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arXiv:cond-mat/0207749 · Physical Review A

Momentum transfer to and elementary excitations of a Bose-Einstein condensate by a time-dependent optical potential

Y. B. Band, M. Sokuler

Published Oct 23, 2002• 4 citations
arXivHTMLPDF

Abstract

We present results of calculations on Bose-Einstein-condensed ${}^{87}\mathrm{Rb}$ atoms subjected to a moving standing-wave light potential of the form ${V}_{L}{(z,t)=V}_{0}(t)\mathrm{cos}(qz\ensuremath{-}\ensuremath{\omega}t).$ We calculate the mean-field dynamics (the order parameter) of the condensate and determine the resulting condensate momentum in the z direction, ${P}_{z}(q,\ensuremath{\omega}{,V}_{0}{,t}_{p}),$ where ${V}_{0}$ is the peak optical potential strength and ${t}_{p}$ is the pulse duration. Although the local-density approximation for the Bogoliubov excitation spectral distribution is a good approximation for very low optical intensities, long pulse duration and sufficiently large values of the wave vector q of the light potential, for small q, short duration pulses, or for not-so-low intensities, the local-density perturbative description of the excitation spectrum breaks down badly, as shown by our results.

DOI
10.1103/physreva.66.043614

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We present results of calculations on Bose-Einstein-condensed 87Rb{}^{87}\mathrm{Rb}87Rb atoms subjected to a moving standing-wave light potential of the form VL(z,t)=V0(t)cos(qz\ensuremath−\ensuremathωt).{V}_{L}{(z,t)=V}_{0}(t)\mathrm{cos}(qz\ensuremath{-}\ensuremath{\omega}t).VL​(z,t)=V0​(t)cos(qz\ensuremath−\ensuremathωt). We calculate the mean-field dynamics (the order parameter) of the condensate and determine the resulting condensate momentum in the z direction, Pz(q,\ensuremathω,V0,tp),{P}_{z}(q,\ensuremath{\omega}{,V}_{0}{,t}_{p}),Pz​(q,\ensuremathω,V0​,tp​), where V0{V}_{0}V0​ is the peak optical potential strength and tp{t}_{p}tp​ is the pulse duration. Although the local-density approximation for the Bogoliubov excitation spectral distribution is a good approximation for very low optical intensities, long pulse duration and sufficiently large values of the wave vector q of the light potential, for small q, short duration pulses, or for not-so-low intensities, the local-density perturbative description of the excitation spectrum breaks down badly, as shown by our results.
The response of a trapped Bose-Einstein condensed gas to a density perturbation generated by a two-photon Bragg pulse is investigated by solving the time-dependent Gross-Pitaevskii equation. We calculate the total momentum imparted to the condensate as a function of both the time duration of the pulse and the frequency difference of the two laser beams. The role of the dynamic response function in characterizing the time evolution of the system is pointed out, with special emphasis to the phonon regime. Numerical simulations are compared with the predictions of local-density approximation. The relevance of our results for the interpretation of current experiments is also discussed.
The dynamic structure factor of a trapped Bose-Einstein condensed gas is investigated at zero temperature in the framework of Bogoliubov theory. Different values of momentum transfer are considered, ranging from the phonon to the single-particle regime. Various approximate schemes are discussed, including the local density approximation, where the system is locally described as a uniform gas, and the impulse approximation, in which the response is fixed by the momentum distribution of the condensate. A comprehensive approach, based on the eikonal expansion, is presented. The predictions of theory are successfully compared with the results of recent two-photon Bragg scattering experiments, at both low and high momentum transfer. Relevant features of the dynamic structure factor are also discussed using the formalism of sum rules and the concept of scaling. Particular emphasis is given to the regime of high momentum transfer, in which the dynamic structure factor is sensitive to the behavior of the order parameter in momentum space, and some instructive examples showing the consequence of long-range coherence are presented.
The non-equilibrium dynamics of the superradiant scattering of light from a BEC is experimentally investigated by monitoring the time evolution of the population in the condensate. In the experiment, a cigar-shaped Bose-Einstein condensate of /sup 87/Rb atoms is produced and is exposed to a single off-resonant laser pulse directed along the condensate symmetry axis. The superradiant back-scattering of light creates a matter wave pulse moving in the same direction of the laser beam at twice the photon recoil velocity with respect to the original condensate. The efficiency of the process is limited by decoherence between the two atomic spatially overlapped wavepackets. The temporal evolution of the condensate observed in the experiment as well as the results of a simple semiclassical model in which the condensate is described by a density operator for the motional states. The population of the two matter wavepackets as a function of the light pulse duration is reported, together with the experimental points with the analytical formula resulting from the theory of collective atomic recoil laser (CARL).
In the Bogoliubov theory, a condensate initially prepared in its ground state described by a stationary Bogoliubov vacuum and later perturbed by a time-dependent potential or interaction strength evolves into a time-dependent excited state which is a dynamical Bogoliubov vacuum. The dynamical vacuum has a simple diagonal form in a time-dependent orthonormal basis of single-particle modes. This diagonal representation leads to a Gaussian probability distribution for possible density-measurement outcomes in position and momentum space.
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.