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arXiv:2110.01678

Energy Full Counting Statistics and Return to Equilibrium

Jane Panangaden

Published Oct 4, 2021• 0 citations
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Abstract

We consider a finite dimensional quantum system $S$ in an arbitrary initial state coupled to an infinitely extended quantum thermal reservoir $R$ in equilibrium at inverse temperature $β$. The coupling is given by a bounded perturbation of the dynamics and the coupling strength is controlled by a parameter $λ$. We assume the system $S + R$ has the property of return to equilibrium, which means that after sufficiently long time, the joint system will have reached equilibrium at inverse temperature $β$. 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 $0$ and time $t$. These measures are called the full counting statistics (FCS). We prove weak convergence of the system and reservoir FCS in the double limit $t \rightarrow \infty$ and $λ\rightarrow 0$.

DOI
10.48550/arxiv.2110.01678

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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.
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.
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.
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Owing to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution-or its Fourier transform known as full counting statistics (FCS)-contains much more information than say the mean value of the measured observable, and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables-the charged moments-that characterize how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a nontrivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out of equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain.
This thesis investigates the dynamics of quantum many-body systems in isolation, in three different setups. We start by considering a paradigmatic quantum chaotic model of magnetism, the Heisenberg antiferromagnet in 2 and 3 spatial dimensions. The focus is on obtaining approximations for the full counting statistics (FCS) of a subsystem order parameter, the staggered magnetization, both in and out of equilibrium (following quantum quenches). We employ mappings of spin variables to bosonic ones, together with self-consistent time-dependent mean-field schemes, to compute the FCS via Gaussian techniques. In particular, we introduce a formula for the FCS of any subsystem observable that is at most quadratic in the bosons. We obtain results both in the presence and absence of long-range magnetic order. We then turn to the first of two ergodicity-breaking scenarios, by considering Hilbert space fragmentation (HSF) in charge- and dipole-conserving quantum chains. We determine the critical charge density at which a "freezing" phase transition from weak to strong fragmentation takes place, for any on-site Hilbert space dimension. To characterize the degree of dynamical connectivity in the system, we introduce a novel framework that enables us to obtain analytic results for the strongly fragmented phase, and numerically explore the weakly fragmented one. We derive some critical exponents, and show that the entanglement entropy of typical eigenstates in the strongly fragmented phase obeys an area law. We also uncover the existence of rare eigenstates whose entropy appears to scale beyond any area law. Finally, we investigate dynamical correlators in the Lieb-Liniger gas, one of the simplest interacting integrable theories solvable by Bethe ansatz. We employ Markov chain Monte Carlo methods to sample matrix elements (form factors) that appear in the Lehmann representation of the finite-temperature single-particle bosonic Green’s function. In the physical regimes we consider, sampling is necessary because a full summation of the relevant form factors is computationally intractable. We benchmark our results against exact solutions available in specific limits, and find them to be very accurate.
The dynamical behavior of many-body systems is often richer than what can be anticipated from their static properties. Here we show that in closed quantum systems this becomes evident by considering the statistics of time-integrated observables. In particular, the analytic properties of their generating functions, as estimated by full counting statistics (FCS), allow one to identify FCS phases, i.e., phases with specific fluctuation properties of time-integrated observables, and to locate transitions between these phases. We discuss in detail the case of the quantum Ising chain in a transverse field. We show that this model displays a continuum of full counting statistics transitions, of which the static transition is just an end point. These singularities are not a consequence of particular choices of initial conditions or other external nonequilibrium protocols such as quenches in coupling constants. They can be probed generically through quantum jump statistics of an associated open problem, and for the case of the quantum Ising chain we outline a possible experimental realization of this scheme by digital quantum simulation with cold ions.