Condensed Matter

Caustics are a striking phenomena in natural optics and hydrodynamics: high-amplitude characteristic patterns that are singular in the short wavelength limit. We use exact numerical and approximate semiclassical analytic methods to study quantum versions of caustics that form in Fock space during far-from-equilibrium dynamics following a quench in the Bose Hubbard dimer and trimer models. Due to interparticle interactions, both models are nonlinear and the trimer is also nonintegrable. The caustics exhibit a hierarchy of morphologies described by catastrophe theory that are stable against perturbations and hence occur generically. The dimer case gives rise to discretized versions of folds and cusps which are the simplest two of Thom's catastrophes. The extra dimension available in the trimer case allows for higher catastrophes including the codimension-3 hyperbolic umbilic and elliptic umbilic catastrophes which we argue are organized by, and projections of, an 8-dimensional corank-2 catastrophe known as X 9 . These results indicate a hitherto unrecognized form of universality in quantum many-body dynamics organized by singularities.

We review the recent developments and the current status in the field of quantum-gas cavity QED. Since the first experimental demonstration of atomic self-ordering in a system composed of a Bose-Einstein condensate coupled to a quantized electromagnetic mode of a high- Q optical cavity, the field has rapidly evolved over the past decade. The composite quantum-gas--cavity systems offer the opportunity to implement, simulate, and experimentally test fundamental solid-state Hamiltonians, as well as to realize non-equilibrium many-body phenomena beyond conventional condensed-matter scenarios. This hinges on the unique possibility to design and control in open quantum environments photon-induced tunable-range interaction potentials for the atoms using tailored pump lasers and dynamic cavity fields. Notable examples range from Hubbard-like models with long-range interactions exhibiting a lattice-supersolid phase, over emergent magnetic orderings and quasicrystalline symmetries, to the appearance of dynamic gauge potentials and non-equilibrium topological phases. Experiments have managed to load spin-polarized as well as spinful quantum gases into various cavity geometries and engineer versatile tunable-range atomic interactions. This led to the experimental observation of spontaneous discrete and continuous symmetry breaking with the appearance of soft-modes as well as supersolidity, density and spin self-ordering, dynamic spin-orbit coupling, and non-equilibrium dynamical self-ordered phases among others. In addition, quantum-gas--cavity setups offer new platforms for quantum-enhanced measurements. In this review, starting from an introduction to basic models, we pedagogically summarize a broad range of theoretical developments and put them in perspective with the current and near future state-of-art experiments.

We identify the chaotic phase of the Bose-Hubbard Hamiltonian by the energy-resolved correlation between spectral features and structural changes of the associated eigenstates as exposed by their generalized fractal dimensions. The eigenvectors are shown to become ergodic in the thermodynamic limit, in the configuration space Fock basis, in which random matrix theory offers a remarkable description of their typical structure. The distributions of the generalized fractal dimensions, however, are ever more distinguishable from random matrix theory as the Hilbert space dimension grows.

We consider a coupled top model describing two interacting large spins, which is studied semiclassically as well as quantum mechanically. This model exhibits variety of interesting phenomena such as quantum phase transition (QPT), dynamical transition and excited state quantum phase transitions above a critical coupling strength. Both classical dynamics and entanglement entropy reveals ergodic behavior at the center of energy density band for an intermediate range of coupling strength above QPT, where the level spacing distribution changes from Poissonian to Wigner-Dyson statistics. Interestingly, in this model we identify quantum scars as reminiscence of unstable collective dynamics even in presence of interaction. Statistical properties of such scarred states deviate from ergodic limit corresponding to random matrix theory and violate Berry's conjecture. In contrast to ergodic evolution, oscillatory behavior in dynamics of unequal time commutator and survival probability is observed as dynamical signature of quantum scar, which can be relevant for its detection.

We show that the Bose-glass phase of a one-dimensional disordered Bose fluid exhibits a chaotic behavior, i.e., an extreme sensitivity to external parameters. Using bosonization, the replica formalism and the nonperturbative functional renormalization group, we find that the ground state is unstable to any modification of the disorder configuration ("disorder" chaos) or variation of the Luttinger parameter ("quantum" chaos, analog to the "temperature" chaos in classical disordered systems). This result is obtained by considering two copies of the system, with slightly different disorder configurations or Luttinger parameters, and showing that inter-copy statistical correlations are suppressed at length scales larger than an overlap length ξ ov ?�|ϵ | ??/α ( |ϵ|?? is a measure of the difference between the disorder distributions or Luttinger parameters of the two copies). The chaos exponent α can be obtained by computing ξ ov or by studying the instability of the Bose-glass fixed point for the two-copy system when ϵ?? . The renormalized, functional, inter-copy disorder correlator departs from its fixed-point value -- characterized by cuspy singularities -- via a chaos boundary layer, in the same way as it approaches the Bose-glass fixed point when ϵ=0 through a quantum boundary layer. Performing a linear analysis of perturbations about the Bose-glass fixed point, we find α=1 .

The mean-field limit of a bosonic quantum many-body system is described by (mostly) non-linear equations of motion which may exhibit chaos very much in the spirit of classical particle chaos, i.e. by an exponential separation of trajectories in Hilbert space with a rate given by a positive Lyapunov exponent λ . The question now is whether λ imprints itself onto measurable observables of the underlying quantum many-body system even at finite particle numbers. Using a Bose-Einstein condensate expanding in a shallow potential landscape as a paradigmatic example for a bosonic quantum many-body system, we show, that the number of non-condensed particles is subject to an exponentially fast increase, i.e. depletion. Furthermore, we show that the rate of exponential depletion is given by the Lyapunov exponent associated with the chaotic mean-field dynamics. Finally, we demonstrate that this chaos-induced depletion is accessible experimentally through the visibility of interference fringes in the total density after time of flight, thus opening the possibility to measure λ , and with it, the interplay between chaos and non-equilibrium quantum matter, in a real experiment.

The paper discusses what characteristic quantities could quantify nonequilibrium states of Bose systems. Among such quantities, the following are considered: effective temperature, Fresnel number, and Mach number. The suggested classification of nonequilibrium states is illustrated by studying a Bose-Einstein condensate in a shaken trap, where it is possible to distinguish eight different nonequilibrium states: weak nonequilibrium, vortex germs, vortex rings, vortex lines, deformed vortices, vortex turbulence, grain turbulence, and wave turbulence. Nonequilibrium states are created experimentally and modeled by solving the nonlinear Schrödinger equation.

We report parametric resonances (PRs) in the mean-field dynamics of a one-dimensional dipolar Bose-Einstein condensate (DBEC) in widely varying trapping geometries. The chief goal is to characterize the energy levels of this system by analytical methods and the significance of this study arises from the commonly known fact that in the presence of interactions the energy levels of a trapped BEC are hard to calculate analytically. The latter characterization is achieved by a matching of the PR energies to energy levels of the confining trap using perturbative methods. Further, this work reveals the role of the interplay between dipole-dipole interactions (DDI) and trapping geometry in defining the energies and amplitudes of the PRs. The PRs are induced by a negative Gaussian potential whose depth oscillates with time. Moreover, the DDI play a role in this induction. The dynamics of this system is modeled by the time-dependent Gross- Pitaevskii equation (TDGPE) that is numerically solved by the Crank-Nicolson method. The PRs are discussed basing on analytical methods: first, it is shown that it is possible to reproduce PRs by the Lagrangian variational method that are similar to the ones obtained from TDGPE. Second, the energies at which the PRs arise are closely matched with the energy levels of the corresponding trap calculated by time-independent perturbation theory. Third, the most probable transitions between the trap energy levels yielding PRs are determined by time-dependent perturbation theory. The most significant result of this work is that we have been able to characterize the above mentioned energy levels of a DBEC in a complex trapping potential.

The narrow s-wave Feshbach resonance of a 6 Li Fermi gas shows strong three-body loss, which is proposed to be used to measure the minute change of a magnetic field around the resonance. However, the eddy current will cause ultracold atom experiencing a magnetic field delayed to the desired magnetic field from the current of the magnetic coils. The elimination of the eddy current effect will play a key role in any experiments that motivated to measure the magnetic field to the precision of a part per million stability. Here, we apply a method to correct the eddy current effect for precision measurement of the magnetic field. We first record the three-body loss influenced by the effect of induced eddy current, then use a certain model to obtain the time constant of the actual magnetic field by fitting the atom loss. This precisely determines the actual magnetic field according to the time response of the three-body loss. After that, we implement the desired magnetic field to the atoms so that we can analyze the three-body loss across the whole narrow Feshbach resonance. The results show that the three-body recombination is the dominated loss mechanism near the resonance. We expect this practical method of correcting the eddy current error of the magnetic field can be further applied to the future studies of quantum few- and many-body physics near a narrow Feshbach resonance.

We model generation of vortex modes in exciton-polariton condensates in semiconductor micropillars, arranged into a hexagonal ring molecule, in the presence of TE-TM splitting. This splitting lifts the degeneracy of azimuthally modulated vortex modes with opposite topological charges supported by this structure, so that a number of non-degenerate vortex states characterized by different combinations of topological charges in two polarization components appears. We present a full bifurcation picture for such vortex modes and show that because they have different energies, they can be selectively excited by coherent pump beams with specific frequencies and spatial configurations. At high pumping intensity, polariton-polariton interactions give rise to the coupling of different vortex resonances and a bistable regime is achieved.