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arXiv:1211.4773 · Physical Review B

Time-integrated observables as order parameters for full counting statistics transitions in closed quantum systems

James M. Hickey, Sam Genway, Igor Lesanovsky, Juan P. Garrahan

Published May 30, 2013• 40 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.87.184303

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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.
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.
Recent research into time-integrated observables has revealed a special class of states which cap- ture the singular features of the generating functions of those observables, as estimated by full counting statistics (FCS). In this work we extend the results of [Phys. Rev. B 87 184303 (2013)] to the 1d anisotropic XY -model and find a set of FCS critical points associated with the time-integrated transverse magnetization and anisotropy. We show dynamical phase transitions (DPTs) as defined in [Phys. Rev. Lett. 110 135704 (2013)] do not emerge on quenching the states associated with the time-integrated anisotropy across FCS critical points. We also study the timescales required to prepare the associated special states of the transverse magnetization using appropriate Markovian baths and find they are independent of the number of spins in the chain. However the probability to evolve to such a state decreases drastically with increasing system size due to dissipation. Thus when preparing such states it is preferable to use few body systems and to reach the thermodynamic limit it is necessary to use an ancillary system in conjunction with the Markovian baths.
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 dynamical properties of equilibrium and nonequilibrium systems, both quantum and classical, under the guise of a thermodynamic formalism. Large deviation functions associated with the generating functions of time-integrated observables play the role of dynamical free energies and thus determine the trajectory phase structure of a system. The 1d Glauber-Ising chain is studied using the time-integrated energy as the dynamical order parameter and a whole curve of second order trajectory transitions are uncovered in the complex counting field plane. The leading dynamical Lee-Yang zeros of the associated generating function are extracted directly from the time dependent high order cumulants. Resolving the cumulants into constituent contributions the motion of each contribution’s leading Lee-Yang zeros pair allows one to infer the positions of the trajectory transition points. Contrastingly if one uses the full cumulants only the positions of those closest to the origin, in the limit of low temperatures, can be inferred. Motivated by homodyne detection schemes this thermodynamic approach to trajectories is extended to the quadrature trajectories of light emitted from open quantum systems. Using this dynamical observable the trajectory phases of a simple “blinking” 3-level system, two weakly coupled 2-level systems and the micromaser are studied. The trajectory phases of this observable are found to either carry as much information as the photon emission trajectories or in some cases capture extra dynamically features of the system (the second example). Finally, the statistics of the time-integrated longitudinal and transverse magnetization in the 1d transverse field quantum Ising model are explored. In both cases no large deviation function exists but the generating functions are still calculable. From the singularities of these generating functions new transition lines emerge. These were shown to be linked to: (a) the survival probability of an associated open system, (b) PT-symmetry, (c) the temporal scaling of the cumulants and (d) the topology of an associated set of states.
We use high-order cumulants to investigate the Lee-Yang zeros of generating functions of dynamical observables in open quantum systems. At long times the generating functions take on a large-deviation form with singularities of the associated cumulant generating functions-or dynamical free energies-signifying phase transitions in the ensemble of dynamical trajectories. We consider a driven three-level system as well as the dissipative Ising model. Both systems exhibit dynamical intermittency in the statistics of quantum jumps. From the short-time behavior of the dynamical Lee-Yang zeros, we identify critical values of the counting field which we attribute to the observed intermittency and dynamical phase coexistence. Furthermore, for the dissipative Ising model we construct a trajectory phase diagram and estimate the value of the transverse field where the stationary state changes from being ferromagnetic (inactive) to paramagnetic (active).
Measuring physical observables requires averaging experimental outcomes over numerous identical measurements. The complete distribution function of possible outcomes or its Fourier transform, known as the full counting statistics, provides a more detailed description. This method captures the fundamental quantum fluctuations in many-body systems and has gained significant attention in quantum transport research. In this Letter, we propose that cusp singularities in the full counting statistics are a novel tool for distinguishing between ordered and disordered phases. As a specific example, we focus on the superfluid-to-Mott transition in the Bose-Hubbard model. Through both analytical analysis and numerical simulations, we demonstrate that the full counting statistics exhibit a cusp singularity as a function of the phase angle in the superfluid phase when the subsystem size is sufficiently large, while it remains smooth in the Mott phase. This discontinuity can be interpreted as a first-order transition between different semiclassical configurations of vortices. We anticipate that our discoveries can be readily tested using state-of-the-art ultracold atom and superconducting qubit platforms.