Necessary conditions under which a classical description will give the correct quantum relaxation behavior are analyzed. Assuming a nonequilibrium preparation, it is shown that the long-time mean values of observables can be expressed in terms of the spectral density and state-specific level densities of the system. Any approximation that reproduces these quantities therefore yields the correct expectation values at long times. Apart from this rigorous condition, a weaker but more practical criterion is established, that is, to require that the total level density is well approximated in the energy range defined by the spectral density. Since the integral level density is directly proportional to the phase-space volume that is energetically accessible to the system, the latter condition means that an appropriate classical approximation should explore the same phase-space volume as the quantum description. In general, however, this is not the case. A well-known example is the unrestricted flow of zero-point energy in classical mechanics. To correct for this flaw of classical mechanics, quantum corrections are derived which result in a restriction of the classically accessible phase space. At the simplest level of the theory, these corrections are shown to correspond to the inclusion of only a fraction of the full zero-point energy into the classical calculation. Based on these considerations, a general strategy for the classical simulation of quantum relaxation dynamics is suggested. The method is (i) dynamically consistent in that it refers to the behavior of the ensemble rather than to the behavior of individual trajectories, (ii) systematic in that it provides (rigorous as well as minimal) criteria which can be checked in a practical calculation, and (iii) practical in that it retains the conceptional and computational simplicity of a standard quasiclassical calculation. Employing various model problems which allow for an analytical evaluation of the quantities of interest, the virtues and limitations of the approach are discussed.
Necessary conditions under which a classical description will give the correct quantum relaxation behavior are analyzed. Assuming a nonequilibrium preparation, it is shown that the long-time mean values of observables can be expressed in terms of the spectral density and state-specific level densities of the system. Any approximation that reproduces these quantities therefore yields the correct expectation values at long times. Apart from this rigorous condition, a weaker but more practical criterion is established, that is, to require that the total level density is well approximated in the energy range defined by the spectral density. Since the integral level density is directly proportional to the phase-space volume that is energetically accessible to the system, the latter condition means that an appropriate classical approximation should explore the same phase-space volume as the quantum description. In general, however, this is not the case. A well-known example is the unrestricted flow of zero-point energy in classical mechanics. To correct for this flaw of classical mechanics, quantum corrections are derived which result in a restriction of the classically accessible phase space. At the simplest level of the theory, these corrections are shown to correspond to the inclusion of only a fraction of the full zero-point energy into the classical calculation. Based on these considerations, a general strategy for the classical simulation of quantum relaxation dynamics is suggested. The method is (i) dynamically consistent in that it refers to the behavior of the ensemble rather than to the behavior of individual trajectories, (ii) systematic in that it provides (rigorous as well as minimal) criteria which can be checked in a practical calculation, and (iii) practical in that it retains the conceptional and computational simplicity of a standard quasiclassical calculation. Employing various model problems which allow for an analytical evaluation of the quantities of interest, the virtues and limitations of the approach are discussed.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
A major difficulty in comparing quantum and classical behavior resides in the struc- tural differences between the corresponding mathematical languages. The Heisenberg equations of motion are operator equations only formally identical to the classical equations of motion. By taking the expectation of these equations, the well-known Ehrenfest theorem provides identities which, however, are not a closed system of equations which allows to evaluate the time evolution of the system. The formalism of the effective action seems to offer a possibility of comparing quantum and classical evolutions in a system- atic and logically consistent way by naturally providing approximation schemes for the expectations of the coordinates which at the zeroth order coincide with the classical evolution [1]. The effective action formalism leads to equations of motion which differ from the classical equations by the addition of terms nonlocal in the time variable. This means that for these equations an initial value problem is not meaningful and they have to be interpreted in an appropriate way. Here we analyze situations in which the nonlocal terms can be reasonably approximated by local ones, so that the quantum corrections do not modify the locality of classical equations. In the simplest approximation, the effective Lagrangian differs from the corresponding classical one by a renormalization of both the potential- and the kinetic-energy terms. We shall not discuss the causal formalism used, for example, in refs. [2-4], as in the approximation considered this would lead to the same local equations. The present contribution describes the beginning of a systematic study of semiclassical evolutions using the effective action formalism. In the first part, after introducing the formalism of the effective action and its expansion in powers of hbar (loop expansion) in the context of quantum mechanics, we concentrate on the structure of the first-order corrections in li. These corrections are evaluated to the second order in the derivative expansion [5], by two different methods. The first is based on a Euclidean approach [6], the second one on an adiabatic approximation in evaluating functional determinants. In the second part of the article we put the formalism at work, choosing as our case study a two-dimensional (2D) anharmonic oscillator of the kind considered in molecular physics. The results of the simulations show that by increasing lithe effective dynamics tends to regularize the classical motion and becomes qualitatively very similar to the quantum evolution provided the energy is sufficiently small. The evaluation of the effective dynamics in more general cases will be presented in a forthcoming paper.
The long term behavior of non-integrable quantum systems is investigated to find a classical counterpart that describes the statistical properties of dynamical fluctuation of quantum observables. On the basis of the quantum-classical correspondence in the integrable system, we introduce the ensemble of classical trajectories (ECT) , which is defined as a set of weighted classical trajectories whose mean density in phase space is one per Planck cell. In the non-integrable system, due to the rapid breakdown of quantum-classical correspondence of the wavepacket dynamics, the ECT is unable to yield the correct temporal behavior of quantum observables. However, numerical experiments reveal that it gives a good approximation of various statistical properties of the quantum dynamical fluctuation. This result strongly suggests that, as far as the statistical properties are concerned, the dominant part of the quantum dynamical fluctuation is determined by classical dynamics, i.e. a set of classical trajectories. The meaning and interpretation of the ECT in relation to the coherent subsystem is also discussed.
The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.
The integrable system is constrained strictly by the conservation law during the time evolution, and the prethermal state from the nearly integrable system is also constrained by the conserved parameters (the constants of motion) with the corresponding generalized Gibbs ensemble (GGE), which is indubitably a powerful tool in the prediction of the relaxation dynamics. For stochastic evolution dynamics with considerable noise, the two-point correlation of local operators (like the density of kinks or transverse magnetization correlators), which do not exhibit the thermal features, display the behaviors of nonthermalization and an asymptotic GGE. In fact it is an asymptotic quasi-steady state with an infinite temperature, therefore the required distance to the nonthermal steady state is in an infinite time average. In this paper, we unambiguously investigate the relaxation of a nonequilibrium system in a canonical ensemble for integrable and nonintegrable systems. Temporal behavior of the many-body quantum system and the corresponding linear-coupling between the harmonic oscillators are discussed. The matrix-method in entropy ensemble is utilized to discuss the boundary and the diagonalization algebraically. The approximation results for nonintegrable system under the considerable perturbations are also presented.
In this PhD thesis we have been concerned with dissipative and decoherence processes which take place specifically for quantum systems in a dissipative bistable potential. This potential describes a Hamiltonian system whose spectrum is composed by a ladder of levels, grouped as doublets under the potential barrier and being almost equally spaced above it. The system interacts with a thermal environment, fully characterized by the so-called bath spectral density. The dynamical quantity we looked at is the so-called reduced density matrix (RDM), obtained by tracing out the full density matrix the environmental degrees of freedom. The matrix elements for the RDM are exactly evaluated by means of the Feynman-Vernon real-time path-integral technique. Depending on which regime of parameters we are interested in, i.e. temperature and coupling strength with the reservoir, a few or several energy levels contribute to the dynamics. In particular, a two-level system (TLS) is first analyzed, corresponding to the regime where only the lowest doublet is energetically accessible. We discussed in the beginning the case of a TLS interacting with Ohmic and super-Ohmic baths and then the situation in which a symmetric TLS is assumed to be coupled with an external detector, like a dc-SQUID, the latter being coupled to an Ohmic environment. This model provides a better description of a real qubit, as frequently discussed in various experiments. In the former case, we developed a theory called Weakly-Interacting Blip Approximation (WIBA), which is nonperturbative in the coupling strength as well as in the tunneling frequency and gives reliable results over a wide range of temperatures and coupling strength to the thermal environment. In the latter case, the dynamics was evaluated upon starting from the Non-Interacting Blip Approximation (NIBA) and approximating the NIBA kernels up to first order in the detector-bath coupling strength. Analytical results, valid also at finite detuning, were obtained and we showed that our theory correctly reproduces the onset of multiple oscillation frequencies in the dynamics due to the entanglement between TLS and detector. Finally, we discussed the more general case of an M-level system coupled with an Ohmic environment. This situation describes the case of a bistable potential where the temperature is high enough that higher-lying energy levels are being populated and the system cannot be reduced to a simple TLS. Many are the experimental situations which can be described by our theory. We developed a novel approximation called Weakly-Interacting VR-Blip Approximation (VR-WIBA) for the dynamics in a driven dissipative bistable potential. In the regimes of low to moderate temperatures and coupling strengths, we exploited a separation of time scales between tunneling and vibrational relaxation events and we could derive analytical kernels for the intra- and inter-well dynamics.