leads to a general form of the Caldeira-Leggett master equation accounting for the angular variation of the differential cross section. The equation describes the decoherence in both the longitudinal and transverse directions with respect to the particle motion. It is shown that, when the differential cross section is concentrated in the forward direction, transverse decoherence dominates. The coherence region off the diagonal of the density matrix is characterized by coherence lengths, which can be deduced, for Gaussian states, from the momentum covariance matrix according to a Heisenberg-type uncertainty relation. Finally, the longitudinal-to-transverse ratio of the coherence lengths is estimated for an $\ensuremath{\alpha}$ particle of a few MeVs. This ratio indicates that the coherence region looks like an ellipsoid elongated in the direction of motion.

On some unexplored decoherence aspects in the Caldeira–Leggett formalism: arrival time distributions, identical particles and diffraction in time

Some unexplored decoherence aspects within the Caldeira-Leggett master equation are analyzed and discussed. The decoherence process is controlled by the two environment parameters, the relaxation rate or friction and the temperature, leading to a gradual transition from the quantum to classical regime. Arrival time distributions, nonminimum-uncertainty-product or stretching Gaussian wave packets, identical particles and diffraction in time display interesting features during the decoherence process undergone by the time dependent interference patterns. We show that the presence of a constant force field does not affect the decoherence, {\it positive} values of the stretching parameter reduces the rate of decoherence, the symmetry of the wave function for identical particles plays no role when open dynamics are considered; and diffraction in time and space is gradually washed out by increasing the temperature and/or relaxation rate in the zero dissipation limit within the so-called quantum shutter problem.

Quantum statistical mechanics, quantum probability and quantum characteristic function

The theory of quantum statistical mechanics is reviewed in the light of the quantum probability theory developed in recent years by mathematicians. We show that—which is our main motivation here—the former is a special model in the framework of the latter, by extending the characteristic function in the classical probability theory to the quantum probability theory based on the requirements of quantum statistical mechanics. We find that due to the fact that fundamental dynamical variables of a quantum system must respect certain commutation relations, one has to reform the classical characteristic function defined on a Euclidean space to a normalized, nonnegative definite function defined on a group which is generated by a Lie algebra formed from these commutation relations. Our key finding is that the group-theoretical characteristic function could be adopted to replace the density matrix commonly used in quantum statistical mechanics for representing a state of the quantum ensemble, and more than that, the new representation has some significant advantages in applications. To show its advantages compared with the conventional density matrix scenario, we illustrate its applications in: solving the master equation that describes the irreversible evolution of a quantum open system; determining the pointer states of a quantum open system; and expanding the density matrix with the pointer states. In particular, thanks to the quantum characteristic function, we are able to write down the solutions of the quantum-optical master equation explicitly, obtain its steady solutions and verify that the pointer states are exactly coherent states. Besides, the Caldeira-Leggett master equation for the quantum Brownian motion and its extension to the Lindblad type are also studied.

Dynamical stability of the weakly nonharmonic propeller-shaped planar Brownian rotator.

Dynamical stability is a prerequisite for control and functioning of desired nanomachines. We utilize the Caldeira-Leggett master equation to investigate dynamical stability of molecular cogwheels modeled as a rigid, propeller-shaped planar rotator. To match certain expected realistic physical situations, we consider a weakly nonharmonic external potential for the rotator. Two methods for investigating stability are used. First, we employ a quantum-mechanical counterpart of the so-called "first passage time" method. Second, we investigate time dependence of the standard deviation of the rotator for both the angle and angular momentum quantum observables. A perturbationlike procedure is introduced and implemented to provide the closed set of differential equations for the moments. Extensive analysis is performed for different combinations of the values of system parameters. The two methods are, in a sense, mutually complementary. Appropriate for the short time behavior, the first passage time exhibits a numerically relevant dependence only on the damping factor as well as on the rotator size. However, the standard deviations for both the angle and angular momentum observables exhibit strong dependence on the parameter values for both short and long time intervals. Contrary to our expectations, the time decrease of the standard deviations is found for certain parameter regimes. In addition, for certain parameter regimes nonmonotonic dependence on the rotator size is observed for the standard deviations and for the damping of the oscillation amplitude. Hence, nonfulfillment of the classical expectation that the size of the rotator can be reduced to the inertia of the rotator. In effect, the task of designing the desired protocols for the proper control of the molecular rotations becomes an optimization problem that requires further technical elaboration.

Non-markovian quantum brownian motion.

We develop a non-Markovian extension to the caldeira-leggett master equation for quantum Brownian motion by using the timescale of the bath as a small parameter around which to perform a perturbative expansion. This framework allows us to deal directly with the time non-locality of the non-Markovian regime. We derive an exact description of the evolution of the reduced density matrix and obtain a new quantum master equa- tion with non-Markovian corrections added perturbatively to the standard Lindblad form. The resulting simulations show new dynamics and a strong deviation from Markovian behaviour. We observe that quantum coherences survive far longer than in the Marko- vian limit. In particular, when the system is fully non-Markovian, quantum phenomena experience a form of decoherence that behaves functionally like an inverse-power law. These results may have important implications in quantum technologies and biology as they suggest environments can be engineered to support long-lived quantum coherence.

Open system dynamics using Gaussian-based multiconfigurational time-dependent Hartree wavefunctions: Application to environment-modulated tunneling.

A variational approach for the quantum dynamics of statistical mixtures is developed, which is based upon the representation of the natural states of the mixture in terms of hybrid Gaussian-based Multiconfiguration Time-Dependent Hartree (G-MCTDH) wavefunctions. The method, termed ρG-MCTDH, is combined with a treatment of dissipation and decoherence based on the nonstochastic open-system Schrödinger equations. The performance and the convergence properties of the approach are illustrated for a two-dimensional tunneling system, where the primary tunneling coordinate, represented by flexible single-particle functions, is resonantly coupled to a second harmonic mode, represented by Gaussian wave packets. The harmonic coordinate is coupled to the environment and two different processes are studied: (i) vibrational relaxation at zero temperature described by a master equation in the Lindblad form and (ii) thermalization induced by the Caldeira-Leggett master equation. In the second case, the evolution from a quantum tunneling regime to a quasistationary classical-limit distribution, driven by the heat bath, is visualized using a flux analysis.

Erratum to: Positivity violations of the density operator in the Caldeira-Leggett master equation

This erratum corrects three mistakes. The first is a typo, the second one is an unclear statement about the states used in the numerical simulations, and the third is a misused reference with respect to an equation.

Positivity violations of the density operator in the Caldeira-Leggett master equation

The Caldeira-Leggett master equation as an example of Markovian master equation without Lindblad form is investigated for mathematical consistency. We explore situations both analytically and numerically where the positivity violations of the density operator occur. We reinforce some known knowledge about this problem but also find new surprising cases. Our analytical results are based on the full solution of the Caldeira-Leggett master equation obtained via the method of characteristics. The preservation of positivity is mainly investigated with the help of the density operator's purity and we give also some numerical results about the violation of the Robertson-Schr\"odinger uncertainty relation.

An entropy production based method for determining the position diffusion’s coefficient of a quantum Brownian motion

Quantum Brownian motion of a harmonic oscillator in the Markovian approximation is described by the respective Caldeira-Leggett master equation. This master equation can be brought into Lindblad form by adding a position diffusion term to it. The coefficient of this term is either customarily taken to be the lower bound dictated by the Dekker inequality or determined by more detailed derivations on the linearly damped quantum harmonic oscillator. In this paper, we explore the theoretical possibilities of determining the position diffusion term's coefficient by analyzing the entropy production of the master equation.

Minimal dissipation model for bipartite quantum systems at finite temperature

We consider the reduced dynamics in a bipartite quantum system (consisting of a central system and an intermediate environment) coupled to a heat bath at finite temperature. To describe this situation, in the simplest possible -- yet physically meaningful way, we introduce the "depolarizing heat bath" as a new minimal dissipation model. We conjecture that at sufficiently strong dissipation, any other dissipation model implemented in the form of a Markovian quantum master equation will yield the same reduced dynamics of the central system, as the minimal model. To support this conjecture, we study a two-level system coupled to an oscillator mode. For the coupling between the two parts, we consider the Jaynes-Cummings or a dephasing coupling, while the coupling to the heat bath is modeled by the quantum optical or the Caldeira-Leggett master equation (neglecting any direct coupling between central system and heat bath). We then provide ample numerical evidence, for both, model-independence and accuracy of the depolarizing heat bath model. Alongside with our study, we investigate different regimes, where the strong coupling condition leads to coherence and/or population stabilization.

Dynamical stability of the one-dimensional rigid Brownian rotator: the role of the rotator’s spatial size and shape

We investigate dynamical stability of a single propeller-like shaped molecular cogwheel modelled as the fixed-axis rigid rotator. In the realistic situations, rotation of the finite-size cogwheel is subject to the environmentally-induced Brownian-motion effect that we describe by utilizing the quantum Caldeira-Leggett master equation. Assuming the initially narrow (classical-like) standard deviations for the angle and the angular momentum of the rotator, we investigate the dynamics of the first and second moments depending on the size, i.e. on the number of blades of both the free rotator as well as of the rotator in the external harmonic field. The larger the standard deviations, the less stable (i.e. less predictable) rotation. We detect the absence of the simple and straightforward rules for utilizing the rotator’s stability. Instead, a number of the size-related criteria appear whose combinations may provide the optimal rules for the rotator dynamical stability and possibly control. In the realistic situations, the quantum-mechanical corrections, albeit individually small, may effectively prove non-negligible, and also revealing subtlety of the transition from the quantum to the classical dynamics of the rotator. As to the latter, we detect a strong size-dependence of the transition to the classical dynamics beyond the quantum decoherence process.

Quantum kicked harmonic oscillator in contact with a heat bath

We consider the quantum harmonic oscillator in contact with a finite temperature bath, modelled by the Caldeira-Leggett master equation. Applying periodic kicks to the oscillator, we study the system in different dynamical regimes between classical integrability and chaos on the one hand, and ballistic or diffusive energy absorption on the other. We then investigate the influence of the heat bath on the oscillator in each case. Phase space techniques allow us to simulate the evolution of the system efficiently. In this way, we calculate high resolution Wigner functions at long times, where the system approaches a quasi-stationary cyclic evolution. Thereby, we are able to perform an accurate study of the thermodynamic properties of a non-integrable, quantum chaotic system in contact with a heat bath.

Dissipaton Equation of Motion with Controlled Truncation

This work aims at a priori accuracy controlled truncation to the dissipaton equation of motion formalism for non-perturbative quantum dissipative dynamics. A new truncation scheme is proposed by adopting the Markovian and high-temperature approximation similar to the Caldeira-Leggett master equation made at the termination level. An accuracy criterion to determine the truncation level is put forward via a Markovianicity analysis. Performances of both the new truncation scheme and the control criterion are illustrated via dynamics simulation of electron transfer systems.

Non-equilibrium effects upon the non-Markovian Caldeira–Leggett quantum master equation

We obtain a non-Markovian quantum master equation directly from the quantization of a non-Markovian Fokker–Planck equation describing the Brownian motion of a particle immersed in a generic environment (e.g. a non-thermal fluid). As far as the especial case of a heat bath comprising of quantum harmonic oscillators is concerned, we derive a non-Markovian Caldeira–Leggett master equation on the basis of which we work out the concept of non-equilibrium quantum thermal force exerted by the harmonic heat bath upon the Brownian motion of a free particle. The classical limit (or dequantization process) of this sort of non-equilibrium quantum effect is scrutinized, as well.

Dissipative dynamics from a stochastic perspective

The stochastic description of dissipative dynamics for both quantum and classical systems is revisited by using Itô calculus. A procedure to work out the master equation for linear systems based on this scheme is outlined. For the Caldeira–Leggett model it is shown that the master equation describing pure dephasing is Markovian-like and the solution is given. The Caldeira–Leggett master equation for Ohmic dissipation, which is valid at high temperatures, is derived.

Semiclassical non-Markovian Brownian motion in anharmonic potentials

The combination of an exact stochastic decomposition of non-Markovian dissipative quantum dynamics with the semiclassical initial value formalism is applied to Brownian motion in a Morse potential. The unified sampling of the stochastic noise and the semiclassical phase space distribution introduced in Koch et al. [W. Koch, F. Grossmann, J.T. Stockburger, J. Ankerhold, Non-Markovian semiclassical dynamics, Phys. Rev. Lett. 100 (2008) 230402] is laid out here in detail. By comparing our numerical results to those obtained by using the Caldeira–Leggett master equation, we show that even in the challenging regime of moderate friction and at low temperatures, where reservoir fluctuations are clearly non-Markovian, this approach allows for the accurate description of dissipative dynamics over many oscillation periods until thermalization is reached.

Effective-mode representation of non-Markovian dynamics: A hierarchical approximation of the spectral density. II. Application to environment-induced nonadiabatic dynamics

The non-Markovian approach developed in the companion paper [Hughes et al., J. Chem. Phys. 131, 024109 (2009)], which employs a hierarchical series of approximate spectral densities, is extended to the treatment of nonadiabatic dynamics of coupled electronic states. We focus on a spin-boson-type Hamiltonian including a subset of system vibrational modes which are treated without any approximation, while a set of bath modes is transformed to a chain of effective modes and treated in a reduced-dimensional space. Only the first member of the chain is coupled to the electronic subsystem. The chain construction can be truncated at successive orders and is terminated by a Markovian closure acting on the end of the chain. From this Mori-type construction, a hierarchy of approximate spectral densities is obtained which approach the true bath spectral density with increasing accuracy. Applications are presented for the dynamics of a vibronic subsystem comprising a high-frequency mode and interacting with a low-frequency bath. The bath is shown to have a striking effect on the nonadiabatic dynamics, which can be rationalized in the effective-mode picture. A reduced two-dimensional subspace is constructed which accounts for the essential features of the nonadiabatic process induced by the effective environmental mode. Electronic coherence is found to be preserved on the shortest time scale determined by the effective mode, while decoherence sets in on a longer time scale. Numerical simulations are carried out using either an explicit wave function representation of the system and overall bath or else an explicit representation of the system and effective-mode part in conjunction with a Caldeira-Leggett master equation.

A finite-difference implementation of the Caldeira–Leggett master equation

The Caldeira-Leggett master equation for dissipative quantum dynamics has predominantly been implemented in phase space, where it plays the role of a quantum Fokker-Planck equation. Here we demonstrate the feasibility of a pure coordinate space implementation in the case of intermediate damping strength and temperature, for times long enough to observe thermal equilibration. After a thorough numerical investigation of the analytically solvable harmonic oscillator case, a Morse oscillator model is studied. This is a case in which a quantal phase space approach would be difficult to implement and we show how analytical results for thermal averages are reproduced to a surprisingly high degree.

A density tensor hierarchy for open system dynamics: retrieving the noise

We develop a density tensor hierarchy for open system dynamics that recovers information about fluctuations (or ‘noise’) lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of products of pure state density matrix elements and can be compactly summarized by a simple generating function. For the case of quantum fluctuations arising when a quantum system interacts with a quantum environment in an overall pure state, the corresponding hierarchy is defined as the environmental trace of products of system matrix elements of the full density matrix. Whereas all members of the classical noise hierarchy are system observables, only the lowest member of the quantum noise hierarchy is directly experimentally measurable. The unit trace and idempotence properties of the pure state density matrix imply descent relations for the tensor hierarchies, that relate the order n tensor, under contraction of appropriate pairs of tensor indices, to the order n − 1 tensor. As examples to illustrate the classical probability distribution formalism, we consider a spatially isotropic ensemble of spin-1/2 pure states, a quantum system evolving by an Itô stochastic Schrödinger equation and a quantum system evolving by a jump process Schrödinger equation. As examples to illustrate the corresponding trace formalism in the quantum fluctuation case, we consider the tensor hierarchies for collisional Brownian motion of an infinite mass Brownian particle and for the weak coupling Born–Markov master equation. In different specializations, the latter gives the hierarchies generalizing the quantum optical master equation and the Caldeira–Leggett master equation. As a further application of the density tensor, we contrast stochastic Schrödinger equations that reduce and that do not reduce the state vector, and discuss why a quantum system coupled to a quantum environment behaves like the latter. The descent relations for our various examples are checked in a series of appendices.

Relaxation dynamics of a quantum Brownian particle in an ideal gas

We show how the quantum analog of the Fokker-Planck equation for describing Brownian motion can be obtained as the diffusive limit of the quantum linear Boltzmann equation. The latter describes the quantum dynamics of a tracer particle in a dilute, ideal gas by means of a translation-covariant master equation. We discuss the type of approximations required to obtain the generalized form of the Caldeira-Leggett master equation, along with their physical justification. Microscopic expressions for the diffusion and relaxation coefficients are obtained by analyzing the limiting form of the equation in both the Schrodinger and the Heisenberg picture.