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

VOLUME 83, NUMBER 15 PHYSICAL REVIEW LETTERS 11OCTOBER 1999

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).

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. 1). Thus, by comparing the axial displacement of the scattered atoms to the radial extent of the expanded FIG. 2. Bragg scattering of phonons and of free particles. condensate, one sees that the Bragg scattering recoil ve- Momentum transfer per particle, in units of hq¯ , is shown vs locity is smaller than the speed of sound in the trapped the frequency difference v͞2p between the two Bragg beams. condensate, i.e., the excitation in the trapped condensate Open symbols represent the phonon excitation spectrum for ͞ ෇ occurs in the phonon regime. a trapped condensate at a chemical potential m h 9.2 kHz. Closed symbols show the free-particle response of an expanded For comparison, Bragg scattering of free particles was cloud. Lines are fits to the difference of two Gaussian line studied by applying a light pulse of equal intensity [12] shapes representing excitation in the forward and backward after allowing the gas to freely expand for 5 ms, during directions. 2877 VOLUME 83, NUMBER 15 PHYSICAL REVIEW LETTERS 11OCTOBER 1999

Neglecting small contributions representing multiparti- cle excitations [3,4], the single quasiparticle contribution to the static structure factor is N S͑q͒ ෇ 0 ͗gj ͑aˆ aˆ y 1 aˆ y aˆ N q q 2q 2q y y ͒ ͘ 1 aˆ 2qaˆ q 1 aˆ qaˆ 2q jg . (2) Substituting the Bogoliubov operators, one obtains [17] ͑ ͒Ӎ͑ 2 2 ͒ ෇ 0͞ B S q uq 1yq 2 2uqyq vq vq . (3) 0 In the limit h¯vq ¿ m, the Bogoliubov excitations become identical to free-particle excitations (uq ! 1, yq ! 0), ͑ ͒ 0 ͑ ͒ ͞ and S q ! 1. For phonons (h¯vq ø m), S q ! hq¯ 2mcs, and the line strength diminishes linearly with q. To the same order of approximation, the quasiparticle resonance is undamped, and the dynamical structure fac- ͑ ͒ ෇ ͑ ͒ ͑ B͒ tor isRS q, v S q d v2vq (satisfying the f sum ͑ ͒ ෇ 0 rule: vS q, v dv vq [3]). Thus, accompanying the diminished line strength, the Bragg resonance is shifted B 0 upward from the free particle resonance by vq 2vq. Equivalently, the suppression of the Bragg resonance FIG. 3. (a) Static structure factor S͑q͒ and (b) shift of the line center from the free-particle resonance. S͑q͒ is the ratio in the phonon regime can be understood in terms of of the line strength at a given chemical potential m to that the many-body condensate wave function. The static observed for free particles. As m increases, the structure factor structure factor is the magnitude of the state vector je͘ ෇ P y p decreases, and the Bragg resonance frequency increases. Solid aˆ aˆ jg͘͞ N. The macroscopic population of the lines are predictions of a local density approximation [Eq. (5)] k k1q k using v0 ෇ 2p31.38 kHz. Dotted line indicates the mean- zero-momentum state picks out two relevant terms in the q summation: field shift of 4m͞7h in the free-particle regime, with data from p [5] shown in open symbols. ͘Ӎ͑ y ͘ y ͒͘͞ ෇ 1͘ 2͘ je aˆ q aˆ 0jg 1 aˆ 0 aˆ 2qjg N je 1 je . (4) These represent two means by which momentum is giving a measured free-particle resonance frequency of imparted to the condensate: either by promoting a zero- 1.38 kHz. The response of trapped condensates was strik- momentum particle to momentum h¯q, or else by demoting ingly different. As the density of the trapped condensates a particle from momentum 2h¯q to zero momentum. was increased, the Bragg scattering resonance was sig- If correlations could be neglected, the total rate of excita- nificantly weakened in strength and shifted upwards in tion would simply be the sum of the independent rates for frequency. This reflects the changing character of the ex- these two processes, proportional to ͗e1 j e1͘ ෇ ͗N0͘ 1 citations created by Bragg scattering as the speed of sound q ෇ 2 ͗ 2 j 2͘ ෇ ͗ 0 ͘ ෇ 2 ͗ 0͘ was increased: at a fixed Bragg scattering momentum, the 1 uq and e e N2q yq, where Nk is the excitations passed from the free-particle to the phonon expected number of atoms of momentum h¯k in the con- regime. densate. This would apply, for example, to a condensate To account for this behavior, we use the zero- in a pure number state, or to an ideal gas condensate with a thermal admixture of atoms with momenta 6h¯q, and temperature Bogoliubov description of a weakly interact- ͑ ͒ ing homogeneous Bose-Einstein condensate [16]. The would always lead to S q . 1. Hamiltonian, Yet, for the many-body ground state of the interacting X X 2 Bose gas, the behavior is dramatically different. Collisions 0 y 2ph¯ a y y H ෇ h¯vk aˆ k aˆ k 1 aˆ k aˆ l aˆ maˆ k1l2m , of zero-momentum atoms admix into the condensate pairs k k,l,m mV of atoms at momenta 6h¯q, the population of which (1) comprises the quantum depletion [18]. As a result, the two momentum transfer mechanisms described above produce for a gas in volume V, where h¯v0 ෇ h¯2k2͞2m, is approxi- k indistinguishable states, and the rate of momentum transfer mated by replacing the zero-momentum operators with p is given by the interference of two amplitudes, not by y ෇ ෇ c-numbers aˆ 0 aˆ 0 N0, where N0 is the number of the sum of two rates. Pair excitations in the condensate atoms with zero momentum. Neglecting terms of or- are correlated so as to minimize the total energy, and 21͞2 der N , the Hamiltonian is diagonalized by a canoni- thereby give destructive interference between the two cal transformation to operators defined by aˆ k ෇ ukbˆk 2 ͑ ͒ ෇ ͑ ͒2 y momentum transfer processes, i.e., S q uq 2yq , ˆ ෇ ෇ ෇ 0 ykb2k, where uk coshfk, yk sinhfk, and tanh2fk 1. For high momentum, ͗N ͘ ø 1 and the interference ͑͞ 0 ͒ q m h¯vk 1m. The energyp of the Bogoliubov excitation plays a minor role. In the phonon regime, while the ˆ y B ෇ 0͑ 0 ͒ 2 2 ͗ 0 ͘ created by bk is h¯vk h¯vk h¯vk 1 2m . independent rates uq and yq (and, hence, N6q ) diverge 2878 VOLUME 83, NUMBER 15 PHYSICAL REVIEW LETTERS 11OCTOBER 1999 as 1͞q, the correlated quantum depletion extinguishes the We thank Yvan Castin for insightful comments. This rate of Bragg excitation. work was supported by the Office of Naval Research, These results for the homogeneous Bose gas can be NSF, JSEP, ARO, NASA, and the David and Lucile applied to trapped, inhomogeneous condensates by a local Packard Foundation. A. P. C. acknowledges additional density approximation since the reduced phonon wave- support from the NSF, A. G. from DAAD, and D. M. S.-K. length q21 (0.4 mm) is much smaller than the condensate from JSEP. size (r . 20 mm) and since the zero-point Doppler width is smaller than the mean-field shift ͑hq¯ ͞mr ø m͞h¯͒ [19,20]. In the Thomas-Fermi regime, the conden- [1] T. J. Greytak, in Quantum Liquids, edited by J. Ruvalds ͑ ͒ ෇ sate hasp a normalized density distribution f n and T. Regge (North-Holland, New York, 1978). 15n͞4n0 1 2 n͞n0, where n0 is the maximum conden- [2] P. E. Sokol, in Bose-Einstein Condensation, edited by sate density. The Bragg excitationv line shape is then A. Griffin, D. W. Snoke, and S. Stringari (Cambridge u University Press, Cambridge, England, 1995), p. 51. 2 02 u 2 02 15 v 2vq t v 2vq I͑v͒dv ෇ 1 2 dv , (5) [3] P. Nozières and D. Pines, The Theory of Quantum Liquids 0͑ ͞ ͒2 0 ͞ 8 vq m h¯ 2vqm h¯ (Addison-Wesley, Redwood City, CA, 1990). [4] A. Griffin, Excitations in a Bose-Condensed Liquid (Cam- from which one can obtain the line strength S͑q͒ and center bridge University Press, Cambridge, England, 1993). frequency. The line strength has the limiting values of [5] J. Stenger et al., Phys. Rev. Lett. 82, 4569 (1999). ͑ ͒ ͞ ͑ 0͞ ͒1͞2 [6] D. S. Jin et al., Phys. Rev. Lett. 77, 420 (1996); M.-O. S q ! 15p 32 h¯vq 2m in the phonon regime and ͑ ͒ ͞ 0 Mewes et al., Phys. Rev. Lett. 77, 988 (1996); D. S. Jin S q ! 1 2 4m 7¯hvq in the free-particle regime [21]. In accordance with the f-sum rule, the center frequency v¯ is et al., Phys. Rev. Lett. 78, 764 (1997); M. R. Andrews et al., Phys. Rev. Lett. 79, 553 (1997); D. M. Stamper- v0͞S͑q͒ given as q . Kurn et al., Phys. Rev. Lett. 81, 500 (1998). q ෇ These predictions are shown in Fig. 3 using v0 [7] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, 2p31.38 kHz. Both the line strength and the shift of Atom-Photon Interactions (Wiley, New York, 1992). the Bragg resonance are well described by our treatment. [8] J. Javanainen and J. Ruostekoski, Phys. Rev. A 52, 3033 For comparison, previous measurements [5] of the mean- (1995). field shift of the Bragg resonance (4m͞7¯h) in the free- [9] M.-O. Mewes et al., Phys. Rev. Lett. 77, 416 (1996). particle regime are also shown, clearly indicating the [10] S. Inouye et al., Science 285, 571 (1999). many-body character of low energy excitations. [11] Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996). Finally, let us discuss finite temperature effects. The [12] The laser beams had diameters of about 2 mm and were directed vertically, so the 120 mm drop of the atomic structure factor at nonzero temperature is increased as sample during the 5 ms time of flight was negligible. ͑ ͒ ෇ ͑ ͒2 ͑ B B ͒ S q uq 2yq 3 1 1 Nq 1 N2q due to the popu- [13] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, in B lations N6q of thermally excited Bogoliubov quasipar- Bose-Einstein Condensation in Atomic Gases, Proceedings ticles at wave vectors 6q. However, in our measurements of the International School of Physics “Enrico Fermi,” using stimulated scattering from two laser beams, the con- edited by M. Inguscio, S. Stringari, and C. E. Wieman (to tribution of the thermal excitations cancels out [14], and be published). thus we extract the zero-temperature structure factor. In [14] Since photons can be absorbed from either beam and then contrast, by measuring light scattering from a single beam emitted into the other, we observe a momentum transfer ͑ ͒ ͑ ͒ one could determine the temperature-dependent structure proportional to S q, v 2 S 2q, 2v . ഠ ± factor. Such a measurement could detect low-momentum [15] A tilt angle of 5 of q with respect to the axial direction introduced Doppler broadening from the radial expansion. thermal excitations, and thus could serve as a thermometer [16] N. N. Bogoliubov, J. Phys. (Moscow) 11, 23 (1947). for a low-temperature gas. [17] Shown explicitly for light scattering from a Bose-Einstein In conclusion, stimulated light scattering was used to condensate in R. Graham and D. Walls, Phys. Rev. Lett. excite phonons in trapped Bose-Einstein condensates with 76, 1774 (1996). wavelengths much smaller than the size of the trapped [18] K. Huang, Statistical Mechanics (Wiley, New York, sample. The static structure factor was shown to be sub- 1987). stantially reduced in the phonon regime. This modifica- [19] E. Timmermans and P. Tommasini (private communica- tion of light-atom interactions arises from the presence of tion). a correlated admixture of momentum excitations in the [20] The discrete spectrum was considered by A. Csordás, condensate. The observed reduction of S͑q͒ also implies R. Graham, and P. Szépfalusy, Phys. Rev. A 54, R2543 (1996); 57, 4669 (1998). a reduction of inelastic Rayleigh scattering of light with 2 3 4 0 [21] S͑q͒ ෇ 15h͞64͑ y 1 4h22yh 1 12h 2 3yh ͒, where wave vector k by a condensate when h¯vk ,m [22]. 2 ෇ 0͞ ෇ ͓͑ 2 ͒͞ ͔ h h¯vq 2m and y p22 arctan h 2 1 2h . This effect may reduce heating in optical dipole traps [22] For a homogeneous Bose-Einstein condensate,p inelas- and reduce the optical density probed in absorption imag- ͑ ͑ ͒ tic scatteringp is reduced by a factor x 1 1 x 2 21 0 ing. For example, the absorption of near-resonant light sinh x ͒͞x, where x ෇ 2¯hvk ͞m. by a homogeneous sodium condensate at a density of [23] D. M. Stamper-Kurn et al., Phys. Rev. Lett. 80, 2702 3 3 1015 cm23 [23] should be reduced by a factor of 2. (1998). 2879