We theoretically analyze interference patterns of parametrically driven one-dimensional ultracold atomic gases. By modulating the interaction strength periodically in time, we propose to excite collective modes in a pair of independent one-dimensional gases at energies corresponding to the drive frequency. The excited collective modes lead to spatial oscillations in the correlations of the interference pattern, which can be analyzed to obtain the sound velocity of the collective modes. We discuss both bosonic and fermionic systems and how such experiments could be used to probe spin-charge separation.
We theoretically analyze interference patterns of parametrically driven one-dimensional ultracold atomic gases. By modulating the interaction strength periodically in time, we propose to excite collective modes in a pair of independent one-dimensional gases at energies corresponding to the drive frequency. The excited collective modes lead to spatial oscillations in the correlations of the interference pattern, which can be analyzed to obtain the sound velocity of the collective modes. We discuss both bosonic and fermionic systems and how such experiments could be used to probe spin-charge separation.
This thesis contains theoretical research on ultracold quantum gases in spatially periodic potentials, featuring high-frequency periodic driving, long-range interactions, or both. The largest part features deep potentials where the behaviour of the gas is well-described by quantum lattice models. The periodic driving is then integrated out to obtain effective time-independent descriptions. One project investigates emergent long-range interactions in a stationary, weak, spatially periodic potential, where a lattice theory is not appropriate.In two short introductory chapters, the topics of ultracold atoms, optical lattice potentials, periodic driving, and long-range interactions, are sketched, without attempting to give a complete overview. Some experimentally relevant length and energy scales are given, but the main focus is on deriving and constructing theoretical descriptions of ultracold gases in various spatially and/or temporally periodic potentials.The first project presented, is focused on how the single-particle spectrum of a Bose gas in a non-separable two-dimensional square lattice is affected by high-frequency periodic driving; the most striking conclusion is that under suitable circumstances, it develops two inequivalent minima, leading to finite-momentum Bose-Einstein condensation. Perturbative calculations indicate that local interactions induce spontaneous time-reversal symmetry breaking (TRSB) in such a system.The second project investigates the interplay between kinetic frustration and long-range interactions in fermionic gases. Both a mean-field approximation and exact diagonalisations predict that such a system, studied in the more specific realisation of a weakly interacting dipolar fermionic gas in a 2D triangular lattice, also leads to spontaneous TRSB. Furthermore, a density wave could form at quarter filling, where the Fermi surface is perfectly nested. Perhaps more interesting yet, a spatially inhomogeneous TRSB pattern is predicted, confined to the low-density sublattice that emerges in the density wave.The third project revolves around the question of supersolidity in the presence of a gauge field. Applying the Bogolyubov approximation to a variation of the extended Bose-Hubbard model, indicates that combining an artificial staggered magnetic field in a 2D square lattice with nearest-neighbour density-density interactions, not only leads to a supersolid with staggered vortices, but also induces an inhomogeneous distribution of the associated currents around the elementary plaquette.In the fourth project, a one-dimensional Bose gas with strong local interactions in a weak lattice at incommensurate densities is shown to feature excitations corresponding to excess or deficit particles. The excitations interact repulsively at long distances, in spite of the fact that the underlying atoms themselves do not. As a consequence, the incommensurability of the density with the lattice can drive a transition to a density wave and even a supersolid.The four above-mentioned research projects combine bosonic and fermionic gases, weak and strong interactions, perturbative and mean-field approximations, effective field theories and exact diagonalisations. The main overall conclusion is that long-range interactions and high-frequency periodic driving lead to a very diverse range of fascinating phenomena in ultracold lattice gases.
Collective oscillations of superfluid mixtures of ultracold fermionic and bosonic atoms are investigated while varying the fermion-boson scattering length. We study the dynamics with respect to excited center of mass modes and breathing modes in the mixture. Parametric resonances are also analyzed when the scattering length varies periodically in time, by comparing partial differential equation models and ordinary differential equation models for the dynamics. An application to the recent experiment with fermionic 6Li and bosonic 7Li atoms, which approximately have the same masses, is discussed.
Periodically driven quantum systems are currently explored in view of realizing novel many-body phases of matter. This approach is particularly promising in gases of ultracold atoms, where sophisticated shaking protocols can be realized and interparticle interactions are well controlled. The combination of interactions and time-periodic driving, however, often leads to uncontrollable heating and instabilities, potentially preventing practical applications of Floquet engineering in large many-body quantum systems. In this work, we experimentally identify the existence of parametric instabilities in weakly interacting Bose-Einstein condensates in strongly driven optical lattices through momentum-resolved measurements, in line with theoretical predictions. Parametric instabilities can trigger the destruction of weakly interacting Bose-Einstein condensates through the rapid growth of collective excitations, in particular in systems with weak harmonic confinement transverse to the lattice axis. Understanding the onset of parametric instabilities in driven quantum matter is crucial for determining optimal conditions for the engineering of modulation-induced many-body systems.
The response of ultracold atomic Bose gases in time-dependent optical lattices is discussed based on direct simulations of the time-evolution of the many-body state in the framework of the Bose-Hubbard model. We focus on small-amplitude modulations of the lattice potential as implemented in several recent experiments and study different observables in the region of the first resonance in the Mott-insulator phase. In addition to the energy transfer we investigate the quasimomentum structure of the system which is accessible via the matter-wave interference pattern after a prompt release. We identify characteristic correlations between the excitation frequency and the quasimomentum distribution and study their structure in the presence of a superlattice potential.
Ultracold gases are of great interest in modern physics. The main reason is that in the systems of ultracold gases the parameters can be easily tuned, thus they can be used as a testing ground for various quantum many-body theories. Interesting macroscopic quantum effects have been observed in the ultracold gas systems, for instance Bose-Einstein condensation. In this thesis, theoretical knowledge of ultracold gases is extended. A summary of the methods used in this thesis is given, including a detailed description of the density response theory and the time-evolving block decimation (TEBD) algorithm. Collective excitations of an ultracold gas in a three-dimensional (3D) spherically symmetric trap are studied in detail in publications II and III. As a result, several collective modes are discovered such as a low energy Higgs-type mode, a second sound-like mode, and a strong mode resembling the Leggett mode. Using the TEBD algorithm, physics of a polaron in a one-dimensional (1D) lattice, and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in 1D are studied in publications IV and I, respectively. In publication I a method to detect the FFLO phase is suggested, namely by the observation of the change in the double occupancy after a lattice depth modulation. Publication IV compares a variational ansatz and the TEBD simulations and finds an excellent agreement, indicating that the variational ansatz can be used to describe the system for a certain range of interactions.
The experimental achievement of Bose-Einstein condensation (BEC) in 1995 has aroused a large interest among the community of the physics of ultracold atomic gases.Recently new topics have attracted the attention of this very active field of research , in particular we could mention low dimensional systems, atoms in lattices, and classical and Fermi gases.As for the case of BEC, the study of the dynamical aspects of these systems provides a very valuable information.In particular time-of-flight experiments and measurements of collective excitations have been successfully realized providing a clear signature of the BEC and the regime of interaction.Following this line, in this thesis, we want to study some dynamical aspects of these new systems nowadays avaliable due to the remarkable development of the experimental techniques.In the first chapter we present the state of the art of the field, providing the historical path and the motivation for the work developed in the next chapters.In the second chapter, we discuss some general concepts in order to introduce the formalism and notation employed throughout the thesis, as well as those topics analyzed in more detail in the further chapters.In chapter three, we explain some general techniques which are useful to deal with one dimensional systems, and analyze in detail the expansion of a one dimensional Bose gas in a guide, which differs significantly from the expansion of a BEC.In particular we are able to predict the range of parameters for which the self-similarity of the expansion is violated.Chapter four is devoted to the physics of atoms in a periodic potential.We start, as in the previous chapter, introducing some general concepts and techniques employed further.First we discuss the physics of the superfluid case in a trap, showing that under certain conditions a three dimensional gas exhibits one dimensional properties.viii In particular, static properties and the lowest collective excitations differ significantly from those of a BEC.We extend the analysis to the insulating phase showing that correlation properties depend on the on-site one-dimensional regime.These correlations can be measured in current time of flight experiments.In the fifth chapter we analyze the dynamics of classical and Fermi gases.We show in detail how expansion and collective excitations depend on the interaction regime and the trap anisotropy.In particular, the expansion can be used to distinguish between a normal Fermi gas and a superfluid one.We summarize in the last chapter the new results presented in this thesis.