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Revista Mexicana de Física

Effective Hamiltonian of an ultracold Bose gas confined in a four-well optical lattice

R. Muñoz-Rodriguez, R. Paredes, R. P. Duarte-Zamorano

Published Jan 1, 2007• 0 citations
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Abstract

In this paper we study an interacting Bose gas at low temperatures, confined in a one-dimensional potential composed of four wells. In order to derive and validate the effective Hamiltonian that describes this system, we study the stationary states of a particle confined in the four-well potential. In particular, we calculate the energies and the corresponding wave functions for the ground state and for the three lowest excited states. It was established that the effective Hamiltonian of a four-well optical lattice is composed of tunneling terms among all the wells, and interaction terms between pairs of particles within the same well.

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In this paper we study an interacting Bose gas at low temperatures, confined in a one-dimensional potential composed of four wells. In order to derive and validate the effective Hamiltonian that describes this system, we study the stationary states of a particle confined in the four-well potential. In particular, we calculate the energies and the corresponding wave functions for the ground state and for the three lowest excited states. It was established that the effective Hamiltonian of a four-well optical lattice is composed of tunneling terms among all the wells, and interaction terms between pairs of particles within the same well.
We study the dynamics of an ultracold gas of interacting bosons confined in a one-dimensional potential composed of nnn wells. We derive an effective NNN-particle Hamiltonian for this system that describes the tunneling of particles between adjacent wells and that includes the interactions among the particles within the same well. The problem is discussed in the light of current research on optical lattices. We demonstrate that the effective Hamiltonian gives the possibility of studying the transition from an ideal insulator to a real conductor when the parameters that modulate the interaction and the tunneling of the particles are varied. To show this we numerically solve Schr\"odinger equation for the case of n=3n=3n=3 wells. We find that for a given number of particles it is possible to establish a weak- and a strong-interaction regime. In the limit of strong interaction the tunneling term can be neglected. One important result in our analysis is that in the weak-interaction regime the system reaches a stationary state as a result of the loss of coherence in the transport of particles through the wells.
We study ultracold atoms in an optical lattice with two local minima per unit cell and show that the low-energy states of a multiband Bose-Hubbard (BH) Hamiltonian with only pairwise interactions are 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 xxx 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 NNN-atom Hamiltonian for each unit cell obtained within the subspace spanned by the Wannier functions of the 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, in both the superfluid and Mott insulator regions.
We discuss interacting and noninteracting one dimensional atomic systems trapped in an optical lattice plus a parabolic potential. We show that, in the tight-binding approximation, the noninteracting problem is exactly solvable in terms of Mathieu functions. We use the analytic solutions to study the collective oscillations of ideal bosonic and fermionic ensembles induced by small displacements of the parabolic potential. We treat the interacting boson problem by numerical diagonalization of the Bose-Hubbard Hamiltonian. From analysis of the dependence upon lattice depth of the low-energy excitation spectrum of the interacting system, we consider the problems of ``fermionization'' of a Bose gas, and the superfluid-Mott insulator transition. The spectrum of the noninteracting system turns out to provide a useful guide to understanding the collective oscillations of the interacting system, throughout a large and experimentally relevant parameter regime.
We study the dynamics of an ultracold interacting Bose-Einstein gas confined in a one-dimensional potential composed of three symmetrical wells. We numerically solve the time-dependent Schrödinger equation of the N -particle Hamiltonian for N = 50, 150, 500, 1000. We demonstrate that the quantum phase transition from a superfluid (SF) to a Mott insulator (MI) phase in the three-well potential depends on the strength of the interactions among the particles, the total number of particles, and the confining potential in which the particles move. We discuss the appearance of population revivals as a function of time and find that, even in the case when the interaction strength among the particles is very small, its effect has the consequence that the system never returns to the initial condition. A stationary state for long times is observed in the SF phase, while the particle population in each well remains almost equal to the initial condition in the MI phase.
We derive closed analytical expressions for the order parameter $\ensuremath{\Phi}(x)$ and for the chemical potential $\ensuremath{\mu}$ of a Bose-Einstein condensate loaded into a harmonically confined, one-dimensional optical lattice for sufficiently weak, repulsive, or attractive interaction and not too strong laser intensities. Our results are compared to exact numerical calculations in order to map out the range of validity of the perturbative analytical approach. We identify parameter values where the optical lattice compensates the interaction-induced nonlinearity, such that the condensate ground state coincides with a simple, single-particle harmonic-oscillator wave function.