Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise

Transport coefficients of a quantum system interacting with a squeezed heat bath

The analytical expressions for the time-dependent friction and diffusion coefficients are presented for the case of coupling in coordinates between the collective subsystem and a squeezed heat bath. The effects of initial phase-sensitive and -insensitive correlations of the heat bath on the diffusion coefficients, fluctuations, and decoherence are studied. The interplay between friction and decoherence is discussed.

Classical limit of master equation for a harmonic oscillator coupled to an oscillator bath with separable initial conditions

The Wigner transform of the master equation describing the reduced dynamics of the system, of a harmonic oscillator coupled to an oscillator bath, was obtained by Karrlein and Grabert [Phys. Rev. E 55, 153 (1997)]. It was shown that for some special correlated initial conditions the master equation reduces, in the classical limit, to the corresponding classical Fokker-Planck equation obtained by Adelman [J. Chem Phys. 64, 124 (1976)]. However, for separable initial conditions the Adelman equations were not recovered. We resolve this problem by showing that, for separable initial conditions, the classical Langevin equations are somewhat different from the one considered by Adelman. We obtain the corresponding Fokker-Planck equation and show that they exactly match the classical limit of the evolution of the Wigner function obtained from the master equation for separable initial conditions. We also discuss why thermal initial conditions correspond to Adelman's solution.

Decoherence, tunneling, and noise-induced activation in a double-potential well at high and zero temperature

We study the effects of the environment on tunneling in an open system described by a static double-well potential. We describe the evolution of a quantum state localized in one of the minima of the potential at t = 0, in both the limits of high and zero environment temperature. We show that the evolution of the system can be summarized in terms of three main physical phenomena--namely, decoherence, quantum tunneling, and noise-induced activation--and we obtain analytical estimates for the corresponding time scales. These analytical predictions are confirmed by large-scale numerical simulations, providing a detailed picture of the main stages of the evolution and of the relevant dynamical processes.

From measurements to quantum friction

We present a quantum theory of friction in which interactions with the surrounding medium are described by generalized measurements of the particle's position and momentum. The theory predicts intrinsically quantum contributions to the particle's steady-state energy and to the associated diffusion in position. We discuss the physical significance of these and demonstrate their significance in ensuring a well behaved theory.

Qubit assisted probing of coherence between mesoscopic states of an apparatus

I present a general scheme through which the evidence of a superposition involving distinct states of a mesoscopic system can be probed. The scheme relies on a single qubit being coupled to a mesoscopic harmonic oscillator in such a way that it can be used to both prepare and probe a superposition of states of the oscillator. Two potentially realizable implementations, one with a single flux qubit coupled to an LC circuit, and the other with a single ion-trap qubit coupled to the collective motion of several ions, are proposed.

Exact Markovian kinetic equation for a quantum Brownian oscillator

We derive an exact Markovian kinetic equation for an oscillator linearly coupled to a heat bath, describing quantum Brownian motion. Our work is based on the subdynamics formulation developed by Prigogine and collaborators. The space of distribution functions is decomposed into independent subspaces that remain invariant under Liouville dynamics. For integrable systems in Poincaré's sense the invariant subspaces follow the dynamics of uncoupled, renormalized particles. In contrast, for nonintegrable systems, the invariant subspaces follow a dynamics with broken time symmetry, involving generalized functions. This result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces. We comment on the relation between our Markovian kinetic equation and the Hu-Paz-Zhang equation.

Electronic decoherence time for non-Born-Oppenheimer trajectories

An expression is obtained for the electronic decoherence time of the reduced density electronic matrix in mixed quantum-classical molecular-dynamics simulations. The result is obtained by assuming that decoherence is dominated by the time dependence of the overlap of minimum-uncertainty packets and then maximizing the rate with respect to the parameters of the wave packets. The expression for the decay time involves quantities readily available in non-Born-Oppenheimer molecular-dynamics simulations, and it is shown to have a reasonable form when compared with two other formulas for the decay time that have been previously proposed.

QUANTUM MECHANICS OF DISSIPATIVE SYSTEMS

Quantum dissipation involves both energy relaxation and decoherence, leading toward quantum thermal equilibrium. There are several theoretical prescriptions of quantum dissipation but none of them is simple enough to be treated exactly in real applications. As a result, formulations in different prescriptions are practically used with different approximation schemes. This review examines both theoretical and application aspects on various perturbative formulations, especially those that are exact up to second-order but nonequivalent in high-order system-bath coupling contributions. Discrimination is made in favor of an unconventional formulation that in a sense combines the merits of both the conventional time-local and memory-kernel prescriptions, where the latter is least favorite in terms of the applicability range of parameters for system-bath coupling, non-Markovian, and temperature. Also highlighted is the importance of correlated driving and dissipation effects, not only on the dynamics under strong external field driving, but also in the calculation of field-free correlation and response functions.

Decoherence effects in reactive scattering

Decoherence effects on quantum and classical dynamics in reactive scattering are examined using a Caldeira-Leggett type model. Through a study of the dynamics of the collinear H + H2 reaction and the transmission over simple one-dimensional barrier potentials, we show that decoherence leads to improved agreement between quantum and classical reactions and transmission probabilities, primarily by increasing the energy dispersion in a well-defined way. Increased potential nonlinearity is seen to require larger decoherence in order to attain comparable quantum-classical agreement.

Strongly coupled matter field and nonanalytic decay rate of dipole molecules in a waveguide

The decay rate gamma of an excited dipole molecule inside a waveguide is evaluated for the strongly coupled matter-field case near a cutoff frequency omegac without using perturbation analysis. Because of the singularity in the density of photon states at the cutoff frequency, we find that gamma depends nonanalytically on the coupling constant g as g4/3, which leads to a vast increase in the decay rate.

Exact quantum master equation via the calculus on path integrals

An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems.

Non-Markovian dynamics of quantum systems. I. Formalism and transport coefficients

Generalized Langevin equations and fluctuation-dissipation relations are derived for the case of a nonlinear non-Markovian noise. The explicit expressions for the time-dependent friction and diffusion coefficients are presented for the case of general and linear couplings in the coordinate and momentum between the collective harmonic oscillator and heat bath. The long-time tails of correlation functions are investigated in the low- and high-temperature regimes of dissipation for different couplings. The Onsager's regression hypothesis is discussed for the non-Markovian dynamics. The Lindblad theory is justified on the basis of the microscopical model.

Towards quantum entanglement in nanoelectromechanical devices

We study arrays of mechanical oscillators in the quantum domain and demonstrate how the motions of distant oscillators can be entangled without the need for control of individual oscillators and without a direct interaction between them. These oscillators are thought of as being members of an array of nanoelectromechanical resonators with a voltage being applicable between neighboring resonators. Sudden nonadiabatic switching of the interaction results in a squeezing of the states of the mechanical oscillators, leading to an entanglement transport in chains of mechanical oscillators. We discuss spatial dimensions, Q factors, temperatures and decoherence sources in some detail, and find a distinct robustness of the entanglement in the canonical coordinates in such a scheme. We also briefly discuss the challenging aspect of detection of the generated entanglement.

Exact decoherence to pointer states in free open quantum systems is universal

In this Letter it is shown that exact decoherence to minimal uncertainty Gaussian pointer states is generic for free quantum particles coupled to a heat bath. More specifically, the Letter is concerned with damped free particles linearly coupled under product initial conditions to a heat bath at arbitrary temperature, with arbitrary coupling strength and spectral densities covering the Ohmic, sub-Ohmic, and supra-Ohmic regime. Then it is true that there exists a time t(c) such that for times t>t(c) the state can always be exactly represented as a mixture (convex combination) of particular minimal uncertainty Gaussian states, regardless of and independent from the initial state. This exact "localization" is hence not a feature specific to high temperature and weak damping limit, but is a generic property of damped free particles.

Stochastic Gravity: Theory and Applications

Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor of quantum fields, stochastic semiclassical gravity is based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel. The noise kernel is the vacuum expectation value of the (operatorvalued) stress-energy bi-tensor which describes the fluctuations of quantum matter fields in curved spacetimes. In the first part, we describe the fundamentals of this new theory via two approaches: the axiomatic and the functional. The axiomatic approach is useful to see the structure of the theory from the framework of semiclassical gravity, showing the link from the mean value of the stress-energy tensor to their correlation functions. The functional approach uses the Feynman-Vernon influence functional and the Schwinger-Keldysh closed-time-path effective action methods which are convenient for computations. It also brings out the open systems concepts and the statistical and stochastic contents of the theory such as dissipation, fluctuations, noise, and decoherence. We then focus on the properties of the stress-energy bi-tensor. We obtain a general expression for the noise kernel of a quantum field defined at two distinct points in an arbitrary curved spacetime as products of covariant derivatives of the quantum field's Green function. In the second part, we describe three applications of stochastic gravity theory. First, we consider metric perturbations in a Minkowski spacetime. We offer an analytical solution of the Einstein-Langevin equation and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. Second, we discuss structure formation from the stochastic gravity viewpoint, which can go beyond the standard treatment by incorporating the full quantum effect of the inflaton fluctuations. Third, we discuss the backreaction of Hawking radiation in the gravitational background of a quasi-static black hole (enclosed in a box). We derive a fluctuation-dissipation relation between the fluctuations in the radiation and the dissipative dynamics of metric fluctuations.

All-forward semiclassical simulations of nonlinear response functions

We propose a quantum trajectory algorithm for computing nonlinear response functions of condensed phase molecular systems based on a time-ordered expansion of the density matrix. The nth-order response function is expressed as a sum of 2(n) impulsive response pathways representing trajectories involving zero, one, and up to n interactions with short external pulses. These are evaluated using a forward propagation algorithm based upon a Liouville space extension of the Bohmian propagation method.

Decoherence and the Loschmidt echo

Decoherence causes entropy increase that can be quantified using, e.g., the purity sigma=Trrho(2). When the Hamiltonian of a quantum system is perturbed, its sensitivity to such perturbation can be measured by the Loschmidt echo M(t). It is given by the squared overlap between the perturbed and unperturbed state. We describe the relation between the temporal behavior of sigma(t) and the average Mmacr;(t). In this way we show that the decay of the Loschmidt echo can be analyzed using tools developed in the study of decoherence. In particular, for systems with a classically chaotic Hamiltonian the decay of sigma and Mmacr; has a regime where it is dominated by the Lyapunov exponents.

Damped harmonic oscillator: pure states of the bath and exact master equations

Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.

Star-unitary transformations: from dynamics to irreversibility and stochastic behavior

We consider a simple model of a classical harmonic oscillator coupled to a field. In standard approaches, Langevin-type equations for bare particles are derived from Hamiltonian dynamics. These equations contain memory terms and are time-reversal invariant. In contrast, the phenomenological Langevin equations have no memory terms (they are Markovian equations) and give a time-evolution split in two branches (semigroups), each of which breaks time symmetry. A standard approach to bridge dynamics with phenomenology is to consider the Markovian approximation of the former. In this paper, we present a formulation in terms of dressed particles, which gives exact Markovian equations. We formulate dressed particles for Poincaré nonintegrable systems, through an invertible transformation operator Lambda introduced by Prigogine and co-workers. Lambda is obtained by an extension of the canonical (unitary) transformation operator U that eliminates interactions for integrable systems. Our extension is based on the removal of divergences due to Poincaré resonances, which breaks time symmetry. The unitarity of U is extended to "star unitarity" for Lambda. We show that Lambda-transformed variables have the same time evolution as stochastic variables obeying Langevin equations, and that Lambda-transformed distribution functions satisfy exact Fokker-Planck equations. The effects of Gaussian white noise are obtained by the nondistributive property of Lambda with respect to products of dynamical variables.

When is quantum decoherence dynamics classical?

A direct classical analog of quantum decoherence is introduced. Similarities and differences between decoherence dynamics examined quantum mechanically and classically are exposed via a second-order perturbative treatment and via a strong decoherence theory, showing a strong dependence on the nature of the system-environment coupling. For example, for the traditionally assumed linear coupling, the classical and quantum results are shown to be in exact agreement.

Quantum and classical correlations in quantum Brownian motion

We investigate the entanglement properties of the joint state of a distinguished quantum system and its environment in the quantum Brownian motion model. This model is a frequent starting point for investigations of environment-induced superselection. Using recent methods from quantum information theory, we show that there exists a large class of initial states for which no entanglement will be created at all times between the system of salient interest and the environment. If the distinguished system has been initially prepared in a pure Gaussian state, then entanglement is created immediately, regardless of the temperature of the environment and the nonvanishing coupling.

Decoherence for classically chaotic quantum maps

We study the behavior of an open quantum system, with an N-dimensional space of states, whose density matrix evolves according to a nonunitary map defined in two steps: A unitary step, where the system evolves with an evolution operator obtained by quantizing a classically chaotic map (baker's map and Harper's map are the two examples we consider). A nonunitary step where the evolution operator for the density matrix mimics the effect of diffusion in the semiclassical (large N) limit. The process of decoherence and the transition from quantum to classical behavior are analyzed in detail by means of numerical and analytic tools. The existence of a regime where the entropy grows with a rate that is independent of the strength of the diffusion coefficient is demonstrated. The nature of the processes that determine the production of entropy is analyzed.

Quantum decoherence, Zeno process, and time symmetry breaking

The complex spectral representation of the Liouville-von Neumann operator outside Hilbert space is applied to the decoherence problem in quantum Brownian motion. In contrast to the path-integral method, often used in the context of quantum decoherence for the case where the environment surrounding the Brownian particle (subsystem) is in thermal equilibrium, our spectral representation is applicable to systems far from equilibrium, including a pure state for the surrounding bath. Starting with this pure initial condition, the subsystem evolves in time obeying a diffusion-type kinetic equation. Hence, the collapse of wave functions is a dynamical phenomenon occurring outside Hilbert space, and is not simply a contamination of the subsystem, a popular view accepted in the so-called "environmental" approach, by the mixed nature of the thermal bath. The essential element in the understanding of quantum decoherence is the "extensivity" of quantities characterizing the thermodynamic limit. Quantum Zeno time is shown to be a lower bound of the decoherence time.

Quantum dissipation in unbounded systems

In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie--Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wave function of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements (Bohm particles) which evolve according to nonclassical equations of motion. Until very recently these schemes have been limited to conservative systems. In this paper, we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme that incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy, we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.

Quantum trajectory analysis of multimode subsystem-bath dynamics

The dynamics of a swarm of quantum trajectories is investigated for systems involving the interaction of an active mode (the subsystem) with an M-mode harmonic reservoir (the bath). Equations of motion for the position, velocity, and action function for elements of the probability fluid are integrated in the Lagrangian (moving with the fluid) picture of quantum hydrodynamics. These fluid elements are coupled through the Bohm quantum potential and as a result evolve as a correlated ensemble. Wave function synthesis along the trajectories permits an exact description of the quantum dynamics for the evolving probability fluid. The approach is fully quantum mechanical and does not involve classical or semiclassical approximations. Computational results are presented for three systems involving the interaction on an active mode with M=1, 10, and 15 bath modes. These results include configuration space trajectory evolution, flux analysis of the evolving ensemble, wave function synthesis along trajectories, and energy partitioning along specific trajectories. These results demonstrate the feasibility of using a small number of quantum trajectories to obtain accurate quantum results on some types of open quantum systems that are not amenable to standard quantum approaches involving basis set expansions or Eulerian space-fixed grids.

Decoherence in a classically chaotic quantum system: entropy production and quantum-classical correspondence

We study the decoherence process for an open quantum system that is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We carefully analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength); for longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is investigated and the issue of quantum to classical correspondence is addressed. Finally, the impact of the interaction with the environment on coherent tunneling is analyzed.

Decoherence and the rate of entropy production in chaotic quantum systems

We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.

Extraction of work from a single thermal bath in the quantum regime

The stationary state of a quantum particle strongly coupled to a quantum thermal bath is known to be non-Gibbsian, due to entanglement with the bath. For harmonic potentials, where the system can be described by effective temperatures, thermodynamic relations are shown to take a generalized Gibbsian form that may violate the Clausius inequality. For the weakly anharmonic case, a Fokker-Planck-type description is constructed. It is shown that then work can be extracted from the bath by cyclic variation of a parameter. These apparent violations of the second law are the consequence of quantum coherence in the presence of the slightly off-equilibrium nature of the bath.

Completely positive quantum dissipation

A completely positive master equation describing quantum dissipation for a Brownian particle is derived starting from microphysical collisions, exploiting a recently introduced approach to subdynamics of a macrosystem. The obtained equation can be cast into Lindblad form with a single generator for each Cartesian direction. Temperature dependent friction and diffusion coefficients for both position and momentum are expressed in terms of the collision cross section.

Decoherence via the dynamical casimir effect

We derive a master equation for a mirror interacting with the vacuum field via radiation pressure. The dynamical Casimir effect leads to decoherence of a superposition state in a time scale that depends on the degree of "macroscopicity" of the state components, and which may be much shorter than the relaxation time scale. Coherent states are selected by the interaction as pointer states.

Purity and decoherence in the theory of a damped harmonic oscillator

For the generalized master equations derived by Karrlein and Grabert for the microscopic model of a damped harmonic oscillator, the conditions for purity of states are written, in particular for different initial conditions and different types of damping, including Ohmic, Drude, and weak coupling cases, and the Agarwal and Weidlich-Haake models. It is shown that the states which remain pure are the squeezed states with variances that are constant in time. For pure states, generalized nonlinear Schrödinger-type equations corresponding to these master equations are also obtained. Then the condition for purity of states of a damped harmonic oscillator is considered in the framework of Lindblad theory for open quantum systems. For a special choice of the environment coefficients, correlated coherent states with constant variances and covariance are shown to be the only states which remain pure all the time during the evolution of the considered system. In Karrlein-Grabert and Lindblad models, as well as in the particular models considered, expressions for the rate of entropy production are written, and it is shown that state which preserve their purity in time are also states which minimize entropy production and, therefore, are the most stable state under evolution in the presence of the environment, and play an important role in the description of decoherence phenomenon.