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arXiv:0807.2602 · New Journal of Physics

Bragg spectroscopy of a strongly interacting Bose–Einstein condensate

J J Kinnunen, M J Holland

Published Jan 20, 2009• 12 citations
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Abstract

We study Bragg spectroscopy of a strongly interacting Bose–Einstein condensate using time-dependent Hartree–Fock–Bogoliubov theory. We include approximatively the effect of the momentum-dependent scattering amplitude which is shown to be the dominant factor in determining the spectrum for large momentum Bragg scattering. The condensation of the Bragg scattered atoms is shown to alter significantly the observed excitation spectrum by creating a novel pairing channel of mobile pairs.

DOI
10.1088/1367-2630/11/1/013030

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We study Bragg spectroscopy of a strongly interacting Bose–Einstein condensate using time-dependent Hartree–Fock–Bogoliubov theory. We include approximatively the effect of the momentum-dependent scattering amplitude which is shown to be the dominant factor in determining the spectrum for large momentum Bragg scattering. The condensation of the Bragg scattered atoms is shown to alter significantly the observed excitation spectrum by creating a novel pairing channel of mobile pairs.
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 develop a general theory of spin-dependent Bragg spectroscopy for spinor Bose-Einstein condensates. This spectroscopy involves using a density and spin-coupled optical probe to excite the system. We show that within the linear-response regime the momentum or energy transferred by the probe is determined by a set of density and spin-density dynamic structure factors. We derive a set of fff-sum rules that provides rigorous constraints for the first energy moments of these structure factors. As an application we compute the dynamic structure factors for cases within all four distinct phases of a spin-1 condensate using Bogoliubov theory. Our results demonstrate that spin-dependent Bragg spectroscopy can be used to selectively investigate the various phonon and magnon excitation branches and will be a useful tool for advancing our understanding of spinor condensates.
Using two-photon Bragg spectroscopy, we study the energy of particlelike excitations in a strongly interacting homogeneous Bose-Einstein condensate, and observe dramatic deviations from Bogoliubov theory. In particular, at large scattering length a the shift of the excitation resonance from the free-particle energy changes sign from positive to negative. For an excitation with wave number q, this sign change occurs at a≈4/(πq), in agreement with the Feynman energy relation and the static structure factor expressed in terms of the two-body contact. For a≳3/q we also see a breakdown of this theory, and better agreement with calculations based on the Wilson operator product expansion. Neither theory explains our observations across all interaction regimes, inviting further theoretical efforts.
We report on measurements of the excitation spectrum of a strongly interacting Bose-Einstein condensate. A magnetic-field Feshbach resonance is used to tune atom-atom interactions in the condensate and to reach a regime where quantum depletion and beyond mean-field corrections to the condensate chemical potential are significant. We use two-photon Bragg spectroscopy to probe the condensate excitation spectrum; our results demonstrate the onset of beyond mean-field effects in a gaseous Bose-Einstein condensate.