1S-2S spectrum of a hydrogen Bose-Einstein condensate

Transcription

1 PHYSICAL REVIEW A, VOLUME 61, S-S spectru of a hydrogen Bose-Einstein condensate Thoas C. Killian* Departent of Physics and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cabridge, Massachusetts 0139 Received August 1999; published 15 February 000 We calculate the two-photon 1S-S spectru of an atoic hydrogen Bose-Einstein condensate in the regie where the cold collision frequency shift doinates the line shape. Wentzel-Kraers-Brillouin and static phase approxiations are ade to find the intensities for transitions fro the condensate to otional eigenstates for S atos. The excited-state wave functions are found using a ean field potential, which includes the effects of collisions with condensate atos. Results agree well with experiental data. This foralis can be used to find condensate spectra for a wide range of excitation schees. PACS nubers: Fi, 3.70.Jz, Jp, s I. ITRODUCTIO In the recent experiental observation of Bose-Einstein condensation BEC in atoic hydrogen 1, the cold collision frequency shift in the 1S-S photoexcitation spectru signaled the presence of a condensate. The shift arises because electronic energy levels are perturbed due to interactions, or collisions, with neighboring atos. In the cold collision regie, the teperature is low enough that the s-wave scattering length, a, is uch less than the theral de Broglie wavelength, T h /k B T, and only s waves are involved in the collisions 3. The cold collision frequency shift has also been studied in the hyperfine spectru of hydrogen in cryogenic asers 4, and cesiu 5 7 and rubidiu 8 in atoic fountains. Theoretical explanations of these results and other work on the hydrogen 1S-S spectru 9,10 have focused on the agnitude of the shift, as opposed to a line shape. In this article we present a calculation of the hydrogen BEC 1S-S spectru. We also describe how the foralis can be used for other atoic systes and experiental conditions. A. The experient The experient is described in Refs. 1 and, and we suarize the iportant aspects here. Hydrogen atos in the 1S, F1, F 1 state are confined in a agnetic trap and evaporatively cooled. The hydrogen condensate is observed in the teperature range K and the condensate fraction never exceeds a few percent. evertheless, the peak density in the noral cloud is alost two orders of agnitude lower than in the condensate and in this study we will neglect the presence of the noncondensed gas. The two-photon transition to the etastable S, F1, F 1 state (1 s is driven by a 43 n laser bea which passes through the saple and is retroreflected. In this configuration, an ato can absorb one photon fro each direction. This results in Doppler-free excitation for which there is no oentu transferred to the ato and no *Present address: ational Institute of Standards and Technology, Gaithersburg, MD Doppler-broadening of the resonance. An ato can also absorb two copropagating photons and receive a oentu kick. This is Doppler-sensitive excitation, and the spectru in this case is recoil shifted and Doppler broadened. The photoexcitation rate is onitored by counting 1 n fluorescence photons fro the excited state. For a typical laser pulse of 500 s, fewer than 1 in 10 4 of the atos are prooted to the S state. S atos experience the sae trapping potential as 1S atos because the agnetic oent is the sae for both states, neglecting sall relativistic corrections. The natural linewidth of the 1S-S transition is 1.3 Hz, but the experiental width, at low density and teperature, is liited by the laser coherence tie. The narrowest observed spectra, obtained when studying a noncondensed gas, have widths of a few khz 11. For the condensate, the cold collision frequency shift is as uch as one MHz and it doinates the line shape. B. Mean field description of the spectru The frequency shift in aser and fountain experients has traditionally been described using the quantu Boltzann equation 4,5,8. In this picture, the frequency shift is the net result of the sall collisional phase shifts arising fro forward scattering events in the gas. A ean field description, however, is ore convenient for studying an inhoogeneous Bose-Einstein condensate. We will derive this picture in detail, but we suarize the results here. Collisions add a ean field energy to the ato s potential energy. ForaS ato excited out of a condensate the ean field ter is E S (r)4 a 1SS n 1S (r)/. Fora1S condensate ato the ean field ter is E 1S (r) 4 a 1S1S n 1S (r)/. The fraction of excited S atos is sall, so S-S interactions can be neglected. The ground state s-wave triplet scattering length has been calculated accurately a 1S-1S n Ref. 1. The 1S-S scattering length, however, is less well known (a 1S-S n fro experient and.3 n fro theory 13. We denote the su of the agnetic trap potential V(r) and the ean field energy E x (r) as the effective potential, V eff x (r) Fig. 1. Here, x is either 1S or S. For 1S conden /000/613/ /$ The Aerican Physical Society

2 THOMAS C. KILLIA PHYSICAL REVIEW A where p j, r j, and H j int are the oentu operator, position operator, and internal state Hailtonian, respectively for particle j. V(r) is the agnetic trapping potential, which is the sae for 1S and S atos. H las is the ato-laser interaction. After aking the rotating wave approxiation, it can be written H las j1 rj S1S j e i4t 1SS j e i4t, FIG. 1. Effective potentials for 1S atos in the condensate and excited S atos. Selected single-particle wave functions are displayed at the height corresponding to their energy. The dashed lines are the agnetic trapping potential V(r), which is identical for 1S and S atos. The thin solid lines are the effective potentials, which include the ean field interaction energy. The vertical lines indicate allowed Doppler-free transitions fro the condensate, which ust preserve irror syetry. The potentials and condensate wave function are for a peak condensate density of c 3 and a agnetic trap oscillation frequency of 4 khz, which are characteristic conditions for a hydrogen BEC and a strong confineent axis of the trap Refs. 1 and. The scattering lengths used in the calculations are a 1S1S n and a 1SS 1.4 n, and the cheical potential is /k B K. The S levels for a near continuu of otional states in an anisotropic three diensional trap. sate atos, the effective potential in the condensate is flat. Because a 1S-S 0 and the condensate density is large, S atos experience a stiff attractive potential in the condensate which supports any bound S otional states. The 1S-S spectru consists of transitions fro the condensate to S otional eigenstates of the effective S potential. For Doppler-free excitation, the final states are bound in the BEC well. Doppler-sensitive excitation populates states that lie about k 0 /k b 643 K above the botto of the S potential, where k 0 is the oentu carried by two laser photons. The latter states extend over a region uch greater than the condensate. Because the excited levels are so different for Doppler-free and Doppler-sensitive excitation, we ust treat the two spectra independently. The rest of this article presents a derivation of the effective potentials and a quantu echanical calculation of the BEC 1S-S spectru. II. 1S-S PHOTOEXCITATIO SPECTRUM OF A HYDROGE BOSE-EISTEI CODESATE A. Hailtonian We start with the any-body Hailtonian for a syste with atos, p j H int j Vr j H las H coll, 1 H j1 where is the frequency of the laser field (he S E 1S E 1S-S on resonance. The laser bea is unifor over the condensate, so we treat the excitation as a standing wave consisting of two counterpropagating plane waves. The effective -photon Rabi frequency for Doppler-free excitation 14, DF r DF M 3 S,1S 3 c R I, 3 is unifor in space. Here, I is the laser intensity in each direction, M S,1S Ref. 15 is a unitless constant, c is the speed of light in vacuu, is the fine structure constant, and R is the Rydberg constant. For Doppler-sensitive excitation, DS r DS e ik 0 z e ik 0 z, where DS DF /. H coll describes the effects of two-body elastic collisions. In the cold collision regie, the interaction can be represented by a shape-independent pseudopotential 16 corresponding to a phase shift per collision of ka, where k is the oentu of each of the colliding particles in the center of ass frae, H coll 4 i j r i r j a 1S-1S 1S1S i 1S1S j a 1S-S e 3 u e 3 u ij a S-S SS i SS j. The su is over (1)/ distinct pairwise interaction ters. The 1S-S interaction projection operator is written in ters of e 3 u ij 1S is j S i 1S j because the doubly spin polarized atos collide on the e 3 u potential during s-wave collisions 13. As entioned above, the S-S scattering ter is negligible for the hydrogen experient, but it is included here for copleteness. Inelastic collisions, such as collisions in which the hyperfine level of one or both of the colliding partners changes,

3 1S-S SPECTRUM OF A HYDROGE BOSE-EISTEI... PHYSICAL REVIEW A will contribute additional shifts, which are not included in this foralis, but these effects are expected to be sall in the experient. B. Syste before laser excitation We ake the approxiation that the syste is at T0, and all atos are initially in the condensate. T0 odels have accurately described any condensate properties 17, and we leave finite teperature effects for future study. The state vector can be written 7 where 1S,0 refers to the single particle electronic and otional state of an ato in a 1S condensate with atos. We use the ket notation (a;b;...;c), in which the entry in the first slot is the state of ato 1, the second entry is the state of ato, etc. Miniization of 0 H 0 leads to the Gross- Pitaevskii, or nonlinear Schrödinger equation 18,19 for the single particle BEC wave function, (r)r0, r V 1S eff rr. 8 The effective potential is V eff 1S (r)v(r)ũn(r), where Ũ 4 a 1S-1S /. Here, n(r)(r) is the density distribution in the -particle condensate. One can interpret (r i ) as the probability of finding condensate particle i at position r i. The kinetic energy is sall and can be neglected. This yields the Thoas-Feri wave function 0, r 1/ n0vr/ũ 1/ Vrn0Ũ 9 0 otherwise, where n(0) is the peak density. The density profile is the inverted iage of the trapping potential. The cheical potential is ()Ũn(0), and it is equal to V 1S inside the eff condensate. The energy of the syste before laser excitation is the iniu of 0 H 0. It satisfies ()E 0 / and is given by E Fro now on, when writing we will drop the explicit dependence on. For a cylindrically syetric haronic trap, it can be shown that n(0) (15 3 w r w z / 3 3/ a 1S-1S ) /5 /8, where w r and w z are the angular frequencies for radial and axial oscillations in the trap. C. Syste after laser excitation To describe the syste after laser excitation we ust find the orthonoral basis of S otional wave functions and their energies. This is done by iniizing q,i H q,i, where 11 is a state with q S atos in S otional level i. The operator Ŝ syetrizes with respect to particle label. We will show below that the state vector of the syste after laser excitation is actually expressed as a superposition of such ters, but for now we need only consider a single q,i. Calculating q,i H q,i involves a soewhat lengthy calculation. Details are given in appendix A and the result is q,i H q,i E 0 q S,i H int p V S eff rs,i E 0 qe 1S-S i. 1 E 0 is the energy of a pure 1S condensate with q atos see Eqs. 10 and A5, i ip /V eff S (r)i, and the effective potential for the S atos is eff rvr 4 a 1S-S n q r. V S 13 The density of 1S atos reaining is n q (r)( q)(r). Finding the S otional states which iniize q,i H q,i, with the requireent that they for an orthonoral basis, is equivalent to finding the eigenstates of the effective S Hailtonian H eff S p V S eff r, 14 and the eigenvalue for state i is i. The effective Hailtonian Eq. 14 is consistent with the two-coponent Hartree-Fock equations used to calculate the single-particle wave functions for double condensates 1. The effective potential and soe S otional states are depicted in Fig. 1. If we denote the iniu of q,i H q,i as E q,i, using Eqs. 10 and 1, the energy supplied by two photons to drive the transition to state i, for q, is h E q,ie 0 qe 1S-S i E 0 E 0 q q E 1S-S i. 15 We have used (E 0 E 0 )/qe 0 / for sall q. ote that i 0 for states bound in the BEC interaction well. Since any S otional levels ay be excited, there will be a distribution of excitation energies in the spectru. When condensate atos are coherently excited to an isolated level i by a laser pulse of duration t, the singleparticle wave functions evolve according to where 1S,0 cos 1S,0sin S,i,

4 THOMAS C. KILLIA PHYSICAL REVIEW A ir0 sin ir0 sin ir0 1/ t/. 17 The detuning fro resonance is. InEq.16, we assue the excitation is weak enough to neglect the change in the single-particle wave function for atos in the condensate 3,4. Depending upon which excitation schee is being described, (r) is either DF (r) or DS (r). The state vector for the syste after excitation can be written 18 where the label q sin is the expectation value of the nuber of S atos excited. Although q is not a good quantu nuber for q,i, the spread in q, given by a binoial distribution, is strongly peaked around q. For short excitation ties, the population in state i grows coherently as t. For the hydrogen experient, however, although the excitation is weak and i(r)0t1, t is longer than the coherence tie of the laser (00 s). This iplies that the nuber of atos excited to level i ust be expressed in a for reiniscent of Feri s Golden Rule. Equation 17 can be rewritten in ters of a delta function using the relation sin (xt)/x t (x) ast. One can neglect i(r)0 copared to because i(r)0 is sall copared to the spread in frequency of the laser excitation. Then q t ir0 he 1S-S i. 19 It is understood that Eq. 19 is to be convolved with the laser spectru or a density of states function. The total S excitation rate is Sh i ir0 he 1S-S i i F i he 1S-S i. 0 Equation 0 defines the overlap factors, F i i(r)/0, which are analogous to Franck-Condon factors in olecular spectroscopy. An expression equivalent to Eq. 0, the strength distribution function or dynaic for factor, is coonly used to describe collective excitations of any body systes 17. The BEC spectru now appears as ties the spectru of a single particle in 0 excited to eigenstates of the effective S potential. The broadening in the 1S-S BEC spectru is hoogeneous because it results fro a spread in the energy of possible excited states, not fro a spread in the energy of initially occupied states. The central results of this calculation are the effective S potential Eq. 13 and the Feri s Golden Rule expression for the excitation rate Eq. 0. Using this foralis we can now calculate the observed spectru for Doppler-free and Doppler-sensitive excitation. D. Doppler-free 1S-S spectru Doppler-free excitation populates states which are bound inside the BEC potential well see Fig. 1. For a condensate in a haronic trap, these states are approxiately eigenstates of a three-diensional haronic oscillator with trap frequencies larger than those of the agnetic trap alone by a factor of 1a 1S-S /a 1S-1S 5 see Eqs. 9 and 13. Because we know the wave functions, we can nuerically evaluate Eq. 0. The result of such a calculation is shown in Fig.. At large red detuning h 4 a 1S-S n(0)/ transitions are to the lowest state in the BEC interaction well. The spectru does not extend to the blue of h 0 because states outside the well have negligible overlap with the condensate and are inaccessible by laser excitation. In the overlap integrals in Fig., wave functions for an infinite haronic trap were used for the S otional states. These deviate fro the actual otional states near the top of the BEC interaction well, introducing sall errors in the stick spectru nearer zero detuning. The envelope of the spectru in Fig. can be derived analytically and reveals soe interesting physics. The S single-particle wave functions ri oscillate rapidly. Thus the transition intensity to state i, governed by the overlap i factor F DF i0, is ost sensitive to the value of (r) r0n(r)/ at the state s classical turning points. At a given laser frequency, the excitation is resonant with all states with otional energy he 1S-S. This suggests the excitation rate is proportional to the integral of the condensate density in a shell at the equipotential surface defined by the classical turning points of S states with otional energy

5 1S-S SPECTRUM OF A HYDROGE BOSE-EISTEI... PHYSICAL REVIEW A Using Eq. 1, the Feri s Golden Rule expression for the spectru Eq. 0 becoes S DF h DF i 4R i nr i V eff S R i he 1S-S i. 3 FIG.. Calculated Doppler-free spectru of a condensate at T 0 in a three-diensional haronic trap. Zero detuning is the unperturbed Doppler-free transition frequency. The stick spectru results fro the su over the transition aplitudes expressed in Eq. 0 using the Thoas-Feri density distribution for a peak condensate density of c 3 4 (a 1S-S a 1S-1S )n(0)/h 0.95 MHz. The trap is spherically syetric with trap 6 khz. The stick heights represent the coefficients of delta functions which ust be convolved with the laser spectru of about 1 khz full width at half axiu. The dashed curve Eq. 5 follows fro the integral over the BEC density distribution, Eq. 4, for the sae peak condensate density. The envelope is independent of the syetry of the trap, but the stick spectru blends into a continuu in a trap with one weak confineent axis such as in the experient described in Refs. 1 and. Resolution of the individual transitions would require a stiff, near spherically syetric trap, very stable experiental conditions, and high signal/ noise. It does not see feasible with the hydrogen experient in the near future. For a spherically syetric trap, we can forally show this by aking Wentzel-Kraers-Brillouin WKB and static phase approxiations 5,6 a technique that has recently been applied to describe s-wave collision photoassociation spectra 7 and quasiparticle excitation in a condensate 8. One uses a WKB expression for the S eigenstate. Then, because of the slow spatial variation of the condensate wave function, the Doppler-free overlap factor only depends on the condensate wave function and the 1S and S potentials where the phase of the upper state is stationary. This yields i F DF i0 4R i nr i D, 1 where R i is the Condon point, or the radius where the local wave vector of the excited state k S i V eff S (r)/ vanishes. R i is equivalent to the classical turning point for state i, and is defined through i V eff S R i. Also, in the liit that we can neglect the slow spatial variation of the BEC wave function, DdV eff S (r)/dr Ri V eff S (R i ) is the slope of the effective S potential at the Condon point. The Doppler-free excitation field and the BEC wave function are spherically syetric, so only S otional states with zero angular oentu are excited. This iplies that in the liit of closely spaced levels, i d in Eq. 3. Using Eq. we can change variables: ddr V eff S (R) and he 1S-S he 1S-S 4 anr/, where aa 1S-S a 1S-1S. This yields S DF h DF 4dr r nr he 1S-S 4 anr. 4 Using the probabilistic interpretation of (r i ) Sec. II B, one can interpret Eq. 4 in the following way. When as excitation is detected at a given frequency, it records the fact that a 1S ato was found at a position that had a 1S density, which brought that ato into resonance with the laser. The rate of excitation is proportional to the probability of finding a condensate ato in a region with the correct density. This is a local density description of the spectru, and it is justified by the slow spatial variation of the condensate wave function. For a Thoas-Feri wave function in a threediensional haronic trap, Eq. 4 reduces to S DF h 15 DF E 1S-S h 8 h ax 1 he 1S-S h ax 1/, 5 for h ax he 1S-S 0, and otherwise S DF (h) 0. Here, h ax 4 an(0)/. Figure shows that for a spherically syetric trap, Eq. 5 agrees with the spectru calculated directly with Feri s Golden Rule Eq. 0 using siple haronic oscillator wave functions. For a trap that has a weak confineent axis, such as described in Refs. 1 and, discrete transitions in the spectru are too closely spaced to be resolved. The envelope given by Eq. 5, however, shows no dependence on the trap frequencies or the syetry or lack thereof of the haronic trap. Theory and experiental data are copared in Fig. 3. Although the statistical error bars for the data are large due to the sall nuber of counted photons, the theoretical BEC

6 THOMAS C. KILLIA PHYSICAL REVIEW A any transition-atrix eleents. In this regie it is possible to odify the derivation of the WKB and static phase approxiations 5 8 to calculate the Doppler-sensitive spectru. We rewrite the Doppler-sensitive Rabi frequency Eq. 4 as DS r DS e ik 0 z e ik 0 z DS leven 4l1i l j l k 0 ry l 0,, 6 FIG. 3. Doppler-free spectru of a condensate: coparison of theory and experient fro Ref. 1. The narrow feature near zero detuning is the spectral contribution fro the noncondensed atos shown 1/40). The broad feature is the spectru of the condensate. The dashed curve is Eq. 5, which coes fro the integral over the BEC density distribution Eq. 4 assuing a Thoas- Feri density distribution for a haronic trap. spectru for a condensate at T0 fits the data reasonably well. The deviations ay indicate nonzero teperature effects or reflect experiental noise. The soothing of the cutoff at large detuning ay be due to shot to shot variation in the peak condensate density for the 10 ato-trapping cycles which contribute to this coposite spectru. Also, at low detuning the BEC spectru is affected by the wing of the Doppler-free line for the noncondensed atos. Using this theory, fro the peak shift in the spectru, the trap oscillation frequencies, and knowledge of a 1S-1S and a 1S-S, one can calculate the nuber of atos in the condensate. Assuing the experiental value of a 1S-S, the result is larger than the nuber deterined fro a odel of the BEC lifetie and loss rates, which is discussed in Ref. 9. The uncertainties are large for these results, but the disagreeent could be due to error in the experiental value of a 1S-S, uncertainty in the gas teperature or trap and laser paraeters, or therodynaic conditions in the trapped gas which are different than assued by the theories. For exaple, we have iplicitly assued local spatial coherence g () (0)1 30 in our for of the BEC wave function Eq. 7. It has not yet been experientally verified that the hydrogen condensate is coherent. E. Doppler-sensitive 1S-S spectru In contrast to the Doppler-free excitation spectru, the Doppler-sensitive spectru in principle reflects the finite oentu spread in the condensate as well as the ean field effects. The relevant oentu spread is given by the uncertainty principle and is /z where z5 isthe length of the condensate along the laser propagation axis. However, in the hydrogen experient the cold collision frequency shift (1 MHz doinates over the Dopplerbroadening in the spectru (k 0 /z100 Hz. We can thus neglect Doppler-broadening, which is equivalent to neglecting the spatial variation of the BEC wave function in where j l (k 0 r) is the spherical Bessel function of order l, and Y l (,) is a spherical haronic. This shows that the Doppler-sensitive laser Hailtonian can excite atos to S otional states with any even value of angular oentu, but with 0. Transitions are to levels with otional energy k 0 / above the botto of the S potential, so we label levels by, their energy deviation fro this value. For siplicity, we consider a spherically syetric trap. This allows us to write a general expression for the S wave functions,l Y 0 l (,)u,l (r)/r where u,l (r)/r satisfies d ll1 V eff dr r S ru,l r E 1S-S k 0 u,l r. Using Eq. 0, the spectru is S DS h,l,l DS r0 he 1S-S k Using Eq. 6, the overlap integral we ust evaluate is,l e ik 0 z e ik 0 z 0 dr ru,l r 4l1i l j l k 0 r nr 9 for l even, and 0 otherwise. Because n(r) varies slowly, one can find an approxiate expression for this atrix eleent. Appendix B gives the details of this derivation and uses the result to reforulate Eq. 8 as S DS h nr DS 4R eff R V S he 1S-S k

7 1S-S SPECTRUM OF A HYDROGE BOSE-EISTEI... PHYSICAL REVIEW A In Ref. 9, experiental data are copared with Eq. 3, and the agreeent is good. III. OTHER APPLICATIOS OF THE FORMALISM FIG. 4. Effective potentials, wave functions and the laser field for Doppler-sensitive excitation of condensate atos. The spatial period of the S wave function, the laser wavelength, and the vertical axes for the potentials are not to scale. The vertical axes for the wave functions and laser field are arbitrary. In the overlap integral for the transition atrix eleent Eq. 9, the only nonzero contribution coes fro the region where the spatial period of the S wave function atches the wavelength of the laser field. This is indicated by the locations of the vertical lines. As the laser frequency is changed, the region of wavelength atch oves. The atrix eleent Eq. 9 gets its ain contribution at R where the classical wave vector of the WKB approxiation for u,l equals the classical wave vector of the WKB approxiation for j l. In effect, R is the point where the spatial period of the wave function atches the wavelength of the laser field, /k 0 see Fig. 4. This leads to a definition for R V eff S R, 31 which is identical to Eq., the definition of the Condon point fro the calculation of the Doppler-free spectru. Because the transition is localized in this way, the atrix eleent Eq. 9 is proportional to n(r ), as is evident in Eq. 30. Using Eq. 31, we can replace the su in Eq. 30 with an integral and change variables, d dr V eff S (R). This yields the Doppler-sensitive line shape S DS h DS 4dr r nr he 1S-S k 0 4 anr. 3 The Doppler-sensitive condensate spectru has the sae shape as the Doppler-free spectru, but it is shifted to the blue by photon oentu-recoil. Because DS DF /, the Doppler-sensitive spectru is half as intense as the Doppler-free. A. Other atoic systes and excitation schees We have specifically considered 1S-S spectroscopy of hydrogen, but the foralis is ore general. For instance, if the ground-excited state interaction were repulsive, this would siply odify the effective S potential Eq. 13 and the for of the otional states excited by the laser would change. Equations 4 and 3 would still be accurate for two-photon excitation to a different electronic state when the ean field interaction doinates the spectru. In the recently observed rf hyperfine spectru of a rubidiu condensate 31, the line shape is deterined by ean field energy and the different agnetic potentials felt by atos in the initial and final states. The theory presented here can be odified to describe this situation as well. For Bragg diffraction or spectroscopy as perfored in Refs. 3 and 33, atos reain in the sae internal state after excitation. Particle exchange syetry of the wave function odifies the ean field interaction energy of the excited atos with the atos reaining in the condensate. In ters of the hydrogen levels, 1S, F1, f 1 atos not in the condensate experience a potential of 8 a 1S-1S n 1S (r)/. This is to be copared with the ean field potential of 4 a 1S-S n 1S (r)/ experienced by S particles excited out of the condensate and 4 a 1S-1S n 1S (r)/ experienced by 1S atos in the condensate. In Appendix A, the point in the derivation where the difference arises is indicated. B. Doppler broadening in the Doppler-sensitive spectru To derive the Doppler-sensitive 1S-S spectru, we neglected the variation of the condensate wave function, which is equivalent to neglecting the atoic oentu spread. This is well justified for the hydrogen experient. The effect of sall but non-negligible oentu is discussed at the end of Appendix B. ow, we briefly describe the Dopplersensitive line shape when Doppler-broadening is doinant. The line shape turns out to be siilar to that which was seen with Bragg spectroscopy of a a condensate 33. When the ean field potential can be neglected, the S otional wave functions are approxiately those of the siple haronic oscillator potential produced by the agnetic trap alone. Because the spatial extent for these otional states is large copared to z, in the region of the condensate the wave functions can be represented as plane wave oentu eigenstates p 34. The spectru becoes S DS h p DS r0 p he 1S-S p DS d 3 p Apk 3 0 ẑ pz

8 THOMAS C. KILLIA PHYSICAL REVIEW A he 1S-S p. 33 The Fourier transfor of the condensate wave function, A(p)d 3 r e ip r/ (r), is nonzero for p x /x, p y /y, and p z /z. The excited states have p z k 0, so we define pp z k 0. Because the laser wavelength is sall copared to the spatial extent of the condensate, p / k 0 / k 0 p/ and the spectru reduces to Sh DS k 0 dp 3 x dp y Ap x xˆp y ŷpẑ, where k 0 p he 1S-S k defines the oentu class that is Doppler shifted into resonance. The spectru is centered at he 1S-S k 0 /, and the line shape depends on the orientation of the condensate wave function with respect to the laser propagation axis. For a Thoas-Feri wave function in a spherically syetric haronic trap A(p) j (pr 0 /)/(pr 0 /) Ref. 0, where r 0 n(0)ũ/w. uerical evaluation of the integral over p x and p y shows that the line shape is approxiately given by the power spectru of the wave function s spatial variation along z, S(h)A(p()ẑ). In recent experients with sall angle light scattering 35, the oentu iparted to atos is sall copared to c s, where c s / is the speed of Bogoliubov sound. In this case one can excite quasiparticles in the condensate as opposed to free particles. The theory described in this paper only treats free particle excitation, but Bogoliubov foralis, cobined with WKB and static phase approxiations, has been used to describe the spectru for quasiparticle excitation 8. IV. DISCUSSIO To ake the proble analytically tractable, we have only derived the BEC spectru for the specific case of a spherically syetric trap. The trap shape does not appear in the final expressions Eqs. 4 and 3, however, and with reasonable confidence we can extend the results to any geoetry. In the experient, the trap aspect ratio is as large as 400 to 1, but the data agrees well with this theory. The physical picture of the transition occurring at the classical turning points, and the probabilistic or local density interpretation of the spectru also support the generalization of Eqs. 4 and 3 to S DF h DF d 3 rn 1S r he 1S-S 4 an 1S r S DS h DS d 3 rn 1S r he 1S-S k an 1S r. 37 Equations 36 and 37 take 4 an 1S (r)/ as a local shift of the transition frequency and ascribe the excitation to a sall region in space where the laser is resonant. This approach is siilar to a quasistatic approxiation in standard spectral line-shape theory 6, which neglects the atoic otion and averages over the distribution of interparticle spacings to find the spectru. Ato pairs at different separations experience different frequency shifts due to atoato interactions. This broadens the line. There are iportant differences between the theory presented here and the quasistatic approxiation, however. For the standard quasistatic treatent to be valid, the lifetie of the excited state should be shorter than a collision tie 36. For a condensate, the classical concept of a collision tie is inapplicable. We have shown that Eq. 36 and 37 result fro a different approxiation: neglecting the slow spatial variation of the BEC wave function. Also, for the condensate spectru, one integrates over ato position in the effective potential, as opposed to integrating over the distribution of ato-ato separations. Finally, the BEC spectral broadening is hoogeneous, which is not norally the case when aking the quasistatic approxiation. It is interesting that although the atos in the condensate are delocalized over a region in which the density varies fro its axiu value to zero, the rapid oscillation of the excited state wave function essentially localizes the transition Eq. and B9. In this way, the excitation probes the condensate wave function spatially. The description of the BEC spectru developed here has provided insight into the excitation process and it is general. We have shown that the foralis of transitions between bound states of the effective potentials can be used when either the ean field or Doppler broadening doinates. It can describe a variety of excitation schees such as twophoton Doppler-free or Doppler sensitive spectroscopy to an excited electronic state, or Bragg diffraction, which leaves the ato in the ground state. ACKOWLEDGMETS We thank D. Kleppner for coents on this anuscript and, along with T. Greytak, for guidance during the course of this study. Discussions of the hydrogen experiental results with D. Fried, D. Landhuis, S. Moss, and in particular L. Willann inspired uch of this theoretical work and pro

9 1S-S SPECTRUM OF A HYDROGE BOSE-EISTEI... PHYSICAL REVIEW A vided valuable feedback. Thoughtful contributions fro W. Ketterle, L. Levitov, M. Oktel, and P. Julienne, and discussions with E. Tiesinga regarding the proper for of the collision Hailtonian, Eq. 5, are gratefully acknowledged. Financial support was provided by the ational Science Foundation and the Office of aval Research. APPEDIX A: EERGY FUCTIOAL FOR THE SYSTEM AFTER LASER EXCITATIO In this appendix, we derive Eq. 1, the energy functional for the syste after excitation which is iniized to find the S wave functions. The Hailtonian and the excited state vector, q,i, are defined in Eqs. 1 and 11. The syetry operator is explicitly written as Ŝ 1/ q P P, where the su runs over the q! q!q! distinct particle label perutations P. The energy functional for q 1S condensate atos and q S atos in state i is q,i H q,i q,i j1 p j Vr jh inth coll j q,i q1s,0 p VrHint1S,0 qs,i p VrHintS,i q,i H coll q,i. We evaluate the interaction ter, q,i H coll q,i S;...;1S;...ŜH coll ŜS;...;1S;... S;...;1S;...H coll ŜŜS;...;1S;... S,...;1S;...H coll ŜS;...;1S;... 1/ q S;...;1S;...H coll P PS;...;1S;..., A1 A where we have used H coll,ŝ0 and ŜŜŜ( q ) 1/.Ofthe (1)/ ters in H coll Eq. 5, (q)(q1)/ of the result in a 1S-1S interaction, (q)q of the result in a 1S-S interaction, and the rest result in a S-S interaction, which we can neglect. For the 1S-1S ters, only the identity perutation contributes. For the 1S-S ters two perutations contribute the identity and switching the labels on the two interacting particles. The expectation value of H coll thus reduces to qq1a 1S-1S0;0r 1 r 0;0 4 qqa 1S-Si;0r 1 r i;0. A3 As entioned in Sec. III A, Eq. A would be odified for Bragg diffraction or spectroscopy as perfored in Refs. 3 and 33 because the internal state is unchanged during laser excitation. We do not explicitly treat this situation because it is not central to this study. Inserting Eq. A3 into Eq. A1, we find the energy functional is where q,i H q,i E 0 q S,i H int p Vr 4 a 1SS n q rs,i E 0 qe 1SS i, E 0 q1s,0 p VrHint1S,0 A4 qq1a 1S-1S 0;0r 1 r 0;0, A5 is the energy for q isolated 1S condensate atos, and i ip /V eff S (r)i. The density in the condensate for q condensate atos (q) is n q (r)( q)0(r 1 r)0. APPEDIX B: WKB AD STATIC PHASE APPROXIMATIOS FOR THE DOPPLER-SESITIVE BEC SPECTRUM In this appendix we calculate the Doppler-sensitive overlap integral, Eq. 9, and siplify Eq. 8. The derivation is siilar to the treatent of Refs. 7 and 8. The overlap integral we ust evaluate is I,l,l e ik 0 z e ik 0 z 0 dr ru,l r4l1i l j l k 0 r nr B1 for l even and 0 otherwise. Because u,l and j l are rapidly varying copared to n it is useful to express u,l and j l in phase-aplitude for through a WKB approxiation. We define the local wave vectors for u,l and j l

10 THOMAS C. KILLIA PHYSICAL REVIEW A k u,l,r k 0 ll1 1/ This is essentially identical to Eq. fro the calculation r V S r eff, of the Doppler-free spectru. We expand the difference in the phases in a Taylor series B around R and write the overlap integral as k j k 0,l,r k 0 ll1 1/. B3 r I,l even 4l1 Then, in the classically allowed region 1 u,l r k u 1/ 1/ nr dx k 0 k j k 0,l,R sin u,l,r, B4,l,r cos u,l,r j k 0,l,R 1 j l k 0 r rk 0 k j k 0,l,r sin jk 0,l,r, B5 V S eff R where r u,l,r dr k,l u,l,r/4, RT r j k 0,l,r k RT 0,ldr k j k 0,l,r/4 B6 B7 k j k 0,l,R x 16l1 nr k 0 k j k 0,l,R V eff S R cos u,l,r j k 0,l,R /4. B10 are the phases. The inner turning points against the centrifugal barriers are denoted by R T. ote that the approxiations are good for (k 0 r) l(l1). For (k 0 r) l(l1), neglecting the sall V S and, the functions behave as daped eff exponentials. The outer turning points are of no concern to the calculation. ow we write I,l even 4l1 dr nr sin u,l,r k u,l,r 1/ sin j k 0,l,r k 0 k j k 0,l,r 4l1 1/ nr drk u,l,rk 0 k j k 0,l,r cos u,l,r j k 0,l,r. B8 We have used the fact that n(r) varies slowly and have dropped rapidly oscillating ters in the integral. We ake the static phase approxiation that the overlap integral will only have contributions fro the point R where the difference in the phase factors is stationary. This point is defined by 0d( u j )/dr R k u (,l,r ) k j (k 0,l,R ), which is equivalent to an l-independent relation defining R for excitation to states with energy defect, V eff S R. B9 To obtain the last line we have used the Fresnel integral dx cos(abx )/b cosa(b/b)/4. Equation B10 only holds for l(l1)(k 0 R ). For l(l1) (k 0 R ), I,l even 0 because j l (k 0 r) is exponentially daped at R. Fro Eqs. 8 and B10, l(l1)(k 0 R ) S DS h DS,l even 16l1nR k 0 k j k 0,l,R V eff S R cos u,l,r j k 0,l,R /4 he 1SS k 0. B11 We can replace the cos function with its average value of 1/ because its phase varies rapidly with l. Thus l(l1)(k 0 R ) leven l1 k 0 k j k 0,l,R cos u,l,r j k 0,l,R R /, l(l1)(k 0 R ) dl l1 k 0 1 ll1 k 0 R B

11 1S-S SPECTRUM OF A HYDROGE BOSE-EISTEI... PHYSICAL REVIEW A and S DS h nr DS 4R eff R V S he 1SS k 0. B13 In the derivation given above, we neglected the variation of the condensate wave function, which is equivalent to neglecting the atoic oentu spread /r, where r is the r extent of the condensate. When ean field effects doinate the spectru, but the atoic oentu is not copletely negligible, the line shape will deviate fro Eq. 3 only for sall detunings, k 0 /r. One can see this fro the overlap integral Eq. B8 by expressing the condensate wave function in ters of the radial Fourier coponents, A r (p)dr e ipr/ (r), to obtain I,l even 4l1 dp A r p dr e ipr/ sin u,l,r k u,l,r 1/ sin j k 0,l,r k 0 k j k 0,l,r. B14 Each oentu coponent will only contribute to the atrix eleent at the point R,l,p where the total phase under the r integral in Eq. B14 is stationary. This leads to a definition of R,l,p for each oentu, p/k u (,l,r,l,p )k j (k 0,l,R,l,p ). When k 0 / r, p/ is negligible and this yields the sae relation as found by neglecting the curvature of the BEC wave function Eq. B9. This iplies S(h k 0 /r) is unaffected by the atoic oentu. When k 0 /r, the oentu spread in the condensate alters I,l even. Thus S(h k 0 /r) will show soe Dopplerbroadening because of finite atoic oentu. This effect is negligible for the hydrogen condensate because the cold collision frequency shift (1 MHz is uch greater than the Doppler width resulting fro a 5--long condensate wave function (k 0 /z100 Hz. 1 D. G. Fried, T. C. Killian, L. Willann, D. Landhuis, S. Moss, D. Kleppner, and T. J. Greytak, Phys. Rev. Lett. 81, T. C. Killian, D. G. Fried, L. Willann, D. Landhuis, S. Moss, D. Kleppner, and T. J. Greytak, Phys. Rev. Lett. 81, P. S. Julienne and F. H. Mies, J. Opt. Soc. A. B 6, J. M. V. A. Koelan, S. B. Crapton, H. T. C. Stoof, O. J. Luiten, and B. J. Verhaar, Phys. Rev. A 38, E. Tiesinga, B. J. Verhaar, H. T. C. Stoof, and D. van Bragt, Phys. Rev. A 45, R K. Gibble and S. Chu, Phys. Rev. Lett. 70, S. Ghezali, Ph. Laurent, S.. Lea, and A. Clairon, Europhys. Lett. 36, S. J. J. M. F. Kokkelans, B. J. Verhaar, K. Gibble, and D. J. Heinzen, Phys. Rev. A 56, R M. Ö. Oktel and L. S. Levitov, Phys. Rev. Lett. 83, M. Ö. Oktel, T. C. Killian, D. Kleppner, and L. Levitov unpublished. 11 C. L. Cesar, D. G. Fried, T. C. Killian, A. D. Polcyn, J. C. Sandberg, I. A. Yu, T. J. Greytak, D. Kleppner, and J. M. Doyle, Phys. Rev. Lett. 77, M. J. Jaieson, A. Dalgarno, and M. Kiura, Phys. Rev. A 51, M. J. Jaieson, A. Dalgarno, and J. M. Doyle, Mol. Phys. 87, R. G. Beausoleil and T. W. Hänsch, Phys. Rev. A 33, F. Bassani, J. J. Forney, and A. Quattropani, Phys. Rev. Lett. 39, K. Huang, Statistical Mechanics Wiley, ew York, 1987, Chap F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, V. L. Ginzburg and L. P. Pitaevskii, Zh. Éksp. Teor. Fiz. 34, Sov. Phys. JETP 7, E. P. Gross, J. Math. Phys. 4, G. Bay and C. J. Pethick, Phys. Rev. Lett. 76, B. D. Esry, C. H. Greene, J. P. Burke, Jr., and J. L. Bohn, Phys. Rev. Lett. 78, M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, Phys. Rev. Lett. 78, M. R. Mathews, D. S. Hall, D. S. Jin, J. R. Ensher, C. E. Wiean, E. A. Cornell, F. Dalfovo, C. Minniti, and S. Stringari, Phys. Rev. Lett. 81, D. S. Hall, M. R. Mathews, J. R. Ensher, C. E. Wiean, and E. A. Cornell, Phys. Rev. Lett. 81, A. Jablonski, Phys. Rev. 68, Allard and J. Kielkopf, Rev. Mod. Phys. 54, P. S. Julienne, J. Res. atl. Inst. Stand. Technol. 101, A. Csordás, R. Graha, and P. Szépfalusy, Phys. Rev. A 57, L. Willann, D. Landhuis, S. Moss, T. C. Killian, D. G. Fried, T. J. Greytak, and D. Kleppner unpublished. 30 W. Ketterle and H.-J. Miesner, Phys. Rev. A 56, I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 8, M. Kozua, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helerson, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. 8,

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