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Physical Review A

Loading of a Bose-Einstein condensate into an optical lattice: The excitation of collective modes

Jesús Plata

Published Mar 10, 2004• 13 citations
Publisher

Abstract

The dynamics of a Bose-Einstein condensate in a harmonic trap with a nonadiabatically loaded optical lattice is studied analytically. As the global effect of the optical potential can be described in terms of a renormalized interaction coupling constant and of an effective mass in the laser direction, a fast loading can be understood as a sudden change of those characteristic parameters. In this approach, a standard scaling transformation is applied to study the resulting dynamics, in particular, the generation of collective modes. The relevance of the excited modes to the interference patterns obtained after free expansion is analyzed. The applicability of trap-frequency adjustments as a strategy for suppressing the loading induced excitations is discussed.

DOI
10.1103/physreva.69.033604

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The dynamics of a Bose-Einstein condensate in a harmonic trap with a nonadiabatically loaded optical lattice is studied analytically. As the global effect of the optical potential can be described in terms of a renormalized interaction coupling constant and of an effective mass in the laser direction, a fast loading can be understood as a sudden change of those characteristic parameters. In this approach, a standard scaling transformation is applied to study the resulting dynamics, in particular, the generation of collective modes. The relevance of the excited modes to the interference patterns obtained after free expansion is analyzed. The applicability of trap-frequency adjustments as a strategy for suppressing the loading induced excitations is discussed.
Abstract We study the structure of quasi-one-dimensional Bogoliubov–de Genes collective excitations of inhomogeneous F = 1 spinor Bose–Einstein condensates confined by a harmonic trap potential and loaded into an optical lattice. Employing a perturbative method, we report general analytical expressions for the polar and ferromagnetic confined collective oscillations. In both cases, the excited eigenfrequencies are given as functions of the effective 1D coupling constants, the trap frequency and the optical lattice parameters. While the presence of the trap yields quantized longitudinal collective modes, whose spectra depends on the collision scattering lengths, the optical lattice shifts their frequencies and, more importantly, yields tunable oscillations of the spin populations, for the even modes, as experimentally observed.
We study the damping of the collective excitations of Bose-Einstein condensates in a harmonic trap potential loaded in an optical lattice. In the presence of a confining potential the system is inhomogeneous and the collective excitations are characterized by a set of discrete confined phononlike excitations. We derive a general convenient analytical description for the damping rate, which takes into account the trapping potential and the optical lattice for the Landau and Beliaev processes at any temperature TTT. At high temperature or weak spatial confinement, we show that both mechanisms display a linear dependence on TTT. In the quantum limit, we find that the Landau damping is exponentially suppressed at low temperatures and the total damping is independent of TTT. Our theoretical predictions for the damping rate under the thermal regime is in complete correspondence with the experimental values reported for the one-dimensional (1D) condensate of sodium atoms. We show that the laser intensity can tune the collision process, allowing a resonant effect for the condensate lifetime. Also, we study the influence of the attractive or repulsive nonlinear terms on the decay rate of the collective excitations. A general expression for the renormalized Goldstone frequency is obtained as a function of the 1D nonlinear self-interaction parameter, laser intensity, and temperature.
We provide an exact analytical model for the dynamics of a 1D Bose–Einstein condensate loaded in a bichromatic optical lattice. Although a host of exact solutions result from this novel method, we mainly concentrate on the solitonic excitations. The trapping potential and its depth of lattice frustration can be varied by tuning the power and the wavelengths of the two overlaying laser beams. Both attractive and repulsive regimes are thoroughly investigated. In the attractive domain, we obtain bright soliton, which reveals interesting variations with the depth of lattice frustration. Localization of the matter wave density is demonstrated as one of the applications in this regime. In the repulsive domain, dark soliton is obtained when the potential resembles an optical lattice. With appropriate tuning of the potential parameter, the dark soliton becomes modulated with an oscillatory background and gradually transforms to bright solitary trains.
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.
We report experimental results on the dynamics and phase evolution of Bose-Einstein condensates in 1D optical lattices. The dynamical behaviour is studied by adiabatically loading the condensate into the lattice and subsequently switching off the magnetic trap. In this case, the condensate is free to expand inside the periodic structure of the optical lattice. The phase evolution of the condensate, on the other hand, can be studied by non-adiabatically switching on the periodic potential. We observe decays and revivals of the interference pattern after a time-of-flight.