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arXiv:2203.04545 · Journal of Statistical Mechanics Theory and Experiment

Quantum states from mixtures of equilibrium distributions

Mihail Mintchev

Published Apr 1, 2022• 0 citations
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Abstract

Abstract We construct and explore a family of states for quantum systems in contact with two or more heath reservoirs. The reservoirs are described by equilibrium distributions. The interaction of each reservoir with the bulk of the system is encoded in a probability, which characterises the particle exchange among them and depends in general on the particle momentum. The convex combination of the reservoir distributions, weighted with the aforementioned probabilities, defines a new distribution. We establish the existence of an emission–absorption regime in which the new distribution generates a non-equilibrium quantum state. We develop a systematic field theory framework for constructing this state and illustrate its physical properties on a simple model. In this context we derive the particle current full counting statistics, the heat current and the Lorenz number. The entropy production and the relative quantum fluctuations are also determined.

DOI
10.1088/1742-5468/ac6252

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Abstract We construct and explore a family of states for quantum systems in contact with two or more heath reservoirs. The reservoirs are described by equilibrium distributions. The interaction of each reservoir with the bulk of the system is encoded in a probability, which characterises the particle exchange among them and depends in general on the particle momentum. The convex combination of the reservoir distributions, weighted with the aforementioned probabilities, defines a new distribution. We establish the existence of an emission–absorption regime in which the new distribution generates a non-equilibrium quantum state. We develop a systematic field theory framework for constructing this state and illustrate its physical properties on a simple model. In this context we derive the particle current full counting statistics, the heat current and the Lorenz number. The entropy production and the relative quantum fluctuations are also determined.
We formulate a full-counting statistics description to study energy exchange in multi-terminal junctions. Our approach applies to quantum systems that are coupled either additively or non-additively (cooperatively) to multiple reservoirs. We derive a Markovian Redfield-type equation for the counting-field dependent reduced density operator. Under the secular approximation, we confirm that the cumulant generating function satisfies the heat exchange fluctuation theorem. Our treatment thus respects the second law of thermodynamics. We exemplify our formalism on a multi-terminal two-level quantum system, and apply it to realize the smallest quantum absorption refrigerator, operating through engineered reservoirs, and achievable only through a cooperative bath interaction model.
We formulate a full-counting statistics description to study energy exchange in multi-terminal junctions from weak to strong system-bath coupling regime. Our approach can describe energy transport in quantum systems that are coupled in a non-additive manner to multiple reservoirs. We derive a Redfield-type equation for the counting-field dependent reduced density operator, valid under the Markovian approximation for the baths. Under the secular approximation, we confirm that the cumulant generating function satisfies the heat exchange fluctuation theorem. Our treatment thus respects the second law of thermodynamics---though for non-additive interaction models the entropy production rate within an individual thermal bath cannot be identified. We exemplify our method on a multi-terminal two-level quantum system. We write down explicit expressions for the energy current and its noise in the additive (weak coupling) and non-additive (strong coupling) cases and verify the so-called thermodynamic uncertainty relation. Lastly, we use our formalism to describe the smallest quantum absorption refrigerator, operating through engineered reservoirs, acting cooperatively.
The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.
We study the entropy production in non-equilibrium quantum systems without dissipation, which is generated exclusively by the spontaneous breaking of time-reversal invariance. Systems which preserve the total energy and particle number and are in contact with two heat reservoirs are analysed. Focussing on point-like interactions, we derive the probability distribution induced by the entropy production operator. We show that all its moments are positive in the zero frequency limit. The analysis covers both Fermi and Bose statistics.
The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open fundamental and practical questions range from how they relax to equilibrium, to how to extract useful work from them. Here, we derive a set of exact results that relate out-of-equilibrium fluctuations in the energy and other observables of a quantum system to its equilibrium properties for a very general family of initial conditions. These quantum fluctuation relations generalise the Jarzynski and Crooks relations to quantum systems with conserved quantities, and can be applied to protocols driving the system between integrable and chaotic regimes, or coupling it to different reservoirs. We illustrate our results with simulations of an integrable model subject to quenches realisable with current technology. Our findings will help guiding research on the interplay of quantum and thermal fluctuations in quantum simulation, and their exploitation in the design of new quantum devices.
We introduce a non-equilibrium version of the Caldeira-Leggett model in which a quantum particle is strongly coupled to a set of engineered reservoirs. The reservoirs are composed by collections of squeezed and displaced thermal modes, in contrast to the standard case in which the modes are assumed to be at equilibrium. The model proves to be very versatile. Strongly displaced/squeezed reservoirs can be used to generate an effective time dependence in the system Hamiltonian and can be identified as sources of pure work. In the case of squeezing, the time dependence is stochastic and breaks the fluctuation-dissipation relation, this can be reconciled with the second law of thermodynamics by correctly accounting for the energy used to generate the initial non-equilibrium conditions. To go beyond the average description and compute the full heat statistics, we treat squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour. As an application of this technique, we show the quantum-classical correspondence between the heat statistics in the non-equilibrium Caldeira-Leggett model and the statistics of a classical Langevin particle under the action of squeezed and displaced colored noises. Finally, we discuss thermodynamic symmetries of the heat generating function, proving a fluctuation theorem for the energy balance and showing that the conservation of energy at the trajectory level emerges in the classical limit.