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arXiv:2107.11547 · Physical Review A

Dynamical preparation of an atomic condensate in a Hofstadter band

Han Fu, Andreas Glatz, F. Setiawan, Kai-Xuan Yao, Zhendong Zhang, Cheng Chin, Kathryn Levin

Published Apr 7, 2022• 2 citations
arXivHTMLPDF

Abstract

The creation of a Hamiltonian in the quantum regime which has nontrivial topological features is a central goal of the cold-atom community, enabling widespread exploration of novel phases of quantum matter. A general scheme to synthesize such Hamiltonians is based on dynamical modulation of optical lattices which thereby generate vector potentials. At the same time, the modulation can lead to heating and serious difficulties with equilibration. Here we show that these challenges can be overcome by demonstrating how a Hofstadter Bose-Einstein condensate (BEC) can be dynamically realized, using experimental protocols. From Gross-Pitaevskii simulations our study reveals a complex, multistage evolution; this includes a chaotic intermediate ``heating'' stage followed by a spontaneous reentrance to the BEC. The observed behavior is reminiscent of evolution in cosmological models.

DOI
10.1103/physreva.105.043301

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The creation of a Hamiltonian in the quantum regime which has nontrivial topological features is a central goal of the cold-atom community, enabling widespread exploration of novel phases of quantum matter. A general scheme to synthesize such Hamiltonians is based on dynamical modulation of optical lattices which thereby generate vector potentials. At the same time, the modulation can lead to heating and serious difficulties with equilibration. Here we show that these challenges can be overcome by demonstrating how a Hofstadter Bose-Einstein condensate (BEC) can be dynamically realized, using experimental protocols. From Gross-Pitaevskii simulations our study reveals a complex, multistage evolution; this includes a chaotic intermediate ``heating'' stage followed by a spontaneous reentrance to the BEC. The observed behavior is reminiscent of evolution in cosmological models.
In cosmological evolution, it is the homogeneous scalar field (inflaton) that drives the universe to expand isotropically and to generate standard model particles. However, to simulate cosmology, atomic gas research has focused on the dynamics of Bose-Einstein condensates (BEC) with continuously applied forces. In this paper we argue that a complementary approach needs also to be pursued; we thus consider the analog BEC experiments in a nondriven and naturally closed atomic system. We implement this using a BEC in an optical lattice which, after a quench, freely transitions from an unstable to a stable state. This dynamical evolution displays the counterpart ``preheating,'' ``reheating,'' and ``thermalization'' phases of cosmology. Importantly, our studies of these analog processes yield tractable analytic models. Additionally, of great utility to the cold atom community, such understanding elucidates the dynamics of nonadiabatic condensate preparation. Indeed, the dynamical processes discussed here are generic and in future cold atom reequilibration experiments it will be important to observe both the preheating stage, corresponding to a fragmented condensate, and the reheating stage, corresponding to a particle cloud.
Shaking optical lattices in a resonant manner offers an efficient and versatile method to devise artificial gauge fields and topological band structures for ultracold atomic gases. This was recently demonstrated through the experimental realization of the Harper–Hofstadter model, which combined optical superlattices and resonant time-modulations. Adding inter-particle interactions to these engineered band systems is expected to lead to strongly-correlated states with topological features, such as fractional Chern insulators. However, the interplay between interactions and external time-periodic drives typically triggers violent instabilities and uncontrollable heating, hence potentially ruling out the possibility of accessing such intriguing states of matter in experiments. In this work, we study the early-stage parametric instabilities that occur in systems of resonantly-driven Bose–Einstein condensates in optical lattices. We apply and extend an approach based on Bogoliubov theory (Lellouch et al 2017 Phys. Rev. X 7 021015) to a variety of resonantly-driven band models, from a simple shaken Wannier–Stark ladder to the more intriguing driven-induced Harper–Hofstadter model. In particular, we provide ab initio numerical and analytical predictions for the stability properties of these topical models. This work sheds light on general features that could guide current experiments to stable regimes of operation.
Phase transitions abound in the physical world, from the subatomic length scales of quark condensation to the decoupling forces in the early universe. In the Bose-Einstein condensation phase transition, a gas of trapped bosonic atoms is cooled to a critical temperature. Below this temperature, a macroscopic number of atoms suddenly starts to occupy a single quantum state; these atoms comprise the Bose-Einstein condensate (BEC). The dynamics of the BEC phase transition are the focus of this dissertation and the experiments described here have provided new information on the details of BEC formation. New theoretical developments are proving to be valuable tools for describing BEC phase transition dynamics and interpreting new experimental results. With their amenability to optical manipulation and probing along with the advent of new microscopic theories, BECs provide an important new avenue for gaining insight into the universal dynamics of phase transitions in general.Spontaneous symmetry breaking in the system's order parameter may be one result of cooling through a phase transition. A potential consequence of this is the spontaneous formation of topological defects, which in a BEC appear as vortices. We experimentally observed and characterized the spontaneous formation of vortices during BEC growth. We attribute vortex creation to coherence length limitations during the initial stages of the phase transition. Parallel to these experimental observations, theory collaborators have used the Stochastic Gross-Pitaevski Equation formalism to simulate the growth of a condensate from a thermal cloud. The experimental and theoretical statistical results of the spontaneous formation of vortex cores during the growth of the condensate are in good quantitative agreement with one another, supporting our understanding of the dynamics of the phase transition. We believe that our results are also qualitatively consistent with the Kibble-Zurek mechanism, a universal model for topological defect formation.Ultimately, our understanding of the dynamics of the BEC phase transition may lead to a broader understanding of phase transitions in general, and provide new insight into the development of coherence in numerous systems.
Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas of quantum mechanics have been realized and proven to be true in experiments, by studying the Bose-Einstein condensates. Amongst them, the observation of macroscopic matter wave interference, the creation of vortices, dark and bright solitons etc, find special mention. The dynamics of this system in presence of an optical lattice reveals a superfluid to Mott insulator quantum phase transition. This has opened a new avenue for research on BEC and its applications. From a theoretical standpoint, the dynamics of BEC can be described by means of an effective mean-field theory. This formalism is much simpler than the many-body Schr¨odinger equation and based on a classical nonlinear evolution equation, known as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc., which are closely related to nonlinear fiber optics. The present thesis deals with nonlinear excitations and different phases of BEC and is organized as follows: after a brief introduction, in chapter two, we start with BEC in a one-dimensional shallow optical lattice, experiencing both cubic and quintic nonlinearity. A superfluid density wave, of twice the periodicity of lattice potential, is identified. This indicates matter redistribution, where atoms from neighbouring sites have been depleted leading to a density wave behaviour. Interestingly, it ceases to exist when only one of these interactions is operative. We predict the loss of superfluidity through a classical dynamical phase transition, where modulational instability leads to the loss of phase coherence. In presence of pure two-body interaction, the superfluid density shows a periodicity commensurate with the lattice potential. Apart from the superfluid state, a density wave insulating phase is also found to exist, possessing two frequency modulations commensurate with the lattice potential. A dynamical phase transition connects the superfluid phase with an insulating phase, which in turn, is connected with the density wave insulating phase. The solutions are found to be marginally stable, as per the Vakhitov-Kolokolov (VK) criterion. Chapter three describes the dynamics of the atomic-molecular Bose-Einstein condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into account the atom-molecular interaction, as well as atom-molecule inter-conversion. The two-body atom-atom, atom-molecule and molecule-molecule interactions are analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic nonlinear term arising from the inter-conversion between atoms and molecules is identical to the scenario in strongly coupled BEC in one dimension. The presence of both weak and strong interactions adds further richness to this system. We find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed in order to see the effect of the both weak and strong interactions on Lieb mode profile and compare it with the pure atomic condensate profile. Another class of solutions exist in this system, where both atomic and molecular condensates possess bright solitons. In chapter four, we investigate the effect of a harmonic trap on the dynamics of grey solitons in Bose-Einstein condensates. It is observed that, these nonlinear excitations of the cigar-shaped Bose-Einstein condensate have strong coupling with the trap at low momenta and hence can be effectively isolated from the Bogoliubov sound modes, which respond weakly to harmonic confinement. This strong coupling with the trap also makes the grey soliton amenable for control and manipulation through trap modulation and temporal variation of the two-body interaction. We have shown that in presence of an expulsive oscillator trap, the grey soliton gets amplified, while having a compression in its width. Exact solutions are also obtained in case of the two component Bose-Einstein condensates (TBECs), where one component is a grey soliton and the other a bright one. The presence of the trap and inter-species interaction significantly affects the grey soliton dynamics. Similar to the single component, in the present case, we have shown that both grey and bright soliton can be compressed simultaneously. A detailed study of the energy and momentum of the soliton is reported, in order to find the effect of the trap on the Lieb mode. In chapter five, we study a BEC-defect complex in a quasi-one dimensional scenario, where the defect atom moves in the condensate with constant velocity. Explicit solutions of the coupled mean-field equations, describing defect-grey soliton dynamics are obtained, demonstrating the coexistence of grey soliton and a localized defect. Unlike the case of dark soliton, where the defect trapping center has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can lower the grey soliton’s maximum velocity, as compared to the defect free case, where the maximum velocity is the sound velocity. We have also carried out the stability analysis of the obtained solutions using renormalized momentum formalism and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable through both the methods. The grey soliton’s energy gets substantially modified through its interaction with the defect, opening up the possibility of its control through defect dynamics. Chapter six describes a parametrically forced Bose-Einstein condensate in the presence of a general moving optical lattice. The interaction between the atoms in the condensate and the time dependent lattice potential leads to a novel propagating superfluid matter wave, which can be controlled through chirp management. This system, when placed in a trap, accelerates and undergoes rapid nonlinear compression, controlled by the chirp. The density achieves its maximum, when the matter wave changes direction. It is also shown that this system exhibits dynamical superfluid insulator transition, where superfluidity breaks down and the condensate transits to an insulating phase. The stability has been investigated using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well as on the nature of the atom-atom interaction. The exact expression for energy is obtained and analyzed in detail to gain physical understanding of the chirp management of the sinusoidal excitations and also the dynamical phase transition.
In this work, we realize the soliton's negative mass regime, when a Bose-Einstein condensate is subjected to a harmonic confinement. We also discuss that this system is favorable for the formation of a bound state when an additional species is considered. When a time modulated optical lattice potential is introduced, the sinusoidal lattice modes lead to the generation of nonlinear resonances. A classical dynamical phase transition is also identified in this case through a superfluid to the insulating phase transition.
In a numerical experiment based on Gross-Pitaevskii formalism, we demonstrate unique topological quantum coherence in optically trapped Bose-Einstein condensates (BECs). Exploring the fact that vortices in a rotating BEC can be pinned by a geometric arrangement of laser beams, we show the parameter range in which vortex-antivortex molecules or multiquantum vortices are formed as a consequence of the optically imposed symmetry. Being low-energy states, we discuss the conditions for spontaneous nucleation of these unique molecules and their direct experimental observation, and provoke the potential use of the phase print of an antivortex or a multiquantum vortex when realized in unconventional circumstances.