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Journal of Physics B Atomic Molecular and Optical Physics

Variational dynamics of Bose-Einstein condensates in deep optical lattices

Andrea Trombettoni, Augusto Smerzi

Published Nov 23, 2001• 27 citations
Publisher

Abstract

We study the dynamics of a Bose-Einstein condensate wavepacket trapped in an optical lattice. In the tight-binding limit, the system maps onto a discrete nonlinear Schrödinger equation. Using a collective coordinate approximation, we investigate different dynamical regimes: diffusive (in which the boson wavepacket spreads out), self-trapped (with a self-induced pinning), breathing (where the centre of mass of the wavepacket moves with an oscillating width), and solitonic (in which the wavepacket shape is exactly preserved during the dynamics).

DOI
10.1088/0953-4075/34/23/315

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We study the dynamics of a Bose-Einstein condensate wavepacket trapped in an optical lattice. In the tight-binding limit, the system maps onto a discrete nonlinear Schrödinger equation. Using a collective coordinate approximation, we investigate different dynamical regimes: diffusive (in which the boson wavepacket spreads out), self-trapped (with a self-induced pinning), breathing (where the centre of mass of the wavepacket moves with an oscillating width), and solitonic (in which the wavepacket shape is exactly preserved during the dynamics).
We study the dynamics of self-trapping in Bose-Einstein condensates (BECs) loaded in deep optical lattices with Gaussian initial conditions, when the dynamics is well described by the discrete nonlinear Schrödinger equation (DNLSE). In the literature an approximate dynamical phase diagram based on a variational approach was introduced to distinguish different dynamical regimes: diffusion, self-trapping, and moving breathers. However, we find that the actual DNLSE dynamics shows a completely different diagram than the variational prediction. We calculate numerically a detailed dynamical phase diagram accurately describing the different dynamical regimes. It exhibits a complex structure that can readily be tested in current experiments in BECs in optical lattices and in optical waveguide arrays. Moreover, we derive an explicit theoretical estimate for the transition to self-trapping in excellent agreement with our numerical findings, which may be a valuable guide as well for future studies on a quantum dynamical phase diagram based on the Bose-Hubbard Hamiltonian.
In the present paper, we analytically and numerically investigate the dynamics of Bose-Einstein condensates (BECs) loaded into deep optical lattices of one dimension (1D), 2D, and 3D. We focus on the self-trapped state and the effect of the lattice dimension. Under the tight-binding approximation, we obtain an analytical criterion for the self-trapped state of BEC using the time-dependent variational method. The phase diagram for self-trapping, soliton, breather, or diffusion of a BEC cloud is obtained accordingly and verified by directly solving a discrete Gross-Pitaevskii equation numerically. In particular, we find that the criterion and the phase diagrams are modified dramatically by the dimension of the optical lattices.
The present work is devoted to the theoretical investigation of the dynamics of Bose –Einstein condensates (BEC) in optical lattice with the long range magnetic dipole-dipole interaction between atoms. The system is described by a nonlocal nonlinear Schrodinger equation (NLSE). We consider the case when the lattice depth is sufficiently large (tight binding approximation). In this case nonlocal NLSE can be reduced to discrete NLSE equation. We show that discrete NLSE equation for some range of parameters can be reduced further to integrable Ablowitz-Ladik equation which supports moving solitons. Taking the one-soliton solution of Ablowitz-Ladik equation as an initial condition we have performed detailed numerical study of a soliton dynamics in both nonlocal NLSE and discrete NLSE. The conditions of existence and stability of moving solitons in dipolar BEC in an optical lattice are numerically revealed. Also the applicability limits for the tight binding approximation for the experimentally achievable range of parameters are investigated.
The dynamics of Bose-Einstein condensates trapped in a deep optical lattice is governed by a discrete nonlinear equation (DNL). Its degree of nonlinearity and the intersite hopping rates are retrieved from a nonlinear tight-binding approximation taking into account the effective dimensionality of each condensate. We derive analytically the Bloch and the Bogoliubov excitation spectra and the velocity of sound waves emitted by a traveling condensate. Within a Lagrangian formalism, we obtain Newtonian-like equations of motion of localized wave packets. We calculate the ground-state atomic distribution in the presence of a harmonic confining potential, the frequencies of small amplitude dipole, and quadrupole oscillations. We finally quantize the DNL, recovering an extended Bose-Hubbard model.
The properties of Bose-Einstein condensates (BEC) can be studied and controlled effectively when trapped in optical lattices. In the limit of tight binding, BEC in optical lattices are well described by the discrete nonlinear Schrödinger equation (DNLSE). Spatially localised modes of excitations, known as discrete breathers or lattice solitons, can occur in the DNLSE. These have been shown to evolve from initial conditions of Gaussian wave-packets. Using this method with the addition of an initial momentum, mobile breathers that travel across the lattice with a constant velocity can also be formed.