Properties of bosonic atoms in small systems with a periodic quasi one-dimensional circular toroidal lattice potential subjected to rotation are examined by performing exact diagonalization in a truncated many body space. The expansion of the many-body Hamiltonian is considered in terms of the first band Bloch functions, and no assumption regarding restriction to nearest-neighbor hopping (tight-binding approximation) is involved. A finite size version of the zero temperature phase diagrams of Fisher et al. \cite{Fisher} is obtained and the results, in remarkable quantitative correspondence with the results available for larger systems, discussed. Ground state properties relating to superfluidity are examined in the context of two-fluid phenomenology. The basic tool, consisting of the intrinsic inertia associated with small rotation angular velocities in the lab frame, is used to obtain ground state `superfluid fractions' numerically. They are analytically associated with one-body, uniform solenoidal currents in the case of the adopted geometry. These currents are in general incoherent superpositions of contributions from each eigenstate of the associated reduced one-body densities, with the corresponding occupation numbers as weights. Full coherence occurs therefore only when only one eigenstate is occupied by all bosons. The obtained numerical values for the superfluid fractions remain small throughout the parameter region corresponding to the `Mott insulator to superfluid' transition, and saturate at unity only as the lattice is completely smoothed out.
Properties of bosonic atoms in small systems with a periodic quasi one-dimensional circular toroidal lattice potential subjected to rotation are examined by performing exact diagonalization in a truncated many body space. The expansion of the many-body Hamiltonian is considered in terms of the first band Bloch functions, and no assumption regarding restriction to nearest-neighbor hopping (tight-binding approximation) is involved. A finite size version of the zero temperature phase diagrams of Fisher et al. \cite{Fisher} is obtained and the results, in remarkable quantitative correspondence with the results available for larger systems, discussed. Ground state properties relating to superfluidity are examined in the context of two-fluid phenomenology. The basic tool, consisting of the intrinsic inertia associated with small rotation angular velocities in the lab frame, is used to obtain ground state superfluid fractions' numerically. They are analytically associated with one-body, uniform solenoidal currents in the case of the adopted geometry. These currents are in general incoherent superpositions of contributions from each eigenstate of the associated reduced one-body densities, with the corresponding occupation numbers as weights. Full coherence occurs therefore only when only one eigenstate is occupied by all bosons. The obtained numerical values for the superfluid fractions remain small throughout the parameter region corresponding to the Mott insulator to superfluid' transition, and saturate at unity only as the lattice is completely smoothed out.
We study properties of ultracold quantum gases trapped in 1D and under the influence of an underlying optical lattice. We analyze both fermionic and bosonic systems following different numerically exact approaches, like quantum Monte Carlo for soft-core bosons and two-species fermions, and a recently developed exact approach for HCB. The exposition is organized as follows. In Chap. 2, we study single species degenerated fermions on 1D lattices. We show that a splitting of the system takes place in a region of the spectrum with eigenstates that have a non-vanishing weight only in a fraction of the system. We also show that if on top of the lattice an alternating potential is introduced, doubling the original periodicity, an additional slicing' of the system can be achieved. The width and number of such regions can be controlled in a given energy range by the amplitude of the new modulation. By filling these systems the ground state may exhibit insulating regions due to the presence of local gaps, which can be observed in the local density of states. As a complement to these 1D results we present in Appendix A results obtained in 2D lattices. In Chap. 3 we review in some detail the two quantum Monte Carlo (QMC) techniques that are relevant to our study of confined soft-core bosons and two-species fermions on optical lattices, the worldline QMC and the zero temperature projector QMC, respectively. The basis of the classical Monte Carlo method are also presented, included a proof of the Central Limit theorem (Appendix B) that establishes its foundations. Properties of soft-core bosons confined on optical lattices are presented in Chap. 4. QMC simulations show that due to the presence of the slowly varying confining potential the concept of commensurability, typical for periodic systems, loses its meaning. Mott domains appears for a continuous range of fillings and always coexist with superfluid phases, as it was mentioned before. The latter feature is reflected by the behavior of the global compressibility, which never vanishes. A local compressibility, proportional to the variance of the local density, is defined in order to characterize the local Mott-insulating phases. Finally, a phase diagram for trapped systems is also presented. In Chap. 5 we study the ground state properties of two-components fermions confined on optical lattices. Like for the bosonic case, Mott domains also appear for a continuous range of fillings and always coexist with compressible phases. We define a local order parameter (that we denominate local compressibility), which characterizes the Mott insulating phases in an unambiguous way. By means of this local compressibility, we study in detail the interphase between the metallic and insulating region finding that critical behavior sets in revealing a new critical exponent. Furthermore, the behavior of the local compressibility and the variance of the density are found to be universal in this case, independently of the confining potential and the strength of the interaction, as usual in critical phenomena. We present in Chap. 6 a recently developed exact numerical approach that allows to study ground state properties of HCB confined on 1-D lattices. This exact treatment is applied to study the off-diagonal behavior of the one-particle density matrix (OPDM) and related quantities in the equilibrium case. We find that the OPDM decays as a power-law x^-1/2 for large-x, irrespective of the confining potential chosen. The power-law above is shown to determine the scaling of the occupation of the lowest natural orbital in the thermodynamic limit. This scaling and its finite size corrections are also studied for arbitrary powers of the confining potential. In addition, we find a power-law decay of the NO occupations at low densities, and show that its exponent is also universal. The low density limit in the lattice, equivalent to continuous systems, is also analyzed in detail. The approach above is generalized in Chap. 7 to study the non-equilibrium dynamics of HCB in 1-D configurations with an underlying lattice. The presence of the lattice enables the creation of pure Fock states of HCB (a HCB per lattice site) where there is no coherence in the system. We show that quasi-long range correlations develop in the equal-time-one-particle density matrix when such states are allowed to evolve freely, and that they lead to the formation of quasi-condensates of HCB at finite momentum. In addition,we obtain an universal power-law describing the population of the quasi-condensate as a function of time. Finally, we discuss how such systems can be used to create atom lasers with a wave-length that can be controlled through the lattice parameter.
For systems of interacting, ultracold spin-zero neutral bosonic atoms, harmonically trapped and subject to an optical lattice potential, we derive an Extended Bose Hubbard (EBH) model by developing a systematic expansion for the Hamiltonian of the system in powers of the lattice parameters and of a scale parameter, the lattice attenuation factor. We identify the dominant terms that need to be retained in realistic experimental conditions, up to nearest-neighbor interactions and nearest-neighbor hoppings conditioned by the on-site occupation numbers. In the mean field approximation, we determine the free energy of the system and study the phase diagram both at zero and at finite temperature. At variance with the standard on site Bose Hubbard model, the zero-temperature phase diagram of the EBH model possesses a dual structure in the Mott insulating regime. Namely, for specific ranges of the lattice parameters, a density wave phase characterizes the system at integer fillings, with domains of alternating mean occupation numbers that are the atomic counterparts of the domains of staggered magnetizations in an antiferromagnetic phase. We show as well that in the EBH model, a zero-temperature quantum phase transition to pair superfluidity is, in principle, possible, but completely suppressed at the lowest order in the lattice attenuation factor. Finally, we determine the possible occurrence of the different phases as a function of the experimentally controllable lattice parameters.
We present a study of the superfluid properties of atomic Bose gases in optical lattice potentials using the Bose-Hubbard model. To do this, we use a microscopic definition of the superfluid fraction based on the response of the system to a phase variation imposed by means of twisted boundary conditions. We compare the superfluid fraction to other physical quantities, i.e., the interference pattern after ballistic expansion, the quasimomentum distribution, and number fluctuations. We have performed exact numerical calculations of all these quantities for small one-dimensional systems. We show that the superfluid fraction alone exhibits a clear signature of the Mott-insulator transition. Observables like the fringe visibility, which probe only ground-state properties, do not provide direct information on superfluidity and the Mott-insulator transition.
We investigate the ground state properties of the repulsive interacting bosons in an one dimensional rotating lattice ring, and reveal that the superfluid density of the system and the mass current it can carry in the rotating coordinate are periodic functions of the velocity of the lattice. In the weakly interacting limit, the vortices are generated at the critical velocities, and in the strongly interacting limit, the phase twist of the bosons induced by the rotating lattice manifests the superfluid density and drive the system to undergo phase transitions between the superfluid and Mott insulator.
We investigate the formation of self-bound quantum droplets in a one-dimensional binary mixture of bosonic atoms, applying the method of numerical diagonalization of the full Hamiltonian. The excitation spectra and ground-state pair correlations signal the formation of a few-boson droplet when crossing the region of critical interspecies interactions. The self-binding affects the rotational excitations, displaying a change in the energy dispersion from negative curvature, associated with superfluidity in the many-body limit, to a nearly parabolic curvature indicative of rigid body rotation. We exploit two global symmetries of the system to further analyze the few-body modes in terms of transition matrix elements and breathing mode dynamics. The ground and excited state energies reported here are exact within the respective Hilbert spaces, which have been adapted according to a given set of single-particle basis states and a specific importance threshold of the employed truncation method. We compare these data to the usual ad hoc inclusion of higher-order contributions in the extended Gross-Pitaevskii equation. For the given parameters we find a remarkable agreement between the few-body regime and the thermodynamic limit in one dimension.
This thesis concerns ultracold two-dimensional atomic gases and the associated novel superfluid phases in an optical lattice subjected to staggered rotations. In the first part, we study the Berezinskii-Kosterlitz-Thouless (BKT) transition in a two-dimensional Bose gas. Motivated by recent experiments on the cold gas in the BKT regime, we employ a modified Popov theory and a renormalization group theory to describe the system. The two-dimensional renormalization group flow equations display unique features, which we interpret as precursors of the BKT physics. We further calculate higher correlation functions close to the BKT transition, which can serve as observables for detecting effects beyond mean-field. Finally, we obtain the density profile under the conditions of the experiment performed in the group of Dalibard in Paris, France. In the second part of the thesis, we study a recent experimental proposal of a time-dependent optical lattice with staggered rotations. We show that in the tight-binding regime, the system can be effectively described by a (Bose-)Hubbard model subjected to an artificial staggered magnetic field. The staggered magnetic field (flux) in the lattice leads to conical points in the band structure which possesses distinct minima for different strength, and can give rise to Dirac-like dispersions. For bosons, we show that besides the homogenous superfluid phase, a staggered-vortex superfluid phase is stabilized for large flux values. We show that the staggered flux renormalizes the phase boundary and find a multi-critical point separating the two superfluid phases and the Mott insulating phase. Finally, the full phase diagram of the generalized Bose-Hubbard model at zero temperature is determined. We then generalize the model for a mixture of ultracold bosons and spin-1/2 fermions. By using the Bogoliubov theory for the bosonic sector, we show that the bosonic superfluid mediates attractive interactions between fermions up to the nearest-neighbor site. The resulting model is an extended Hubbard model in the presence of a staggered magnetic field, where the on-site and the nearest-neighbor attractive interactions can be tuned independently. We then employ the Bardeen-Cooper-Schrieffer theory for superconductivity, by considering an on-site pairing order parameter competing with a nearest-neighbor resonating-valence-bond order parameter. Due to conical points in the energy band, the system possesses quantum critical property and the resulting phase diagram displays features which are strikingly similar to that of high-critical-temperature superconductors or heavy-fermion superconductors.