Entropy, chaos, and excited-state quantum phase transitions in the Dicke model

Quantum Information Scrambling in Adiabatically Driven Critical Systems

Quantum information scrambling refers to the spread of the initially stored information over many degrees of freedom of a quantum many-body system. Information scrambling is intimately linked to the thermalization of isolated quantum many-body systems, and has been typically studied in a sudden quench scenario. Here, we extend the notion of quantum information scrambling to critical quantum many-body systems undergoing an adiabatic evolution. In particular, we analyze how the symmetry-breaking information of an initial state is scrambled in adiabatically driven integrable systems, such as the Lipkin-Meshkov-Glick and quantum Rabi models. Following a time-dependent protocol that drives the system from symmetry-breaking to a normal phase, we show how the initial information is scrambled, even for perfect adiabatic evolutions, as indicated by the expectation value of a suitable observable. We detail the underlying mechanism for quantum information scrambling, its relation to ground- and excited-state quantum phase transitions, and quantify the degree of scrambling in terms of the number of eigenstates that participate in the encoding of the initial symmetry-breaking information. While the energy of the final state remains unaltered in an adiabatic protocol, the relative phases among eigenstates are scrambled, and so is the symmetry-breaking information. We show that a potential information retrieval, following a time-reversed protocol, is hindered by small perturbations, as indicated by a vanishingly small Loschmidt echo and out-of-time-ordered correlators. The reported phenomenon is amenable for its experimental verification, and may help in the understanding of information scrambling in critical quantum many-body systems.

Excited-state quantum phase transitions and the entropy of the work distribution in the anharmonic Lipkin-Meshkov-Glick model

Studying the implications and characterizations of the excited-state quantum phase transitions (ESQPTs) would enable us to understand various phenomena observed in quantum many-body systems. In this work, we delve into the affects and characterizations of the ESQPTs in the anharmonic Lipkin-Meshkov-Glick (LMG) model by means of the entropy of the quantum work distribution. The entropy of the work distribution measures the complexity of the work distribution and behaves as a valuable tool for analyzing nonequilibrium work statistics. We show that the entropy of the work distribution captures salient signatures of the underlying ESQPTs in the model. In particular, a detailed analysis of the scaling behavior of the entropy verifies that it not only acts as a witness of the ESQPTs but also reveals the difference between different types of ESQPTs. We further demonstrate that the work distribution entropy also behaves as a powerful tool for understanding the features and differences of ESQPTs in the energy space. Our results provide further evidence of the usefulness of the entropy of the work distribution for investigating various phase transitions in quantum many-body systems and open up a promising way for experimentally exploring the signatures of ESQPTs.

Mixed eigenstates in the Dicke model: Statistics and power-law decay of the relative proportion in the semiclassical limit

How the mixed eigenstates vary when approaching the semiclassical limit in mixed-type many-body quantum systems is an interesting but still less known question. Here, we address this question in the Dicke model, a celebrated many-body model that has a well defined semiclassical limit and undergoes a transition to chaos in both quantum and classical cases. Using the Husimi function, we show that the eigenstates of the Dicke model with mixed-type classical phase space can be classified into different types. To quantitatively characterize the types of eigenstates, we study the phase space overlap index, which is defined in terms of the Husimi function. We look at the probability distribution of the phase space overlap index and investigate how it changes with increasing system size, that is, when approaching the semiclassical limit. We show that increasing the system size gives rise to a power-law decay in the behavior of the relative proportion of mixed eigenstates. Our findings shed more light on the properties of eigenstates in mixed-type many-body systems and suggest that the principle of uniform semiclassical condensation of Husimi functions should also be valid for many-body quantum systems.

Analysis of chaos and regularity in the open Dicke model

We present an analysis of chaos and regularity in the open Dicke model, when dissipation is due to cavity losses. Due to the infinite Liouville space of this model, we also introduce a criterion to numerically find a complex spectrum which approximately represents the system spectrum. The isolated Dicke model has a well-defined classical limit with two degrees of freedom. We select two case studies where the classical isolated system shows regularity and where chaos appears. To characterize the open system as regular or chaotic, we study regions of the complex spectrum taking windows over the absolute value of its eigenvalues. Our results for this infinite-dimensional system agree with the Grobe-Haake-Sommers (GHS) conjecture for Markovian dissipative open quantum systems, finding the expected 2D Poisson distribution for regular regimes, and the distribution of the Ginibre unitary ensemble (GinUE) for the chaotic ones, respectively.

Spectral kissing and its dynamical consequences in the squeeze-driven Kerr oscillator

Transmon qubits are the predominant element in circuit-based quantum information processing, such as existing quantum computers, due to their controllability and ease of engineering implementation. But more than qubits, transmons are multilevel nonlinear oscillators that can be used to investigate fundamental physics questions. Here, they are explored as simulators of excited state quantum phase transitions (ESQPTs), which are generalizations of quantum phase transitions to excited states. We show that the spectral kissing (coalescence of pairs of energy levels) experimentally observed in the effective Hamiltonian of a driven SNAIL-transmon is an ESQPT precursor. We explore the dynamical consequences of the ESQPT, which include the exponential growth of out-of-time-ordered correlators, followed by periodic revivals, and the slow evolution of the survival probability due to localization. These signatures of ESQPT are within reach for current superconducting circuits platforms and are of interest to experiments with cold atoms and ion traps.

Theory of Dynamical Phase Transitions in Quantum Systems with Symmetry-Breaking Eigenstates

We present a theory for the two kinds of dynamical quantum phase transitions, termed DPT-I and DPT-II, based on a minimal set of symmetry assumptions. In the special case of collective systems with infinite-range interactions, both are triggered by excited-state quantum phase transitions. For quenches below the critical energy, the existence of an additional conserved charge, identifying the corresponding phase, allows for a nonzero value of the dynamical order parameter characterizing DPTs-I, and precludes the main mechanism giving rise to nonanalyticities in the return probability, trademark of DPTs-II. We propose a statistical ensemble describing the long-time averages of order parameters in DPTs-I, and provide a theoretical proof for the incompatibility of the main mechanism for DPTs-II with the presence of this additional conserved charge. Our results are numerically illustrated in the fully connected transverse-field Ising model, which exhibits both kinds of dynamical phase transitions. Finally, we discuss the applicability of our theory to systems with finite-range interactions, where the phenomenology of excited-state quantum phase transitions is absent. We illustrate our findings by means of numerical calculations with experimentally relevant initial states.

Chaos and Thermalization in the Spin-Boson Dicke Model

We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called "efficient basis" over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis.

Signatures of excited-state quantum phase transitions in quantum many-body systems: Phase space analysis

Using the Husimi quasiprobability distribution, we investigate the phase space signatures of excited-state quantum phase transitions (ESQPTs) in the Lipkin-Meshkov-Glick and coupled top models. We show that the ESQPT is evinced by the dynamics of the Husimi function, that exhibits a distinct time dependence in the different ESQPT phases. We also discuss how to identify the ESQPT signatures from the long-time averaged Husimi function and its associated marginal distributions. Moreover, from the calculated second moment and Wherl entropy of the long-time averaged Husimi function, we estimate the critical points of the ESQPT in both models, obtaining a good agreement with analytical (mean field) results. We provide a firm evidence that phase space methods are both a new probe for the detection and a valuable tool for the study of ESQPTs.

Constant of Motion Identifying Excited-State Quantum Phases

We propose that a broad class of excited-state quantum phase transitions (ESQPTs) gives rise to two different excited-state quantum phases. These phases are identified by means of an operator C[over ^], which is a constant of motion in only one of them. Hence, the ESQPT critical energy splits the spectrum into one phase where the equilibrium expectation values of physical observables crucially depend on this constant of motion and another phase where the energy is the only relevant thermodynamic magnitude. The trademark feature of this operator is that it has two different eigenvalues ±1, and, therefore, it acts as a discrete symmetry in the first of these two phases. This scenario is observed in systems with and without an additional discrete symmetry; in the first case, C[over ^] explains the change from degenerate doublets to nondegenerate eigenlevels upon crossing the critical line. We present stringent numerical evidence in the Rabi and Dicke models, suggesting that this result is exact in the thermodynamic limit, with finite-size corrections that decrease as a power law.

Characterizing the Lipkin-Meshkov-Glick model excited-state quantum phase transition using dynamical and statistical properties of the diagonal entropy

Using the diagonal entropy, we analyze the dynamical signatures of the Lipkin-Meshkov-Glick model excited-state quantum phase transition (ESQPT). We first show that the time evolution of the diagonal entropy behaves as an efficient indicator of the presence of an ESQPT. We also compute the probability distribution of the diagonal entropy values over a certain time interval and we find that the resulting distribution provides a clear distinction between the different phases of ESQPT. Moreover, we observe that the probability distribution of the diagonal entropy at the ESQPT critical point has a universal form, well described by a beta distribution, and that a reliable detection of the ESQPT can be obtained from the diagonal entropy central moments.

Statistical properties of the localization measure of chaotic eigenstates in the Dicke model

The quantum localization is one of the remarkable phenomena in the studies of quantum chaos and plays an important role in various contexts. Thus, an understanding of the properties of quantum localization is essential. In spite of much effort dedicated to investigating the manifestations of localization in the time-dependent systems, the features of localization in time-independent systems are still less explored, particularly in quantum systems which correspond to the classical systems with smooth Hamiltonian. In this work, we present such a study for a quantum many-body system, namely, the Dicke model. The classical counterpart of the Dicke model is given by a smooth Hamiltonian with two degrees of freedom. We examine the signatures of localization in its chaotic eigenstates. We show that the entropy localization measure, which is defined in terms of the information entropy of Husimi distribution, behaves linearly with the participation number, a measure of the degree of localization of a quantum state. We further demonstrate that the localization measure probability distribution is well described by the β distribution. We also find that the averaged localization measure is linearly related to the level repulsion exponent, a widely used quantity to characterize the localization in chaotic eigenstates. Our findings extend the previous results in billiards to the quantum many-body system with classical counterpart described by a smooth Hamiltonian, and they indicate that the properties of localized chaotic eigenstates are universal.

Anomalous Thermalization in Quantum Collective Models

We show that apparently thermalized states still store relevant amounts of information about their past, information that can be tracked by experiments involving nonequilibrium processes. We provide a condition for the microcanonical quantum Crook's theorem, and we test it by means of numerical experiments. In the Lipkin-Meshkov-Glick model, two different procedures leading to the same equilibrium states give rise to different statistics of work in nonequilibrium processes. In the Dicke model, two different trajectories for the same nonequilibrium protocol produce different statistics of work. Microcanonical averages provide the correct results for the expectation values of physical observables in all the cases; the microcanonical quantum Crook's theorem fails in some of them. We conclude that testing quantum fluctuation theorems is mandatory to verify if a system is properly thermalized.

Probing the excited-state quantum phase transition through statistics of Loschmidt echo and quantum work

By analyzing the probability distributions of the Loschmidt echo (LE) and quantum work, we examine the nonequilibrium effects of a quantum many-body system, which exhibits an excited-state quantum phase transition (ESQPT). We find that depending on the value of the controlling parameter the distribution of the LE displays different patterns. At the critical point of the ESQPT, both the averaged LE and the averaged work show a cusplike shape. Furthermore, by employing the finite-size scaling analysis of the averaged work, we obtain the critical exponent of the ESQPT. Finally, we show that at the critical point of ESQPT the eigenstate is a highly localized state, further highlighting the influence of the ESQPT on the properties of the many-body system.

Nonergodicity in the Anisotropic Dicke Model

We study the ergodic-nonergodic transition in a generalized Dicke model with independent corotating and counterrotating light-matter coupling terms. By studying level statistics, the average ratio of consecutive level spacings, and the quantum butterfly effect (out-of-time correlation) as a dynamical probe, we show that the ergodic-nonergodic transition in the Dicke model is a consequence of the proximity to the integrable limit of the model when one of the couplings is set to zero. This can be interpreted as a hint for the existence of a quantum analogue of the classical Kolmogorov-Arnold-Moser theorem. In addition, we show that there is no intrinsic relation between the ergodic-nonergodic transition and the precursors of the normal-superradiant quantum phase transition.

Thermodynamics and superradiant phase transitions in a three-level Dicke model

We analyze the thermodynamic properties of a generalized Dicke model, i.e., a collection of three-level systems interacting with two bosonic modes. We show that at finite temperatures the system undergoes first-order phase transitions only, which is in contrast to the zero-temperature case where a second-order phase transition exists as well. We discuss the free energy and prominent expectation values. The limit of vanishing temperature is discussed as well.