Elimination of Feshbach loss in a Bose Einstein condensate

Transcription

1 Optcs Communcatons 34 (004) Elmnaton of Fesbac loss n a Bose Ensten condensate D.D. Yavuz * Edward L. Gnzton Laboratory, Stanford Unversty, Stanford, CA 94305, USA Receved 3 January 003; receved n revsed form 4 July 003; accepted 0 February 004 Abstract We suggest a tecnque to elmnate nelastc losses n an atomc condensate wen tuned close to a Fesbac resonance. Te key dea s to couple te quas-bound molecular state to a bound molecular state wt an electromagnetc feld. Suc couplng forces te populaton of te Fesbac state to zero, tereby elmnatng all of te losses assocated wt ts state. Ó 004 Elsever B.V. All rgts reserved. PACS: )s; 3.80.Qk; 3.80.Pj In recent experments on Bose Ensten condensates large loss rates were observed wen a quas-bound molecular state was tuned slowly close to a Fesbac resonance wt an atomc condensate [,]. Altoug te precse mecansm for ts loss s n understood, t s lkely tat te loss s assocated wt te populaton of te quasbound molecular state. As suggested by Yurovsky et al. [3], one mecansm wc causes loss s tree body recombnaton. In ts work we extend te suggeston of Harrs [4] to coupled, zero-temperature, atomc and molecular condensates, and demonstrate a tecnque to elmnate Fesbac loss. Te key dea s to * Present address: UW at Madson Pyscs Department, 50 Unversty Avenue Sterlng Hall, Madson, WI 53705, USA. Tel.: ; fax: E-mal address: dyavuz@stanfordalumn.org (D.D. Yavuz). couple te quas-bound molecular state to a bound molecular state wt an electromagnetc feld. Due to destructve nterference, te quas-bound molecular state accumulates no populaton and all of te losses assocated wt ts state are elmnated. Te nature of ts nterference as strong smlartes wt electromagnetcally nduced transparency (EIT) for lgt felds, were an ncdent optcal beam tuned to wat was prevously lne center as near zero absorpton [5]. A scematc of te system to be studed s sown n Fg.. Te energy of te atomc condensate s magnetcally tuned close to a Fesbac resonance wt te quas-bound datomc molecules. We assume tat te quas-bound molecules decay wt a rate C. Te quantty X n Fg. s te Rab frequency of te electromagnetc feld tat couples te quas-bound and bound molecular states. In symmetrc molecules ts transton s dpole forbdden and X s te effectve two-pon /$ - see front matter Ó 004 Elsever B.V. All rgts reserved. do:0.06/j.optcom

2 54 D.D. Yavuz / Optcs Communcatons 34 (004) Γ Ω ω a between coupled atomc and molecular condensates ave been dscussed by Henzen et al. [3] and Holland and Kokkelmans [4]. Coerent oscllatons between atomc and molecular condensates ave been observed by te Weman group [5]. Te formaton of a molecular BEC as been recently demonstrated [6,7]. We proceed by usng te mean feld approac [8 0]. Followng Fg., te coupled, Gross-Ptaevsk equatons for te atomc condensate feld, w a, te molecular condensate feld for te quasbound state, w, and te molecular condensate feld for te bound state, w, are: Fg.. Energy-level scematc for te studed system. Te atomc condensate s magnetcally tuned close to a Fesbac resonance wt te quas-bound datomc molecules. Te couplng feld wt te Rab frequency, X, couples te bound and quas-bound molecular states. Rab frequency. Wtout te couplng feld, te atoms form quas-bound molecules and, due to te large decay rate of tese molecules, te atomc condensate experences loss. Wt te couplng feld, tere s destructve nterference n two quantum pats tat form te quas-bound molecules. In analogy wt EIT for lgt felds, perfect nterference s obtaned only for te deal case of C ¼ 0. Te atomc condensate experences no loss on te exact Fesbac resonance and te bandwdt of te transparency s determned by te ntensty of te couplng feld. Pertnent pror work ncludes general formalsms tat descrbe laser-asssted, electron atom scatterng [6] and cold-atom collsons n te presence of one or more resonant poassocaton lasers [7,8]. Van Abeleen and Veraar [9] ave suggested a possble loss mecansm wen a Fesbac resonance s crossed rapdly wt g ramp speed of te magnetc feld. Te scematc of Fg. as been suggested by several autors as an effcent way to convert an atomc condensate to a molecular condensate [0,]. Drummond et al. [] as sown ow stmulated Raman adabatc passage (STIRAP) can be used to form a molecular condensate by Raman-couplng te free atomc state to a bound molecular state. Dynamcs ow a ow ow ¼ V a w a þ X g a jw j!w a þ a w w a ; þ C w ¼ V w þ X g jw j!w þ aw a þ Xw ; þ C w ¼ V w þ X ðaþ ðbþ g jw j!w þ Xw : ðcþ Here, to smplfy wat follows, we assume all mean-feld ampltudes to be spatally unform. Te quanttes C and C are te decay rates, V are te unform pentals (ncludng te magnetc feld contrbutons and resonance detunngs for zero magnetc feld) seen by eac spece, X s te Rab frequency of te couplng feld tat couples molecular states, and te couplng coeffcent a descrbes te process tat converts atoms to quasbound molecules (Fesbac process). g j are te self- and cross-nteracton constants between te t and jt spece and are related to background scatterng lengt troug te relaton g j ¼ 4p a j =m. Tese equatons ave been used by many autors to descrbe coerent atomc and molecular condensate dynamcs [3,0,]. Workng n nteracton pcture, we expand mean-feld quanttes as: w a ðtþ ¼/ a ðtþe xat ; w ðtþ ¼/ ðtþe Dxt e xt ; and w ðtþ ¼/ ðtþe Dxt e xt. Te energes of te states are defned as: x a ¼ V a þ P g aj/ ð0þj,

3 D.D. Yavuz / Optcs Communcatons 34 (004) x ¼ V þ P g j/ ð0þj, and x ¼ V þ P g j/ ð0þj ; Dx ¼ðx a x Þ s te detunng from te Fesbac resonance. Wt tese defntons, te coupled equatons are: o/ a ¼ a / / a þ X g a ðj/ ðtþj j/ ð0þj Þ / a ; ðaþ Dx þ C / þ o/ ¼ a/ a þ X / þ X g ðj/ ðtþj j/ ð0þj Þ / ; ðbþ Dx þ C / þ o/ ¼ X / þ X g ðj/ ðtþj j/ ð0þj Þ / : ðcþ In te above equatons, we ave taken te couplng feld to be on exact resonance wt te molecular states and used te ratng-wave approxmaton. For smplcty, we ave taken C to be a constant, and neglected te densty dependent decay of te atomc and molecular condensates due to atom molecule and molecule molecule collsons. Te ncluson of tese decay terms do n cange our results sgnfcantly. From Eqs. (), wt C ¼ C ¼ 0, we obtan te conservaton condton for te tal atom number: o ðj/ aj þ j/ j þ j/ j Þ¼0. We proceed by numercally solvng Eqs. () wt te parameters of an MIT experment [,3]. We consder te 907 G Fesbac resonance n a Na atomc condensate wt j/ a ð0þj ¼ / cm 3, a aa ¼ 3:4 nm, and a ¼ 4: 0 5 (mks). We take C ¼ 0:0 Hz [], consder an on-resonance exctaton (Dx ¼ 0), and take C =p ¼ 0:3 MHz. Te coce of C s consstent wtn an order of magntude to te estmated rates of van Abeleen et al. [9], Yurovsky et al. [3] and Tmmermans et al. [9]. We also assume / ð0þ ¼/ ð0þ ¼0. Altoug te atom-molecule and molecule molecule nteracton constants are n known, our results are nsenstve to ter exact values. In Fg., te normalzed densty of te atomc condensate, j/ a j, s plted vs. tme for X ¼ 0andX=p¼ MHz, respectvely. Wtout te couplng feld, te condensate decays very rapdly; te loss s due to te formaton of te quas-bound molecules wtn te condensate, followed by te nelastc decay of tese molecules. Wt te couplng feld on, due to destructve nterference, te loss to te atomc condensate s elmnated. Te ntal decrease n te densty s due to te producton of bound molecules, wc s essental for creatng nterference. Ts penomenon s n te sprt of te preparaton energy for EIT wt lgt felds []. Te last terms on te rgt-and sde of Eqs. () are a densty-dependent frequency sft of te mean-feld ampltudes. We now proceed wt te analytcal nterpretatons of te numercal results of Fg. and neglect te frequency sfts n Eqs. (). We assume o/ C / and o/ jxj C / n Eqs. (b) and (c) to obtan expressons for te molecular feld ampltudes: / ¼ / ¼ aðdx þ C Þ ðdx þ C ÞðDx þ C Þ jxj / a ; ax ðdx þ C ÞðDx þ C Þ jxj / a : ð3aþ ð3bþ Wt tese expressons, te dfferental equaton for te atomc mean feld ampltude (Eq. (a)) s: Fg.. Te normalzed densty of te atomc condensate, j/ a j, vs. tme for X ¼ 0 and X=p ¼ MHz. Wt te couplng feld on, te decay of te atomc condensate s elmnated.

4 56 D.D. Yavuz / Optcs Communcatons 34 (004) o/ a 4jaj ðdx þ C Þ ¼ ðdx þ C ÞðDx þ C Þ jxj j/ aj / a 4p m aðdxþj/ aj / a ; ð4þ were aðdxþ s te complex scatterng lengt for te atomc condensate (wtout te background contrbuton). Te magnary part of aðdxþ determnes te loss to te condensate and s plted vs. frequency n Fg. 3(a) for te parameters of te numercal smulaton of Fg.. Wt te couplng feld, te loss profle s double-peaked and s zero on te exact Fesbac resonance (prevous maxmum). By coosng te ntensty of te couplng feld, te nterference profle may be eter narrower or wder tan te decay wdt, C. Te real part of aðdxþ s te cange n te atom atom nteracton energy due to te Fesbac nteracton and s plted versus frequency n Fg. 3(b) []. As seen from Fg. 3(a), perfect transparency s only obtaned on exact Fesbac resonance. Due to te nonlnear frequency sfts of Eqs. (), an Fg. 3. (a) Te normalzed magnary part of te scatterng lengt. Wt te couplng feld on, tere s no loss on exact Fesbac resonance. (b) Te real part of te scatterng lengt normalzed to ts background value. atomc condensate wc s ntally tuned to te exact Fesbac resonance wll go n and out of te resonance durng coerent nteractons and te transparency wll be degraded. One way to elmnate tese sfts s to bound te mnmum ntensty of te couplng feld tereby lmtng te number of generated molecules. From Eqs. (3), ts mnmum ntensty bound s: jxj 4jaj j/ a j : ð5þ Ts restrcton mgt be overcome by trackng te resonance usng a tme varyng magnetc feld or a frequency-crped couplng laser. As ned above, trougout ts paper, for smplcty, we ave taken C to be constant and neglected densty dependent decay of te atomc and molecular condensates due to atom molecule collsons. Wt te ncluson of tese decay terms, Eqs. () read: o/ a þ cj/ j / a ¼ a / / a þ X g a ðj/ ðtþj j/ ð0þj Þ / a ; Dx þ C þ cj/ aj / þ o/ ¼ a/ a þ X / þ X g ðj/ ðtþj j/ ð0þj Þ / ; Dx þ C / þ o/ ¼ X / þ X g ðj/ ðtþj j/ ð0þj Þ / ; ð6þ were c s te atom molecule state cangng collson rate [3,9]. In Eq. (6), C represents any er unknown loss mecansm of te fragle quas-bound state. Numercally solvng Eq. (6), wt c cm 3 /s [3,9], and wt all te er parameters dentcal, we fnd tat te results of Fg. practcally reman uncanged. In our formalsm, for smplcty, we ave n ncluded an upper electronc molecular state, wc s necessary for two-pon couplng te quas-bound and bound molecular states n symmetrc molecules. For te numercal smulatons of

5 D.D. Yavuz / Optcs Communcatons 34 (004) Fg., te ncluson of suc a state doesnõt cange te results sgnfcantly as long as te detunng from ts state s larger tan 0 tmes te decay wdt of ts state. Strong Raman couplng for tese condtons requre laser power denstes of about 0 kw/cm. Tunng closer to te upper electronc state reduces te laser ntensty requrement, owever degrades transparency. Te amount of degradaton can be found by adabatcally elmnatng te mean-feld ampltude of te upper electronc state and substtutng te relevant quanttes n Eq. (4) wt ter effectve values. Tese substtutons are: X! X X =ðdx þ C 3Þ, C! C Imag½jX j =ðdx þ C 3 ÞŠ, and C! C Imag½jX j =ðdx þ C 3 ÞŠ. Here X and X are te Rab frequences of te two drvng lasers, dx s te detunng from upper state, and C 3 s te decay wdt of ts state. Several recent papers dscuss te mportance of te noncondensate modes n te coupled condensate dynamcs [4,3 5]. We beleve tat te ncluson of tese modes wll n cange te results of ts letter sgnfcantly snce te quasbound state does n accumulate sgnfcant populaton. In summary, we ave suggested a tecnque to elmnate losses to an atomc condensate near a Fesbac resonance. Te predctons of ts letter can be realzed wtn current expermental condtons. By couplng te quas-bound state, te double-peaked nterference profle of Fg. 3(a) sould be readly observable. Te atomc condensate experences dsperson of te scatterng lengt at te pont of zero loss. Suc a unque dspersve feature may fnd possble applcatons n nonlnear matter wave nteractons. Acknowledgements I would lke to tank Steve Harrs for suggestng te problem and for many frutful dscussons and Zacary Dutton for an early suggeston. Ts work was supported by te OSD Multdscplnary Unversty Researc Intatve Program (MURI), te US Army Researc Offce, te US Offce of Naval Researc, and te US Ar Force Offce of Scentfc Researc. References [] S. Inouye, M.R. Andrews, J. Stenger, H.-J. Mesner, D.M. Stamper-Kurn, W. Ketterle, Nature (London) 39 (998) 5. [] J. Stenger, S. Inouye, M.R. Andrews, H.-J. Mesner, D.M. Stamper-Kurn, W. Ketterle, Pys. Rev. Lett. 8 (999) 4. [3] V.A. Yurovsky, A. Ben-Reuven, P.S. Julenne, C.J. Wllams, Pys. Rev. A. 60 (999) R765. [4] S.E. Harrs, Pys. Rev. A 66 (00) [5] K.-J. Boller, A. Imamoglu, S.E. Harrs, Pys. Rev. Lett. 66 (99) 593; M.O. Scully, M.S. Zubary, Quantum Optcs, Cambrdge Unversty Press, Cambrdge, England, 997. [6] N.J. Kylstra, C.J. Joacan, Europys. Lett. 36 (996) 657; N.J. Kylstra, C.J. Joacan, Pys. Rev. A 57 (998) 4. [7] P.O. Fedcev, Y. Kagan, G.V. Slyapnkov, T.M. Walraven, Pys. Rev. Lett. 77 (996) 93. [8] J.L. Bon, P.S. Julenne, Pys. Rev. A 56 (997) 486; J.L. Bon, P.S. Julenne, Pys. Rev. A 60 (999) 44. [9] F.A. van Abeleen, B.J. Veraar, Pys. Rev. Lett. 83 (999) 550. [0] M. Macke, R. Kowalsk, J. Javananen, Pys. Rev. Lett. 84 (000) 3803; M. Macke, Pys. Rev. A 66 (00) [] S.J.J.M.F. Kokkelmans, H.M.J. Vssers, B.J. Veraar, Pys. Rev. A 63 (00) [] P.D. Drummond, K.V. Keruntsyan, D.J. Henzen, R.H. Wynar, Pys. Rev. A 65 (00) [3] D.J. Henzen, R. Wynar, P.D. Drummond, K.V. Keruntsyan, Pys. Rev. Lett. 84 (000) 509. [4] S.J.J.M.F. Kokkelmans, M.J. Holland, Pys. Rev. Lett 89 (00) [5] E.A. Donley, N.R. Claussen, S.T. Tompson, C.E. Weman, Nature (London) 47 (00) 59. [6] S. Jocm, M. Bartensten, A. Altmeyer, G. Hendl, S. Redl, C. Cn, J.H. Densclag, R. Grmm, Scence 30 (003) 0. [7] M. Grener, C.A. Regal, D.S. Jn, Nature 46 (003) 537. [8] F. Dalfovo, S. Gorgn, L.P. Ptaevsk, S. Strngar, Rev. Mod. Pys. 7 (999) 463. [9] E. Tmmermans, P. Tommasn, M. Hussen, A. Kerman, Pys. Rep. 35 (999) 99. [0] E. Tmmermans, P. Tommasn, R. Ce, M. Hussen, A. Kerman, e-prnt cond-mat/ [] S.E. Harrs, Z.F. Luo, Pys. Rev. A 5 (995) 99. [] Te expresson for te scatterng lengt s dentcal to tat obtaned n Ref. [4]. [3] M. Macke, K. Suomnen, J. Javananen, Pys. Rev. Lett. 89 (00) [4] V.A. Yurovsky, A. Ben-Reuven, Pys. Rev. A 67 (003) [5] T. Koler, T. Gasenzer, K. Burnett, Pys. Rev. A 67 (003) 0360.