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

VOLUME 83, NUMBER 15 PHYSICAL REVIEW LETTERS 11OCTOBER 1999 Excitation of Phonons in a Bose-Einstein Condensate by Light Scattering D. M. Stamper-Kurn, A. P. Chikkatur, A. Görlitz, S. Inouye, S. Gupta, D. E. Pritchard, and W. Ketterle Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received 3 June 1999) 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 for the homogeneous Bose gas using a local density approximation. PACS numbers: 03.75.Fi, 05.30.–d, 32.80.Pj, 65.50.+m Spectroscopic studies have been used to assemble a thereby “optically imprinting” phonons into the gas. The complete understanding of the structure of atoms and momentum imparted to the condensate was measured by simple molecules. Similarly, neutron and light scattering a time-of-flight analysis. This study is the first to explore have long been used to probe the microscopic excitations phonons with wavelengths much smaller than the size of of liquid helium [1–4], and can be regarded as the the trapped sample, allowing a direct connection to the spectroscopy of a many-body quantum system. With theory of the homogeneous Bose gas. We show the ex- the realization of gaseous Bose-Einstein condensates, the citation of phonons to be significantly weaker than that spectroscopy of this new quantum fluid has begun. of free particles, providing dramatic evidence for corre- The character of excitations in a weakly interacting lated momentum excitations in the many-body condensate Bose-Einstein condensed gas depends on the relation be- wave function. tween the wave vectorp of the excitation q and the inverse In optical Bragg spectroscopy, an atomic sample is 21 healing length j ෇ 2 mcs͞h¯, which is the wavep vec- illuminated by two laser beams with wave vectors k1 and tor related to the speed of Bogoliubov sound cs ෇ m͞m, k2 and a frequency difference v which is much smaller 2 where m ෇ 4ph¯ an0͞m is the chemical potential, a is than their detuning D from an atomic resonance. The the scattering length, n0 is the condensate density, and m intersecting beams create a periodic, traveling intensity 21 is the atomic mass. For large wave vectors (q ¿ j ), modulation Imod͑r, t͒ ෇ I cos͑q ? r 2vt͒, where q ෇ the excitations are particlelike with a quadratic disper- k1 2 k2. The atoms experience a potential due to the ac 2 sion relation. Excitations in the free-particle regime have Stark effect of strength Vmod ෇ h¯G ͞8D3Imod͞Isat [7], been accessed by near-resonant light scattering [5]. For from which they may scatter. Here, G is the linewidth of 21 small wave vectors (q ø j ), the gas responds collec- the atomic resonance, and Isat is the saturation intensity. tively and density perturbations propagate as phonons at The response of an N-particle system to this per- the speed of Bogoliubov sound. Such quasiparticle ex- turbation can be evaluated using Fermi’s golden rule. citations have been observed at wavelengths comparable We express Vmod in second-quantized notation Vˆmod ෇ ͞ ͓ y͑ ͒ 2ivt y͑ ͒ 1ivt͔ y͑ ͒ ෇ to the size of the trapped condensate [6] and thus were PV 2 rˆ q e 1 rˆ 2q e , where rˆ q strongly influenced by boundary conditions. y k aˆ k1qaˆ k is the Fourier transform of the atomic density In this Letter, we use Bragg spectroscopy to probe ex- y operator at wave vector q, and aˆ (aˆ ) is the destruction citations in the phonon regime. Two laser beams inter- k k (creation) operator for an atom with momentum h¯k. For secting at a small angle were used to create excitations in the ground state jg͘ with energy E , the excitation rate a Bose-Einstein condensate with wave vector q ,j21, g per particle is then µ ∂ 2 X 2p V ͗ y͑ ͒ ͘ 2 ͑ ͑ ͒͒͒ ෇ 2 ͑ ͒ j fjrˆ q jg j d h¯v2 Ef 2 Eg 2pvRS q, v , Nh¯ 2 f where excited states j f͘ have energy Ef , and vR ෇ V͞2¯h is the two-photon Rabi frequency. Thus, light scattering created by laser and evaporative cooling and stored in a directly measures the dynamical structure factor, S͑q, v͒, cigar-shaped magnetic trap with trapping frequencies of which is the Fourier transform of density correlations in vr ෇ 2p3150 Hz and vz ෇ 2p318 Hz in the radial state jg͘ [3,8]. Integrating over v gives the static structure and axial directions, respectively [9]. factor S͑q͒ ෇ ͗gjrˆ ͑q͒rˆ y͑q͒jg͘͞N. The condensate was then exposed to two laser beams In this work, measurements were performed on both which intersected at an angle of ഠ14± and were aligned magnetically trapped and freely expanding Bose-Einstein symmetrically about the radial direction, so that the dif- condensates of sodium. Condensates of ഠ107 atoms were ference wave vector q was directed axially (Fig. 1a). 2876 0031-9007͞99͞83(15)͞2876(4)$15.00 © 1999 The American Physical Society VOLUME 83, NUMBER 15 PHYSICAL REVIEW LETTERS 11OCTOBER 1999 which the atomic density was reduced by a factor of 23 and the speed of sound by a factor of 5 from that of the trapped condensate. Thus, Bragg scattering in the expanded sample occurred in the free-particle regime. The momentum transferred to the atomic sample was determined by the average axial position in time-of-flight images. To extract small momentum transfers, the images were first fitted (in regions where the Bragg scattered atoms were absent) to a bimodal distribution which cor- rectly describes the free expansion of a condensate in the FIG. 1. Observation of momentum transfer by Bragg scat- Thomas-Fermi regime, and of a thermal component [13]. tering. (a) Atoms were exposed to laser beams with wave The chemical potential m of the trapped condensate was vectors k1 and k2 and frequency difference v, imparting momentum h¯q along the axis of the trapped condensate. The determined from the radial width of the condensate dis- Bragg scattering response of trapped condensates [(b) and (d)] tribution [11]. The noncondensate distribution (typically was much weaker than that of condensates after a 5 ms free less than 20% of the total population) was subtracted from expansion [(c) and (e)]. Absorption images [(b) and (c)] af- the images before evaluating the momentum transfer. ter 70 ms time of flight show scattered atoms distinguished By varying the frequency difference v, the Bragg scat- from the denser unscattered cloud by their axial displacement. Curves (d) and (e) show radially averaged (vertically in image) tering spectrum was obtained for trapped and for freely ex- profiles of the optical density after subtraction of the thermal panding condensates (Fig. 2). The momentum transfer per distribution. The Bragg scattering velocity is smaller than the atom, shown in units of the recoil momentum hq¯ , is anti- speed of sound in the condensate (position indicated by circle). symmetric about v ෇ 0 as condensate atoms are Bragg Images are 3.3 3 3.3 mm. scattered in either the forward or the backward direction, depending on the sign of v [14]. Both beams were derived from a common source, and From these spectra, we determined the total line strength then passed through two acousto-optical modulators op- and the center frequency (Fig. 3) by fitting the momentum erated with the desired frequency difference v, giving the transfer to the difference of two Gaussian line shapes, rep- beams a detuning of 1.6 GHz below the jF ෇ 1͘ !jF0 ෇ resenting excitation in the forward and the backward direc- 0, 1, 2͘ optical transitions. Thus, at the optical wave- tion. Since S͑q͒ ෇ 1 for free particles, we obtain the static length of 589 nm, the Bragg recoil velocity was hq¯ ͞m Ӎ structure factor as the ratio of the line strengths for the 7 mm͞s, giving a predicted Bragg resonance frequency of trapped and the expanded atomic samples. Spectra were 0 ෇ 2͞ Ӎ vq hq¯ 2m 2p31.5 kHz for free particles. The taken for trapped condensates at three different densities beams were pulsed on at an intensity of about 1 mW͞cm2 by compressing or decompressing the condensates in the for a duration of 400 ms. To suppress super-radiant magnetic trap prior to the optical excitation. Rayleigh scattering [10], both beams were linearly polar- The Bragg resonance for the expanded cloud was cen- ized in the plane defined by the condensate axis and the tered at 1.54(15) kHz with an rms width of 900 Hz con- wave vector of the light. sistent with Doppler broadening [15]. This frequency The Bragg scattering of a trapped condensate was ana- includes an expected 160 Hz residual mean-field shift, lyzed by switching off the magnetic trap 100 ms after the end of the light pulse, and allowing the cloud to freely evolve for 70 ms. During the free expansion, the density of the atomic cloud dropped and quasiparticles in the condensate transformed into free particles and were then imaged by resonant absorption imaging (Fig. 1). Bragg scattered atoms were distinguished from the unscattered atoms by their axial displacement. The speed of Bo- goliubov sound at the center of the trapped condensate ෇ is relatedp to the velocity of radial expansion yr as cs yr ͞ 2 [11] (cs ෇ 11 mm͞satm͞h ෇ 6.7 kHz as shown in Fig.

Load more