We present a unifying description of dynamical phase transitions in the unitary evolution of an isolated quantum particle and the dissipative relaxation of a classical Brownian particle, based on a dynamical generat- ing function. In the semiclassical and weak-noise limits, Fisher zeros of this function condense in the complex- time plane and reach the corresponding physical axes, producing dynamical phase transitions through a competition between return trajectories. This establishes a direct connection between quantum and classical (finite-time) dynamical phase transitions, where the semiclassical limit plays the role of the thermodynamic limit. In particular, the established link naturally provides a classical version of the Loschmidt amplitude and shows how dynamical quantum phase transitions, typically associated with isolated many-body systems, can arise in a single-particle quantum system. The dynamical phases are distinguished by a unifying, trajectory- based order parameter, realized as a classical correlation function and a sequential quantum weak value.
We present a unifying description of dynamical phase transitions in the unitary evolution of an isolated quantum particle and the dissipative relaxation of a classical Brownian particle, based on a dynamical generat- ing function. In the semiclassical and weak-noise limits, Fisher zeros of this function condense in the complex- time plane and reach the corresponding physical axes, producing dynamical phase transitions through a competition between return trajectories. This establishes a direct connection between quantum and classical (finite-time) dynamical phase transitions, where the semiclassical limit plays the role of the thermodynamic limit. In particular, the established link naturally provides a classical version of the Loschmidt amplitude and shows how dynamical quantum phase transitions, typically associated with isolated many-body systems, can arise in a single-particle quantum system. The dynamical phases are distinguished by a unifying, trajectory- based order parameter, realized as a classical correlation function and a sequential quantum weak value.
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.
This chapter addresses phase transitions and dynamic scaling occurring in systems comprised of interacting indistinguishable quantum particles, for which entanglement correlations are crucial. It first describes how the dynamics (in real time) and thermodynamics (in imaginary time) of quantum many-particle Hamiltonians can be mapped onto field theories based on coherent-state path integrals. While bosons are described by complex-valued fields, fermions are represented by anticommuting Grassmann variables. Since quantum-mechanical systems are of inherently dynamical nature, the corresponding field theory action entails d + 1 dimensions, with time playing a special role. For Hamiltonians that incorporate only two-particle interactions, we can make contact with the previously studied Langevin equations, yet with effectively multiplicative rather than additive noise. As an illustration, this formalism is applied to deduce fundamental properties of weakly interacting boson superfluids. Whereas Landau–Ginzburg theory already provides a basic hydrodynamic description, the Gaussian approximation allows the computation of density correlations, the Bose condensate fraction, and the normal- and superfluid densities from the particle current correlations. We next establish that quantum fluctuations are typically irrelevant for thermodynamic critical phenomena, provided that T c > 0, and readily extend finite-size scaling theory to the imaginary time axis to arrive at general scaling forms for the free energy. Intriguing novel phenomena emerge in the realm of genuine quantum phase transitions at zero temperature, governed by other control parameters such as particle density, interaction or disorder strengths.
This thesis addresses different aspects of the dynamics of isolated quantum many-body systems far from equilibrium, ranging from transient phenomena at short times to questions about relaxation and equilibration at late times. The first part deals with nonequilibrium critical phenomena. Dynamical quantum phase transitions are characterized in a two-dimensional spin system by studying the distribution of Fisher zeros in the complex time plane. Moreover, a new topologically driven transition in the nonequilibrium steady state of quenched Chern insulators is investigated, resulting in the identification of universal nonanalytic behavior of the Hall conductance as a function of the quench parameter. The second part of the thesis comprises a study of transient dynamics of the transverse-field Ising model in two and three dimensions based on a perturbative derivation of classical networks to encode the time-evolved wave function; this construction is also applicable to represent the state as an artificial neural network. In the last part of the thesis a possible definition of the irreversibility of quantum many-body dynamics is introduced, which is based on the echo dynamics of observables under imperfect effective time reversal. Echo dynamics is studied in generic and integrable quantum systems as well as in the semiclassical limit. The relation to out-of-time-order correlators is explored, which were recently introduced to probe the quantum butterfly effect.
Phase transitions have recently been formulated in the time domain of quantum many-body systems, a phenomenon dubbed dynamical quantum phase transitions (DPTs), whose phenomenology is often divided in two types. One refers to distinct phases according to long-time averaged order parameters, while the other is focused on the nonanalytical behavior emerging in the rate function of the Loschmidt echo. Here we show that such DPTs can be found in systems with few degrees of freedom; i.e., they can take place without resorting to the traditional thermodynamic limit. We illustrate this by showing the existence of the two types of DPTs in a quantum Rabi model---a system involving a spin-21 and a bosonic mode. The dynamical criticality appears in the limit of an infinitely large ratio of the spin frequency with respect to the bosonic one. We determine its dynamical phase diagram and study the long-time averaged order parameters, whose semiclassical approximation yields a jump at the transition point. We find the critical times at which the rate function becomes nonanalytical, showing its associated critical exponent as well as the corrections introduced by a finite frequency ratio. Our results open the door for the study of DPTs without the need to scale up the number of components, thus allowing for their investigation in well controllable systems.
We present a general theory of classical metastability in open quantum systems. Metastability is a consequence of a large separation in timescales in the dynamics, leading to the existence of a regime when states of the system appear stationary, before eventual relaxation toward a true stationary state at much larger times. In this work, we focus on the emergence of classical metastability, i.e., when metastable states of an open quantum system with separation of timescales can be approximated as probabilistic mixtures of a finite number of states. We find that a number of classical features follow from this approximation, for the manifold of metastable states, long-time dynamics between them, and symmetries of the dynamics. Namely, those states are approximately disjoint and thus play the role of metastable phases, the relaxation toward the stationary state is approximated by a classical stochastic dynamics between them, and weak symmetries correspond to their permutations. Importantly, the classical dynamics is observed not only on average but also at the level of individual quantum trajectories: We show that time coarse-grained continuous measurement records can be viewed as noisy classical trajectories, while their statistics can be approximated by that of the classical dynamics. Among others, this explains how first-order dynamical phase transitions arise from metastability. Finally, to verify the presence of classical metastability in a given open quantum system, we develop an efficient numerical approach that delivers the set of metastable phases together with the effective classical dynamics. Since the proximity to a first-order dissipative phase transition manifests as metastability, the theory and tools introduced in this work can be used to investigate such transitions---which occur in the large size limit---through the metastable behavior of many-body systems of moderate sizes accessible to numerics.
We discuss the emergence of the classical description from quantum mechanics for chaotic systems. We consider a specific model, which corresponds to an ion trapped in a harmonic potential and submitted to a sequence of laser pulses: the kicked harmonic oscillator, under conditions that lead to chaotic behavior for the classical system. We show that noise plays an essential role in the transition from quantum to classical behavior. This transition is described in terms of the separation between the classical phase space distribution and the Wigner function corresponding to the quantum system, obtained by integrating over the whole phase space the magnitude of the difference between the two distributions. It is shown that, in the semiclassical limit, this separation is governed by a single parameter, which depends on the diffusion coefficient, the Lamb-Dicke parameter, and the kick strength. The Lamb-Dicke parameter plays the role of a dimensioneless Planck constant. As this parameter goes to zero, we show that both distributions remain close together for all times.