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Conference on Lasers and Electro-Optics

Towards homogeneously broadened spectroscopy of Bogoliubov excitations in Bose condensates

Nadav Y. Katz, Erez Gershnabel, Eitan E. Rowen, Nir Davidson

Published Jan 1, 2004• 0 citations
Publisher

Abstract

Bose-Einstein condensates (BEC) serve as a model system for the study of the dynamic and spectroscopic response of a quantum system to momentum excitation. However, high -resolution spectroscopy of BEC has suffered from various finite-size (Doppler broadening) and inhomogeneous density effects [1]. We propose and demonstrate experimentally [2] a momentum space echo method using two and four -photon Bragg pulses, in order to overcome these broadening mechanisms. This method is pre sented in Fig. 1, where the excitation scheme and resulting populations in momentum space are shown. We demonstrate this scheme experimentally by using a four -photon echo pulse to reflect the initi ally populated excitations. The spectroscopic response of t he echo pulse is narrower than that of a regular Bragg pulse, as shown in Fig. 2.

DOI
10.1364/iqec.2004.imm7

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Bose-Einstein condensates (BEC) serve as a model system for the study of the dynamic and spectroscopic response of a quantum system to momentum excitation. However, high -resolution spectroscopy of BEC has suffered from various finite-size (Doppler broadening) and inhomogeneous density effects [1]. We propose and demonstrate experimentally [2] a momentum space echo method using two and four -photon Bragg pulses, in order to overcome these broadening mechanisms. This method is pre sented in Fig. 1, where the excitation scheme and resulting populations in momentum space are shown. We demonstrate this scheme experimentally by using a four -photon echo pulse to reflect the initi ally populated excitations. The spectroscopic response of t he echo pulse is narrower than that of a regular Bragg pulse, as shown in Fig. 2.
We propose and demonstrate an echo method to reduce the inhomogeneous linewidth of Bogoliubov excitations, in a harmonically-trapped Bose-Einstein condensate. Our proposal includes the transfer of excitations with momentum +q+q+q to $\ensuremath{-}q$ using a double two photon Bragg process, in which a substantial reduction of the inhomogeneous broadening is calculated. Furthermore, we predict an enhancement in the method's efficiency for low momentum due to many-body effects. As a proof of principle, the echo can also be implemented by using a four photon process, as is demonstrated experimentally.
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.
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.
This paper presents Bragg spectroscopy of the multi-branch Bogoliubov spectrum of elongated Bose-Einstein condensates. The condensate consists of N=10/sup 5/ atoms of /sup 87/Rb which was excited at wavenumber k by using two Bragg beams. Results are analyzed by numerically solving the Gross-Pitaevskii (GP) equation for the order parameter of the condensate. This study shows that the response of the condensate to a Bragg pulse is indeed significantly affected by the radial degrees of freedom. By using Bragg pulses longer than in previous measurements, this multi-branch spectrum is resolved, finding good agreement with the predictions of Gross-Pitaevskii theory.