The Role of Interspecies Interactions in the Preparation of a Low-entropy Gas of Polar Molecules in a Lattice

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1 The Role of nterspeces nteractons n the Preparaton of a Low-entropy Gas of Polar Molecules n a Lattce A. Safav-Nan, M. L. Wall, and A. M. Rey JLA, NST and Department of Physcs, Unversty of Colorado, 44 UCB, Boulder, Colorado 839, USA The preparaton of a quantum degenerate gas of heteronuclear molecules has been an outstandng challenge. We use path ntegral Quantum Monte Carlo smulatons to understand the role of nteractons and fnte temperature effects n the protocol currently employed to adabatcally prepare a low-entropy gas of polar molecules n a lattce startng from an ultracold Bose-Ferm mxture. We fnd that nterspeces nteractons affect the fnal temperature of the mxture after the adabatc loadng procedure and detrmentally lmt the molecular peak fllng. Our conclusons are n agreement wth recent expermental measurements [] and therefore are of mmedate relevance for the myrad experments that am to form molecules from dual-speces atomc gases. arxv: v [quant-ph] 4 Jul 5 ntroducton. Polar molecules, nteractng va longrange and ansotropc dpolar nteractons, hold great promse as quantum smulators hostng exotc quantum phases [, 3], as well as a dverse range of phenomena, rangng from quantum magnetsm [4], to many-body localzaton [5], to synthetc spn-orbt couplng [6, 7]. A necessary ngredent for smulatng these these behavors s to the ablty to reach low entropy condtons. However, despte the rapd expermental progress, a relable method to form such a state has remaned out of reach [8 ]. The two man obstacles to realzng a low-entropy state are the napplcablty of standard atomc coolng technques to polar molecules and the requrement to suppress chemcal reactons. A proposed soluton to these problems s to form the molecules drectly n a deep optcal lattce through assocaton after optmally loadng a degenerate atomc gas mxture [ ]. Ths protocol s shown schematcally n Fg.. However, n order for ths scheme to be a relable pathway, t s necessary to have a hgh atom/molecule converson effcency by creatng a large regon where the denstes of the two atomc speces overlap and correspond to exactly one atom of each speces per lattce ste. Temperature, nteractons, and loadng condtons can sgnfcantly lmt the achevement of ths requrement. n ths Letter, we use path ntegral Monte Carlo smulatons (QMC) based on the worm algorthm [], as well as ts two-worm extenson, to nvestgate the effects of fnte temperature and nterspeces nteractons durng adabatc loadng from a dpole trap nto an optcal lattce. We show that the fnal temperature of the lattce system followng adabatc loadng depends strongly on the strength and sgn of nterspeces nteractons, and can be far from the deal, zero-temperature regme. n contrast to prevous studes, whch do not treat the adabatc loadng procedure and fnd that attractve nterspeces nteractons enhance the on-ste denstes [, 3], our analyss predcts that both attractve and repulsve nteractons can lead to substantal depleton of on-ste denstes of the bosonc speces and takes nto account the contrbutons of dmensonalty, effectve mass mbalance, Adabatc loadng STRAP FG.. (Color onlne) Producton of ground state molecules from a dual-speces mxture. (Top lne) A fnte-temperature dual-speces gas of bosonc speces A atoms (red) and fermonc speces B atoms (blue), s adabatcally loaded from a harmonc trap nto an optcal lattce n the presence of nterspeces nteractons. The ntal temperature and nteractons determne the temperature, densty overlap, and peak fllng of the mxture. (Bottom lne) Followng STRAP, only those stes wth exactly one atom of each speces are converted to a ground state molecule, wth some probablty to excte the molecule to a hgher lattce band (schematc wavefunctons n green box). and quantum statstcs [4 6]. Addtonally, we dscuss the mpact of mperfectons n stmulated Raman Adabatc Passage (STRAP), whch s used to convert Feshbach molecules to ground state molecules, on the ground state molecule producton [, 7]. We study the adabatcty of ths protocol, and provde approxmate analytcal formulas whch can be used to determne the probablty of promoton of molecules to hgher bands durng the STRAP procedure. Snce molecules n hgher bands have a much larger tunnelng rate, an apprecable hgher band populaton can greatly mpact dynamcs nvolvng molecular moton. We fnd that for expermentally relevant parameters there s no sgnfcant populaton transfer, provded the Lamb-Dcke crteron s satsfed. Our conclusons are corroborated by the recent expermental observatons reported n [] and thus of fundamental mportance for ongong expermental efforts to acheve a hgh-fllng lattce gas of ground state polar molecules. Sngle-component gases n the lattce We begn by

2 a,a Here, (a,a ) s the bosonc creaton (annhlaton) operator for speces A and n,a = a,a a,a. The frst and second terms n the Hamltonan Eq. () are the tunnelng JA and the on-ste nteracton, assumed repulsve wthpstrength U >, respectvely. Fnally, µ,a = µ ξ=x,y,z wξ ξ, where wξ and ξ are the strength of harmonc confnement and the coordnate of ste along axs ξ, respectvely, and µ s the chemcal potental. n order to facltate comparsons wth the expermental data n [], we wll use 87 Rb for speces A, but we stress that our conclusons apply for generc bosonc speces descrbed by the Bose-Hubbard model. The effectve external harmonc confnement ncludes the addtonal confnement due to the curvature of the lattce beams. The A atoms are assumed to experence a lattce potental V = EA, where EA s the recol energy of the A atoms. To characterze the densty as a functon of atom number, we frst match the entropy of the gas n the dpole trap to the one n the lattce and fnd the fnal temperature Tfn /JA. The bottom rght nset of Fg. shows an example of ths procedure for 3 atoms. The sold green lne shows the entropy of a weakly nteractng Bose gas, whch descrbes the A atoms n the harmonc trap. Blue trangles denote the entropy of the A atoms after loadng nto the lattce, gven by Z T E(T ) E() E(T ) E() Sf (T ) = + dt, () T T where E(T ) s the system energy at temperature T. Both E(T ) and E() are drectly measured n our QMC smulatons. An example of the entropy matchng procedure for an ntal temperature T =.4Tc, Tc the crtcal temperature for Bose-Ensten condensaton, s shown wth dashed arrows. Next we determne the peak fllng, fp. To compare wth the expermental procedure of extractng the peak fllng, we frst ntegrate along z and mmc the expermental magng resoluton by applyng a Gaussan flter wth a wdth of 4 stes to the ntegrated densty. We ft the resultng densty to a Thomas-Ferm (TF) dstr 3/. The buton n(x, y) = 43 fp σz (x/σx ) (y/σy ) extracted fp values are plotted n Fg. as a functon of 3 T /JA = NRb 4 8 T /JA = 5.5 T /Tc S(T ) 3.5 Peak Fllng (fp ) studyng trapped sngle-speces gases of bosons or fermons. We assume the bosons are trapped n a deep lattce where they can be descrbed by the Bose-Hubbard Hamltonan wth an addtonal external harmonc confnement: X UX n,a (n,a ) H= JA (a,a aj,a + h.c.) + h j X µ,a n,a. () Double Occupancy T /JA = T /J.5 T /JA = NA FG.. (Color onlne) Peak fllng of speces A vs. atom number. (Man panel) The peak fllng fp as a functon of number of 87 Rb atoms, used as an example of bosonc speces A, wth a lattce depth of V = EA gven for harmonc confnements of ωr (ωz ) = π 4(6) Hz (flled trangles) and π 35(7) Hz (flled crcles). Empty damonds are expermental data []. The dashed (sold) lne shows the onste densty at the center of the trap at T /JA = 5 for the weaker (stronger) harmonc confnement. (Lower nset) Entropy matchng procedure for 87 Rb followng adabatc lattce loadng wth strong harmonc confnement. Sold green lne s weakly nteractng gas and blue trangles are 3 lattceconfned Rb. (Upper nset) Double occupancy at T /JA = (flled upsde-down trangles) and T /JA = 5 (crosses) for the strong harmonc confnement. (Rght panels) ntegrated densty n(x, y) at ponts marked and at T /JA = (upper) and T /JA = 5 (lower) for strong harmonc confnement. atom number, NA, after the adabatc loadng procedure, wth flled crcles (trangles) for two dfferent trappng condtons. The fllng at the center of the trap s dsplayed wth dashed (sold) lnes for the weaker (stronger) harmonc confnement at T /JA = 5. The empty damonds are the expermental results []. The rghtmost panels of Fg. gve examples of the resultng dstrbutons ntegrated along the z-axs, n(x, y), for NA 4 (labeled ) and NA 8 (labeled ). For these two values of NA we show the dstrbuton at T /JA = (top), whch s the low-temperature result, followed by the ntegrated densty dstrbuton at T /JA = 5 (bottom), whch s closer to the expermental temperature. t s worth notng that whle the densty at the center of the trap s strongly dependent on temperature, fp does not dsplay a strong dependence on T /JA wthn the expermentally-relevant temperature range. However, ths does not mean that the effcency of the for-

3 3 maton of molecules s unaffected by temperature. Ths s evdent f one probes the number of doubly occuped stes wth ncreasng temperature. Stes wth double (or hgher) occupances do not result n molecule formaton. The upper nset of Fg. shows the double occupancy at T/J A = (nverted trangles) and T/J A = 5 (crosses). n the low-temperature regme (T/J A = ) the frst Mott shell extends to N A 5 partcles, after whch a superflud regon forms at the center of the trap, before transtonng to the second Mott shell at N A 8 (see the dashed blue lne n Fg ). However close to or above T/J A = 5, 5 % of the A atoms are n doubly occuped stes and hence wll not partcpate n molecule formaton. Next, we consder the fermonc speces B of the mxture, takng 4 K for comparson wth recent expermental data []. However, as before, our conclusons are vald for generc fermonc speces that can be descrbed by a tghtbndng model. The dfference n the polarzablty wth respect to 87 Rb means that the 4 K atoms feel a lattce potental of depth V = 9E B, where E B s the B speces recol energy. At low temperatures, p-wave nteractons n the spn-polarzed gas can be neglected, and the determnaton of the densty reduces to a non-nteractng problem. As the harmonc trap s separable, ths problem can be straghtforwardly treated by drect dagonalzaton for the sngle-partcle egenenerges E nξ and correspondng wavefunctons ψ nξ. Wth these sngle-partcle quanttes we can evaluate the grand canoncal partton functon, from whch we fnd the entropy per partcle S/N, as well as the on-ste densty n( r) = n(x)n(y)n(z), n(ξ) = n ξ +exp(β(e nξ µ)) ψ n ξ (ξ), (3) where ξ = x, y, z and β = /T s the nverse temperature. n Fg. 3 we show the peak fllng of speces B as a functon of atom number. For fermons we use a Gaussan ft wth n(x, y) = πσ z f p exp [ x /(σx) y /(σy) ] after ntegraton along z. Here, we also account for the expermental magng resoluton and pxelaton by applyng a Gaussan flter. The band, delmted by the blue trangles and orange squares, shows the range of peak fllngs for < T fn /J B < 5, wth J B the tunnelng of the B atoms. As the temperature of the B atoms n the lattce ncreases, so does the wdth of the cloud, resultng n a decrease n the peak fllng. The empty damonds show the expermental data []. The plateau ndcates the atoms formng an ncompressble band nsulator. On the rght panel we show the ntegrated densty dstrbuton at ponts marked and. At, the B atoms form a band nsulator n a large regon of the lattce. Lattce-confned two-speces mxture We use a twocomponent mxture of soft-core ( A ) and hard-core ( B ) bosons to study the combned effect of nterspeces nteractons and fnte temperature on the densty of Peak Fllng (fp) N B FG. 3. (Color onlne) Peak fllng of the fermonc speces B vs. atom number. (Man panel) The peak fllng of 4 K, used as an example for the fermonc speces B, at T fn /J B = (flled squares) and T fn /J B = 5 (flled trangles) extracted from the ntegrated densty dstrbutons n(x, y), ncludng mmcked resoluton and pxelaton effects. Results are for harmonc trap frequences (4 4 6) Hz and a 9E B deep lattce. Empty damonds show expermental results []. nsets show the ntegrated densty dstrbutons at ponts and. the A speces. Hard-core bosons act as a stand-n for fermonc B speces, as path-ntegral QMC cannot smulate fermons due to the sgn problem. Whle there s no drect mappng from hard-core bosons to fermons for local observables n three dmensons, we compared the hard-core and fermonc profles for the sngle-speces case and found excellent agreement. Based on ths, we expect our results for the local densty n the mxture to also be vald. The two-component mxture s descrbed by the Hamltonan H = J γ (a,γ a j,γ + h.c.) + U n,a (n,a ) j,γ + U AB n,a n,b µ γ, n,γ. (4),γ Here γ =A, B and U AB s the nterspeces on-ste nteracton whch can be tuned to be repulsve or attractve. To study the effect of nterspeces nteractons we have performed smulatons wth 4 < U AB /J A < 4 wth 5 A atoms and 5 B atoms. Prevous studes have shown that at a gven T/J A, the presence of attractve nteractons enhances the converson effcency [3]. Ths corresponds to an enhancement of f p measured at the same T/J A for every U AB <. n Fg. 4 we show f p (T/J A =.) and f p (T/J A = ) as a functon of U AB /J A usng flled squares and flled crcles connected by a lne, respectvely. We normalze the values to f p,

4 4 a) b) fp(t )/f p U AB /J A S/kB c) Tf /J T/TF T/J A T/TF FG. 4. (Color onlne) Dependence of speces A peak fllng on U AB. (a) Peak fllng normalzed to zero-temperature value, f p(t )/fp, vs. U AB/J A at T/J A=. (empty squares) and T/J A= (empty crcles), gnorng the nteracton dependence of the fnal temperature due to adabatc loadng. Accountng for nteracton effects durng loadng at ntal temperatures T /T F =. (.3) are gven wth flled squares (trangles). (b) The entropy of speces A as a functon of T/J A s shown for U AB/J A = 4 and -4 usng crcles and squares, respectvely. The dashed lne s the entropy of nonnteractng fermons n a harmonc trap, whch we use as an estmate for S. (c) Fnal temperature vs. ntal temperature for U AB/J A = 4, -,, and 4 (sold, dash, dot-dash, and dot-dot-dash, respectvely). We use a harmonc confnement of ω r(ω z) = π3(5) Hz the peak fllng of A at T/J A =. n the absence of speces B. t s clear that attractve nteractons lead to an enhancement of f p f T/J A s kept constant. However, ths smple analyss does not descrbe current experments, snce startng at the same ntal temperature, T /T F, T F the Ferm temperature, leads to a fnal temperature T fn /J A after adabatc loadng that depends on the sgn and magntude of U AB. n Fg. 4 (c) we show T fn /J A for U AB /J A = 4, -,, and 4 usng sold, dash, dot-dash, and dot-dot-dash lnes, respectvely. From Fg. 4(c) t s clear that attractve nteractons tend to cause more severe heatng durng the adabatc loadng. For the range of ntal entropes consdered n ths work, ths addtonal heatng nhbts and counteracts the beneft ganed by makng the two speces attract. n Fg. 4 (a) we show f p as a functon of U AB /J A for T /T F =. and.3 usng flled squares and flled trangles, respectvely. Followng adabatc loadng, both attractve and repulsve nteractons destablze the A speces Mott nsulator and reduce the peak fllng. STRAP adabatcty and hgher-band transfer. Pars of A and B atoms n an optcal lattce ste are transferred to weakly bound Feshbach molecules followng a magneto []- or photo [8]-assocaton step. These Feshbach molecules are converted nto ground state molecules va a two-photon STRAP sequence nvolvng an ntermedate electroncally excted molecular state. As the STRAP process must remove THz of molecular energy, the dfference of the wavevectors of the two photons nvolved n the STRAP sequence, denoted by k u and k d, can have sgnfcant varaton on the lattce scale of a few mcrons. The resultng momentum transfer can excte the resultng ground state molecules to hgher lattce bands. Here, we nvestgate the adabatcty of the STRAP procedure, as well as the rate of transfer of molecules to hgher bands. We consder a STRAP process whch couples the Feshbach molecule (FBM) ( f ) and ground molecular (GSM) g states through an excted level ( e ) [7]. We assume that these are the only states nvolved and neglect the populaton of other molecular levels or atomc scatterng states durng the STRAP. To match current experments, we study the case of vanshng sngle- and two-photon detunngs. Usng adabatc perturbaton theory [8 3] we fnd that the momentum transfer followng the process f g has the same form as the case of large sngle-photon detunng, e ( k u k d ) r. Typcal ST- RAP lnewdths are khz, sgnfcantly greater than the khz spacng between bands, and so the band structure s fully unresolved. Thus, the relatve populaton of molecules n the dfferent bands followng STRAP are determned by ratos of Rab frequences n the bass of Wanner states wth band ndex n. We use w n ( r) and w n ( r) to denote the Wanner states for FBM and GSM, respectvely. For FBM n the lowest band, the relatve populaton of GSM n the frst excted band along the drecton ξ can thus be estmated by w e kξξ w w e kξξ w (k ξ a) α π ( + α) V/E R, (5) where we have used a harmonc approxmaton for the Wanner functons, wth k ξ ( k u k d ) ξ the momentum transfer along drecton ξ, a the lattce spacng, α the polarzablty rato of the GSM to the FBM, V the lattce depth for the FBM, and E R the molecular recol energy. For the parameters of the JLA experment [], a = 53 nm, k u = π/(968 nm) and k d = π/(689 nm) co-propagatng at a 45 angle wth respect to the x and y lattce axes, and α.9, we fnd the total populaton n the frst excted bands to be %. n general, our results ndcate that for experments n the Lamb-Dcke regme k ξ a/(v/e R ) /4, STRAP does not nduce apprecable populaton transfer to excted bands. n concluson, we studed the combned effects of nterspeces nteractons, temperature, and adabatc loadng on the successful preparaton of a low entropy gas of polar molecules. We have shown that nterspeces nteractons have a sgnfcant effect on the fnal temperature of the lattce gas followng adabatc loadng, whch n turn can lead to the depleton of the peak fllng of the bosonc speces and a lower effcency of molecule formaton. Based on our results, molecule formaton eff-

5 5 cency s greatest when the lattce loadng s performed wth a non-nteractng mxture. For lower ntal temperatures where the fnal temperature after loadng s T fn /J A 5, attractve nteractons may lead to a hgher rate of molecule formaton; ths temperature s around T/T F.5 for the parameters consdered n ths work. Moreover, we consdered the role of the STRAP procedure used to convert the resultng Feshbach molecules to ground state molecules n the converson effcency. We fnd that the probablty of promotng molecules to hgher bands durng STRAP s not a sgnfcant concern provded the Lamb-Dcke crteron s satsfed. Acknowledgments We would lke thank B. Capogrosso- Sansone for useful dscussons. Ths work was supported by the NSF (PF-94 and PFC- 5844), AFOSR, AFOSR-MUR, NST and ARO ndvdual nvestgator awards. MLW thanks the NRC for support. Part of the computng for ths project was performed at the OU Supercomputng Center for Educaton & Research (OSCER) at the Unversty of Oklahoma (OU). Ths work also utlzed the Janus supercomputer, whch s supported by the Natonal Scence Foundaton (award number CNS-8794) and the Unversty of Colorado Boulder. The Janus supercomputer s a jont effort of the Unversty of Colorado Boulder, the Unversty of Colorado Denver and the Natonal Center for Atmospherc Research. [] Steven A. Moses, Jacob P. Covey, Matthew T. Mecnkowsk, Bo Yan, Bryce Gadway, Jun Ye, and Deborah S. Jn, arxv: [] L. D. Carr, D. Demlle, R. V. Krems, and J. Ye, New J. Phys., 5549 (9). [3] M. A. Baranov, M. Dalmonte, G. Pupllo, and P. Zoller, Chemcal Revews, 5 (). [4] M. L. Wall, K. R. A. Hazzard, and A. M. Rey, arxv: (4). [5] Norman Y. Yao, Chrs R. Laumann, Sarang Gopalakrshnan, Mchael Knap, Markus Muller, Eugene A. Demler, and Mkhal D. Lukn, Phys. Rev. Lett. 3, 43 (4). [6] Sergey V. Syzranov, Mchael L. Wall, Vctor Gurare, and Ana Mara Rey, Nature Communcatons 5, 539 (4). [7] D. Peter, N. Y. Yao, N. Lang, S. D. Huber, M. D. Lukn, and H. P. Büchler, arxv: (4). [8] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jn, and J. Ye, Nature 5, 5 (3). [9] B. Zhu, B. Gadway, M. Foss-Feg, J. Schachenmayer, M. L. Wall, K. R. A. Hazzard, B. Yan, S. A. Moses, J. P. Covey, D. S. Jn, J. Ye, M. Holland, and A. M. Rey, Phys. Rev. Lett., 744 (4). [] K. R. A. Hazzard, B. Gadway, M. Foss-Feg, B. Yan, S. A. Moses, J. P. Covey, N. Y. Yao, M. D. Lukn, J. Ye, D. S. Jn, and A. M. Rey, Phys. Rev. Lett. 3, 953 (4). [] K.-K. N, S. Ospelkaus, M. H. G. de Mranda, A. Pe er, B. Neyenhus, J. J. Zrbel, S. Kotochgova, P. S. Julenne, D. S. Jn, and J. Ye, Scence 3, 3 (8). [] H. Cho, D. McCarron, D. L. Jenkn, M. P. Köppnger, and S. L. Cornsh, European Physcal Journal D (),.4/epjd/e-76-. [3] M. Debatn, T. Takekosh, R. Rameshan, L. Rechsöllner, F. Ferlano, R. Grmm, R. Vexau, N. Bouloufa, O. Duleu, and H.-C. Nagerl, Phys. Chem. Chem. Phys. 3, 896 (). [4] T. Takekosh, M. Debatn, R. Rameshan, F. Ferlano, R. Grmm, H.-C. Nägerl, C. R. Le Sueur, J. M. Hutson, P. S. Julenne, S. Kotochgova, and E. Temann, Phys. Rev. A 85, 356 (). [5] T. Takekosh, L. Rechsöllner, A. Schndewolf, J. M. Hutson, C. R. Le Sueur, O. Duleu, F. Ferlano, R. Grmm, and H.-C. Nägerl, Phys. Rev. Lett. 3, 53 (4). [6] C.-H. Wu, J. W. Park, P. Ahmad, S. Wll, and M. W. Zwerlen, Phys. Rev. Lett. 9, 853 (). [7] M.-S. Heo, T. T. Wang, C. A. Chrstensen, T. M. Rvachov, D. A. Cotta, J.-H. Cho, Y.-R. Lee, and W. Ketterle, Phys. Rev. A 86, 6 (). [8] S. Dutta, J. Lorenz, A. Altaf, D. S. Ellott, and Y. P. Chen, Phys. Rev. A 89, 7 (4). [9] J. W. Park, S. A. Wll, and M. W. Zeerlen, arxv: (5). [] G. Quéméner and P. S. Julenne, Chemcal Revews, 4949 (). [] N. V. Prokof ev, B. V. Svstunov, and. S. Tuptsyn, Phys. Lett. A 38, 53 (998); Sov. Phys. JETP 87, 3 (998). [] B. Damsk, L. Santos, E. Temann, M. Lewensten, S. Kotochgova, P. Julenne, and P. Zoller, Phys. Rev. Lett. 9, 4 (3). [3] J. K. Freercks, M. M. Maśka, Anz Hu, Thomas M. Hanna, C. J. Wllams, P. S. Julenne, and R. Lemańsk, Phys. Rev. A 8, 65(R), (). [4] D. Jaksch, V. Ventur, J.. Crac, C.J. Wllams, and P. Zoller, Phys. Rev. Lett. 89, 44 (). [5] Lode Pollet, Cornna Kollath, Ulrch Schollwoeck, and Matthas Troyer, Phys. Rev. A 77, 368 (8). [6] Lode Pollet, Cornna Kollath, Krs Van Houcke, and Matthas Troyer, New J. Phys. 65 (8). [7] S. Ospelkaus, A. Pe er, K. K. N, J. J. Zrbel, B. Neyenhus, S. Kotochgova, P. S. Julenne, J. Ye, and D. S. Jn, Nature Physcs, 4, 6-66 (8). [8] See the supplementary materal at [xxx] for detals of the adabatc perturbaton theory calculaton. [9] A. Polkovnkov, Phys. Rev. B 7, 6(R) (5). [3] G. Rgoln, G. Ortz, and V. H. Ponce, Phys. Rev. A 78, 558 (8).