Quantum Systems In Equilibrium Research Articles

Comparing metrics for mixed quantum states: Sjöqvist and Bures

It is known that there are infinitely many distinguishability metrics for mixed quantum states. This freedom, in turn, leads to metric-dependent interpretations of physically meaningful geometric quantities such as complexity and volume of quantum states. In this paper, we first present an explicit and unabridged mathematical discussion on the relation between the Sjoqvist metric and the Bures metric for arbitrary nondegenerate mixed quantum states, using the notion of decompositions of density operators by means of ensembles of pure quantum states. Then, to enhance our comprehension of the difference between these two metrics from a physics standpoint, we compare the formal expressions of these two metrics for arbitrary thermal quantum states specifying quantum systems in equilibrium with a reservoir at non-zero temperature. For illustrative purposes, we show the difference between these two metrics in the case of a simple physical system characterized by a spin-qubit in an arbitrarily oriented uniform and stationary external magnetic field in thermal equilibrium with a finite-temperature bath. Finally, we compare the Bures and Sjoqvist metrics in terms of their monotonicity property.

Emission and collisional correlation in far-off equilibrium quantum systems

We propose a scheme to describe dynamical correlations in finite fermion systems which are open in the sense that they can lose particles, electrons in the present case. It is built as an extension of recently developed schemes to describe dissipative dynamics in finite fermion systems, namely stochastic time-dependent adiabatic local-density approximation (STDLDA) and its averaged version, ASTDLDA, so far being applied only in closed systems. STDLDA and ASTDLDA are based on real-time real-space dynamics in terms of time-dependent density functional theory and add dynamical two-body collisions in a stochastic manner. The extension to systems that can emit electrons is achieved by complementing the complete (numerical) description of wave functions in “inner” space (inside the computation box) through a global description in “outer” space (outside the computation box) and a careful bookkeeping of flow between inner and outer regions. We test the method in a 1D model system, mimicking simple molecules or clusters. Two test cases are investigated: a metal-like system excited by an instantaneous dipole boost, and a covalent-like system excited by an instantaneous one-particle–one-hole transition. The dynamics is analyzed in terms of one-body observables (dipole moment, ionization, entropy). STDLDA and ASTDLDA exhibit clearly dissipative features at variance with mean-field dynamics. Unlike the pattern in closed systems, differences between STDLDA and ASTDLDA show up for ionization. They can be attributed to significantly larger fluctuations of the mean field in STDLDA.

Quantum corrections to the classical field approximation for one-dimensional quantum many-body systems in equilibrium

We present a semiclassical treatment of one-dimensional many-body quantum systems in equilibrium, where quantum corrections to the classical field approximation are systematically included by a renormalization of the classical field parameters. Our semiclassical approximation is reliable in the limit of weak interactions and high temperatures. As a specific example, we apply our method to the interacting Bose gas and study experimentally observable quantities, such as correlation functions of bosonic fields and the full counting statistics of the number of particles in an interval. Where possible, our method is checked against exact results derived from integrability, showing excellent agreement.

Quantum critical behaviour at the many-body localization transition.

Phase transitions are driven by collective fluctuations of a system's constituents that emerge at a critical point1. This mechanism has been extensively explored for classical and quantum systems in equilibrium, whose critical behaviour is described by the general theory of phase transitions. Recently, however, fundamentally distinct phase transitions have been discovered for out-of-equilibrium quantum systems, which can exhibit critical behaviour that defies this description and is not well understood1. A paradigmatic example is the many-body localization (MBL) transition, which marks the breakdown of thermalization in an isolated quantum many-body system as its disorder increases beyond a critical value2-11. Characterizing quantum critical behaviour in an MBL system requires probing its entanglement over space and time4,5,7, which has proved experimentally challenging owing to stringent requirements on quantum state preparation and system isolation. Here we observe quantum critical behaviour at the MBL transition in a disordered Bose-Hubbard system and characterize its entanglement via its multi-point quantum correlations. We observe the emergence of strong correlations, accompanied by the onset of anomalous diffusive transport throughout the system, and verify their critical nature by measuring their dependence on the system size. The correlations extend to high orders in the quantum critical regime and appear to form via a sparse network of many-body resonances that spans the entire system12,13. Our results connect the macroscopic phenomenology of the transition to the system's microscopic structure of quantum correlations, and they provide an essential step towards understanding criticality and universality in non-equilibrium systems1,7,13.

Experimentally Accessible Witnesses of Many-Body Localization

The phenomenon of many-body localized (MBL) systems has attracted significant interest in recent years, for its intriguing implications from a perspective of both condensed-matter and statistical physics: they are insulators even at non-zero temperature and fail to thermalize, violating expectations from quantum statistical mechanics. What is more, recent seminal experimental developments with ultra-cold atoms in optical lattices constituting analog quantum simulators have pushed many-body localized systems into the realm of physical systems that can be measured with high accuracy. In this work, we introduce experimentally accessible witnesses that directly probe distinct features of MBL, distinguishing it from its Anderson counterpart. We insist on building our toolbox from techniques available in the laboratory, including on-site addressing, super-lattices, and time-of-flight measurements, identifying witnesses based on fluctuations, density–density correlators, densities, and entanglement. We build upon the theory of out of equilibrium quantum systems, in conjunction with tensor network and exact simulations, showing the effectiveness of the tools for realistic models.

Local quantum thermometry using Unruh–DeWitt detectors

We propose an operational definition for the local temperature of a quantum field employing Unruh–DeWitt detectors, as used in the study of the Unruh and Hawking effects. With this definition, an inhomogeneous quantum system in equilibrium can have different local temperatures, in analogy with the Tolman–Ehrenfest theorem from general relativity. We study the local temperature distribution on the ground state of hopping fermionic systems on a curved background. The observed temperature tends to zero as the thermometer-system coupling g vanishes. Yet, for small but finite values of g, we show that the product of the observed local temperature and the logarithm of the local speed of light is approximately constant. Our predictions should be testable on ultracold atomic systems.

Efficient computations of quantum canonical Gibbs state in phase space.

The Gibbs canonical state, as a maximum entropy density matrix, represents a quantum system in equilibrium with a thermostat. This state plays an essential role in thermodynamics and serves as the initial condition for nonequilibrium dynamical simulations. We solve a long standing problem for computing the Gibbs state Wigner function with nearly machine accuracy by solving the Bloch equation directly in the phase space. Furthermore, the algorithms are provided yielding high quality Wigner distributions for pure stationary states as well as for Thomas-Fermi and Bose-Einstein distributions. The developed numerical methods furnish a long-sought efficient computation framework for nonequilibrium quantum simulations directly in the Wigner representation.

Canonical versus noncanonical equilibration dynamics of open quantum systems.

In statistical mechanics, any quantum system in equilibrium with its weakly coupled reservoir is described by a canonical state at the same temperature as the reservoir. Here, by studying the equilibration dynamics of a harmonic oscillator interacting with a reservoir, we evaluate microscopically the condition under which the equilibration to a canonical state is valid. It is revealed that the non-Markovian effect and the availability of a stationary state of the total system play a profound role in the equilibration. In the Markovian limit, the conventional canonical state can be recovered. In the non-Markovian regime, when the stationary state is absent, the system equilibrates to a generalized canonical state at an effective temperature; whenever the stationary state is present, the equilibrium state of the system cannot be described by any canonical state anymore. Our finding of the physical condition on such noncanonical equilibration might have significant impact on statistical physics. A physical scheme based on circuit QED is proposed to test our results.

Theory of time-resolved nonresonant x-ray scattering for imaging ultrafast coherent electron motion

Future ultrafast x-ray light sources might image ultrafast coherent electron motion in real-space and in real-time. For a rigorous understanding of such an imaging experiment, we extend the theory of non-resonant x-ray scattering to the time-domain. The role of energy resolution of the scattering detector is investigated in detail. We show that time-resolved non-resonant x-ray scattering with no energy resolution offers an opportunity to study time-dependent electronic correlations in non- equilibrium quantum systems. Furthermore, our theory presents a unified description of ultrafast x-ray scattering from electronic wave packets and the dynamical imaging of ultrafast dynamics using inelastic x-ray scattering by Abbamonte and co-workers. We examine closely the relation of the scattering signal and the linear density response of electronic wave packets. Finally, we demonstrate that time-resolved x-ray scattering from a crystal consisting of identical electronic wave packets recovers the instantaneous electron density.

Universal time fluctuations in near-critical out-of-equilibrium quantum dynamics.

Out-of-equilibrium quantum systems display complex temporal patterns. Such time fluctuations are generically exponentially small in the system volume and therefore can be safely ignored in most of the cases. However, if one consider small quench experiments, time fluctuations can be greatly enhanced. We show that time fluctuations may become stronger than other forms of equilibrium quantum fluctuations if the quench is performed close to a critical point. For sufficiently relevant operators the full distribution function of dynamically evolving observable expectation values becomes a universal function uniquely characterized by the critical exponents and the boundary conditions. At regular points of the phase diagram and for nonsufficiently relevant operators the distribution becomes Gaussian. Our predictions are confirmed by an explicit calculation on the quantum Ising model.

Energy and transverse momentum fluctuations in the equilibrium quantum systems

The fluctuations in the ideal quantum gases are studied using the strongly intensive measures Δ[A,B] and Σ[A,B] defined in terms of two extensive quantities A and B. In the present Letter, these extensive quantities are taken as the motional variable, A=X, the system energy E or transverse momentum PT, and number of particles, B=N. This choice is most often considered in studying the event-by-event fluctuations and correlations in high energy nucleus–nucleus collisions. The recently proposed special normalization ensures that Δ and Σ are dimensionless and equal to unity for fluctuations given by the independent particle model. In statistical mechanics, the grand canonical ensemble formulation within the Boltzmann approximation gives an example of independent particle model. Our results demonstrate the effects due to the Bose and Fermi statistics. Estimates of the effects of quantum statistics in the hadron gas at temperatures and chemical potentials typical for thermal models of hadron production in high energy collisions are presented. In the case of massless particles and zero chemical potential the Δ and Σ measures are calculated analytically.

Variational Principle in the Quantum Statistical Theory

In the present paper, a generalization of the method of partial summation of the expansion of the thermodynamical potential is proposed. This generalization allows one to obtain the corresponding equations for higher-order correlation matrices, as well as to formulate the variational method for their solution. We show that correlation matrices of equilibrium quantum system satisfy a variational principle for thermodynamic potential which is functional of these matrices that provides a thermodynamic consistency of the theory. This result is similar to a variational principle for correlation functions of classical systems.

Holographic evolution of the mutual information

We compute the time evolution of the mutual information in out of equilibrium quantum systems whose gravity duals are Vaidya spacetimes in three and four dimensions, which describe the formation of a black hole through the collapse of null dust. We find the holographic mutual information to be non monotonic in time and always monogamous in the ranges explored. We also find that there is a region in the configuration space where it vanishes at all times. We show that the null energy condition is a necessary condition for both the strong subadditivity of the holographic entanglement entropy and the monogamy of the holographic mutual information.

On the use of non-canonical quantum statistics

We develop a method using a coarse graining of the energy fluctuations of an equilibrium quantum system which produces simple parameterizations for the behaviour of the system. As an application, we use these methods to gain more understanding on the standard Boltzmann-Gibbs statistics and on the recently developed Tsallis statistics. We conclude on a discussion of the role of entropy and the maximum entropy principle in thermodynamics.

Using lattice methods in non-canonical quantum statistics

We define a natural coarse-graining procedure which can be applied to any closed equilibrium quantum system described by a density matrix ensemble and we show how the coarse-graining leads to the Gaussian and canonical ensembles. After this motivation, we present two ways of evaluating the Gaussian expectation values with lattice simulations. The first one is computationally demanding but general, whereas the second employs only canonical expectation values but it is applicable only for systems which are almost thermodynamical.

A Mathematical Relation Between the Potential of the Rotating Wave Approximation and an Estimation of the Fluctuation in Mori′s Theory

We show a mathematical relation between the potential of the rotating wave approximation and an estimation of the fluctuation in Mori′s theory of generalized Brownian motion. We use this relation to prove that, when an autocorrelation function R A ( t), t ∈ R concerning the Bogoliubov scalar product is given for a certain quantum observable A in an equilibrium quantum system in finite volume, we can obtain a Hamiltonian H RWA by making the rotating wave approximation from R A ( t) such that we reconstruct the autocorrelation function R A( t) in the system governed by H RWA.

Mori′s Memory Kernel Equation in Equilibrium Quantum Systems in Finite Volumes

For any observable in an equilibrium quantum system in a finite volume, we show a method of deriving Mori′s memory kernel equation, which consists of three components, Mori′s frequency, memory function, and fluctuation, i.e., we give a concrete mathematical expression for the three components in terms of an autocorrelation function of the observable.

Thermodynamic properties of an anharmonic fermionic oscillator

An exactly soluble fermionic system (anharmonic oscillator) is presented. Exact expressions for average quantities such as the number of particles, energy, and magnetization of an ensemble of quantum systems in equilibrium with a reservoir of particles and heat are derived. Critical curves of the average magnetization as a function of temperature are obtained for a lattice of interacting anharmonic oscillators within the mean field approximation. To clarify the role played by the parameters of the present model, the Ising model is studied as a limiting case.

On the exact mean-field description of continuous quantum systems in equilibrium II

The thermodynamic limit of free energy density of continuous quantum systems with bounded separable 2-body interactions, confined to a compact Riemann space, is investigated. It is shown that such systems, similarly as n-particle quantum systems in a bounded region in R v , which were discussed in an earlier article, can be exactly described in this limit as noninteracting systems subject to a mean field, equal to the averaged 2-body interaction. The form of equations for the mean field depends on the statistics obeyed by the particles. Conditions for the values of inverse temperature, at which the mean-field description may be invalid, are derived. As an application of the method, a system of coupled oscillators on a sphere in R v is investigated.

On the exact mean-field description of continuous quantum systems in equilibrium

On the exact mean-field description of continuous quantum systems in equilibrium