Quantum master equation for the microcanonical ensemble

Modeling the dynamics of quantum systems coupled to large-dimensional baths using effective energy states

The quantum dynamics of a low-dimensional system in contact with a large but finite harmonic bath is theoretically investigated by coarse-graining the bath into a reduced set of effective energy states. In this model, the couplings between the system and the bath are obtained from statistically averaging over the discrete, degenerate effective states. Our model is aimed at intermediate bath sizes in which non-Markovian processes and energy transfer between the bath and the main system are important. The method is applied to a model system of a Morse oscillator coupled to 40 harmonic modes. The results are found to be in excellent agreement with the direct quantum dynamics simulations presented in the work of Bouakline et al. [J. Phys. Chem. A 116, 11118-11127 (2012)], but at a much lower computational cost. Extension to larger baths is discussed in comparison to the time-convolutionless method. We also extend this study to the case of a microcanonical bath with finite initial internal energies. The computational efficiency and convergence properties of the effective bath states model with respect to relevant parameters are also discussed.

Stochastic Thermodynamics of a Quantum Dot Coupled to a Finite-Size Reservoir

In nanoscale systems coupled to finite-size reservoirs, the reservoir temperature may fluctuate due to heat exchange between the system and the reservoirs. To date, a stochastic thermodynamic analysis of heat, work, and entropy production in such systems is, however, missing. Here we fill this gap by analyzing a single-level quantum dot tunnel coupled to a finite-size electronic reservoir. The system dynamics is described by a Markovian master equation, depending on the fluctuating temperature of the reservoir. Based on a fluctuation theorem, we identify the appropriate entropy production that results in a thermodynamically consistent statistical description. We illustrate our results by analyzing the work production for a finite-size reservoir Szilard engine.

Ensemble dependence of information-theoretic contributions to the entropy production

The entropy production of an open system coupled to a reservoir initialized in a canonical state can be expressed as a sum of two microscopic information-theoretic contributions: the system-bath mutual information and the relative entropy measuring the displacement of the environment from equilibrium. We investigate whether this result can be generalized to situations where the reservoir is initialized in a microcanonical or in a certain pure state (e.g., an eigenstate of a nonintegrable system), such that the reduced dynamics and thermodynamics of the system are the same as for the thermal bath. We show that while in such a case the entropy production can still be expressed as a sum of the mutual information between the system and the bath and a properly redefined displacement term, the relative weight of those contributions depends on the initial state of the reservoir. In other words, different statistical ensembles for the environment predicting the same reduced dynamics for the system give rise to the same total entropy production but to different information-theoretic contributions to the entropy production.

Open quantum systems coupled to finite baths: A hierarchy of master equations

An open quantum system in contact with an infinite bath approaches equilibrium, while the state of the bath remains unchanged. If the bath is finite, the open system still relaxes to equilibrium but it induces a dynamical evolution of the bath state. In this paper, we study the dynamics of open quantum systems in contact with finite baths. We obtain a hierarchy of master equations that improve their accuracy by including more dynamical information of the bath. For instance, as the least accurate but simplest description in the hierarchy, we obtain the conventional Born-Markov-secular master equation. Remarkably, our framework works even if the measurements of the bath energy are imperfect, which not only is more realistic but also unifies the theoretical description. Also, we discuss this formalism in detail for a particular noninteracting environment where the Boltzmann temperature and the Kubo-Martin-Schwinger relation naturally arise. Finally, we apply our hierarchy of master equations to study the central spin model.

Kinetics and thermodynamics of a driven open quantum system

Redfield theory provides a closed kinetic description of a quantum system in weak contact with a very dense reservoir. Landau-Zener theory does the same for a time-dependent driven system in contact with a sparse reservoir. Using a simple model, we analyze the validity of these two theories by comparing their predictions with exact numerical results. We show that despite their a priori different range of validity, these two descriptions can give rise to an identical quantum master equation. Both theories can be used for a nonequilibrium thermodynamic description, which we show is consistent with exact thermodynamic identities evaluated in the full system-reservoir space. We emphasize the importance of properly accounting for the system-reservoir interaction energy and of operating in regimes where the reservoir can be considered as close to ideal.

Entropy increase in K-step Markovian and consistent dynamics of closed quantum systems

We consider sequences of measurements implemented by positive operator valued measures (POVMs). Starting from the assumption that these sequences may be described as consistent and Markovian, even and especially for closed quantum systems, we identify properties of the equilibrium state that coincide with the properties of typical pure quantum states. We define a physical entropy that converges against the standard entropies in the approach to equilibrium. Furthermore, strict limits to its possible decrease are derived on the basis of Renyi entropies. It is demonstrated that Landauer's principle follows directly from these limits. Since the above assumptions are rather strong, we exemplify the fact that they may nevertheless apply by checking them numerically for some transition paths in a concrete model.

Heat generation and transport due to time-dependent forces

We study heat generation and transport properties for solids in the presence of arbitrary time-dependent forces. Using a nonequilibrium Green's function (NEGF) approach we present an exact analytical expression of heat current for the linear system. We found that the current can be expressed in terms of the displacement of the atoms in the center and the self-energy of the heat bath. We carry out the calculation for a periodic driving force and study the dependence of steady state current on frequency and system size for one- and two-dimensional systems. We obtain an explicit solution of current for a one-dimensional linear chain connected with a Rubin bath. We found that the heat current is related to the density of states of the system and is independent of the bath temperature in ballistic transport. The baths can absorb energy only when the external frequency lies within the phonon band frequency. We also discuss the effect due to nonlinear interactions in the center.

Emergence of canonical ensembles from pure quantum states

We consider a system weakly interacting with a bath as a thermodynamic setting to establish a quantum foundation of statistical physics. It is shown that even if the composite system is initially in an arbitrary nonequilibrium pure quantum state, the unitary dynamics of a generic weak interaction almost always drives the subsystem into the canonical ensemble, in the usual sense of typicality. A crucial step is taken by assuming that the matrix elements of the interaction Hamiltonian have random phases, while their amplitudes are left unrestricted.

Fluctuation-dissipation theorem for the microcanonical ensemble

A derivation of the fluctuation-dissipation theorem for the microcanonical ensemble is presented using linear response theory. The theorem is stated as a relation between the frequency spectra of the symmetric correlation and response functions. When the system is not in the thermodynamic limit, this result can be viewed as an extension of the fluctuation-dissipation relations to a situation where dynamical fluctuations determine the response. Therefore, the relation presented here between equilibrium fluctuations and response can have a very different physical nature from the usual one in the canonical ensemble. These considerations imply that the fluctuation-dissipation theorem is not restricted to the context of the canonical ensemble, where it is usually derived. Dispersion relations and sum rules are also obtained and discussed in the present case. Although analogous to the Kramers-Kronig relations, they are not related to the frequency spectrum but to the energy dependence of the response function.