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arXiv:cond-mat/0309584 · Physical Review A

Echo spectroscopy of bulk Bogoliubov excitations in trapped Bose-Einstein condensates

Erez Gershnabel, Nadav Y. Katz, Roee Ozeri, Eitan E. Rowen, Jeff Steinhauer, Nir Davidson

Published Apr 23, 2004• 2 citations
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Abstract

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 $+q$ 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.

DOI
10.1103/physreva.69.041604

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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\ensuremath{-}q\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.
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 study the linearized Bogoliubov excitation spectrum of infinitely long anharmonically trapped Bose–Einstein condensates, with the aim of overcoming inhomogeneous broadening. We compare the Bogoliubov spectrum of a harmonic trap with that of a theoretical flat-bottom trap and find a significant reduction in the inhomogeneous broadening of the lineshape of Bogoliubov excitations. While the Bragg excitation spectrum for a condensate in a harmonic trap supports a number of radial modes, the flat trap is found to significantly support just one mode. We also study the excitation spectrum of realistic anharmonic traps with potentials of finite power dependence on the radial coordinate. We observe a correlation between the number of radial modes and the number of bound states in the effective potential of the quasi-particles. Finally, we compare a full numerical Gross–Pitaevskii simulation of a finite-length condensate to our model of infinite, linearized Gross–Pitaevskii excitations. We conclude that our model captures the essential physics.
We study the linearized Bogoliubov excitation spectrum of infinitely long anharmonically trapped Bose-Einstein condensates, with the aim of overcoming inhomogeneous broadening. We compare the Bogoliubov spectrum of a harmonic trap with that of a theoretical flat-bottom trap and find a dramatic reduction in the inhomogeneous broadening of the lineshape of Bogoliubov excitations. While the Bragg excitation spectrum for a condensate in a harmonic trap supports a number of radial modes, the flat trap is found to significantly support just one mode. We also study the excitation spectrum of realistic anharmonic traps with potentials of finite power dependence on the radial coordinate. We observe a correlation between the number of radial modes and the number of bound states in the effective potential of the quasi-particles. Finally we compare a full numerical Gross-Pitaevskii simulation of a finite-length condensate to our model of infinite, linearized Gross-Pitaevskii excitations. We conclude that our model captures the essential physics.
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.
In a Bose-Einstein condensate, the excitation of a Bogoliubov phonon with low momentum (e.g., by a two-photon Bragg process) is strongly suppressed due to destructive interference between two indistinguishable excitation pathways. Here we show that scattering of this sound excitation into a double-momentum mode is strongly enhanced due to constructive interference. This enhancement yields an inherent amplification of second-order sound excitations of the condensate, as we confirm experimentally. We further show that due to parity considerations, this effect is extended to higher-order excitations.
We present a detailed finite-temperature Hartree-Fock-Bogoliubov (HFB) treatment of the two-dimensional trapped Bose gas. We highlight the numerical methods required to obtain solutions to the HFB equations within the Popov approximation, the derivation of which we outline. This method has previously been applied successfully to the three-dimensional case and we focus on the unique features of the system which are due to its reduced dimensionality. These can be found in the spectrum of low-lying excitations and in the coherence properties. We calculate the Bragg response and the coherence length within the condensate in analogy with experiments performed in the quasi-one-dimensional regime [Richard et al., Phys. Rev. Lett. 91, 010405 (2003)] and compare to results calculated for the one-dimensional case. We then make predictions for the experimental observation of the quasicondensate phase via Bragg spectroscopy in the quasi-two-dimensional regime.