Dynamical quantum phase transitions: a review

Dynamical Transition Due to Feedback-Induced Skin Effect

The traditional dynamical phase transition refers to the appearance of singularities in an observable with respect to a control parameter for a late-time state or singularities in the rate function of the Loschmidt echo with respect to time. Here, we study the many-body dynamics in a continuously monitored free fermion system with conditional feedback under open boundary conditions. We surprisingly find a novel dynamical transition from a logarithmic scaling of the entanglement entropy to an area-law scaling as time evolves. The transition, which is noticeably different from the conventional dynamical phase transition, arises from the competition between the bulk dynamics and boundary skin effects. In addition, we find that while quasidisorder or disorder cannot drive a transition for the steady state, a transition occurs for the maximum entanglement entropy during the time evolution, which agrees well with the entanglement transition for the steady state of the dynamics under periodic boundary conditions.

Scaling and universality at ramped quench dynamical quantum phase transitions

The nonequilibrium dynamics of a periodically driven extended XY model, in the presence of linear time dependent magnetic field, is investigated using the notion of dynamical quantum phase transitions (DQPTs). Along the similar lines to the equilibrium phase transition, the main purpose of this work is to search fundamental concepts such as scaling and universality at the ramped quench DQPTs. We have shown that the critical points of the model, where the gap closing occurs, can be moved by tuning the driven frequency and consequently the presence of or absence of DQPTs can be flexibly controlled by adjusting the driven frequency. We have uncovered that, for a ramp across the single quantum critical point, the critical mode at which DQPTs occur is classified into three regions: the Kibble-Zurek (KZ) region, where the critical mode scales linearly with the square root of the sweep velocity, the pre-saturated (PS) region, and the saturated (S) region where the critical mode makes a plateau versus the sweep velocity. While for a ramp that crosses two critical points, the critical modes disclose just the KZ and PS regions. On the basis of numerical simulations, we find that the dynamical free energy scales linearly with time, as approaches to DQPT time, with the exponentν=1±0.01for all sweep velocities and driven frequencies.

Chebyshev polynomial approach to Loschmidt echo: Application to quench dynamics in two-dimensional quasicrystals

The understanding of quantum phase transitions in disordered or quasicrystal media is a central issue in condensed matter physics. In this paper we investigate localization properties of the two-dimensional Aubry-André model. We find that the system exhibits self-duality for the transformation between position and momentum spaces at a critical quasiperiodic potential, leading to an energy-independent Anderson transition. Most importantly, we present the implementation of an efficient and accurate algorithm based on the Chebyshev polynomial expansion of the Loschmidt echo, which characterizes the nonequilibrium dynamics of quantum quenched quasiperiodic systems. We analytically prove that the system under quench dynamics displays dynamical quantum phase transitions and further provide numerical verification by computing the polynomial expansion of the Loschmidt echo. Our results may provide insight into the realization of electronic transport in experiments.

Exact zeros of fidelity in finite-size systems as a signature for probing quantum phase transitions

The fidelity is widely used to detect quantum phase transitions, which is characterized by either a sharp change of fidelity or the divergence of fidelity susceptibility in the thermodynamical limit when the phase-driving parameter is across the transition point. In this work, we unveil that the occurrence of exact zeros of fidelity in finite-size systems can be applied to detect quantum phase transitions. In general, the fidelity F(γ,γ[over ̃]) always approaches zero in the thermodynamical limit, due to the Anderson orthogonality catastrophe, no matter whether the parameters of two ground states (γ and γ[over ̃]) are in the same phase or different phases, and this makes it difficult to distinguish whether an exact zero of fidelity exists by finite-size analysis. To overcome the influence of orthogonality catastrophe, we study finite-size systems with twist boundary conditions, which can be introduced by applying a magnetic flux, and demonstrate that exact zeros of fidelity can be always accessed by tuning the magnetic flux when γ and γ[over ̃] belong to different phases. On the other hand, no exact zero of fidelity can be observed if γ and γ[over ̃] are in the same phase. We demonstrate the applicability of our theoretical scheme by studying concrete examples, including the Su-Schrieffer-Heeger model, Creutz model, and Haldane model. Our work provides a practicable way to detect quantum phase transitions via the calculation of fidelity of finite-size systems.

The dynamical bulk boundary correspondence and dynamical quantum phase transitions in the Benalcazar-Bernevig-Hughes model

In this article we demonstrate that dynamical quantum phase transitions (DQPTs) occur for an exemplary higher order topological insulator, the Benalcazar-Bernevig-Hughes model, following quenches across a topological phase boundary. A dynamical bulk boundary correspondence is also seen both in the eigenvalues of the Loschmidt overlap matrix and the boundary return rate. The latter is found from a finite size scaling analysis for which the relative simplicity of the model is crucial. Contrary to the usual two dimensional case the DQPTs in this model show up as cusps in the return rate, as for a one dimensional model, rather than as cusps in its derivative as would be typical for a two dimensional model. We explain the origin of this behaviour.

Experimental Observation of the Yang-Lee Quantum Criticality in Open Quantum Systems

The Yang-Lee edge singularity was originally studied from the standpoint of mathematical foundations of phase transitions. However, direct observation of anomalous scaling with the negative scaling dimension has remained elusive due to an imaginary magnetic field required for the nonunitary criticality. We experimentally implement an imaginary magnetic field with an open quantum system of heralded single photons, directly measure the partition function, and demonstrate the Yang-Lee edge singularity via the quantum-classical correspondence. We also demonstrate unconventional scaling laws for finite-temperature quantum dynamics.

Indication of critical scaling in time during the relaxation of an open quantum system

Near continuous phase transitions, universal power-law scaling, characterized by critical exponents, emerges. This behavior reflects the singular responses of physical systems to continuous control parameters like temperature or external fields. Universal scaling extends to non-equilibrium dynamics in isolated quantum systems after a quench, where time takes the role of the control parameter. Our research unveils critical scaling in time also during the relaxation dynamics of an open quantum system. Here we experimentally realize such a system by the spin of individual Cesium atoms dissipatively coupled through spin-exchange processes to a bath of ultracold Rubidium atoms. Through a finite-size scaling analysis of the entropy dynamics via numerical simulations, we identify a critical point in time in the thermodynamic limit. This critical point is accompanied by the divergence of a characteristic length, which is described by critical exponents that turn out to be unaffected by system specifics.

Complex phase diagram and supercritical matter

The supercritical region is often described as uniform with no definite transitions. The distinct behaviors of the matter therein, e.g., as liquidlike and gaslike, however, suggest "supercritical boundaries." Here we provide a mathematical description of these phenomena by revisiting the Yang-Lee theory and introducing a complex phase diagram, specifically a four-dimensional (4D) one with complex T and p. While the traditional 2D phase diagram with real temperature T and pressure p values (the physical plane) lacks Lee-Yang (LY) zeros beyond the critical point, preventing the occurrence of criticality, the off-plane zeros in this 4D scenario still induce critical anomalies in various physical properties. This relationship is evidenced by the correlation between the Widom line and LY edges in van der Waals, 2D Ising model, and water. The diverged supercritical boundaries manifest the high-dimensional feature of the phase diagram: e.g., when LY zeros of complex T or p are projected onto the physical plane, boundaries defined by isobaric heat capacity C_{p} or isothermal compression coefficient K_{T} emanates. These results demonstrate the incipient phase transition nature of the supercritical matter.

Dynamical Detection of Topological Spectral Density

Local density of states (LDOS) is emerging as powerful means of exploring classical-wave topological phases. However, the current LDOS detection method remains rare and merely works for static situations. Here, we introduce a generic dynamical method to detect both the static and Floquet LDOS, based on an elegant connection between dynamics of chiral density and local spectral densities. Moreover, we find that the Floquet LDOS allows to measure out Floquet quasienergy spectra and identify topological π modes. As an example, we demonstrate that both the static and Floquet higher-order topological phase can be universally identified via LDOS detection, regardless of whether the topological corner modes are in energy gaps, bands, or continuous energy spectra without band gaps. Our study opens a new avenue utilizing dynamics to detect topological spectral densities and provides a universal approach of identifying static and Floquet topological phases.

Aperiodic dynamical quantum phase transition in multi-band Bloch Hamiltonian and its origin

We investigate the dynamical quantum phase transition (DQPT) in the multi-band Bloch Hamiltonian of the one-dimensional periodic Kitaev model, focusing on quenches from a Bloch band. By analyzing the dynamical free energy and Pancharatnam geometric phase (PGP), we show that the critical times of DQPTs deviate from periodic spacing due to the multi-band effect, contrasting with results from two-band models. We propose a geometric interpretation to explain this non-uniform spacing. Additionally, we clarify the conditions needed for DQPT occurrence in the multi-band Bloch Hamiltonian, highlighting that a DQPT only arises when the quench from the Bloch states collapses the band gap at the critical point. Moreover, we establish that the dynamical topological order parameter, defined by the winding number of the PGP, is not quantized but still exhibits discontinuous jumps at DQPT critical times due to periodic modulation. Additionally, we extend our analysis to mixed-state DQPT and find its absence at non-zero temperatures.

Effects of third-neighbor interactions on the frustrated quantum Ising model

We investigate thermal and quantum phase transitions of the J_{1}-J_{2}-J_{3} transverse Ising model on the square lattice. The model is studied within a cluster mean-field decoupling, which allows us to describe phase diagrams and the free-energy landscape in the neighborhood of phase transitions. Our findings indicate that the third-neighbor coupling (J_{3}) can affect the nature of phase transitions of the model. In particular, ferromagnetic third-neighbor couplings favor the onset of continuous order-disorder phase transitions, eliminating the tricritical point of the superantiferromagnetic-paramagnetic (SAFM-PM) phase boundary. On the other hand, the enhancement of frustration introduced by weak antiferromagnetic J_{3} gives rise to the staggered dimer phase favoring the onset of discontinuous classical phase transitions. Moreover, we find that quantum annealed criticality (QAC), which takes place when the classical discontinuous phase transition becomes critical by the enhancement of quantum fluctuations introduced by the transverse magnetic field, is eliminated from the SAFM-PM phase boundary by a relatively weak ferromagnetic J_{3}. Nevertheless, this change in the nature of phase transitions can still be observed in the presence of antiferromagnetic third-neighbor couplings being also found in the staggered-dimer phase boundary. Therefore, our findings support that QAC persists under the presence of frustrated antiferromagnetic third-neighbor couplings and is suppressed when these couplings are ferromagnetic, suggesting that frustration plays a central role in the onset of QAC.

Inhibition of non-Hermitian topological phase transitions in sliding photonic quasicrystals

Non-Hermitian (NH) quasicrystals have been a topic of increasing interest in current research, particularly in the context of NH topological physics and materials science. Recently, it has been suggested and experimentally demonstrated using synthetic photonic lattices that a class of NH quasicrystals can feature topological spectral phase transitions. Here we consider a NH quasicrystal with a uniformly-drifting (sliding) incommensurate potential and show that, owing to violation of Galilean invariance, the topological phase transition is washed out and the quasicrystal is always in the delocalized phase with an entirely real-energy spectrum. The results are illustrated by considering quantum walks in synthetic photonic lattices.

Quantum Mpemba Effect in a Quantum Dot with Reservoirs

We demonstrate the quantum Mpemba effect in a quantum dot coupled to two reservoirs, described by the Anderson model. We show that the system temperatures starting from two different initial values (hot and cold) cross each other at finite time (and thereby reverse their identities; i.e., hot becomes cold and vice versa) to generate thermal quantum Mpemba effect. The slowest relaxation mode believed to play the dominating role in Mpemba effect in Markovian systems does not contribute to such anomalous relaxation in the present model. In this connection, our analytical result provides necessary condition for producing quantum Mpemba effect in the density matrix elements of the quantum dot, as a combined effect of the remaining relaxation modes.

Anomalous correlation-induced dynamical phase transitions

The nonanalyticity of the Loschmidt echo at critical times in quantum quenched systems is termed as the dynamical quantum phase transition, extending the notion of quantum criticality to a nonequilibrium scenario. In this paper, we establish a new paradigm of dynamical phase transitions driven by a sudden change in the internal spatial correlations of the disorder potential in a low-dimensional disordered system. The quench dynamics between prequenched pure and postquenched random system Hamiltonian reveals an anomalous dynamical quantum phase transition triggered by an infinite disorder correlation in the modulation potential. The physical origin of the anomalous phenomenon is associated with the overlap between the two distinctly different extended states. Furthermore, we explore the quench dynamics between the prequenched random and postquenched pure system Hamiltonian. Interestingly, the quenched system undergoes dynamical quantum phase transitions for the prequench white-noise potential in the thermodynamic limit. In addition, the quench dynamics also shows a clear signature of the delocalization phase transition in the correlated Anderson model.

Dynamical Quantum Phase Transitions of the Schwinger Model: Real-Time Dynamics on IBM Quantum

Simulating the real-time dynamics of gauge theories represents a paradigmatic use case to test the hardware capabilities of a quantum computer, since it can involve non-trivial input states' preparation, discretized time evolution, long-distance entanglement, and measurement in a noisy environment. We implemented an algorithm to simulate the real-time dynamics of a few-qubit system that approximates the Schwinger model in the framework of lattice gauge theories, with specific attention to the occurrence of a dynamical quantum phase transition. Limitations in the simulation capabilities on IBM Quantum were imposed by noise affecting the application of single-qubit and two-qubit gates, which combine in the decomposition of Trotter evolution. The experimental results collected in quantum algorithm runs on IBM Quantum were compared with noise models to characterize the performance in the absence of error mitigation.

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.

Quantum Rabi dynamics of trapped atoms far in the deep strong coupling regime

The coupling of a two-level system with an electromagnetic field, whose fully quantized version is the quantum Rabi model, is among the central topics of quantum physics. When the coupling strength becomes large enough that the field mode frequency is reached, the deep strong coupling regime is approached, and excitations can be created from the vacuum. Here we demonstrate a periodic variant of the quantum Rabi model in which the two-level system is encoded in the Bloch band structure of cold rubidium atoms in optical potentials. With this method we achieve a Rabi coupling strength of 6.5 times the field mode frequency, which is far in the deep strong coupling regime, and observe a subcycle timescale raise in bosonic field mode excitations. In a measurement recorded in the basis of the coupling term of the quantum Rabi Hamiltonian, a freezing of dynamics is revealed for small frequency splittings of the two-level system, as expected when the coupling term dominates over all other energy scales, and a revival for larger splittings. Our work demonstrates a route to realize quantum-engineering applications in yet unexplored parameter regimes.

Stable Many-Body Resonances in Open Quantum Systems

Periodically driven quantum many-body systems exhibit novel nonequilibrium states, such as prethermalization, discrete time crystals, and many-body localization. Recently, the general mechanism of fractional resonances has been proposed that leads to slowing the many-body dynamics in systems with both U(1) and parity symmetry. Here, we show that fractional resonance is stable under local noise models. To corroborate our finding, we numerically study the dynamics of a small-scale Bose–Hubbard model that can readily be implemented in existing noisy intermediate-scale quantum (NISQ) devices. Our findings suggest a possible pathway toward a stable nonequilibrium state of matter, with potential applications of quantum memories for quantum information processing.

Statistical complexity and the road to equilibrium in many-body chaotic quantum systems

In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., the road to equilibrium. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally, by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states.

Classical theory of universal quantum work distribution in chaotic and disordered non-interacting Fermi systems

We present a universal theory of quantum work statistics in generic disordered non-interacting Fermi systems, displaying a chaotic single-particle spectrum captured by random matrix theory. We consider quantum quenches both within a driven random matrix formalism and in an experimentally accessible microscopic model, describing a two-dimensional disordered quantum dot. By extending Anderson’s orthogonality determinant formula to compute quantum work distribution, we demonstrate that work statistics is non-Gaussian and is characterized by a few dimensionless parameters. At longer times, quantum interference effects become irrelevant and the quantum work distribution is well-described in terms of a purely classical ladder model with a symmetric exclusion process in energy space, while bosonization and mean field methods provide accurate analytical expressions for the work statistics. Our results demonstrate the universality of work distribution in generic chaotic Fermi systems, captured by the analytical predictions of a mean field theory, and can be verified by calorimetric measurements on nanoscale circuits.

Critical Phenomena in Light-Matter Systems with Collective Matter Interactions

We study the quantum phase diagram and the onset of quantum critical phenomena in a generalized Dicke model that includes collective qubit-qubit interactions. By employing semiclassical techniques, we analyze the corresponding classical energy surfaces, fixed points, and the smooth Density of States as a function of the Hamiltonian parameters to determine quantum phase transitions in either the ground (QPT) or excited states (ESQPT). We unveil a rich phase diagram, the presence of new phases, and new transitions that result from varying the strength of the qubits interactions in independent canonical directions. We also find a correspondence between the phases emerging due to qubit interactions and those in their absence but with varying the strength of the non-resonant terms in the light-matter coupling. We expect our work to pave the way and stimulate the exploration of quantum criticality in systems combining matter-matter and light-matter interactions.

Reconstructing the dynamical quantum phase transitions via dimensional expansion in a generalized Su-Schrieffer-Heeger model

We know that a one-dimensional (1D) modulated system can simulate 2D topological states by expanding the dimension. This scenario provides a justifiable avenue to test the dilatation of the dynamical quantum phase transition (DQPT). Through a generalized Su-Schrieffer-Heeger model, we have shown how the Loschmidt echo, Fisher zero, and Dynamical topological order parameter (DTOP) transit from one to two dimensions. Owing to the introduced pseudomomentum, the derivative of the return rate does not always capture the DQPT well, but the Fisher zero and the DTOP can be treated as faithful indicators. A topology-independent parameter will also affect the occurrence of the DQPTs for quenches inside a given phase. Moreover, a comparison with the Haldane model owning the same phase diagram implies that a pair of fixed points will lead to different critical momentum distributions, thus different robustness, further reminding us that the correspondences between the equilibrium and dynamical phases transitions are multifarious.

Detection of Quantum Phase Boundary at Finite Temperatures in Integrable Spin Models

Quantum phase transitions occur when quantum fluctuation destroys order at zero temperature. With an increase in temperature, normally the thermal fluctuation wipes out any signs of this transition. Here, we identify a physical quantity that shows nonanalytic behavior at finite temperatures, when an interaction parameter is quenched across the line of quantum phase transition. This quantity under consideration is the long time limit of a form of quantum fidelity. Our treatment is analytic for XY chain and 2D Kitaev model and is numerical for a 3D Hamiltonian applicable to Weyl semimetals.

Numerical renormalization group study of the Loschmidt echo in Kondo systems

We study the dynamical properties of the one-channel and two-channel spin-1/2 Kondo models after quenching in Hamiltonian variables. Eigen spectrum of the initial and final Hamiltonians is calculated by using the numerical renormalization group method implemented within the matrix product states formalism. We consider multiple quench protocols in the considered Kondo systems, also in the presence of external magnetic field of different intensities. The main emphasis is put on the analysis of the behavior of the Loschmidt echo L(t), which measures the ability of the system's revival to its initial state after a quench. We show that the decay of the Loschmidt echo strongly depends on the type of quench and the ground state of the system. For the one-channel Kondo model, we show that L(t) decays as, [Formula: see text], where [Formula: see text] is the Kondo temperature, while for the two-channel Kondo model, we demonstrate that the decay is slower and given by [Formula: see text]. In addition, we also determine the dynamical behavior of the impurity's magnetization, which sheds light on identification of the relevant time scales in the system's dynamics.

Finite-Time Dynamical Phase Transition in Nonequilibrium Relaxation

We uncover a finite-time dynamical phase transition in the thermal relaxation of a mean-field magnetic model. The phase transition manifests itself as a cusp singularity in the probability distribution of the magnetization that forms at a critical time. The transition is due to a sudden switch in the dynamics, characterized by a dynamical order parameter. We derive a dynamical Landau theory for the transition that applies to a range of systems with scalar, parity-invariant order parameters. Close to criticalilty, our theory reveals an exact mapping between the dynamical and equilibrium phase transitions of the magnetic model, and implies critical exponents of mean-field type. We argue that interactions between nearby saddle points, neglected at the mean-field level, may lead to critical, spatiotemporal fluctuations of the order parameter, and thus give rise to novel, dynamical critical phenomena.

Connecting Scrambling and Work Statistics for Short-Range Interactions in the Harmonic Oscillator

We investigate the relationship between information scrambling and work statistics after a quench for the paradigmatic example of short-range interacting particles in a one-dimensional harmonic trap, considering up to five particles numerically. In particular, we find that scrambling requires finite interactions, in the presence of which the long-time average of the squared commutator for the individual canonical operators is directly proportional to the variance of the work probability distribution. In addition to the numerical results, we outline the mathematical structure of the N-body system which leads to this outcome. We thereby establish a connection between the scrambling properties and the induced work fluctuations, with the latter being an experimental observable that is directly accessible in modern cold-atom experiments.

Work statistics across a quantum critical surface

We study the universality of work statistics of a system quenched through a quantum critical surface. By using the adiabatic perturbation theory, we obtain the general scaling behavior for all cumulants of work. These results extend the studies of Kibble-Zurek mechanism scaling of work statistics from an isolated quantum critical point to a critical surface. As an example, we study the scaling behavior of work statistics in the two-dimensional (2D) Kitaev honeycomb model featured with a critical line. By utilizing the trace formula for quadratic fermionic Hamiltonian, we obtain the exact characteristic function of work of the 2D Kitaev model at zero temperature. The results confirm our prediction.

The Classical–Quantum Passage: A van der Waals Description

We undertake a van der Waals inquiry at very low temperatures so as to find signs of a classical–quantum frontier. We investigate the relation of such signs with the celebrated van der Waals gas–liquid transition. We specialize the discussion with respect to the noble gases. For such purpose, we use rather novel thermal statistical quantifiers such as the disequilibrium, the statistical complexity, and the thermal efficiency. Fruitful insights are thereby gained.

Scaling Laws for the Sensitivity Enhancement of Non-Gaussian Spin States

We identify the large-N scaling of the metrological quantum gain offered by over-squeezed spin states that are accessible by one-axis twisting, as a function of the preparation time. We further determine how the scaling is modified by relevant decoherence processes and predict a discontinuous change of the quantum gain at a critical preparation time that depends on the noise. Our analytical results provide recipes for optimal and feasible implementations of quantum enhancements with non-Gaussian spin states in existing experiments, well beyond the reach of spin squeezing.

Topological holographic quench dynamics in a synthetic frequency dimension

The notion of topological phases extended to dynamical systems stimulates extensive studies, of which the characterization of nonequilibrium topological invariants is a central issue and usually necessitates the information of quantum dynamics in both the time and momentum dimensions. Here, we propose the topological holographic quench dynamics in synthetic dimension, and also show it provides a highly efficient scheme to characterize photonic topological phases. A pseudospin model is constructed with ring resonators in a synthetic lattice formed by frequencies of light, and the quench dynamics is induced by initializing a trivial state, which evolves under a topological Hamiltonian. Our key prediction is that the complete topological information of the Hamiltonian is encoded in quench dynamics solely in the time dimension, and is further mapped to lower-dimensional space, manifesting the holographic features of the dynamics. In particular, two fundamental time scales emerge in the dynamical evolution, with one mimicking the topological band on the momentum dimension and the other characterizing the residue time evolution of the state after the quench. For this, a universal duality between the quench dynamics and the equilibrium topological phase of the spin model is obtained in the time dimension by extracting information from the field evolution dynamics in modulated ring systems in simulations. This work also shows that the photonic synthetic frequency dimension provides an efficient and powerful way to explore the topological nonequilibrium dynamics.

Exceptional dynamical quantum phase transitions in periodically driven systems

Extending notions of phase transitions to nonequilibrium realm is a fundamental problem for statistical mechanics. While it was discovered that critical transitions occur even for transient states before relaxation as the singularity of a dynamical version of free energy, their nature is yet to be elusive. Here, we show that spontaneous symmetry breaking can occur at a short-time regime and causes universal dynamical quantum phase transitions in periodically driven unitary dynamics. Unlike conventional phase transitions, the relevant symmetry is antiunitary: its breaking is accompanied by a many-body exceptional point of a nonunitary operator obtained by space-time duality. Using a stroboscopic Ising model, we demonstrate the existence of distinct phases and unconventional singularity of dynamical free energy, whose signature can be accessed through quasilocal operators. Our results open up research for hitherto unknown phases in short-time regimes, where time serves as another pivotal parameter, with their hidden connection to nonunitary physics.

Analytic approaches to periodically driven closed quantum systems: methods and applications

We present a brief overview of some of the analytic perturbative techniques for the computation of the Floquet Hamiltonian for a periodically driven, or Floquet, quantum many-body system. The key technical points about each of the methods discussed are presented in a pedagogical manner. They are followed by a brief account of some chosen phenomena where these methods have provided useful insights. We provide an extensive discussion of the Floquet-Magnus (FM) expansion, the adiabatic-impulse approximation, and the Floquet perturbation theory. This is followed by a relatively short discourse on the rotating wave approximation, a FM resummation technique and the Hamiltonian flow method. We also provide a discussion of some open problems which may possibly be addressed using these methods.

Floquet dynamical quantum phase transitions in periodically quenched systems

Dynamical quantum phase transitions (DQPTs) are characterized by nonanalytic behaviors of physical observables as functions of time. When a system is subject to time-periodic modulations, the nonanalytic signatures of its observables could recur periodically in time, leading to the phenomena of Floquet DQPTs. In this work, we systematically explore Floquet DQPTs in a class of periodically quenched one-dimensional system with chiral symmetry. By tuning the strength of quench, we find multiple Floquet DQPTs within a single driving period, with more DQPTs being observed when the system is initialized in Floquet states with larger topological invariants. Each Floquet DQPT is further accompanied by the quantized jump of a dynamical topological order parameter, whose values remain quantized in time if the underlying Floquet system is prepared in a gapped topological phase. The theory is demonstrated in a piecewise quenched lattice model, which possesses rich Floquet topological phases and is readily realizable in quantum simulators like the nitrogen-vacancy center in diamonds. Our discoveries thus open a new perspective for the Floquet engineering of DQPTs and the dynamical detection of topological phase transitions in Floquet systems.

Quantum aging and dynamical universality in the long-range O(N→∞) model

Quantum quenches to or near criticality give rise to the phenomenon of aging, manifested by glassylike dynamics at short times and far from equilibrium. The recent surge of interest in the dynamics of quantum many-body systems has rejuvenated interest in this phenomenon. Motivated by the ubiquitous long-range interactions in emerging experimental platforms, it is vital to study quantum aging in such settings. In this paper, we investigate the dynamical universality and aging in the d-dimensional O(N) model with the long-range coupling 1/x^{d+σ} and in the mean-field limit N→∞ that allows an exact treatment. An immediate consequence of long-range coupling is the emergence of nonlinear light cones. We focus on the correlation and response functions, and identify a rich scaling behavior depending on how the corresponding space-time positions are located relative to each other, via a local light cone, and to the time of the quench via a global quench light cone. We determine the initial-slip exponent that governs the short-time dependence of two-point functions. We highlight the qualitative features of aging due to the long-range coupling, in particular in the region outside the light cones. As an important consequence of long-range coupling, the correlation function decays as 1/x^{d+σ} outside the quench light cone while increasing polynomially with the total time after quench. This is while, for short-time differences, the two-time response function "equilibrates" at all distances even outside this light cone. Our analytic findings are in excellent agreement with exact numerics, and provide a useful benchmark for modern experimental platforms with long-range interactions.

Observing Dynamical Quantum Phase Transitions through Quasilocal String Operators

We analyze signatures of the dynamical quantum phase transitions in physical observables. In particular, we show that both the expectation value and various out of time order correlation functions of the finite length product or string operators develop cusp singularities following quench protocols, which become sharper and sharper as the string length increases. We illustrated our ideas analyzing both integrable and nonintegrable one-dimensional Ising models showing that these transitions are robust both to the details of the model and to the choice of the initial state.

Cavity-QED Quantum Simulator of Dynamical Phases of a Bardeen-Cooper-Schrieffer Superconductor

We propose to simulate dynamical phases of a BCS superconductor using an ensemble of cold atoms trapped in an optical cavity. Effective Cooper pairs are encoded via the internal states of the atoms, and attractive interactions are realized via the exchange of virtual photons between atoms coupled to a common cavity mode. Control of the interaction strength combined with a tunable dispersion relation of the effective Cooper pairs allows exploration of the full dynamical phase diagram of the BCS model as a function of system parameters and the prepared initial state. Our proposal paves the way for the study of the nonequilibrium features of quantum magnetism and superconductivity by harnessing atom-light interactions in cold atomic gases.

Classification of dark solitons via topological vector potentials

Dark solitons are some of the most interesting nonlinear excitations and are considered to be the one-dimensional topological analogs of vortices. However, in contrast to their two-dimensional vortex counterparts, the topological characteristics of a dark soliton are far from fully understood because the topological charge defined according to the phase jump cannot reflect its essential property. Here, similar to the complex extension used in the exploration of the partition-function zeros to depict thermodynamic states, we extend the complex coordinate space to explore the density zeros of dark solitons. Surprisingly we find that these zeros constitute some pointlike magnetic fields, each of which has a quantized magnetic flux of elementary π. The corresponding vector potential fields demonstrate the topology of the Wess-Zumino term and can depict the essential characteristics of dark solitons. Then we classify the dark solitons according to the Euler characteristic of the topological manifold of the vector potential fields. Our study not only reveals the topological features of dark solitons but can also be applied to explore and identify new dark solitons with high topological complexity.

Experimental optimal generation of hybrid entangled states in photonic quantum walks

While the existence of disorders is commonly believed to weaken the unique properties of quantum systems, recent progress has predicted that it can exhibit a counterintuitive enhanced effect on the behavior of entanglement generation, which is even independent of the chosen initial conditions and physical platforms. However, to achieve a maximally entangled state in such disordered quantum systems, the key limitation of this is the scarcity of an infinite coherence time, which makes its experimental realization challenging. Here, we experimentally investigate the entanglement entropy dynamics in a photonic quantum walk with disorders in time. Through the incorporation of a classic optimization algorithm, we experimentally demonstrate that such disordered systems can relax to a high-entanglement hybrid state at any given time step. Moreover, this prominent entangling ability is universal for a wide variety of initial conditions. Our results may inspire achieving a well-controlled entanglement generator for quantum computation and information tasks.

Work statistics and symmetry breaking in an excited-state quantum phase transition

We examine how the presence of an excited-state quantum phase transition manifests in the dynamics of a many-body system subject to a sudden quench. Focusing on the Lipkin-Meshkov-Glick model initialized in the ground state of the ferromagnetic phase, we demonstrate that the work probability distribution displays non-Gaussian behavior for quenches in the vicinity of the excited-state critical point. Furthermore, we show that the entropy of the diagonal ensemble is highly susceptible to critical regions, making it a robust and practical indicator of the associated spectral characteristics. We assess the role that symmetry breaking has on the ensuing dynamics, highlighting that its effect is only present for quenches beyond the critical point. Finally, we show that similar features persist when the system is initialized in an excited state and briefly explore the behavior for initial states in the paramagnetic phase.

Entanglement View of Dynamical Quantum Phase Transitions

The analogy between an equilibrium partition function and the return probability in many-body unitary dynamics has led to the concept of dynamical quantum phase transition (DQPT). DQPTs are defined by nonanalyticities in the return amplitude and are present in many models. In some cases, DQPTs can be related to equilibrium concepts, such as order parameters, yet their universal description is an open question. In this Letter, we provide first steps toward a classification of DQPTs by using a matrix product state description of unitary dynamics in the thermodynamic limit. This allows us to distinguish the two limiting cases of "precession" and "entanglement" DQPTs, which are illustrated using an analytical description in the quantum Ising model. While precession DQPTs are characterized by a large entanglement gap and are semiclassical in their nature, entanglement DQPTs occur near avoided crossings in the entanglement spectrum and can be distinguished by a complex pattern of nonlocal correlations. We demonstrate the existence of precession and entanglement DQPTs beyond Ising models, discuss observables that can distinguish them, and relate their interplay to complex DQPT phenomenology.

Dynamic Observation of Topological Soliton States in a Programmable Nanomechanical Lattice

Topological soliton states, existing in the topological structures with edge defect or interface defect, are usually studied under steady state. Here, we experimentally observe the dynamic processes of the generation and the extinction of such soliton states in the Su-Schrieffer-Heeger model. The different topological structures are implemented on a programmable nanomechanical lattice, consisting of eight adjacent string resonators which are parametrically coupled by manipulation voltages. Moreover, the beating and localization behaviors at different topological interfaces are also observed in the same device. These results explicitly exhibit the dynamic processes of topological soliton states, which reveal real potential toward integrated multifunctional topological materials.

Feedback induced magnetic phases in binary Bose-Einstein condensates

Weak measurement in tandem with real-time feedback control is a new route toward engineering novel nonequilibrium quantum matter. Here we develop a theoretical toolbox for quantum feedback control of multicomponent Bose-Einstein condensates (BECs) using backaction-limited weak measurements in conjunction with spatially resolved feedback. Feedback in the form of a single-particle potential can introduce effective interactions that enter into the stochastic equation governing system dynamics. The effective interactions are tunable and can be made analogous to Feshbach resonances-spin independent and spin dependent-but without changing atomic scattering parameters. Feedback cooling prevents runaway heating due to measurement backaction and we present an analytical model to explain its effectiveness. We showcase our toolbox by studying a two-component BEC using a stochastic mean-field theory, where feedback induces a phase transition between easy-axis ferromagnet and spin-disordered paramagnet phases. We present the steady-state phase diagram as a function of intrinsic and effective spin-dependent interaction strengths. Our result demonstrates that closed-loop quantum control of Bose-Einstein condensates is a powerful tool for quantum engineering in cold-atom systems.

Unified Theory to Characterize Floquet Topological Phases by Quench Dynamics

The conventional characterization of periodically driven systems usually necessitates the time-domain information beyond Floquet bands, hence lacking universal and direct schemes of measuring Floquet topological invariants. Here we propose a unified theory, based on quantum quenches, to characterize generic d-dimensional Floquet topological phases in which the topological invariants are constructed with only minimal information of the static Floquet bands. For a d-dimensional phase that is initially static and trivial, we introduce the quench dynamics by suddenly turning on the periodic driving. We show that the quench dynamics exhibits emergent topological patterns in (d-1)-dimensional momentum subspaces where Floquet bands cross, from which the Floquet topological invariants are directly obtained. This result provides a simple and unified characterization in which one can extract the number of conventional and anomalous Floquet boundary modes and identify the topologically protected singularities in the phase bands. These applications are illustrated with one- and two-dimensional models that are readily accessible in cold-atom experiments. Our study opens a new framework for the characterization of Floquet topological phases.

Dynamical Phase Transitions in Dissipative Quantum Dynamics with Quantum Optical Realization

We study dynamical phase transitions (DPT) in the driven and damped Dicke model, realizable for example by a driven atomic ensemble collectively coupled to a damped cavity mode. These DPTs are characterized by nonanalyticities of certain observables, primarily the overlap of time evolved and initial state. Even though the dynamics is dissipative, this phenomenon occurs for a wide range of parameters and no fine-tuning is required. Focusing on the state of the "atoms" in the limit of a bad cavity, we are able to asymptotically evaluate an exact path integral representation of the relevant overlaps. The DPTs then arise by minimization of a certain action function, which is related to the large deviation theory of a classical stochastic process. Finally, we present a scheme which allows a measurement of the DPT in a cavity-QED setup.

Experimental Observation of Equilibrium and Dynamical Quantum Phase Transitions via Out-of-Time-Ordered Correlators

The out-of-time-ordered correlators (OTOC), a fundamental concept for quantifying quantum information scrambling, has recently been suggested to be an order parameter to dynamically detect both equilibrium quantum phase transitions (EQPTs) and dynamical quantum phase transitions (DQPTs). Here we report the first experimental observation of EQPTs and DQPTs in a quantum spin chain via quench dynamics of OTOC on a nuclear magnetic resonance quantum simulator. We observe that the quench dynamics of the OTOC can unambiguously detect the DQPTs and the equilibrium critical point, while conventional order parameters such as the longitudinal magnetization can not. Moreover, we investigate the two-body correlations throughout the quench dynamics, and find that OTOC can extract the equilibrium critical point with higher accuracy and is more robust to decoherence than that of two-body correlation. Our experiment paves a way for experimentally investigating DQPTs through OTOCs and for studying the EQPTs through the nonequilibrium quantum quench dynamics with quantum simulators.

Sub-picosecond thermalization dynamics in condensation of strongly coupled lattice plasmons

Bosonic condensates offer exciting prospects for studies of non-equilibrium quantum dynamics. Understanding the dynamics is particularly challenging in the sub-picosecond timescales typical for room temperature luminous driven-dissipative condensates. Here we combine a lattice of plasmonic nanoparticles with dye molecule solution at the strong coupling regime, and pump the molecules optically. The emitted light reveals three distinct regimes: one-dimensional lasing, incomplete stimulated thermalization, and two-dimensional multimode condensation. The condensate is achieved by matching the thermalization rate with the lattice size and occurs only for pump pulse durations below a critical value. Our results give access to control and monitoring of thermalization processes and condensate formation at sub-picosecond timescale.

Topological Invariants for Quantum Quench Dynamics from Unitary Evolution

Recent experiments began to explore the topological properties of quench dynamics, i.e., the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well-established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. However, it only works for a topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench, regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.

Exact Nonequilibrium Steady State of Open XXZ/XYZ Spin-1/2 Chain with Dirichlet Boundary Conditions

We investigate a dissipatively driven XYZ spin-1/2 chain in the Zeno limit of strong dissipation, described by the Lindblad master equation. The nonequilibrium steady state is expressed in terms of a matrix product ansatz using novel site-dependent Lax operators. The components of Lax operators satisfy a simple set of linear recurrence equations that generalize the defining algebraic relations of the quantum group U_{q}(sl_{2}). We reveal connection between the nonequilibrium steady state of the nonunitary dynamics and the respective integrable model with edge magnetic fields, described by coherent unitary dynamics.