Narrow Results

In this paper, we study the Gross–Pitaevskii energy of a Bose–Einstein condensate in the presence of an optical lattice, modeled by a periodic potential V(x₃) in the third direction. We study a simple case where the wells of the potential V correspond to regions where V vanishes, and are separated by small intervals of size δ where V is large. According to the intensity of V, we determine the limiting energy as δ tends to 0. In the critical case, the periodic potential approaches a sum of delta functions and the limiting energy has a contribution due to the value of the wave function between the wells. The proof relies on Gamma convergence type techniques.

Our aim is to analyze the various energy functionals appearing in the physics literature and describing the behavior of a Bose–Einstein condensate in an optical lattice. We want to justify the use of some reduced models and control the error of approximation. For that purpose, we will use the semi-classical analysis developed for linear problems related to the Schrödinger operator with periodic potential or multiple wells potentials. We justify, in some asymptotic regimes, the reduction to low dimensional problems and analyze the reduced problems.

In terms of the Green's function method we study the energy spectrum of dipolar boson atoms with the dipole-dipole interaction in an optical lattice. The Superfluid-Mott-Insulator phase transition condition of the dipolar bosons is determined from the energy-band structure of the excitation spectrum as a function of interatomic repulsion, dipolar bosons interaction and the tunnel coupling constants. The superfluid phase is explained explicitly from the energy spectrum derived in terms of the Bogoliubov approach as well.

We investigate analytically the stability of Bloch waves at the boundary of the first Brillouin zone in two coupled Bose–Einstein condensates confined in an optical lattice. Contrary to the single component case, we find here two critical density regimes which determine the stability of the Bloch waves. Breakdown of Bloch oscillations appear when n₁/n₂<N_C1 or when n₁/n₂>N_C2, here N_C1 and N_C2 are some critical values of n₁/n₂. There is an intermediate regime between N_C1 and N_C2 where the Bloch oscillations are stable and the condensates behave like single particles.

The advent of coherent matter waves in the form of Bose-Einstein condensates, coupled with periodic potentials in the form of optical lattices, has established a new area of research on the boundary between atomic and condensed matter physics. This article is a brief review of the recent experimental progress in the area of degenerate Bose gases loaded into optical lattices, including strongly correlated systems and the role of dimensionality. Future prospects will also be outlined.

We study a confined mixture of Rb and K atoms in a one dimensional optical lattice, at low temperature, in the quanta1 degeneracy regime. This mixture exhibits an attractive boson-fermion interaction, and thus above certain values of the number of particles the mixture collapses. We investigate, in the mean-field approximation, the curve for which this phenomenon occurs, in the space of number of particles of both species. This is done for different types of optical lattices.

We describe a simple model of fermions in quasi-one dimension that features interaction-induced deconfinement (a phase transition where the effective dimensionality of the system increases as interactions are turned on) and which can be realised using dipolar fermions in an optical lattice¹. The model provides a relisation of a "soft quantum matter" phase diagram of strongly-correlated fermions, featuring meta-nematic, smectic and crystalline states, in addition to the normal Fermi liquid. In this paper we review the model and discuss in detail the mechanism behind each of these transitions on the basis of bosonization and detailed analysis of the RPA susceptibility.

From the discrete nonlinear Schrödinger equation and the linear stability analysis, the modulational instability (MI) of dipolar Bose–Einstein Condensates (BECs) in an optical lattice is studied. The MI relation of the dipolar BECs in the optical lattice with the on-site interaction and the inter-site interaction is obtained. The results show that there is a great influence of inter-site interaction on MI of dipolar BEC in the optical lattice. This gives us some useful information for manipulating dipolar BECs in practice.

We show how the remotest sites of a finite lattice can be entangled, with the amount of entanglement exceeding that of a singlet, solely through the dynamics of an ideal Bose gas in a special initial state in the lattice. When additional occupation number measurements are made on the intermediate lattice sites, then the amount of entanglement and the length of the lattice separating the entangled sites can be significantly enhanced. The entanglement generated by this dynamical procedure is found to be higher than that for the ground state of an ideal Bose gas in the same lattice. A second dynamical evolution is shown to verify the existence of these entangled states, as well entangle qubits belonging to well separated quantum registers.

We study asymmetric two-component mixtures of hard-core bosons in one dimension by carrying out quantum Monte Carlo calculations. We have found that the background bosons have different roles in suppressing superfluidity depending on the degree of asymmetry and the interspecies interactions. For weak asymmetry, the superfluid stiffness is solely governed by the density of bosons with a renormalized chemical potential, whereas in the other limit with heavy background bosons, localization effects due to the interspecies interactions appear. Particularly, strong asymmetry in hopping amplitudes induces a solid phase transition at commensurate filling.

We have studied the ground state properties of hard-core bosons in a two-leg optical ladder in the presence of uniform and staggered frustrations due to an artificial gauge field. By calculating the ground state via the Lanczos method, we find first-order phase transitions tuned by the staggered gauge field between the Meissner and the vortex states. The momentum distributions show that the Meissner state has edge and staggered currents, while the vortex states have vortex-solid or vortex-glass phases in the presence of a staggered field.

In this paper, we study the static and dynamic properties of vortices formation in a rotating Bose–Einstein condensate in synthetic magnetic field. The condensate is confined in a harmonic potential and an optical lattice potential. We numerically solve the time-dependent Gross–Pitaevskii equation in two dimensions and compare the vortices formation process with rotating frame method. We derive the two expressions about vortex number N${}_{\mathrm{\text{v}}}$ and chemical potential $\mu$ as a function of rotational frequency $\mathrm{\Omega }$, which show that the number of vortices N${}_{\mathrm{\text{v}}}$ increases linearly with $\mathrm{\Omega }$, while the chemical potential $\mu$ remains nearly unchanged with increasing $\mathrm{\Omega }$. The analytical results are in qualitative agreement with the numerical ones. Moreover, we also investigate the vortex dynamics. Numerical results indicate that the time spent for the formation of steady state vortices is reduced drastically if the optical lattice is considered. Nevertheless, the synthetic magnetic field approach is still low efficient to generate more vortices.

Motivated by recent developments in experimental studies of Bose–Fermi mixtures, we investigate monopole oscillation dynamics of Bose–Fermi mixtures in a three-dimensional (3D) optical lattice and an external isotropic harmonic trap potential. We use dynamical Gutzwiller approximation and calculate time dependence of the average spatial extent of particles in a coexisting phase where superfluid bosons and metal fermions coexist. With the trap potential, the bosons concentrate at and around the center of the potential and the fermions surround the bosons. They begin to demonstrate monopole oscillations when the trap potentials are suddenly changed to the ones with smaller curvature. In particular, the correlation between the oscillations of the bosons and fermions is different depending on the condition of the initial confinement. We also show the case where only one of the two trap potentials is changed.

In this paper, we investigate the spectral functions of the Higgs mode in $O(2)$ model, which can be experimentally realized in a two-dimensional Bose gas. Zero temperature limit is considered. Our calculation fully includes the 2-loop contributions. Peaks show up in the spectral functions of both the longitudinal and the scalar susceptibilities. Thus, this model cannot explain the disappearance of the response at the weak interaction limit. Neither it can explain the similarity between the longitudinal and the scalar susceptibilities in the visibility of the Higgs mode. A possible lower peak at about $2m_{\sigma }$ is also noted.

We predict the collapse and recurrence of Rabi oscillations in a two-component Bose–Einstein condensate trapped in an optical lattice. The optical trap creates a periodic potential with a periodic coupling. The collapse and revival sequence is explained on the basis of wave-packet dynamics. The collapse and revival sequence is found to be a consequence of competition between the potential energy difference and the driving frequency.

In this paper we review recent progress in studying quantum phase transitions in one- and two-component Bose–Einstein condensates (BEC) in optical lattices. These phase transitions involve the emergence and disappearance of quantum coherence over the whole optical lattice and of linear superposition of macroscopic quantum states. The latter may provide new means to engineer and manipulate novel macroscopic quantum states and novel coherent atomic beams for quantum information processing, quantum computing and other purposes.

In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose–Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross–Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent structures in them. Trains of solitons in one dimension and vortex arrays in two dimensions are prototypical examples of the resulting nonlinear waveforms, upon which we briefly touch at the end of this review.

We consider several effects of the matter wave dynamics which can be observed in Bose–Einstein condensates embedded into optical lattices. For low-density condensates, we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross–Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models, we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force and lattice defects affect the nonlinear matter waves.

Introducing the next-nearest-neighbor hopping t′ into the Bose–Hubbard model, we study its effects on the phase diagram, on the ground-state energy, and on the quasiparticle and quasihole dispersion relations of the Mott insulating phase in optical lattices. We have found that a negative value of t′ enlarges the Mott-insulating region on the phase diagram, while a positive value of t′ acts oppositely. We have also found that the effects of t′ are dependent on the dimensionality of optical lattices with its effects largest in three-dimensional optical lattices.

We propose a scheme to simulate the t–J–U model and investigate the quantum phase transitions using cold atoms confined in a two-dimensional optical lattice. By adjusting the parameters of external laser fields, we find a phase transition from antiferromagnetic phase to Gossamer superfluid. The Gossamer superfluid has a d-wave or mixed d- and s-wave symmetry, depending on the hopping parameters which can be adjusted by tuning the laser field. Experimental protocol to observe these new phenomena for future experiments has also been briefly discussed.