We study the formation and collision-aided decay of an ultra-cold atomic Bose-Einstein condensate in the first excited band of a double-well 2D-optical lattice with weak harmonic confinement in the perpendicular z direction. This lattice geometry is based on an experiment by Wirth et al. [1]. The double well is asymmetric, with the local ground state in the shallow well nearly degenerate with the first excited state of the adjacent deep well. We compare the band structure obtained from a tight-binding model with that obtained numerically using a plane wave basis. We find the tight binding model to be in quantitative agreement for the lowest two bands, qualitative for next two bands, and inadequate for even higher excited bands. The band widths of the excited bands are much larger than the harmonic oscillator energy spacing in the z direction. We then study the thermodynamics of a non-interacting Bose gas in the first excited band. We estimate the condensate fraction and critical temperature, Tc, as functions of lattice parameters. For typical atom numbers, the critical energy kBTc, with kB the Boltzmann constant, is larger than the excited band widths and harmonic oscillator energy. Using conservation of total energy and atom number, we show that the temperature increases after the lattice transformation. Finally, we estimate the time scale for a two-body collision-aided decay of the condensate as a function of lattice parameters. The decay involves two processes, the dominant one in which both colliding atoms decay to the ground band, and the second involving excitation of one atom to a higher band. For this estimate, we have used tight binding wave functions for the lowest four bands, and numerical estimates for higher bands. The decay rate rapidly increases with lattice depth, but close to the critical temperature, it stays smaller than the tunneling rate between the s and p orbitals in adjacent wells.
We show that for ultra-cold neutral bosonic atoms held in a three-dimensional periodic potential or optical lattice, a Hubbard model with dominant, attractive three-body interactions can be generated. In fact, we derive that the effect of pair-wise interactions can be made small or zero starting from the realization that collisions occur at the zero-point energy of an optical lattice site and the strength of the interactions is energy dependent from effective-range contributions. We determine the strength of the two-and three-body interactions for scattering from van-der-Waals potentials and near Fano-Feshbach resonances. For van-der-Waals potentials, which for example describe scattering of alkaline-earth atoms, we find that the pair-wise interaction can only be turned off for species with a small negative scattering length, leaving the 88 Sr isotope a possible candidate. Interestingly, for collisional magnetic Feshbach resonances this restriction does not apply and there often exist magnetic fields where the two-body interaction is small. We illustrate this result for several known narrow resonances between alkali-metal atoms as well as chromium atoms. Finally, we compare the size of the three-body interaction with hopping rates and describe limits due to three-body recombination.In 1998 Jaksch et al. [1] suggested that laser-cooled atomic samples can be held in optical lattices, periodic potentials created by counter-propagating laser beams. These three-dimensional lattices have spatial periods between 400 nm and 800 nm and depths V 0 as high as V 0 /h ∼ 1 MHz, where h is Planck's constant. An ensemble of atoms then realize either the fermionic or bosonic Hubbard model, where atoms hop from site to site and interact only when on the same site. The interaction driven quantum phase transition of this model was first realized by Ref. [2].Today, optical lattices are seen as a natural choice in which to simulate other many-body Hamiltonians. These include Hamiltonians with complex band structure such as double-well lattices [3-6], two-dimensional hexagonal lattices [6][7][8][9], as well as those with spin-momentum couplings possibly leading to topological matter [10,11]. Quantum phase transitions in these Hamiltonians enable ground-state wavefunctions with unusual order parameters, such as pair superfluids and striped phases [12][13][14]. Phase transitions in Hamiltonians with long-range dipole-dipole interactions using atoms or molecules with large magnetic or electric dipole moments can also be studied. Finally, atoms in optical lattices can be used to measure gravitational acceleration (little-g) [15][16][17], shed light on non-linear measurements [18][19][20][21], and be used for quantum information processing.Over the last ten years ultra-cold atom experiments have also investigated few-body phenomena. In particular, three-body interactions have been studied through Efimov physics of strongly interacting atoms observed as resonances in three-body recombination, where three colliding atoms create a dimer and a...
We study ultracold atoms in an optical lattice with two local minima per unit cell and show that the low energy states of a multi-band Bose-Hubbard (BH) Hamiltonian with only pair-wise interactions is equivalent to an effective single-band Hamiltonian with strong three-body interactions. We focus on a double-well optical lattice with a symmetric double well along the x axis and single well structure along the perpendicular directions. Tunneling and two-body interaction energies are obtained from an exact band-structure calculation and numerically-constructed Wannier functions in order to construct a BH Hamiltonian spanning the lowest two bands. Our effective Hamiltonian is constructed from the ground state of the N -atom Hamiltonian for each unit cell obtained within the subspace spanned by the Wannier functions of two lowest bands. The model includes hopping between ground states of neighboring unit cells. We show that such an effective Hamiltonian has strong three-body interactions that can be easily tuned by changing the lattice parameters. Finally, relying on numerical mean-field simulations, we show that the effective Hamiltonian is an excellent approximation of the two-band BH Hamiltonian over a wide range of lattice parameters, both in the superfluid and Mott insulator regions.
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