The Kibble–Zurek mechanism (KZM) successfully predicts the density of topological defects deposited by the phase transitions, but it is not clear why. Its key conjecture is that, near the critical point of the second-order phase transition, critical slowing down will result in a period when the system is too sluggish to follow the potential that is changing faster than its reaction time. The correlation length at the freeze-out instant t ̂ when the order parameter catches up with the posttransition broken symmetry configuration is then decisive, determining when the mosaic of broken symmetry domains locks in topological defects. To understand why the KZM works so well, we analyze the Landau–Ginzburg model and show why temporal evolution of the order parameter plays such a key role. The analytical solutions we obtain suggest experimentally accessible observables that can shed light on symmetry-breaking dynamics while testing the conjecture on which the KZM is based.

According to the Kibble-Zurek mechanism (KZM), the density of topological defects created during a second-order phase transition is determined by the correlation length at the freeze-out time. This suggests that the final configuration of topological defects in such a transition is largely established during the impulse regime, soon after the critical point is traversed. Motivated by this, we conjecture that machine learning (ML) can predict the final configuration of topological defects based on the time evolution of the order parameter over a short interval in the vicinity of the critical point, well before the order parameter settles into the emerging new minima resulting from spontaneous symmetry breaking. Furthermore, we show that the predictability of ML also follows the power law scaling dictated by KZM. We demonstrate these using a Recurrent Neural Network.

We identify persistent oscillations in a nonintegrable quantum Ising chain. In the integrable chain with nearest-neighbor interactions, the nature, origin, and decay of post-transition oscillations are tied to the Kibble-Zurek mechanism. Remarkably, when coupling to the next-nearest neighbor is added, the resulting nonintegrable ”zigzag” chain (still in the quantum Ising universality class) supports persistent oscillation: Topological defects (kinks) appear as a result of the quantum phase transition. However, in a ”zigzag” Ising chain defects can form Cooper pairs. The oscillation of the Cooper-pair condensate has a frequency that depends on the binding energy gap between the paired and the unpaired defects, so it can be excited by resonant driving. While one might have expected that the integrability-breaking ”zigzag” coupling causes relaxation, the oscillations we identify are persistent: Their longevity in our simulations is likely limited only by numerical accuracy. This oscillation of the Cooper-pair condensate of kinks is a manifestation of quantum coherence and should be experimentally accessible.

Quantum Darwinism recognizes that decoherence imprints redundant records of preferred quasi-classical pointer states on the environment. These redundant records are then accessed by observers. We show how redundancy enables and even implies consensus between observers who use fragments of that decohering environment to acquire information about systems of interest. We quantify consensus using information-theoretic measures that employ mutual information to assess the correlation between the records available to observers from distinct – hence, independently accessible – fragments of the environment. We prove that when these fragments have enough information about a system, observers that access them will attribute the same pointer state to that system. Thus, those who know enough about the system agree about what they know. We then test proposed measures of consensus in a solvable model of decoherence as well as in numerical simulations of a many-body system. These results provide detailed understanding of how our classical everyday world arises from within the fundamentally quantum Universe we inhabit.

The Kibble-Zurek mechanism (KZM) describes the non-equilibrium dynamics and topological defect formation in systems undergoing second-order phase transitions. KZM has found applications in fields such as cosmology and condensed matter physics. However, it is generally not suitable for describing first-order phase transitions. It has been demonstrated that transitions in systems like superconductors or charged superfluids, typically classified as second-order, can exhibit weakly first-order characteristics when the influence of fluctuations is taken into account. Moreover, the order of the phase transition (i.e., the extent to which it becomes first rather than second order) can be tuned. We explore quench-induced formation of topological defects in such tunable phase transitions and propose that their density can be predicted by combining KZM with nucleation theory.

The formation of topological defects in second-order phase transitions can be investigated by solving partial differential equations for the evolution of the order parameter in space and time, such as the Langevin equation. We demonstrate that the ordinary differential equations governing either the temporal or spatial dependence in the Langevin equation provide surprisingly substantial insights into the dynamics of the phase transition. The temporal evolution of the order parameter predicts the essence of the adiabatic-impulse scenario, including the scaling of the freeze-out time, which is crucial to the Kibble-Zurek mechanism (KZM). In particular, Bernoulli differential equations that arise in the overdamped case can be solved analytically. The spatial part of the evolution, in turn, leads to the characteristic size of domains that choose the same broken symmetry. Apart from the fundamental insights into the KZM, this finding enables the exploration of Kibble-Zurek scaling using ordinary differential equations over a large range of quench timescales, which would otherwise be difficult to achieve with numerical simulations of the full partial differential equations.

A quantum phase transition from the miscible to the immiscible phase of a quasi-one-dimensional binary Bose-Einstein condensate is driven by ramping down the coupling amplitude of its two hyperfine states. It results in a random pattern of spatial domains where the symmetry is broken separated by defects. In distinction to previous studies [J. Sabbatini et al., Phys. Rev. Lett. 107, 230402 (2011), New J. Phys. 14 095030 (2012)], we include nonzero detuning between the light field and the energy difference of the states, which provides a bias towards one of the states. Using the truncated Wigner method, we test the biased version of the quantum Kibble-Zurek mechanism [M. Rams et al., Phys. Rev. Lett. 123, 130603 (2019)] and observe a crossover to the adiabatic regime when the quench is sufficiently fast to dominate the effect of the bias. We verify a universal power law for the population imbalance in the nonadiabatic regime both at the critical point and by the end of the ramp. Shrinking and annihilation of domains of the unfavourable phase after the ramp, that is, already in the broken symmetry phase, enlarges the defect-free sections by the end of the ramp. The consequences of this phase-ordering effect can be captured by a phenomenological power law.

"How much information about a system S can one extract from a fragment F of the environment E that decohered it?" is the central question of Quantum Darwinism. To date, most answers relied on the quantum mutual information of SF, or on the Holevo bound on the channel capacity of F to communicate the classical information encoded in S. These are reasonable upper bounds on what is really needed but much harder to calculate-the accessible information in the fragment F about S. We consider a model based on imperfect c-not gates where all the above can be computed, and discuss its implications for the emergence of objective classical reality. We find that all relevant quantities, such as the quantum mutual information as well as various bounds on the accessible information exhibit similar behavior. In the regime relevant for the emergence of objective classical reality this includes scaling independent of the quality of the imperfect c-not gates or the size of E, and even nearly independent of the initial state of S.

We establish bounds on quantum correlations in many-body systems. They reveal what sort of information about a quantum system can be simultaneously recorded in different parts of its environment. Specifically, independent agents who monitor environment fragments can eavesdrop only on amplified and redundantly disseminated-hence, effectively classical-information about the decoherence-resistant pointer observable. We also show that the emergence of classical objectivity is signaled by a distinctive scaling of the conditional mutual information, bypassing hard numerical optimizations. Our results validate the core idea of quantum Darwinism: objective classical reality does not need to be postulated and is not accidental, but rather a compelling emergent feature of quantum theory that otherwise-in the absence of decoherence and amplification-leads to "quantum weirdness." In particular, a lack of consensus between agents that access environment fragments is bounded by the information deficit, a measure of the incompleteness of the information about the system.

We identify and investigate the origin and nature of the transition between Gaussian and exponential forms of decoherence: the decoherence factor (that controls the time dependence of the off-diagonal terms of the density matrix expressed in the pointer basis representation) is the convolution of the Fourier transforms of the spectral density and of the overlap (between the eigenstates the environment with and without couplings to the system). Spectral density alone tends to lead to the (approximately) Gaussian decay of coherence while the overlap alone results in a (largely) exponential decay. We show that these two contributions combine as a convolution, their relative importance controlled by the strength of the system-environment coupling. The resulting decoherence factor in the strong and weak coupling limits leads to predominantly Gaussian or exponential decay, respectively, as is demonstrated with two paradigmatic examples of decoherence—a spin-bath model and the quantum Brownian motion.

Core quantum postulates including the superposition principle and the unitarity of evolutions are natural and strikingly simple. I show that—when supplemented with a limited version of predictability (captured in the textbook accounts by the repeatability postulate)—these core postulates can account for all the symptoms of classicality. In particular, both objective classical reality and elusive information about reality arise, via quantum Darwinism, from the quantum substrate. This approach shares with the Relative State Interpretation of Everett the view that collapse of the wavepacket reflects perception of the state of the rest of the Universe relative to the state of observer’s records. However, our “let quantum be quantum” approach poses questions absent in Bohr’s Copenhagen Interpretation that relied on the preexisting classical domain. Thus, one is now forced to seek preferred, predictable, hence effectively classical but ultimately quantum states that allow observers keep reliable records. Without such (i) preferred basis relative states are simply “too relative”, and the ensuing basis ambiguity makes it difficult to identify events (e.g., measurement outcomes). Moreover, universal validity of quantum theory raises the issue of (ii) the origin of Born’s rule, pk=|ψk|2, relating probabilities and amplitudes (that is simply postulated in textbooks). Last not least, even preferred pointer states (defined by einselection—environment—induced superselection)—are still quantum. Therefore, unlike classical states that exist objectively, quantum states of an individual system cannot be found out by an initially ignorant observer through direct measurement without being disrupted. So, to complete the ‘quantum theory of the classical’ one must identify (iii) quantum origin of objective existence and explain how the information about objectively existing states can appear to be essentially inconsequential for them (as it does for states in Newtonian physics) and yet matter in other settings (e.g., thermodynamics). I show how the mathematical structure of quantum theory supplemented by the only uncontroversial measurement postulate (that demands immediate repeatability—hence, predictability) leads to preferred states. These (i) pointer states correspond to measurement outcomes. Their stability is a prerequisite for objective existence of effectively classical states and for events such as quantum jumps. Events at hand, one can now enquire about their probability—the probability of a pointer state (or of a measurement record). I show that the symmetry of entangled states—(ii) entanglement—assisted invariance or envariance—implies Born’s rule. Envariance also accounts for the loss of phase coherence between pointer states. Thus, decoherence can be traced to symmetries of entanglement and understood without its usual tool—reduced density matrices. A simple and manifestly noncircular derivation of pk=|ψk|2 follows. Monitoring of the system by its environment in course of decoherence typically leaves behind multiple copies of its pointer states in the environment. Only pointer states can survive decoherence and can spawn such plentiful information-theoretic progeny. This (iii) quantum Darwinism allows observers to use environment as a witness—to find out pointer states indirectly, leaving systems of interest untouched. Quantum Darwinism shows how epistemic and ontic (coexisting in epiontic quantum state) separate into robust objective existence of pointer states and detached information about them, giving rise to extantons—composite objects with system of interest in the core and multiple records of its pointer states in the halo comprising of environment subsystems (e.g., photons) which disseminates that information throughout the Universe.

It is now widely accepted that quenches through the critical region of quantum phase transitions result in post-transition states populated with topological defects-analogs of the classical topological defects. However, consequences of the very nonclassical fact that the state after a quench is a superposition of distinct, broken-symmetry vacua with different numbers and locations of defects have remained largely unexplored. We identify coherent quantum oscillations induced by such superpositions in observables complementary to the one involved in symmetry breaking. These oscillations satisfy Kibble-Zurek dynamical scaling laws with the quench rate, with an instantaneous oscillation frequency set primarily by the gap of the system. In addition to the obvious fundamental significance of a superposition of different broken symmetry states, quantum coherent oscillations can be used to verify unitarity and test for imperfections of the experimental implementations of quantum simulators.

The quantum Kibble-Zurek mechanism (QKZM) predicts universal dynamical behavior near the quantum phase transitions (QPTs). It is now well understood for the one-dimensional quantum matter. Higher-dimensional systems, however, remain a challenge, complicated by the fundamentally different character of the associated QPTs and their underlying conformal field theories. In this work, we take the first steps toward theoretical exploration of the QKZM in two dimensions for interacting quantum matter. We study the dynamical crossing of the QPT in the paradigmatic Ising model by a joint effort of modern state-of-the-art numerical methods, including artificial neural networks and tensor networks. As a central result, we quantify universal QKZM behavior close to the QPT. We also note that, upon traversing further into the ferromagnetic regime, deviations from the QKZM prediction appear. We explain the observed behavior by proposing an extended QKZM taking into account spectral information as well as phase ordering. Our work provides a testing platform for higher-dimensional quantum simulators.

We investigate the effect of coupling between translational and internal degrees of freedom of composite quantum particles on their localization in a random potential. We show that entanglement between the two degrees of freedom weakens localization due to the upper bound imposed on the inverse participation ratio by purity of a quantum state. We perform numerical calculations for a two-particle system bound by a harmonic force in a 1D disordered lattice and a rigid rotor in a 2D disordered lattice. We illustrate that the coupling has a dramatic effect on localization properties, even with a small number of internal states participating in quantum dynamics.

Markus Schmitt,1 Marek M. Rams,2 Jacek Dziarmaga,2 Markus Heyl,3 and Wojciech H. Zurek4 Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany Jagiellonian University, Institute of Theoretical Physics, Łojasiewicza 11, PL-30348 Kraków, Poland Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, Dresden 01187, Germany Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: June 18, 2021)

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We explore nonadiabatic quantum phase transitions in an Ising spin chain with a linearly time-dependent transverse field and two different spins per unit cell. Such a spin system passes through critical points with gapless excitations, which support nonadiabatic transitions. Nevertheless, we find that the excitations on one of the chain sublattices are suppressed in the nearly adiabatic regime exponentially. Thus, we reveal a coherent mechanism to induce exponentially large density separation for different quasiparticles.

A system gradually driven through a symmetry-breaking phase transition is subject to the Kibble-Zurek mechanism (KZM). As a consequence of the critical slowing down, its state cannot follow local equilibrium, and its evolution becomes nonadiabatic near the critical point. In the simplest approximation, that stage can be regarded as an ``impulse'' where the state of the system remains unchanged. It leads to the correct KZM scaling laws. However, such a ``freeze-out'' might suggest that the coherence length of the nascent order parameter remains unchanged as the critical region is traversed. By contrast, the original causality-based discussion emphasized the role of the sonic horizon: Domains of the broken symmetry phase can expand with a velocity limited by the speed of the relevant sound. This effect was demonstrated in the quantum Ising chain where the dynamical exponent $z=1$ and quasiparticles excited by the transition have a fixed speed of sound. To elucidate the role of the sonic horizon, in this paper we study two systems with $z>1$ where the speed of sound is no longer fixed, and the fastest excited quasiparticles set the size of the sonic horizon. Their effective speed decays with increasing transition time. In the extreme case, the dynamical exponent $z$ can diverge such as in the Griffiths region of the random Ising chain where the localization of excited quasiparticles freezes the growth of the correlation range when the critical region is traversed. Of particular interest is an example with $z<1$, the long-range extended Ising chain, where there is no upper limit to the velocity of excited quasiparticles with small momenta. Initially, the power-law tail of the correlation function grows adiabatically, but in the nonadiabatic stage it lags behind the adiabatic evolution.