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.
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.
Abstract The Kibble–Zurek mechanism (KZM) describes the dynamics across a phase transition leading to the formation of topological defects, such as vortices in superfluids and domain walls in spin systems. Here, we experimentally probe the distribution of kink pairs in a one-dimensional quantum Ising chain driven across the paramagnet-ferromagnet quantum phase transition, using a single trapped ion as a quantum simulator in momentum space. The number of kink pairs after the transition follows a Poisson binomial distribution, in which all cumulants scale with a universal power law as a function of the quench time in which the transition is crossed. We experimentally verified this scaling for the first cumulants and report deviations due to noise-induced dephasing of the trapped ion. Our results establish that the universal character of the critical dynamics can be extended beyond KZM, which accounts for the mean kink number, to characterize the full probability distribution of topological defects.
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.
Kibble-Zurek theory (KZ) stands out as the most robust theory of defect generation in the dynamics of phase transitions. KZ utilizes the structure of equilibrium states away from the transition point to estimate the excitations due to the transition using adiabatic and impulse approximations. Here we show, the actual nonequilibrium dynamics lead to a qualitatively different scenario from KZ, as far correlations between the defects (rather than their densities) are concerned. For a quantum Ising chain, we show, this gives rise to a Gaussian spatial decay in the domain wall (kinks) correlations, while KZ would predict an exponential fall. We propose a simple but general framework on top of KZ, based on the ``quantum coarsening'' dynamics of local correlators in the supposed impulse regime. We outline how our picture extends to generic interacting situations.
Quantum phase transitions provide a controlled route for generating many-body excitations, but the dynamics after the critical point can be as important as the initial defect creation. Recent progress in quantum annealing has made it possible to access coherent nonequilibrium dynamics in programmable Ising systems with thousands of superconducting qubits. Here we use this capability to study a longitudinally biased quantum Ising chain, where Kibble--Zurek defect creation is followed by nonintegrable post-critical dynamics. The longitudinal bias confines kink--antikink excitations into mesonic bound states, so that the final spin configurations encode both the production of defects near the critical point and the subsequent evolution of the confined excitations. Using energy-scale rescaling and zero-noise extrapolation, we find that the defect density follows the expected biased Kibble--Zurek/Landau--Zener crossover and agrees with matrix-product-state simulations with uniform bias. In contrast, magnetization, spatial profiles, and minority-domain statistics reveal that mesonic evolution is interrupted by localization of the post-critical domain pattern. Matrix-product-state simulations with disorder reproduce this separation between robust defect creation and localized post-critical dynamics. Our results show that large-scale quantum annealers can probe the fate of critical excitations beyond defect counting.
We investigate the dynamics of formation of topological defects in ultra-cold quantum gases. Using classical field methods, such as the truncated Wigner approximation and the stochastic Gross-Pitaevskii equation, we explore the connection between the rate of a phase transition and the number of topological defects formed as a consequence of it. We show that in various systems such defects form in number comparable with the predictions of the Kibble-Zurek theory, known for thirty years but as yet unverified experimentally. We also consider the spontaneous production of entangled vortices following a quantum phase transition in a multi-component spin-1 Bose-Einstein condensate (BEC). To begin, we provide a literature review and describe some of the background theory of multi-component condensates, stochastic methods and Kibble-Zurek theory. Part of this thesis focus on a two component Bose-Einstein condensates of the same atomic species. In particular on the quantum critical point separating the miscible from the immiscible phase of such systems. We show that such phase transition cannot only be achieved with the tuning of the interaction constants between the components, but also through changes in the intensity of the coupling between atoms in different hyperfine states. We first determine the value of the critical coupling, and also the critical exponents defining the universality class of the coupling—induced phase transition. We then proceed to simulate the dynamics of the transition in effective 1D ring and harmonically trapped condensates, and test the results against predictions of the Kibble-Zurek (KZ) mechanism. We find good agreement between the KZ theory and the numerical results in the ring geometry, and we motivate the origin of the discrepancies between the two in harmonically trapped BECs. We use the ”equivalence” of inhomogeneous phase transitions in anisotropic systems and spatially inhomogeneous quenches to engineer coupling quenches and systematically study key aspects of the inhomogeneous KZ mechanism. The coupling induced miscible—immiscible transition results in the formation of a stable pattern of domains that easily be detected and counted in current ultra-cold gases experiment, and the scheme is a viable candidate for the first experimental testing of the KZ predictions. We then examine the spontaneous formation of a superflow as result of the thermal Bose-Einstein phase transition in ring traps. According to the Kibble-Zurek theory, the fast production of a condensate in a ring trap should result in the formation of solitons, with the number depending on the rate of the temperature quench. Solitons are short-living topological defects and their observation in ultra-cold gases experiments remains challenging. We show here that even after they decay, solitons leave a signature in the variance of the final winding number of the condensate. Solitons fragment a 1D condensate into regions of random phase, hence the larger the number of solitons, the higher the probability that the resulting phase wraps around the ring by a multiple of 2π. We numerically simulate the BEC transition in this system with the stochastic Gross-Pitaevskii equation, and observe that the winding number scales with the quenching time as predicted by the KZ mechanism. We examine the effect of inhomogeneity on the expectation value of the winding number, and find that the tilting of the ring greatly suppress the spontaneous formation of vortices. This has implications for the necessary horizontal alignment of any experiment on this system. Finally, we consider the polar to anti-ferromagnetic phase transition in magnetically quenched spin-1 BEC. In this quantum phase transition a pure condensate of atoms in the mF = 0 Zeeman sublevel is converted into a three component BEC with atoms in mF = 0, ±1 through spin changing collisions. An analogous mechanism, four—wave mixing, in quantum optics produces entangled photons, and it has been proposed in ultra-cold gases as a method to obtain Einstein-Podolski-Rosen entanglement of macroscopic atomic ensembles. We begin with a rotating 1D spinor condensate of atoms in the mF = 0 component, and perform a quench of the magnetic field through the critical point. We show how the transition produces a pair of entangled components with different winding numbers, and quantify the entanglement as a function of the modes considered and of the type of magnetic quench, sudden or instantaneous. Lastly, we propose a method to use the original spin zero component as coherent source to perform homodyne detection and consider the two rotating entangled components as a resource for the sensing of rotations.
We study the collisionless dynamics of two classes of nonintegrable pairing models. One is a Bardeen-Cooper-Schrieffer model with separable energy-dependent interactions, and the other is a two-dimensional topological superconductor with spin-orbit coupling and a band-splitting external field. The long-time quantum quench dynamics at integrable points of these models are well understood. Namely the squared magnitude of the time-dependent order parameter $\mathrm{\ensuremath{\Delta}}(t)$ can vanish (Phase I), reach a nonzero constant (Phase II), or periodically oscillate as an elliptic function (Phase III). We demonstrate that nonintegrable models, too, exhibit some or all of these nonequilibrium phases. Remarkably, elliptic periodic oscillations persist, even though both their amplitude and functional form change drastically with integrability breaking. Striking new phenomena accompany loss of integrability. First, an extremely long timescale emerges in the relaxation to Phase III, such that short-time numerical simulations risk erroneously classifying the asymptotic state. This timescale diverges near integrable points. Second, an entirely new Phase IV of quasiperiodic oscillations of $|\mathrm{\ensuremath{\Delta}}|$ emerges in the quantum quench phase diagrams of nonintegrable pairing models. As integrability techniques do not apply for the models we study, we develop the concept of asymptotic self-consistency and a linear stability analysis of the asymptotic phases. With the help of these new tools, we determine the phase boundaries, characterize the asymptotic state, and clarify the physical meaning of the quantum quench phase diagrams of BCS superconductors. We also propose an explanation of these diagrams in terms of bifurcation theory.