Nonlinear Tight-Binding Approximation for Bose-Einstein Condensates in a Lattice

TL;DR

This paper develops a nonlinear tight-binding approximation for Bose-Einstein condensates in optical lattices, deriving analytical spectra, wave dynamics, and quantized models to better understand their behavior.

Contribution

It introduces a novel nonlinear tight-binding approach that accounts for effective dimensionality, providing analytical spectra and quantized models for BECs in lattices.

Findings

Derived analytical Bloch and Bogoliubov spectra.

Calculated sound velocity and wavepacket dynamics.

Quantized the model to extend the Bose-Hubbard framework.

Abstract

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 wavepackets. We calculate the ground-state atomic distribution in the presence of an harmonic confining potential, and the frequencies of small amplitude dipole and quadrupole oscillations. We finally quantize the DNL, recovering an extended Bose-Hubbard model.