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arXiv:1409.8610

Energy conservation, counting statistics, and return to equilibrium

Vojkan Jakšić, Jane Panangaden, Annalisa Panati, Claude‐Alain Pillet

Published Sep 30, 2014• 0 citations
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Abstract

We study a microscopic Hamiltonian model describing an N-level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at temperature T. Assuming that the coupled system S+R is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers S->R and R->S in the process of return to equilibrium. The first FCS describes the increase of the energy of the system S. It is an atomic probability measure, denoted $P_{S,λ,t}$, concentrated on the set of energy differences $σ(H_S)-σ(H_S)$ ($σ(H_S)$ is the spectrum of the Hamiltonian of S, $t$ is the length of the time interval during which the measurement of the energy transfer is performed, and $λ$ is the strength of the interaction between S and R). The second FCS, $P_{R,λ,t}$, describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit $t\rightarrow\infty$ of these two measures followed by the weak coupling limit $λ\rightarrow 0$ and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of $P_{R,λ,t}$ by quantum transfer operators.

DOI
10.48550/arxiv.1409.8610

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•Feb 1, 2018•0 citations
We study a microscopic Hamiltonian model describing an N-level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at temperature T. Assuming that the coupled system S+R is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers S->R and R->S in the process of return to equilibrium. The first FCS describes the increase of the energy of the system S. It is an atomic probability measure, denoted PS,λ,tP_{S,λ,t}PS,λ,t​, concentrated on the set of energy differences σ(HS)−σ(HS)σ(H_S)-σ(H_S)σ(HS​)−σ(HS​) (σ(HS)σ(H_S)σ(HS​) is the spectrum of the Hamiltonian of S, ttt is the length of the time interval during which the measurement of the energy transfer is performed, and λλλ is the strength of the interaction between S and R). The second FCS, PR,λ,tP_{R,λ,t}PR,λ,t​, describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit t→∞t\rightarrow\inftyt→∞ of these two measures followed by the weak coupling limit λ→0λ\rightarrow 0λ→0 and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of PR,λ,tP_{R,λ,t}PR,λ,t​ by quantum transfer operators.
We consider a finite dimensional quantum system SSS in an arbitrary initial state coupled to an infinitely extended quantum thermal reservoir RRR in equilibrium at inverse temperature βββ. The coupling is given by a bounded perturbation of the dynamics and the coupling strength is controlled by a parameter λλλ. We assume the system S+RS + RS+R has the property of return to equilibrium, which means that after sufficiently long time, the joint system will have reached equilibrium at inverse temperature βββ. In this context, we prove a refinement to the first law of thermodynamics, which states that the total energy of the system and reservoir is conserved. Specifically, we define two measures which encode all the information about the fluctuations of the system and reservoir energy when two measurements are made at time 000 and time ttt. These measures are called the full counting statistics (FCS). We prove weak convergence of the system and reservoir FCS in the double limit t→∞t \rightarrow \inftyt→∞ and λ→0λ\rightarrow 0λ→0.
The first law of thermodynamics states that the average total energy current between different reservoirs vanishes at large times. In this paper we examine this fact at the level of the full statistics of two-time measurement protocols, also known as the Full Counting Statistics. Under very general conditions, we establish a tight form of the first law asserting that the fluctuations of the total energy current computed from the energy variation distribution are exponentially suppressed in the large time limit. We illustrate this general result using two examples: the Anderson impurity model and a two-dimensional spin lattice model.
By solving the exact master equation of open quantum systems, we formulate the quantum thermodynamics from weak to strong couplings. The open quantum systems exchange matters, energies, and information with their reservoirs through quantum particle tunnelings that are described by the generalized Fano-Anderson Hamiltonians. We find that the exact solution of the reduced density matrix of these systems approaches a Gibbs-type state in the steady-state limit for the systems in arbitrary initial states as well as for both the weak and strong system-reservoir coupling strengths. When the couplings become strong, thermodynamic quantities of the system must be renormalized. The renormalization effects are obtained nonperturbatively after exactly tracing over all reservoir states through the coherent state path integrals. The renormalized system Hamiltonian is characterized by the renormalized system energy levels and interactions, corresponding to the quantum work done by the system. The renormalized temperature is introduced to characterize the entropy production counting the heat transfer between the system and the reservoir. We further find that only with the renormalized system Hamiltonian and other renormalized thermodynamic quantities can the exact steady state of the system be expressed as the standard Gibbs state. Consequently, the corresponding exact steady-state particle occupations in the renormalized system energy levels obey the Bose-Einstein and the Fermi-Dirac distributions for bosonic and fermionic systems, respectively. In the very weak system-reservoir coupling limit, the renormalized system Hamiltonian and the renormalized temperature are reduced to the original bare Hamiltonian of the system and the initial temperature of the reservoir. Thus, the conventional statistical mechanics and thermodynamics are thereby rigorously deduced from quantum dynamical evolution. In the last, this nonperturbative renormalization method is also extended to general interacting open quantum systems.
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.
A general thermodynamic framework is presented for open quantum systems in fixed contact with a thermal reservoir. The first and second law are obtained for arbitrary system-reservoir coupling strengths, and including both factorized and correlated initial conditions. The thermodynamic properties are adapted to the generally strong coupling regime, approaching the ones defined for equilibrium, and their standard weak-coupling counterparts as appropriate limits. Moreover, they can be inferred from measurements involving only system observables. Finally, a thermodynamic signature of non-Markovianity is formulated in the form of a negative entropy production rate.
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.