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arXiv:cond-mat/0111438 · Physical Review Letters

Excitation Spectrum of a Bose-Einstein Condensate

Jeff Steinhauer, Roee Ozeri, Nadav Y. Katz, Nir Davidson

Published Mar 12, 2002• 343 citations
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Abstract

We report a measurement of the excitation spectrum omega(k) and the static structure factor S(k) of a Bose-Einstein condensate. The excitation spectrum displays a linear phonon regime, as well as a parabolic single-particle regime. The linear regime provides an upper limit for the superfluid critical velocity, by the Landau criterion. The excitation spectrum agrees well with the Bogoliubov spectrum in the local density approximation, even close to the long-wavelength limit of the region of applicability. Feynman's relation between omega(k) and S(k) is verified, within an overall constant.

DOI
10.1103/physrevlett.88.120407

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The elementary excitation spectrum of a two-component Bose—Einstein condensate in different hyperfine states is obtained by Green's function method. It is found to have two branches. In the long wave-length limit, the two branches of the excitation spectrum are reduced to one phonon excitation and one single-particle excitation. The single-particle one has an energy gap. When the energy gap exists, we study the Landau critical velocity and the depletion of the condensate. With the obtained Green's functions, we calculate the structure factor of a two-component condensate. It is found that the static structure factor comprises only the branch of the phonon excitation and the single-particle excitation makes no contribution to the structure factor.
Using Bogoliubov theory we calculate the excitation spectrum of a spinor Bose-Einstein condensed gas with an equal Rashba and Dresselhaus spin-orbit coupling in the stripe phase. The emergence of a double gapless band structure is pointed out as a key signature of Bose-Einstein condensation and of the spontaneous breaking of translational invariance symmetry. In the long wavelength limit the lower and upper branches exhibit, respectively, a clear spin and density nature. For wave vectors close to the first Brillouin zone, the lower branch acquires an important density character responsible for the divergent behavior of the structure factor and of the static response function, reflecting the occurrence of crystalline order. The sound velocities are calculated as functions of the Raman coupling for excitations propagating orthogonal and parallel to the stripes. Our predictions provide new perspectives for the identification of supersolid phenomena in ultracold atomic gases.
We measure the time oscillations of a freely evolving standing wave of phonons in a Bose-Einstein condensate. We present the technique of short Bragg pulses, which stimulates the standing wave. The subsequent oscillations are observed in situ. The frequency of the oscillations gives the dispersion relation, the amplitude gives the static structure factor, and the decay gives the dephasing time. The new technique gives orders of magnitude more sensitivity than Bragg spectroscopy, allowing for the observation of deviations from the local density approximation. Specifically, it is seen that the phonons undergo a transition from three dimensions to one dimension, when their wavelength becomes longer than the transverse radius of the condensate. The one-dimensional regime contains an inflection point in the dispersion relation, a decrease in the superfluid critical velocity, a minimum in the group velocity, and an increase in the lifetime of the standing wave oscillations.
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.
Summary form only given. The excitation spectrum of a Bose-Einstein condensate should give much insight into this system. For wavelengths much shorter than the radius of the condensate, the excitation spectrum is an essentially continuous function, which characterizes the intrinsic bulk properties of the condensate Previously, isolated points on the excitation spectrum and their dependence on the chemical potential were measured We report the first measurement of the excitation spectrum, as well as the static structure factor.