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arXiv:cond-mat/0410168 · Physical Review A

Bragg spectroscopy of an accelerating condensate with solitary-wave behavior

Katharine J. Challis, R. J. Ballagh

Published Mar 18, 2005• 0 citations
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Abstract

We present a theoretical treatment of Bragg spectroscopy of an accelerating condensate in a solitary-wave state. Our treatment is based on the Gross-Pitaevskii equation with an optical potential representing the Bragg pulse and an additional external time-dependent potential generating the solitary-wave behavior. By transforming to a frame translating with the condensate, we derive an approximate set of equations that can be readily solved to generate approximate Bragg spectra. Our analytic method is accurate within a well-defined parameter regime and provides physical insight into the structure of the spectra. We illustrate our formalism using the example of Bragg spectroscopy of a condensate in a time-averaged orbiting potential trap.

DOI
10.1103/physreva.71.033620

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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.
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.
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 measure the response of an elongated Bose-Einstein condensate to a two-photon Bragg pulse. If the duration of the pulse is long, the total momentum transferred to the condensate exhibits a nontrivial behavior which reflects the structure of the underlying Bogoliubov spectrum. It is thus possible to perform a spectroscopic analysis in which axial phonons with a different number of radial nodes are resolved. The local density approximation is shown to fail in this regime, while the observed data agree well with the results of simulations based on the numerical solution of the Gross-Pitaevskii equation.