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arXiv:cond-mat/9912422 · Journal of Physics B Atomic Molecular and Optical Physics

Mean-field treatment of Bragg scattering from a Bose-Einstein condensate

P. B. Blakie, R. J. Ballagh

Published Sep 14, 2000• 39 citations
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Abstract

A unified semiclassical treatment of Bragg scattering from Bose-Einstein condensates is presented. The formalism is based on Gross-Pitaevskii equations driven by a classical light field, and leads to a single-component equation incorporating the effects of mean-field nonlinearity and spatial inhomogeneity. Three-dimensional numerical simulations of this equation in cylindrical symmetry are used to investigate Bragg scattering for a number of cases. Scattering from a condensate released from a trap produces characteristic cycling of the atomic momentum, and an analytic description is given using a linear model incorporating spatial non-uniformity. Simple expressions obtained for the momentum packet cycling are shown to accurately describe the full nonlinear behaviour within a well specified validity range and a transition linewidth is derived. For the case of a trapped condensate, a numerical study of momentum spectroscopy is carried out and the shift of the Bragg resonance condition due to mean-field nonlinearity is investigated.

DOI
10.1088/0953-4075/33/19/311

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Physical Review A•May 13, 2004•19 citations
A unified semiclassical treatment of Bragg scattering from Bose-Einstein condensates is presented. The formalism is based on Gross-Pitaevskii equations driven by a classical light field, and leads to a single-component equation incorporating the effects of mean-field nonlinearity and spatial inhomogeneity. Three-dimensional numerical simulations of this equation in cylindrical symmetry are used to investigate Bragg scattering for a number of cases. Scattering from a condensate released from a trap produces characteristic cycling of the atomic momentum, and an analytic description is given using a linear model incorporating spatial non-uniformity. Simple expressions obtained for the momentum packet cycling are shown to accurately describe the full nonlinear behaviour within a well specified validity range and a transition linewidth is derived. For the case of a trapped condensate, a numerical study of momentum spectroscopy is carried out and the shift of the Bragg resonance condition due to mean-field nonlinearity is investigated.
We present a detailed theoretical analysis of Bragg spectroscopy from a Bose-Einstein condensate at T=0K.T=0 \mathrm{K}.T=0K. We demonstrate that within the linear-response regime, both a quantum-field-theory treatment and a mean-field Gross-Pitaevskii treatment lead to the same value for the mean evolution of the quasiparticle operators. The observable for Bragg spectroscopy experiments, which is the spectral response function of the momentum transferred to the condensate, can therefore be calculated in a mean-field formalism. We analyze the behavior of this observable by carrying out numerical simulations in axially symmetric three-dimensional cases and in two dimensions. An approximate analytic expression for the observable is obtained and provides a means for identifying the relative importance of three broadening and shift mechanisms (mean field, Doppler, and finite pulse duration) in different regimes. We show that the suppression of scattering at small values of q observed by Stamper-Kurn et al. [Phys. Rev. Lett. 83, 2876 (1999)] is accounted for by the mean-field treatment, and can be interpreted in terms of the interference of the u and vvv quasiparticle amplitudes. We also show that, contrary to the assumptions of previous analyses, there is no regime for trapped condensates for which the spectral response function and the dynamic structure factor are equivalent. Our numerical calculations can also be performed outside the linear-response regime, and show that at large laser intensities a significant decrease in the shift of the spectral response function can occur due to depletion of the initial condensate.
We present semiclassical descriptions of Bose-Einstein condensates for configurations with spatial symmetry, e.g., cylindrical symmetry, and without any symmetry. The description of the cylindrical case is quasi-one-dimensional (Q1D), in the sense that one only needs to solve an effective 1D nonlinear Schr\"odinger equation, but the solution incorporates 3D aspects of the problem, as a result of which the 1D equation is supplemented by a noncanonical (quartic) normalization condition. The solution in classically allowed regions is matched onto that in classically forbidden regions by a connection formula that properly accounts for the nonlinear mean-field interaction. Special cases for vortex solutions are treated too. Comparisons of the Q1D solution with the full 3D and Thomas-Fermi ones are presented, and conditions for the applicability of the effective low-dimensional equations are obtained.
The behaviour of the momentum transferred to a trapped Bose-Einstein condensate by a two-photon Bragg pulse reflects the structure of the underlying Bogoliubov spectrum. In elongated condensates, axial phonons with different numbers of radial nodes give rise to a multibranch spectrum which can be resolved in Bragg spectroscopy, as shown by Steinhauer et al (2003 Phys. Rev. Lett. 90 060404). Here we present a detailed theoretical analysis of this process. We calculate the momentum transferred by numerically solving the time-dependent Gross-Pitaevskii (GP) equation. In the case of a cylindrical condensate, we compare the results with those obtained by linearizing the GP equation and using a quasiparticle projection method. This analysis shows how the axial-phonon branches affect the momentum transfer, in agreement with our previous interpretation of the observed data. We also discuss the applicability of this type of spectroscopy to typical available condensates, as well as the role of nonlinear effects.
We extend classical-field methods [Blakie et al., Adv. Phys. 57, 363 (2008)] to provide a description of atom-molecule systems. We use a model of Bose-Einstein condensation of atoms close to a Feshbach resonance, in which the tunable scattering length of the atoms is described using a system of coupled atom and molecule fields [Holland et al., Phys. Rev. Lett. 86, 1915 (2001)]. We formulate the basic theoretical methods for a coupled atom-molecule system, including the determination of the phenomenological parameters in our system, the Thomas-Fermi description of Bose-Einstein condensate, the Bogoliubov--de Gennes equations, and the Bogoliubov excitation spectrum for a homogenous condensed system. We apply this formalism to the special case of Bragg scattering from a uniform condensate and find that for moderate and large scattering lengths, there is a dramatic difference in the shift of the peak of the Bragg spectra, compared to that based on a structureless atom model. The result is compatible with the experimental results of Papp et al. [Phys. Rev. Lett 101, 135301 (2008)] for Bragg scattering from a nonuniform condensate.
We present semianalytical many-body results for energies and excitation frequencies for an inhomogeneous Bose-Einstein condensate over a wide range of atom numbers NNN for both small sss-wave scattering lengths, typical of most laboratory experiments, and large scattering lengths, achieved by tuning through a Feshbach resonance. Our dimensional perturbation treatment includes two-body correlations at all orders and yields analytical results through first order by taking advantage of the high degree of symmetry of the condensate at the zeroth-order limit. Because NNN remains a parameter in our analytical results, the challenge of calculating energies and excitation frequencies does not rise with the number of condensate atoms. In this proof-of-concept paper the atoms are confined in a spherical trap and are treated as hard spheres. Our many-body calculations compare well to Gross-Pitaevskii results in the weakly interacting regime and depart from the mean-field approximation as the density approaches the strongly interacting regime. The excitation frequencies provide a particularly sensitive test of beyond-mean-field corrections. For example, for N=2000N=2000N=2000 atoms and an experimentally realized large scattering length of $a=0.433{a}_{ho}\phantom{\rule{0.3em}{0ex}}({a}_{ho}=\sqrt{\ensuremath{\hbar}∕m{\ensuremath{\omega}}_{ho}})$ we predict a 7575%75 shift from the mean-field breathing mode frequency.