We study nonequilibrium signatures of tilt-induced localization in a one-dimensional Bose-Einstein condensate loaded in a shallow optical lattice. The tilt strength acts as a control parameter for the localization-delocalization crossover. We also consider the effects of repulsive interactions, which tend to delocalize the condensate. We first characterize localized and delocalized regimes through sudden quenches of the interaction strength and the external tilt. The resulting dynamics is analyzed using the survival probability and its power spectral density. Localized condensates exhibit strong memory retention, pronounced revivals, regular dynamics and a narrow spectral response, whereas delocalized condensates show suppressed recurrences, irregular dynamics and a broader distribution of spectral weight over many frequencies. We then investigate finite-rate ramps of the tilt strength across the localization threshold. Using the localization length and the Bogoliubov excitation gap, we extract the relevant critical exponents and perform Kibble-Zurek scaling analysis in the driven dynamics. Our results establish quench response and finite-rate scaling as complementary dynamical probes of localization in interacting Bose gases, with direct relevance to cold-atom experiments in tilted optical lattices.
We study nonequilibrium signatures of tilt-induced localization in a one-dimensional Bose-Einstein condensate loaded in a shallow optical lattice. The tilt strength acts as a control parameter for the localization-delocalization crossover. We also consider the effects of repulsive interactions, which tend to delocalize the condensate. We first characterize localized and delocalized regimes through sudden quenches of the interaction strength and the external tilt. The resulting dynamics is analyzed using the survival probability and its power spectral density. Localized condensates exhibit strong memory retention, pronounced revivals, regular dynamics and a narrow spectral response, whereas delocalized condensates show suppressed recurrences, irregular dynamics and a broader distribution of spectral weight over many frequencies. We then investigate finite-rate ramps of the tilt strength across the localization threshold. Using the localization length and the Bogoliubov excitation gap, we extract the relevant critical exponents and perform Kibble-Zurek scaling analysis in the driven dynamics. Our results establish quench response and finite-rate scaling as complementary dynamical probes of localization in interacting Bose gases, with direct relevance to cold-atom experiments in tilted optical lattices.
We investigate localization transitions in interacting Bose-Einstein condensates (BECs) confined in tilted optical lattices, focusing on both the continuum limit accessed via shallow lattice depths and the tight-binding limit realized in the deep lattice regime. Utilizing the Gross-Pitaevskii equation (GPE) and the many-body Bose-Hubbard model, we analyze the scaling behavior of localization indicators, such as the root mean square width and fidelity susceptibility, as a function of the applied tilt. Our results reveal clear signatures of a localization-delocalization transition driven by the linear potential, with scaling properties that characterize criticality even in the presence of interactions within the GPE description. Despite the single-mode nature of the condensate wavefunction, we demonstrate that it can effectively probe quantum criticality. Building on this, we propose the use of interacting BECs in tilted lattices as a platform for quantum critical sensing, where the condensate wavefunction serves both as a sensitive probe of localization and a practical resource for quantum-enhanced metrology. This approach opens new avenues for precision gradient sensing based on localization phenomena in bosonic systems.
We study the effect of atomic interaction on the localization and the associated dynamics of Bose-Einstein condensates in a one-dimensional quasiperiodic optical lattice and random disordered potentials. When the interactions are absent, the condensates exhibit localization, which weakens as we increase the interaction strength beyond a threshold value for both potential types. We inspect the localized and delocalized states by perturbing the system via quenching the interaction strength instantaneously to zero and studying the dynamics of the condensate, which we further corroborate using the out-of-time-order correlator. The temporal behavior of the time correlator displays regular dynamics for the localized state, while it shows temporal chaos for the delocalized state. We confirm this dynamical behavior by analyzing the power spectral density of the time correlator. We further identify that the condensate admits a quasiperiodic route to chaotic dynamics for both the potentials. Finally, we present the variation of the maximal Lyapunov exponents for different nonlinearity and disorder strengths that have a positive value in the regime where the time-correlator function shows chaotic behavior. Through this, we establish the strong connection between the spatially delocalized state of the condensate and its temporal chaos.
The center of mass dynamics of cold atoms and the Bose-Einstein condensate in one dimensional optical lattice is considered both in the absence and in the presence of external forcing. We discuss three situations for matter waves: first, the cold atoms; second, sufficiently dilute condensate where the dynamics are governed by the single particle wave packet dynamics; third, strong interaction regime, where, inter-atomic interaction can no longer be ignored. The analytical formalism developed for the two regimes, namely, deep optical lattice and shallow optical lattice. Parametric dependencies of energy spectrum and classical period, revival time and super revival are explained for the two regimes. The dynamics of condensate in driven optical lattice crystal are analyzed by studying dynamical stability of the condensate. The stability is determined by the dispersion behavior of the condensate excited in driven optical lattice. The recurrence behavior of the condensate close to the nonlinear resonances is analyzed as a function of time for delicate recurrences which take place for instance when lattice is weakly perturbed and robust recurrences which may manifest themselves for sufficiently strong external driving force. The analysis is not only valid for dilute condensate but also applicable for strongly interacting homogeneous condensate provided, the external modulation causes no significant change in density profile of the condensate. We explain parametric dependence of the dynamical recurrence times which can easily be realized in laboratory experiments. In addition, we find a good agreement between the obtained analytical results and numerical calculations. The stability of condensate is also explored in driven optical lattice numerically.
Quantum simulators are easily controllable quantum systems that allow us to physically model the behavior of more complex systems, that are intractable with even the most advanced supercomputers, to gain insight into their behavior. Using lattice confined spinor Bose-Einstein condensates (BEC) as quantum simulators, we explore the quantum critical dynamics of the superfluid to Mott insulator phase transition in a three-dimensional cubic optical lattice. Our observations reveal that, in a quantum version of the Kibble-Zurek mechanism, the dynamics can be frozen by a sufficiently quick ramp across the transition. Additionally, we utilize discrete energy signatures in nonequilibrium spin mixing dynamics generated by atoms that are tightly confined to single lattice sites after a quench across this transition to understand the spatial dynamics of a BEC during this transition. This "indirect microscopy" method is required to determine the spatial distribution because achieving single lattice site resolved imaging of three dimensional lattice confined quantum gases remains experimentally challenging even with advanced quantum gas microscopy methods. The observed number distributions additionally demonstrate that it may be possible to design lattice ramp sequences to manipulate the distribution of atoms to enhance desired effects.
When a second-order phase transition is crossed at finite speed, domains with independent order parameters can appear in the system, with the consequent formation of defects at the domain boundaries. The Kibble-Zurek theory provides a description for this universal phenomenon, which applies to many different systems in nature, and it predicts a power-law dependence of the defect density on the quench rate. This thesis reports on the results of the experimental study of the Kibble-Zurek mechanism in elongated Bose-Einstein condensates of atomic sodium gases, following the observations on the spontaneous formation of defects after temperature quenches across the BEC transition. The power-law scaling of the defect number with the quench speed was observed and characterized for the first time in ultracold gases. The characterization of the density and phase profiles of the defects allowed their identification as solitonic vortices, representing the first direct experimental evidence for this kind of long living excitation, which sets a link between solitons and vortices. The measurements reported in this thesis provide a novel approach to the study of the critical phenomena happening at phase transitions, and introduce to the possibility of exploring the turbulent dynamics of quenched systems through the spontaneous production of solitonic vortices.
The problem of understanding how a coherent, macroscopic Bose- Einstein condensate (BEC) emerges from the cooling of a thermal Bose gas has attracted significant theoretical and experimental interest over several decades. The pioneering achievement of BEC in weakly interacting dilute atomic gases in 1995 was followed by a number of experimental studies examining the growth of the BEC number, as well as the development of its coherence. More recently, there has been interest in connecting such experiments to universal aspects of nonequilibrium phase transitions, in terms of both static and dynamical critical exponents. Here, the spontaneous formation of topological structures such as vortices and solitons in quenched cold-atom experiments has enabled the verification of the Kibble-Zurek mechanism predicting the density of topological defects in continuous phase transitions, first proposed in the context of the evolution of the early universe. This chapter reviews progress in the understanding of BEC formation and discusses open questions and future research directions in the dynamics of phase transitions in quantum gases. Introduction The equilibrium phase diagram of the dilute Bose gas exhibits a continuous phase transition between condensed and noncondensed phases. The order parameter characteristic of the condensed phase vanishes above some critical temperature T c and grows continuously with decreasing temperature below this critical point. However, the dynamical process of condensate formation has proved to be a challenging phenomenon to address both theoretically and experimentally. This formation process is a crucial aspect of Bose systems and of direct relevance to all condensates discussed in this book, despite their evident system-specific properties. Important questions leading to intense discussions in the early literature include the time scale for condensate formation and the role of inhomogeneities and finite-size effects in “closed” systems. These issues are related to the concept of spontaneous symmetry breaking, its causes, and implications for physical systems (see, for example, Chapter 5 by Snoke and Daley). In this chapter, we give an overview of the dynamics of condensate formation and describe the present understanding provided by increasingly well-controlled cold-atom experiments and corresponding theoretical advances over the past twenty years. We focus on the growth of BECs in cooled Bose gases, which, from a theoretical standpoint, requires a suitable nonequilibrium formalism.