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arXiv:gr-qc/9302029 · Presented at

Quantum and Thermal Fluctuations, Uncertainty Principle, Decoherence and Classicality*

B. L. Hu, Yuhong Zhang

Published Feb 22, 1993• 3 citations
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Abstract

We scrutize the commonly used criteria for classicality and examine their underlying issues. The two major issues we address here are that of decoherence and fluctuations. We borrow the insights gained in the study of the semiclassical limit of quantum cosmology to discuss the three criteria of classicality for a quantum closed system: adiabaticity, correlation and decoherence. We then use the Brownian model as a paradigm of quantum open systems to discuss the relation of quantum and thermal fluctuations and their role in the transition from quantum to classical. We derive the uncertainty relation at finite temperature. We study how the fluctuations of a quantum system evolve after it is brought in contact with a heat bath and analyse the decoherence and relaxation processes. From the effect of fluctuations on decoherence we show the relation between these two sets of criteria of classicality. Finally, we briefly comment on the issue of nonintegrability in quantum open systems.

DOI
10.48550/arxiv.gr-qc/9302029

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We scrutize the commonly used criteria for classicality and examine their underlying issues. The two major issues we address here are that of decoherence and fluctuations. We borrow the insights gained in the study of the semiclassical limit of quantum cosmology to discuss the three criteria of classicality for a quantum closed system: adiabaticity, correlation and decoherence. We then use the Brownian model as a paradigm of quantum open systems to discuss the relation of quantum and thermal fluctuations and their role in the transition from quantum to classical. We derive the uncertainty relation at finite temperature. We study how the fluctuations of a quantum system evolve after it is brought in contact with a heat bath and analyse the decoherence and relaxation processes. From the effect of fluctuations on decoherence we show the relation between these two sets of criteria of classicality. Finally, we briefly comment on the issue of nonintegrability in quantum open systems.
We derive the uncertainty relation for a quantum open system consisting of a Brownian particle interacting with a bath of quantum oscillators at finite temperature. We examine how the quantum and thermal fluctuations of the environment contribute to the uncertainty in the canonical variables of the system. We show that upon contact with the bath (assumed to be ohmic in this paper) the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time in similar models in the discussion of quantum to classical transition. This offers some insight into the physical mechanisms involved in the environment-induced decoherence process. We obtain closed analytic expressions for this generalized uncertainty relation under the conditions of high temperature and weak damping, separately. We also consider under these conditions an arbitrarily squeezed initial state and show how the squeeze parameter enters in the generalized uncertainty relation. Using these results we examine the transition of the system from a quantum pure state to a nonequilibrium quantum statistical state and to an equilibrium quantum statistical state. The three stages are marked by the decoherence time and the relaxation time, respectively. With these observations we explicate the physical conditions under which the two basic postulates of quantum statistical mechanics become valid. We also comment on the inappropriate usage of the word “classicality” in many decoherence studies of quantum to classical transition.
We use the quantum Brownian model to derive the uncertainty relation for a quantum open system in an arbitrarily-squeezed initial state interacting with an environment at finite temperature. We examine the relative importance of the quantum and thermal fluctuations in the evolution of the system towards equilibrium with the aim of clarifying the meaning of quantum, classical and thermal. We show that upon contact with the bath the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time calculated before in the context of quantum to classical transitions. We also use these results to deduce the conditions when the two basic postulates of quantum statistical mechanics become valid.
We use the quantum Brownian model to derive the uncertainty relation for a quantum open system in an arbitrarily-squeezed initial state interacting with an environment at finite temperature. We examine the relative importance of the quantum and thermal fluctuations in the evolution of the system towards equilibrium with the aim of clarifying the meaning of quantum, classical and thermal. We show that upon contact with the bath the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time calculated before in the context of quantum to classical transitions. We also use these results to deduce the conditions when the two basic postulates of quantum statistical mechanics become valid.
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.
According to the inflationary scenario of cosmology, all structure in the Universe can be traced back to primordial fluctuations during an accelerated (inflationary) phase of the very early Universe. A conceptual problem arises due to the fact that the primordial fluctuations are quantum, while the standard scenario of structure formation deals with classical fluctuations. In this essay we present a concise summary of the physics describing the quantum-to-classical transition. We first discuss the observational indistinguishability between classical and quantum correlation functions in the closed system approach (pragmatic view). We then present the open system approach with environment-induced decoherence. We finally discuss the question of the fluctuations' entropy for which, in principle, the concrete mechanism leading to decoherence possesses observational relevance.
This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. It is in part a continuation of an earlier paper on the uncertainty principle for quantum-mechanical histories. We consider histories characterized by position samplings at n moments of time. This includes the case of position sampling continuous in time in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, F=mx\ifmmode\ddot\else\textasciidieresis\fi{}+V'(x), at n-2 times. We find that it is peaked around F=0; thus classical correlations are exhibited. We show that the width of the peak \ensuremath{\Delta}F is largely independent of the initial state. We show that the uncertainty principle takes the form 2${\mathrm{\ensuremath{\sigma}}}^{2}$(\ensuremath{\Delta}F)2{)}^{2})2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/t2{\mathit{t}}^{2}t2, where \ensuremath{\sigma} is the width of the position samplings, and t is the time scale between projections. We determine the modifications to this result when the system is coupled to a thermal environment, thus obtaining a measure of the comparative sizes of quantum and thermal fluctuations. We show that the thermal fluctuations become comparable to the quantum fluctuations under the same conditions that decoherence effects come into play, in agreement with earlier work. We also study an alternative measure of classical correlations, namely, the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred.We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. Under appropriate conditions, which we describe, they admit an interpretation as a statistical ensemble of classical solutions, with the probability of each individual classical solution given by a smeared Wigner function of its initial data. We study the decoherence properties of histories characterized by the value of the field equations, F, at a sequence of times. We argue that they will be decoherent if their initial data are decoherent.