Excitation of Phonons in a Bose-Einstein Condensate by Light Scattering

Abstract

Stimulated small-angle light scattering was used to measure the structure factor of a Bose-Einstein condensate in the phonon regime. The excitation strength for phonons was found to be significantly reduced from that of free particles, revealing the presence of correlated pair excitations and quantum depletion in the condensate. The Bragg resonance line strength and line shift agreed with predictions of a local density approximation.Spectroscopic studies have been used to assemble a complete understanding of the structure of atoms and simple molecules. Similarly, neutron and light scattering have long been used to probe the microscopic excitations of liquid helium [1][2][3][4], and can be regarded as the spectroscopy of a many-body quantum system. With the experimental realization of gaseous Bose-Einstein condensates, the spectroscopy of this new quantum fluid has begun.The character of excitations in a weakly-interacting Bose-Einstein condensed gas depends on the relation between the wavevector of the excitation q and the inverse healing length ξ −1 = √ 2mc s /h which is the wavevector related to the speed of Bogoliubov sound c s = µ/m where µ = 4πh 2 an 0 /m is the chemical potential, a the scattering length, n 0 the condensate density, and m the atomic mass. For large wavevectors (q ≫ ξ −1 ), the excitations are particlelike with a quadratic dispersion relation. Excitations in the free-particle regime have been accessed by near-resonant light scattering [5]. For small wavevectors (q ≪ ξ −1 ), the gas responds collectively and density perturbations propagate as phonons at the speed of Bogoliubov sound. Such quasi-particle excitations have been observed at wavelengths comparable to the size of the trapped gas [6] and thus were strongly influenced by boundary conditions.In this Letter, we describe the use of Bragg spectroscopy to probe excitations in the phonon regime. Using two laser beams intersecting at a small angle, excitations in a Bose-Einstein condensate were created with wavevector q < ξ −1 , thus optically "imprinting" phonons into the gas. The momentum imparted to the condensate in the form of quasi-particles was directly measured by a time-of-flight analysis. This study is the first to explore phonons with wavelengths much smaller than the size of the trapped sample, allowing a direct connection to the theory of the homogeneous Bose gas. By direct comparison, we show the excitation of phonons to be significantly weaker than that of free particles. This provides dramatic evidence for the presence of correlated momentum excitations in the many-body condensate wavefunction.In optical Bragg spectroscopy, an atomic sample is illumined by two laser beams with wavevectors k 1 and k 2 and a frequency difference ω which is much smaller than their overall detuning ∆ from an atomic resonance. The intersecting beams create a periodic, traveling intensity modulation I mod (r, t) = I cos(q · r − ωt) where q = k 1 − k 2 . Atoms exposed to this intensity modulation experience a potential due to the ac Stark effect o...

Spinor Condensates and Light Scattering from Bose-Einstein Condensates

“…Using the density distribution of a condensate in the Thomas-Fermi regime (Eq. 2.19) and the predicted S( q, ω) for a homogeneous condensate at the local value of the density, the Bragg resonance line shape is calculated to be [11] …”

“…Phonon [11] BEC Free-particle [10] Velocity q/m = 7 mm/s c = 10 mm/s q/m = 60 mm/s Energy ω 0 q = h × 1.5 kHz µ = h × 6 kHz ω 0 q = h × 100 kHz Table 1. Conditions for Bragg scattering in the phonon and the free-particle regimes.…”

“…In this paper, we describe two applications of Bragg spectroscopy to study excitations of a Bose-Einstein condensate in either the free-particle [10] (large momentum transfer) or the phonon [11] (small momentum transfer) regime. The discussion includes a description of the dynamic structure factor of a Bose-Einstein condensate which leads to the interpretation of our measurements as an observation of the zero-point momentum distribution of trapped condensates, as a measurement of the energies of free-particle and phonon excitations, and as evidence for correlations in the many-body condensate wavefunction introduced by interatomic interactions.…”

Bragg scattering and the spin structure factor of two-component atomic gases

“…to measure the structure factors (i.e. the Fourier transform of the density-density correlation function) in clouds of spinless bosonic atoms [10].…”

“…After a brief review of the light-matter interaction for spin-1/2 atoms, we shall show that a stimulated Bragg scattering experiment analogous to the one of [10] is able to measure the response function to perturbation operators which are a combination of the total and the spin densities with polarization-dependent weights. The fluctuation-dissipation theorem then relates the absorbed energy to the structure factor.…”