Statistical fluctuations in linear quantum-mechanical systems are shown to result from a projection of the total quantum system onto a restricted subspace. The resulting equations of motion are of the generalized Langevin form, with fluctuating and dissipative terms. These terms are related by a quantum-mechanical fluctuation-dissipation relation that ensures thermal equilibration. We analyze the dynamical behavior of the subsystem and elucidate the meaning and interrelation of several ubiquitous concepts in the following context: weak-coupling limit, Markovian limit, rotating-wave approximation (RWA), and low-temperature behavior. The three most salient consequences of our analysis are as follows: (1) The time scale for the correlation of fluctuations and the dissipation can be quite distinct, (2) the traditional implementation of the RWA only gives valid results in the strict weak-coupling limit, and (3) a reformulation of the RWA valid at arbitrary coupling strengths, and hence at arbitrarily low temperatures, is possible.
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.
Building upon the work of Hu, Paz, and Zhang [1,2] on open quantum systems we consider the quantum Brownian motion (QBM) model with one oscillator (position variable
x) as the system, {\it nonlinearly} coupled to an environment of
N harmonic oscillators (with mass
mn, natural frequency
ωn, position
qn and momentum
pn variables) in the form
∑n(vn1(x)qnk+vn2(x)pnl) where
k,l are integers (the present work only considers the
k=l=2 cases). The vertex functions $v_{n1}, v_{n2} $ are of the form
vn1=λCn1f(x),vn2(x)=−λCn2mn−2ωn−2f(x) where
Cn1,2 are the coupling constants with the
nth oscillator,
f(x) is any arbitrary function of
x, and
λ is a dimensionless constant. Employing the closed-time-path formalism the influence action
SIF is calculated using a perturbative expansion in
λ. It is possible to identify the terms in
SIF quadratic or higher in
Δ(s)≡f(x+(s))−f(x−(s)) to constitute the noise kernel, while terms linear in
Δ to that of the dissipation kernel. The non-Gaussian noise kernel gives rise to non-zero three-point correlation function of the corresponding stochastic force. The pathway presented here should be useful for the exploration of \textit{non-Gaussian properties of systems nonlinearly coupled with their environments}; examples in early universe cosmology and in quantum optomechanics (QOM) are mentioned. A modified fluctuation-dissipation relation (FDR) is also established, which ensures the consistency of the model and the accuracy of results even at higher perturbative orders. Another result of significance is the derivation of a nonlinear Langevin equation which is expected to be useful for many open quantum system applications.
The non-Markovian Langevin approach is applied to study the dynamics of fermionic (bosonic) oscillator linearly coupled to a fermionic (bosonic) environment. The analytical expressions for occupation numbers in two different types of couplings (rotating-wave approximation and fully coupled) are compared and discussed. The weak-coupling and high- and low-temperature limits are considered as well. The conditions under which the environment imposes its thermal equilibrium on the collective subsystem are discussed. The sameness of the results, obtained with both the Langevin approach and the discretized environment method are shown. Short- and long-time nonequilibrium dynamics of fermionic and bosonic open quantum systems are analyzed both analytically and numerically.
The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schrödinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.
The dynamical aspects of quantum Brownian motion at low temperatures are investigated. Exact calculations of quantum entanglement among two Brownian oscillators are given without invoking the Born-Markov or Born approximation widely used for the study of open systems. Nonlocality due to finite oscillator separation is studied. Our approach is applicable for arbitrary time scale, in particular suitable to probe the short-time regime at cold temperatures where many experiments on quantum information processing are performed. We study separability criteria based on the uncertainty relation, negativity, and entanglement of formation. We found a crossover behavior in a disentanglement process between quantum- and thermal-fluctuation-dominated regimes. The fluctuation-dissipation theorem is violated for two Brownian particles interacting with a common environment. The deviation from the relation originates in the interaction mediated by the environment, which drives two particles into a steady oscillatory state, preventing the system from thermalizing. Consequently there is a residual entanglement at low temperatures for arbitrary coupling strength. This entanglement is generated from the environment nonperturbatively even when two modes are not entangled initially. When the distance between two oscillators is varied, competition between entanglement induced from the environment and modified decoherence due to finite separation leads to the characteristic distance where entanglement is minimized.
Quantum diffusion equations featuring explicitly time-dependent transport coefficients are derived from generalized non-Markovian Langevin equations. Generalized fluctuation-dissipation relations and analytic expressions for calculating the friction and diffusion coefficients in nuclear processes are obtained. The asymptotic behavior of the transport coefficients and correlation functions for a damped harmonic oscillator that is linearly coupled in momentum to a heat bath is studied. The coupling to a heat bath in momentum is responsible for the appearance of the diffusion coefficient in coordinate. The problem of regression of correlations in quantum dissipative systems is analyzed.