Superfluid dynamics of a Bose–Einstein condensate in a periodic potential

Abstract

We investigate the superfluid properties of a Bose-Einstein condensate (BEC) trapped in a one dimensional periodic potential. We study, both analytically (in the tight binding limit) and numerically, the Bloch chemical potential, the Bloch energy and the Bogoliubov dispersion relation, and we introduce two different, density dependent, effective masses and group velocities. The Bogoliubov spectrum predicts the existence of sound waves, and the arising of energetic and dynamical instabilities at critical values of the BEC quasi-momentum which dramatically affect its coherence properties. We investigate the dependence of the dipole and Bloch oscillation frequencies in terms of an effective mass averaged over the density of the condensate. We illustrate our results with several animations obtained solving numerically the time-dependent Gross-Pitaevskii equation. I. INTRODUCTIONThe study of the superfluid properties of Bose-Einstein condensates (BECs) trapped in periodic potentials are attracting a fast growing interest. The main reason is that the control parameters of such systems are widely tunable in realistic experiments, allowing for the investigation of different and fundamental issues of quantum mechanics, ranging from quantum phase transitions [1] and atom optics [2,3] to the dynamics of Bloch and Josephson oscillations [4][5][6]. Several efforts are also focusing on the realization of new technological devices as BEC interferometers working at the Heisenberg limit [3], and quantum information processors [2].The dynamics of BECs in lattices is highly non-trivial, essentially because of the competition/interplay between the discrete translational invariance of the periodic potential and the nonlinearity arising from the interatomic interactions. For deep enough optical potentials, interactions induce a quantum transition from the superfluid to a Mott-insulator phase [1,7,8]. In this work we will study the system in a region of parameters such that its ground state stands deeply in the superfluid phase, with the dynamics governed by the Gross-Pitaevskii equation (GPE). Because of the discrete translational invariance, the excitation spectrum of the system exhibits a band structure which has several analogies with the electron Bloch bands in metals [9][10][11]. On the other hand, the coexistence of Bloch bands and nonlinearity allows, for instance, solitonic structures [12][13][14] and dynamical instabilities [15][16][17] which do not have an analog neither in metals, nor in Galilean invariant systems.Exact, time-dependent solutions of the GPE with an external periodic potential, Eq.( 1), can be written as Bloch states, namely as plane waves modulated by functions having the same periodicity of the lattice. The dynamics of small amplitude perturbations on top of these states satisfies two coupled, linear Bogoliubov equations, which can be solved numerically. However, when the interwell barriers of the periodic potential are high enough, the system can be described in a nonlinear tight binding approximation and ...

Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices

“…Remark 1 : When the optical lattice has deep wells (large |V 1 | or |V 2 |), one can also obtain an analytical description of BECs in terms of Wannier wave functions using the so-called "tight-binding approximation" [42]. In this regime, the BEC dynamics is governed by a discrete nonlinear Schrödinger equation, which is derived by expanding the field operator in a Wannier basis of localized wave functions at each lattice site.…”

Optical Lattices: Theory

“…The periodic potential is generally realized with two counterpropagating laser beams [4,5,6,7,8,9,10,11,12], so as to create an optical lattice (OL). As expected in the mean-field GPE limit, the BEC Bogoliubov excitation spectrum has a band structure, in analogy with the electronic Bloch bands [13,14,15,16,17,18,19,20,21]. When the the power of the laser is fairly larger than the chemical potential, the lowest band dynamics maps on a discrete nonlinear Schrödinger (DNLS) equation [22].…”

“…We also present a brief discussion of the comparison with numerical results for the excitation spectra of the continuous GPE [20].…”

“…Note that the ψ j are plane waves in the lattice, but do not correspond to plane waves in real space. Within the DNL equation framework, the energy per particle ε(k) and chemical potential µ(k) corresponding to these solutions are found to be [20] …”

“…The propagation of sound in a harmonically trapped condensate without OL has been observed experimentally [32] and studied theoretically [33,34,35,36]. In the presence of a periodic potential, the condensate wavefunction can be expanded in Bloch waves, having amplitudes modulated with the periodicity of the OL, which can also become energetically unstable when the group velocity is larger than the sound velocity [17,20,37,38,39,40]. The energetic instability manifests itself with the emission of quasi-particles out of a condensate flowing against a small obstacle.…”