Dynamic structure factor of a Bose-Einstein condensate in a one-dimensional optical lattice

Abstract

We study the effect of a one dimensional periodic potential on the dynamic structure factor of an interacting Bose Einstein condensate at zero temperature. We show that, due to phononic correlations, the excitation strength towards the first band develops a typical oscillating behaviour as a function of the momentum transfer, and vanishes at even multiples of the Bragg momentum. The effects of interactions on the static structure factor are found to be significantly amplified by the presence of the optical potential. Our predictions can be tested in stimulated photon scattering experiments.When a Bose-Einstein condensate is loaded into an optical lattice, its properties change in a very strong way [1]. For deep potential wells, it even happens that the coherence of the sample is lost and one observes the transition to the Mott-insulator phase [2,3]. Interesting phenomena occur also for low optical potential depth, for instance Bloch oscillations [4] and tunneling effects [5][6][7] can be investigated in this regime. In 1D optical lattices the transition to the insulator phase is expected to take place for very large intensities of the optical lattice, so that there is a very extended range of parameters where the gas can be described as a fully coherent system.In this Letter, we study the elementary excitations of an interacting Bose gas in the presence of a periodic potential and discuss how these states can be excited via inelastic processes using, for example, Bragg spectroscopy [8,9]. To this purpose we develop the formalism of the dynamic structure factor, a quantity directly related to the linear response of the system. We will restrict ourselves to the case of a system in the presence of a 1-dimensional optical potentialcreated by two counterpropagating laser beams. In Eq.(1) d is the lattice spacing and s is a dimensionless parameter which denotes the intensity of the laser in units of the recoil energy E R = q 2 B /2m. Here q B =hπ/d is the Bragg momentum denoting the boundary of the first Brillouin zone and m is the atomic mass. The inclusion of an additional harmonic potential produced, for example, by magnetic trapping does not modify the excitation spectrum in a profound way, unless the wave length of the excitation is comparable with the size of the sample. Along the tranverse directions we assume uniform confinement, so that the 3D Gross-Pitaevkii (GP) equation for the ground state order parameter ϕ(z) takes the 1D formwhere n is the average 3D density and the order parameter ϕ is normalized according to d/2 −d/2 |ϕ(z)| 2 dz = 1. As usual g = 4πh 2 a/m is the interaction coupling constant fixed by the scattering length a.The elementary excitations correspond to the solutions of the linearized time-dependent GP-equation and are described by the Bogoliubov equationswhere ϕ is the ground state solution of Eq.(2) and the amplitudes u jq and v jq satisfy the normalization conditionThe solutions u jq (z) and v jq (z) are Bloch waves: u jq (z) = exp(iqz/h)ũ jq (z) whereũ jq is periodic in space with peri...

Matter-wave scattering from interacting bosons in an optical lattice

“…(38), which shows that to this level of the approximation the inelastic cross section stems from the excitation of density fluctuations only [44]. In fact, the quantity ǫ q /ω q corresponds to the dynamic structure factor in Bogoliubov approximation, which characterizes the system's response to a density perturbation [23,45].…”

Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator

“…Emphasis is put on the measurement of excitation spectra which characterize the transition from the superfluid [2,3] to the Mott insulating state [1,4,5]. Several features observed in the spectra go beyond the description of current theoretical models.…”

Spin Waves in One Dimensional Magnetic Material

“…Anisotropy in z direction is clearly visible in the plot. In order to avoid the anisotropy, the Marsaglia's method [9] was adopted to generate random orientation in spin in Monte Carlo simulation. Figure 2(b) shows 1000 randomly generated spin orientations using Marsaglia's method.…”

“…S  q and it is related to the static structure factor () S q by the following integral equation [9]:…”