Global and local condensate and superfluid fraction of a few hard core bosons in a cubic optical lattice plus external harmonic confinement

We explore the global and local condensate and superfluid (SF) fractions in a system of a few hard core (HC) bosons (N=8 and N=40) trapped inside a combined harmonic optical cubic lattice (CHOCL) at T=0 K. The condensate fraction (CF) is computed for individual lattice wells by separating the one-body density matrix (OBDM) of the whole system into components at the various lattice sites. Then each "lattice-site" component is diagonalized to find its eigenvalues. The eigenvalues are obtained by a method presented earlier [Dubois and Glyde, Phys. Rev. A {\bf 63}, 023602 (2001)]. The effects of interference between the condensates in the lattice wells on the CF in one well is also investigated. The SF fraction (SFF) is calculated for N=40 by using the diffusion formula of Pollock and Ceperley [Pollock and Ceperley, Phys. Rev. B {\bf 36}, 8343 (1987)]. Our chief result is an opposing behavior of the global CF and SFF with increasing lattice wave vector $k$. In addition, the CF in a lattice well is enhanced by the interference with its neighbor wells beyond the result when the interference is neglected. The global SF is depleted with a rise of the repulsion between the bosons, yet at very strong interaction superfluidity is still present. The global CF remains almost constant with increasing HC repulsion. A reduction in the lattice dimension, i.e. an increase in the lattice wave vector, increases the local CF in each lattice well, but reduces the corresponding local SFF. At large HC repulsion, a coexisting SF-(vacuum)MI phase is established.Comment: The tables in the previous withdrawn version (v2) have been corrected and the article updated. This is the final version which has been submitted to Phys. Rev.

Conditions for order and chaos in the dynamics of a trapped Bose-Einstein condensate in coordinate and energy space

We investigate numerically conditions for order and chaos in the dynamics of an interacting Bose- Einstein condensate (BEC) confined by an external trap cut off by a hard-wall box potential. The BEC is stirred by a laser to induce excitations manifesting as irregular spatial and energy oscillations of the trapped cloud. Adding laser stirring to the external trap results in an effective time-varying trapping frequency in connection with the dynamically changing combined external+laser potential trap. The resulting dynamics are analyzed by plotting their trajectories in coordinate phase space and in energy space. The Lyapunov exponents are computed to confirm the existence of chaos in the latter space. Quantum effects and trap anharmonicity are demonstrated to generate chaos in energy space, thus confirming its presence and implicating either quantum effects or trap anharmonicity as its generator. The presence of chaos in energy space does not necessarily translate into chaos in coordinate space. In general, a dynamic trapping frequency is found to promote chaos in a trapped BEC. An apparent means to suppress chaos in a trapped BEC is achieved by increasing the characteristic scale of the external trap with respect to the condensate size.Comment: 19 pages, 14 page

Tunneling of a few strongly repulsive hard-sphere bosons in an optical lattice with tight external harmonic confinement: A quantum Monte Carlo investigation in continuous space

The effect of strongly repulsive interactions on the tunneling amplitude of hard-sphere (HS) bosons confined in a simple cubic (sc) optical lattice plus tight external harmonic confinement in continuous space is investigated. The quantum variational Monte Carlo (VMC) and the variational path integral Monte Carlo (VPI) techniques are used at zero temperature. The effects of the lattice spacing $\pi/k$ on the tunneling amplitude is also considered. The occupancies of the lattice sites as a function of the repulsion between the bosons are further revealed. Our chief result is, that for a small number of bosons (N=8) the overlap of the wave functions in neighboring wells does not change with an increase of the repulsive interactions and changes only minimally for a larger number of particles (N=40). The tunneling amplitude rises with a reduction in the lattice spacing. In addition, the occupancy of the center of the trap decreases in favor of a rise in the occupancy of the lattice sites at the edges of the trap with increasing HS repulsion. Further, it was found that the energy per particle at certain optical depths is insensitive to the number of particles and variations in the HS diameter of the bosons. In order to support our results, we compare the VMC results with corresponding VPI results.Comment: 16 pages, 24 figures. This is an improvement of the previous version in which we consider the presence of pair-tunneling in the strongly interacting regime. The tunneling amplitude is measured in terms of the overlap of wave functions in neighboring well

Thermodynamic properties of an interacting hard-sphere Bose gas in a trap using the static fluctuation approximation

A hard-sphere (HS) Bose gas in a trap is investigated at finite temperatures in the weakly-interacting regime and its thermodynamic properties are evaluated using the static fluctuation approximation (SFA). The energies are calculated with a second-quantized many-body Hamiltonian and a harmonic oscillator wave function. The specific heat capacity, internal energy, pressure, entropy and the Bose-Einstein (BE) occupation number of the system are determined as functions of temperature and for various values of interaction strength and number of particles. It is found that the number of particles plays a more profound role in the determination of the thermodynamic properties of the system than the HS diameter characterizing the interaction, that the critical temperature drops with the increase of the repulsion between the bosons, and that the fluctuations in the energy are much smaller than the energy itself in the weakly-interacting regime.Comment: 34 pages, 24 Figures. To appear in the International Journal of Modern Physics

Generalized Bose-Einstein Condensation

Generalized Bose-Einstein condensation (GBEC) involves condensates appearing simultaneously in multiple states. We review examples of the three types in an ideal Bose gas with different geometries. In Type I there is a discrete number of quantum states each having macroscopic occupation; Type II has condensation into a continuous band of states, with each state having macroscopic occupation; in Type III each state is microscopically occupied while the entire condensate band is macroscopically occupied. We begin by discussing Type I or "normal" BEC into a single state for an isotropic harmonic oscillator potential. Other geometries and external potentials are then considered: the {}"channel" potential (harmonic in one dimension and hard-wall in the other), which displays Type II, the {}"cigar trap" (anisotropic harmonic potential), and the "Casimir prism" (an elongated box), the latter two having Type III condensations. General box geometries are considered in an appendix. We particularly focus on the cigar trap, which Van Druten and Ketterle first showed had a two-step condensation: a GBEC into a band of states at a temperature $T_{c}$ and another "one-dimensional" transition at a lower temperature $T_{1}$ into the ground state. In a thermodynamic limit in which the ratio of the dimensions of the anisotropic harmonic trap is kept fixed, $T_{1}$ merges with the upper transition, which then becomes a normal BEC. However, in the thermodynamic limit of Beau and Zagrebnov, in which the ratio of the boundary lengths increases exponentially, $T_{1}$ becomes fixed at the temperature of a true Type I phase transition. The effects of interactions on GBEC are discussed and we show that there is evidence that Type III condensation may have been observed in the cigar trap.Comment: 17 pages; 6 figures. Intended for American Journal of Physic