Many-body theory of excitation dynamics in an ultracold Rydberg gas

Transcription

1 PHYSICAL REVIEW A 76, Many-body theory of exctaton dynamcs n an ultracold Rydberg gas C. Ates, 1 T. Pohl, 2 T. Pattard, 1, * and J. M. Rost 1 1 Max Planck Insttute for the Physcs of Complex Systems, Nöthntzer Straße 38, D-1187 Dresden, Germany 2 ITAMP, Harvard-Smthsonan Center for Astrophyscs, 6 Garden Street, MS14, Cambrdge, Massachusetts 2138, USA Receved 25 May 27; publshed 2 July 27 We develop a theoretcal approach for the dynamcs of Rydberg exctatons n ultracold gases,wth a realstcally large number of atoms. We rely on the reducton of the sngle-atom Bloch equatons to rate equatons, whch s possble under varous expermentally relevant condtons. Here, we explctly refer to a two-step exctaton scheme. We dscuss the condtons under whch our approach s vald by comparng the results wth the soluton of the exact quantum master equaton for two nteractng atoms. Concernng the emergence of an exctaton blockade n a Rydberg gas, our results are n qualtatve agreement wth experment. Possble sources of quanttatve dscrepancy are carefully examned. Based on the two-step exctaton scheme, we predct the occurrence of an antblockade effect and propose possble ways to detect ths exctaton enhancement expermentally n an optcal lattce, as well as n the gas phase. DOI: 1.113/PhysRevA PACS number s : 32.8.Rm, 32.7.Jz, 34.2.Cf I. INTRODUCTION The possblty to routnely create samples of ultracold gases n the -Kelvn regme has opened a new avenue to the nvestgaton of nteractng many-partcle systems. At such temperatures, the thermal veloctes of the atoms are low enough that the atoms move a neglgble dstance over the duraton of the experment. Hence, thermal collsons are not relevant and t s possble to study quasstatc nteractons between the partcles. For denstes of a dlute ultracold but nondegenerate gas typcal for atoms n magneto-optcal traps, the nteracton between ground state atoms s very weak. Rydberg atoms, on the other hand, can strongly nteract among each other, even n a dlute gas, due to ther large polarzablty, whch scales wth the prncpal quantum number n as n 7. Ths scalng allows accurate control over ther nteractons 1 over a huge range by varyng n. In contrast to amorphous solds, wth whch ultracold Rydberg gases share some smlartes, the atoms are practcally statonary on the tme scale of electronc dynamcs because of ther low thermal knetc energy 2,3. A strkng consequence of the strong Rydberg-Rydberg nteracton s the so-called dpole blockade,.e., a suppresson of Rydberg exctatons due to an nduced dpole couplng of the Rydberg atoms to ther envronment. Ths phenomenon was frst consdered theoretcally n proposals to buld fast quantum logc gates 4, to mprove the resoluton of atomc clocks 5, and to create sngle-atom and snglephoton sources 6. It was expermentally verfed for second-order dpole-dpole or van der Waals couplng between the Rydberg atoms 7,8 by measurng the densty of the Rydberg atoms as a functon of ncreasng laser ntensty, atomc densty, or prncpal quantum number,.e., as a functon of ncreasng nteracton strength. By applyng and varyng an external electrc feld the blockade effect was also *Present address: APS Edtoral Offce, 1 Research Road, Rdge, NY 11961, USA. demonstrated for a drect.e., frst-order dpole-dpole nteracton of the Rydberg atoms, and t was shown that the suppresson of exctatons s partcularly pronounced at the so-called Förster resonances 9. Furthermore, t was shown that the blockade effect also leads to a quenchng of the probablty dstrbuton for the exctaton to the Rydberg state The theoretcal descrpton of ths laser-drven, nteractng many-partcle system s challengng. In 7 a mean feld approach was used and the Bloch equatons for a sngle Rydberg atom n a sphere were solved. Wthn the sphere, embedded n a constant background densty of Rydberg atoms, no further exctatons were allowed. Wth the help of a ft parameter the expermental results of 7 could be reproduced. The system was also nvestgated by solvng the manypartcle Schrödnger equaton numercally 13. There, ntellgent use was made of the fact that the blockade tself reduces the number of atoms that can be excted, whch allows a substantal reducton n the number of states that had to be consdered for the calculatons. Yet, the number of atoms that could be smulated was stll so small that approprate boundary condtons had to be used to establsh contact wth the experments. However, experments usng a two-step threelevel exctaton scheme could not be descrbed snce mportant effects, such as radatve decay, were not ncluded. Here, we focus, n partcular, on the two-step exctaton scheme, used n the experments 8,1, where the ntermedate level decays radatvely. As we wll show, ths leads to a reducton of the descrpton of the Rydberg exctaton dynamcs n a sngle atom to a rate equaton, whch n turn enables us to formulate a quasclasscal approach takng fully nto account all atoms n the exctaton volume and all nteractons of the Rydberg atoms. Expermentally, a gas of atoms s prepared n a magnetooptcal trap MOT wth peak denstes up to 1 11 cm 3 at temperatures of about 1 K. Under these condtons the gas s far from the quantum degenerate regme and can be vewed as a classcal deal gas. Furthermore, the laser ntenstes used n 8,1 are relatvely low, so that coherent couplng of the atoms by the laser feld, e.g., through stmulated /27/76 1 / The Amercan Physcal Socety

2 ATES et al. FIG. 1. Sketch of the two-step exctaton scheme for rubdum. emsson and reabsorpton of photons, s neglgble. However, the nteracton of the ndvdual atoms wth the laser felds has to be treated quantum mechancally. Our approach s based on the observaton that, under the condtons of the experments 8 and 1, the descrpton of the sngle-atom exctaton dynamcs can be reduced substantally to a sngle rate equaton usng an adabatc approxmaton for the coherences. Despte the approxmatons made, the rate equaton accurately descrbes the populaton dynamcs of the Rydberg state, ncludng nontrval effects such as the Autler-Townes splttng of the exctaton lne. Ths smplfcaton n the descrpton of the sngle-atom dynamcs s the key that ultmately allows us to fully account for the correlated many-partcle dynamcs wth a smple Monte Carlo samplng, thereby reducng greatly the complexty of a full quantum treatment. The paper s organzed as follows. In Sec. II we present the approach, whch enables us to descrbe the dynamcs n an ultracold gas of nteractng three-level atoms usng a many-body rate equaton. Startng from the full quantum master equaton, we justfy our approxmatons frst on the sngle-atom level Sec. II A, then for the nteractng system Sec. II B and fnally, descrbe how the Rydberg-Rydberg nteracton s ncluded n our descrpton Sec. II C. For two nteractng atoms, we compare the results of our rate equaton wth the soluton of the quantum master equaton Sec. III. In Sec. IV we compare the results of the smulatons for a realstc number of atoms wth expermental data and comment on the possblty to expermentally observe an nteracton nduced enhancement of Rydberg exctaton antblockade. Secton V summarzes the results. Throughout the paper atomc unts wll be used unless stated otherwse. II. TWO-STEP RYDBERG EXCITATION IN AN ULTRACOLD GAS A. Dynamcs of the nonnteractng system In what follows, we wll dscuss a two-step cw-exctaton scheme for the Rydberg state see Fg. 1, as typcally used n experments. In the frst step, the atom s excted from ts ground state g to an ntermedate level m wth a transton strength gven by the Rab frequency. The photon for ths step s typcally provded by the MOT lasers, whch are tuned on resonance wth the transton g m durng the tme of Rydberg exctaton. In the second step, a separate tunable laser drves the transton between the ntermedate level and the desred Rydberg state e wth Rab frequency, where n ths step we allow for a detunng from resonant exctaton. The decay of the ntermedate level wth rate has to be taken nto account, as ts radatve lfetme s typcally shorter than the pulse duraton. On the other hand, the lfetme of the Rydberg state s much longer so that ts decay can be neglected. The coherent dynamcs of N nonnteractng three-level atoms coupled to the two laser felds s governed n the nteracton pcture by the Hamltonan H, where PHYSICAL REVIEW A 76, N H = H + H AL h = e e, N h + h AL, h AL = 2 m g + g m + 2 e m + m e, 1 2a 2b descrbe the nteracton of the levels of atom wth the laser beams. The tme evoluton of the system ncludng the decay of the ntermedate level s then gven by a quantum master equaton for the N-partcle densty matrx ˆ N ˆ, d dt ˆ = H, ˆ + L ˆ, 3 where the spontaneous decay of level m s ncluded va the Lndblad operator L. In general, the rate of spontaneous decay of an atom s nfluenced by the presence of other atoms through a couplng medated by the radaton feld, whch can account for collectve effects such as superradance. The strength of ths couplng s determned by the dmensonless quantty x j 2 r r j /, whch measures the atom-atom dstance n unts of the wavelength of the g m transton. For x j 1 the spontaneous decay of an atom s strongly affected by ts neghbors, whle for x j 1 the atoms radate ndependently. In typcal experments wth ultracold gases, the mean atomc dstance between atoms s a 5 m. For the 5s 5p transton of Rb ths corresponds to x j 4. Hence, the collectve decay s neglgble and the Lndblad operator can be wrtten as a sum of sngle-atom operators, N L = L ˆ L 1 2 L L ˆ 1 2 ˆ L L, 4 wth L = g m and L = m g. 5 Hence, the dynamcs of the atoms s completely decoupled 1 and the N-atom densty matrx factorzes, ˆ = ˆ 1 ˆ 1 N. The tme evoluton of a nonnteractng gas of three-level atoms s therefore completely determned by the master equaton for the sngle-atom densty matrx ˆ 1 k ˆ,.e., the optcal Bloch equatons OBE for a three-level atom,

3 MANY-BODY THEORY OF EXCITATION DYNAMICS IN AN PHYSICAL REVIEW A 76, gg = 2 gm mg + mm, mm = 2 gm mg + 2 me em mm, 6a 6b = for. 8 Solvng the algebrac equatons arsng from Eqs. 6 and 8 for the populatons, makng use of Eq. 7, and nsertng nto the dfferental equaton for mm and ee, one arrves at mm = q 1 mm + q 2 ee + q 3, 9a ee = 2 me em, gm = 2 mm gg + 2 ge 2 gm, me = me 2 ee mm 2 ge 2 me, ge = ge 2 me + 2 gm, 6c 6d 6e 6f = for. 6g As usual, the level populatons are descrbed by the dagonal elements of the densty matrx, whereas the off-dagonal elements,.e., the coherences, contan the nformaton about the transton ampltudes between the levels. Conservaton of probablty leads to the sum rule =1 for the populatons so that eght ndependent varables reman to be solved for. Ths sngle-atom descrpton s too complex to serve as the bass of a tractable descrpton for the many-partcle system. Fortunately, under the set of relevant expermental parameters, 6 smplfes substantally. In the experments 8,1, the upper transton s much more weakly drven than the lower one due to the dfferent transton dpole moments of the respectve exctatons. Ths defnes two well separated tme scales, such that the Rydberg transton m e s slow compared to the pump transton g m. Thus, the tme evoluton of the system s governed by the slow Rydberg transton n the sense that the coherences of the fast pump transton wll adabatcally follow the slow dynamcs of the Rydberg transton. Furthermore, the decay rate of the ntermedate level s much larger than the Rab frequency of the upper transton mplyng that the populatons wll evolve only slghtly over a tme 1. Hence, dephasng of the atomc transton dpole moments,.e., dampng of the oscllatons of coherences, s fast compared to the dynamcs of the Rydberg populaton. Under these condtons, the coherences can be expressed as a functon of the populatons at each nstant of tme,.e., ther dynamcs can be elmnated adabatcally 14 by settng 7 ee = q 4 mm + q 5 ee + q 6, 9b where the coeffcents q k =q k,,, are some functons of the parameters descrbng the exctaton dynamcs of the three-level system. To smplfy further, we note that wthn the adabatc approxmaton 8 the dynamcs of the populaton dfference mm gg can be neglected for tmes t 1/2. Ths can be verfed by drect ntegraton of mm gg from the OBE, whch shows that the dynamcs of the populaton dfference s proportonal to 1 exp 2 t and thus reaches ts saturaton lmt at a tme scale on the order of 1/2. Usng the sum rule 7 ths leads to the relaton 2 mm + ee =, 1 whch can be used to elmnate the populaton of the ntermedate level occurrng n Eq. 9. Fnally, one arrves at a sngle dfferental equaton for ee, ee = ee +, ee whch can readly be solved to gve ee t = ee 1 exp t, ee where ee = ee,,, denotes the steady-state occupaton of level e and =,,, s the rate for populatng the Rydberg level for short tmes. The expressons for and ee are gven n the Appendx. Here we note only that n the lmt they reduce to = 2 / / 2 2, 1 ee = / 2, 13 whch shows that the resonant exctaton rate s proportonal to / 2. Introducng an effectve ground state eff gg =1 ee, one can wrte Eq. 11 n the form of a rate equaton RE for an effectve two-level atom ee t = eff gg ee, 14 wth deexctaton rate = 1 ee. 15 ee A comparson of the solutons of the OBE 6 and the RE 14 for the Rydberg populatons as a functon of the detunng s shown n Fg. 2 for dfferent pulse lengths. The parameters correspond to those of the experments 1,8. The agreement of the solutons s generally good and becomes even better for longer pulses. For the parameters of

4 ATES et al. ρ ee (a) ρ ee.4.2 (b) FIG. 2. Color onlne Populaton of the Rydberg level for the three-level system of Fg. 1 accordng to the RE 14 sold lnes and OBE 6 dashed lnes for dfferent pulse lengths:.3 s lowest par of curves, 1. s mddle par, and 2. s. The parameters n MHz are,, = 4,.2,6 n a and 22.1,.8,6 n b. experment 1 the convergence of the RE soluton to that of the OBE n the regon around = s slower as a functon of pulse length. Ths s due to, whch ndcates that t s not fully justfed to neglect the nonlnear short-tme populaton dynamcs. The RE reproduces the Autler-Townes splttng of the ntermedate level m manfest n a splttng of the Rydberg lne, proportonal to for short tmes. The splttng s transent, as the steady state wth ts sngle central peak s approached for long tmes when the Rydberg populaton reaches the saturaton lmt. A detaled analyss of the peak structure of the Rydberg populatons n ths system, especally the occurrence of the Autler-Townes splttng and ts mpact on the exctaton blockade, has been gven n 15. For future reference, we wll cast the sngle-atom RE 14 nto a form that wll be used for the smulaton of the nteractng many-partcle system. To ths end, we denote the state of the atom by, where =1 f the atom s n the Rydberg state, and = otherwse. Furthermore, we defne the rate of change for the state,, 1 +, 16 whch descrbes exctaton of the atom f t s n the effectve ground state = and deexctaton f t s n the excted state =1. Usng these defntons, we can combne Eq. 14, whch determnes ee t, and the correspondng equaton for eff gg t n the form of an evoluton equaton for the sngle-atom state dstrbuton functon p, dp = T, p, 17 dt wth p =1 ee, p 1 = ee, and the transton rate matrx PHYSICAL REVIEW A 76, T, =,, +,1 1,. 18 The frst term of Eq. 18 descrbes the transton 1, through whch the system can leave the state, whle the opposte process 1, whch brngs the system nto the state, s descrbed by the second term. Proceedng to the case of N nonnteractng atoms, we defne the many-partcle state as the confguraton contanng all sngle-atom states,.e., 1,...,,... N and as the many-body confguraton whch s dentcal to except for the state of atom,.e., 1,...,1,..., N. If we fnally use the notaton,, and, 1, 1 N, N, the matrx of the transton rates generalzes to T, =,, +,1,, 19 and the evoluton equaton for the many-body state dstrbuton functon P can be wrtten n a closed form as dp = T, P. 2 dt For nonnteractng partcles the rate depends besdes on the laser detunng only on the state of partcle,.e., on. However, ths s no longer true n the nteractng case and wll depend on the entre many-body confguraton. B. Correlated many-partcle dynamcs In order to study the correlated dynamcs of the nteractng many-partcle system, we have to add the Hamltonan descrbng the Rydberg-Rydberg nteracton H RR = 1 U j e,e j e,e j 21 2,j j to H cf. Eq. 1, where U j s the nteracton energy of a par of Rydberg atoms at a dstance r j r r j. The quantum master equaton 3 then reads d dt ˆ = H + H RR, ˆ + L ˆ, 22 wth the Lndblad operator gven n Eq. 4. To see whch terms of the master equaton are affected by the ncluson of the Rydberg-Rydberg nteracton we consder the commutator H +H RR, n the many-body bass 1,..., N = 1 N, where denotes the state of atom, U H + H RR, = + j j j 2 j,e,e U + j j j 2 j,e,e, 23 and rewrte t usng the conservaton of probabltes for each atom,.e., 1= k,g+ k,m+ k,e k,g + k,e, and the

5 MANY-BODY THEORY OF EXCITATION DYNAMICS IN AN symmetry of the Rydberg-Rydberg nteracton U j =U j as H + H RR, =,e,g,g,e + j U j j,e j,e U + j,j j 2,e j,e,g j,g,g j,g,e j,e. 24 In the frst term of Eq. 24 the Rydberg-Rydberg nteracton shows up as an addtonal local detunng of an atom at r, whenever the atom at r j s n the Rydberg state.e., f j = j =e. In partcular, no addtonal coherences are generated by the Rydberg-Rydberg nteracton and, therefore, ths term does not change the structure of the master equaton as compared to the nonnteractng case. The second term descrbes drect transtons between states where atoms and j are not n the Rydberg state and the state where the atoms form a Rydberg par. These transtons requre the smultaneous absorpton or emsson of at least two photons and are thus hgher-order processes. The dynamcs of these multphoton processes s very slow compared to all other transtons n the system, therefore t can be neglected see also the dscusson n Secs. III and IV A,.e., the commutator 24 can be approxmated as H + H RR,,e,g,g,e + j U j j,e j,e. 25 Thus, wthn ths approxmaton, we recover the smple pcture, whch s commonly used for the explanaton of the dpole blockade effect, namely, that a hghly excted atom shfts the Rydberg levels of nearby atoms out of resonance wth the exctaton laser. By neglectng multphoton transtons, the structure of the master equaton s not changed compared to the nonnteractng system and we can perform the adabatc approxmaton dscussed above. Identfyng fnally j,e j,e wth j,ts straghtforward to generalze Eq. 19 to the nteractng case, T, =,, +,1,, 26 where now, =,, and all atoms are coupled by the energetc shft caused by the Rydberg-Rydberg nteracton = + + j U j, 27 j so that n the nteractng case the rate for a state change,, for the atom depends on the entre many-body confguraton through the local detunng. The above approxmatons smplfy the descrpton of the correlated many-partcle dynamcs to a hgh degree, snce a partcular many-partcle confguraton s drectly coupled to only N confguratons by the transton rate matrx T,, whch has to be compared to the avalable number of 2 N many-partcle states. To explctly show ths smplfcaton, we nsert Eq. 26 nto the evoluton equaton 2 of the state dstrbuton functon, perform the sum over, and fnally arrve at dp dt N N =,, P +,,1 P. 28 Knowng U j, Eq. 28 can be solved wth standard Monte Carlo samplng technques, allowng us to treat systems up to several 1 5 atoms. We emphasze that the descrpton presented above s not restrcted to the three-level scheme consdered n ths work. It can, e.g., also be appled for a drect exctaton of the Rydberg state from the ground state two-level scheme provded that the atomc coherences are damped out fast enough to not sgnfcantly affect the populaton dynamcs of the Rydberg state e.g., f the bandwdth of the exctaton laser s larger than the Rab frequency of the transton. For a sngle-step exctaton scheme the de exctaton rates are gven by = = , where s the measured wdth of the exctaton lne. C. Determnaton of the Rydberg-Rydberg nteracton An accurate determnaton of the nteracton potental U j s challengng due to the mxng of a large number of electroncally excted molecular potental curves. Results from a perturbatve treatment exst for the r j asymptote of the alkal-metal atoms 16 and for the level shfts of Rb 17 as well as calculatons for Cs based on the dagonalzaton of the nteracton Hamltonan of two hghly excted atoms usng a large number 5 of par states as bass 18. In the latter sprt, a smple pcture was formulated n 19 for Rb that allows for an ntutve understandng of the basc dependence of U j on r j and on the prncpal quantum number n of the Rydberg state. Followng 19, a par of Rydberg atoms n states a and b at dstance r j experences a shft U j of ts electronc energy due to an nduced dpole couplng V j = aa bb /r 3 j to an energetcally close par of states a and b. The shft s gven by the egenvalues of the two-state Hamltonan matrx PHYSICAL REVIEW A 76, U j = 1 2 ± 2 +4V 2 j

6 ATES et al. f e ρ ee;ee (a) (c) n H = V j V j, where s the asymptotc r j dfference between the energes of the two pars. For a par ns,ns of two atoms n the ns state, the relevant dpole couplng s to the energetcally close par n 1 p 3/2,np 3/2. For an arbtrary but fxed quantum number n we may defne 2 n n s n 1 p n sn p. The nteracton strength for other Rydberg levels n then follows from the scalng 1 2 n = 2 n n* n = n n * 4 n * * 3 n, 3a, 3b where n * =n ncludes the approprate quantum defect for the ns states of Rb =3.13. For r j one recovers the famlar van der Waals r 6 dependence and the domnant n 11 scalng for the par nteracton U j. For Rb we wll use n the followng the values 2 n =8438 a.u. and n =.378 a.u. for n =48 from 19. III. ACCURATE TREATMENT OF TWO INTERACTING ATOMS As a test for our rate equaton approach n the case of nteractng atoms, we have numercally solved the full quantum master equaton 22 and the rate equaton 28 for two nteractng atoms separated by an nteratomc dstance r. The quantty drectly accessble n the experments s the fracton of excted atoms f e. It s shown n Fgs. 3 a and 3 b as a (b) (d) n FIG. 3. Color onlne Comparson of the solutons of the master equaton 22 dashed lnes and the rate equaton 28 sold lnes for two nteractng atoms at dstance r=5 m. Upper graphs a and b show the fracton of excted atoms f e, lower graphs c and d the probablty ee;ee that both atoms are n the Rydberg state as a functon of the prncpal quantum number n for a pulse length =2 s. The parameters of a and c and b and d are those of Fgs. 2 a and 2 b, respectvely. PHYSICAL REVIEW A 76, functon of the prncpal quantum number n for exctaton parameters used n the experments 1,8, respectvely. The overall agreement between the exact result and our approxmaton s very good, and the dscrepancy of only a few percent between the solutons s comparable to that of the sngle-atom calculatons cf. Fg. 2; note the dfferent scalng of the ordnate and practcally ndependent of the nteracton strength. Ths ndcates that most of the devaton s a consequence of the approxmatons already ntroduced at the sngle-atom level. For both parameter sets we see a suppresson n f e for large n,.e., an exctaton blockade. Addtonally, n the case where the sngle-atom exctaton spectrum shows a doublepeak structure Fg. 3 b, there s an exctaton enhancement for a certan n. Its actual value depends on the separaton r of the atoms, so that n a gas ths antblockade wll be smeared out due to the wde dstrbuton of mutual atomc dstances. However, for atoms regularly arranged n space,.e., on a lattce where the nteratomc dstances are fxed, the antblockade should be clearly vsble 15. To verfy that the observed ant blockade n f e s really a suppresson enhancement of Rydberg pars we have plotted the probablty ee;ee that both atoms are n the Rydberg state. Indeed, we observe a complete suppresson of the par state n the blockade regme Fg. 3 c and the antblockade peak Fg. 3 d as well as a good agreement between the solutons of the master and the rate equaton n both cases. Neglectng two-photon transtons the second term n Eq. 24 s the central approxmaton, whch we make n the descrpton of the dynamcs of the nteractng system. In fact, these processes can be domnant, f the two-photon detunng vanshes far away from resonance,.e., f 2ph 2 +U r = for, U r. Ths s clearly seen n Fg. 4 a, where ee;ee s shown as a functon of the laser detunng for two atoms separated by r=5 m. The soluton of the master equaton exhbts a trple-peak structure wth the central peak located at = U r /2 cf. Eq. 24, whch s not present n the soluton of the rate equaton. However, the probablty for ths two-photon transton s too small to be vsble n the sgnal of the total probablty f e that the atoms are n the Rydberg state see nset. Increasng the nteratomc dstance to r=7 m,.e., decreasng the nteracton strength, we expect that the blockade mechansm becomes neffectve and the contrbuton of Rydberg pars to f e becomes relevant. Ths s ndeed reflected n the fact that the peak of ee;ee n Fg. 4 b s orders of magntude hgher than n Fg. 4 a. Here, however, the atoms are successvely excted to the Rydberg state by two snglephoton transtons. Hence, the peak n ee;ee s correctly reproduced by the rate equaton. IV. RYDBERG EXCITATION IN LARGE ENSEMBLES AND COMPARISON WITH THE EXPERIMENT A. Dpole blockade 1. Densty of Rydberg atoms We have calculated the densty of Rydberg atoms as a functon of the peak densty of a Rb gas n a MOT accordng

7 MANY-BODY THEORY OF EXCITATION DYNAMICS IN AN PHYSICAL REVIEW A 76, ρ ee;ee (a) f e ρ e [cm -3 ] ρ [cm -3 ] ρ e [cm -3 ] ρ [cm -3 ] ρ ee;ee (b) f e FIG. 4. Color onlne Probablty ee;ee that both atoms are n the Rydberg state 82S as a functon of the laser detunng after an exctaton tme of =2 s at an nteratomc dstance of r=5 m a and r=7 m b. Sold lnes are the solutons of Eq. 28, the dashed lnes of Eq. 22. The exctaton parameters are those of Fg. 2 b. The nsets show the correspondng fracton of excted atoms f e. FIG. 5. Densty of Rydberg atoms as a functon of the peak densty n the MOT for a pulse length of =2 s for the 82S black and the 62S state gray of Rb. Crcles: expermental data taken from 8. Lnes: Calculatons usng dfferent models for the par nteractons potental: two-state model of Ref. 19 sold and pure van der Waals nteracton from perturbatve treatment 16 dashed. to Eq. 28 for exctatons to the 62S and 82S state va the two-step exctaton scheme as measured n 8. More specfcally, we have determned the Rab frequency of the frst exctaton step by usng the data for the 5S 1/2 F=2 5P 3/2 F=3 trappng transton of 87 Rb 2 and by takng the ntensty of the MOT lasers from the experment 21. The measurement of as a functon of the ntensty of the MOT lasers usng the Autler-Townes splttng of a Rydberg lne 21 s n very good agreement wth our result. To obtan the couplng strength of the Rydberg transton we have ftted t to the low-ntensty measurements n 8 usng our rate equaton and scaled the result to hgh ntenstes and/or exctatons to dfferent prncpal quantum numbers. Fgure 5 shows the results of our calculatons and the experment. Although we see a qualtatve agreement, we predct Rydberg denstes about twce as large as the measured ones. As the curves for both prncpal quantum numbers exhbt the same devaton from the measured data, t s temptng to scale our results to the expermental ponts usng a common factor. Note, however, that wthout other nfluences n the experment, there s no free parameter n our descrpton that would justfy such a scalng. In the followng we estmate the quanttatve nfluence that several effects could have on the results presented. 2. Influence of dfferent Rydberg-Rydberg nteractons The exact Rydberg-Rydberg nteracton may dffer from the one we have used n our descrpton. To assess the mpact of such a dfference, we have performed our calculatons wth the smple two-state model dscussed above sold lnes n Fg. 5 and assumng a pure van der Waals nteracton C 6 /r 6 between the Rydberg atoms dashed lnes n Fg. 5. The nteracton coeffcents C 6 n for the latter are calculated n second-order perturbaton theory for r and have been taken from 16. The nteracton strength for the ns states calculated n ths way s consderably larger than the one from the two-state model e.g., for the 82S state the dfference n U r at r=1 m s roughly a factor of 2.5 and ncreases wth decreasng r. Yet, the fnal results for the Rydberg populaton dffer only slghtly see Fg. 5. We conclude that e s relatvely robust aganst changes n the nteracton strength. Ths s due to the fact that the measurement of the Rydberg densty as a functon of the ground state densty does not probe the exact shape of the nteracton potental but rather the crtcal dstance r c at whch the energetc shft caused by the nteracton becomes larger than half the wdth of the spectral lne 2 MHz. For U r determned n perturbaton theory and estmated by the two-state approxmaton r c 8 m and r c =7 m, respectvely, for the 82S state, so that sgnfcant dfferences emerge only for large denstes. 3. Influence of ons Another effect, so far not accounted for, s the presence of ons. The exctaton pulse length used n 8 was 2 s. For pulse duratons that long, t was shown that a sgnfcant amount of Rydberg atoms can undergo onzng collsons even for a repulsve Rydberg-Rydberg nteracton 19,

8 ATES et al. PHYSICAL REVIEW A 76, N p FIG. 6. Estmated average number N p of n=82 Rydberg pars excted by multphoton transtons as a functon of the laser detunng after =2 s for a ground state peak densty =1 1 cm 3. The presence of ons n the system nfluences the exctaton dynamcs due to the polarzng effect of the electrc feld of the ons on the hghly susceptble Rydberg atoms. The Rydberg-on nteracton r 4, therefore, leads to an addtonal energetc shft of the Rydberg levels and, thus, can lead to an enhanced exctaton suppresson. To see f the presence of ons can account for the dfference between our results and the measured data, we have performed calculatons n whch we have replaced up to 2% of the Rydberg atoms by ons. The change n the results compared to the stuaton wthout ons s comparable to that of stronger Rydberg-Rydberg nteracton dscussed above. Therefore, ons can be ruled out as a source for the dscrepancy between our and the expermental results. 4. Influence of multphoton transtons The exctaton lne profles presented n 8 showed an enormous broadenng for measurements at hgh denstes. In contrast, the lne profles that we have calculated wth the present approach are much narrower, n accordance wth the smulatons reported n Ref. 13. The strong lne broadenng n the experment could be due to nonresonant effects, such as multphoton transtons, not ncluded n our rate descrpton see dscusson n Sec. III. To estmate ther possble nfluence, we have to determne frst the number of Rydberg pars, whch could be excted by these transtons. To ths end, we have determned the number of ground state atoms n p r r, whch form a par wth a dstance between r and r+ r n the exctaton volume, from the par densty n p r. Furthermore, we have calculated the probablty ee;ee for a par of atoms to be n the Rydberg state after =2 s by solvng the quantum master equaton w ME p and the rate equaton w RE p for two atoms as a functon of the laser detunng and nteratomc dstance r cf. Fg. 4. The dfference w p r, =w ME p r, w RE p r, should gve a rough estmate for the probablty of a Rydberg par beng excted by a multphoton transton. The average number of such pars as a functon of can then be estmated by N p = w p r, n p r r. Fgure 6 shows that for a sample wth ground state peak densty =1 1 cm 3 our estmate yelds a neglgble number of Rydberg pars excted by multphoton transtons after 2 s. Although these estmates are rather crude, the result shows that multphoton effects are too small to explan the broadenng of the exctaton lne profle n the experment 8. In summary, the unexplaned lne broadenng and the dfference between experment and theory n the Rydberg populatons make t lkely that some addtonal, presently not known process, has contrbuted sgnfcantly to the results obtaned n 8. B. Antblockade 1. Lattce confguratons The dscusson n Sec. III has shown that the structure of the sngle-atom exctaton lne strongly nfluences the exctaton dynamcs n the nteractng system. Even on resonance, the Rydberg-Rydberg nteracton can cause an exctaton enhancement, f the spectral lne exhbts a double-peak structure. Ths antblockade occurs whenever the nteractonnduced energetc shft for an atom at poston r matches the detunng max at whch the sngle-atom exctaton probablty has ts maxmum value. In the gas phase, where the mutual atomc dstances are broadly dstrbuted, the antblockade can hardly be observed by measurng the fracton of excted atoms f e, as the condton = max s only met by relatvely few atoms 15. In contrast, f the atoms are regularly arranged n space, e.g., wth the help of an optcal lattce produced by CO 2 lasers 23, one should clearly observe peaks n f e for certan n see Fg. 7 a. The peak postons can easly be determned by analyzng the geometry of the underlyng lattce. Moreover, the effect s qute robust aganst lattce defects unoccuped lattce stes and should therefore be expermentally realzable. A more detaled dscusson can be found n 15. The underlyng lattce structure allows for a statstcal nterpretaton of the antblockade as clusterng of Rydberg atoms. Usng the termnology of percolaton theory, we defne a cluster of sze s as a group of s nearest neghbor stes occuped by Rydberg atoms. For neglgble Rydberg- Rydberg nteracton the exctaton of atoms on a lattce s analogous to the stuaton encountered n classcal ste- percolaton theory. Ths s seen n Fg. 7 b, where a hstogram of the average number n s of s-clusters per lattce ste as a functon of the cluster sze normalzed to the number of 1-clusters,.e., solated Rydberg atoms s shown for atoms excted to the state n=4. The shaded area represents the predcton of percolaton theory 24 for the same number of solated Rydberg atoms per ste and shows good agreement wth the measured data. In the antblockade regme n =65, Fg. 7 c we observe a broadenng of the cluster sze dstrbuton and a sgnfcant enhancement of larger Rydberg clusters, whle n the blockade regme n=68, Fg. 7 d a quenchng of the dstrbuton and an enhancement of the probablty to excte solated Rydberg atoms s evdent. 2. Random gases Based on the soluton of a many-body rate equaton usng Monte Carlo samplng, the present approach s partcularly well suted to determne statstcal propertes of nteractng Rydberg gases

9 MANY-BODY THEORY OF EXCITATION DYNAMICS IN AN PHYSICAL REVIEW A 76, f e (a) Q n s /n n s (b) (c) (d) s s FIG. 7. Color onlne a Fracton of excted atoms for atoms on a smple cubc lattce wth 2% unoccuped stes as a functon of the prncpal quantum number n. The lattce constant s a=5 m; all other parameters are those of Fg. 2 b. b d Correspondng number of Rydberg clusters per lattce ste n s normalzed to the number of 1-clusters.e., solated Rydberg atoms n 1 as a functon of the cluster sze s for prncpal quantum number n=4 b, n =65 c, and n=68 d. The shaded areas represent predctons from percolaton theory 24 for a system wth the same number of solated Rydberg atoms 1-clusters per lattce ste. In 1 the dstrbuton of the number of Rydberg atoms was measured as a functon of the nteracton strength. The dstrbutons obtaned were quantfed by Mandel s Q-parameter Q = N 2 e N e 2 1, 31 N e where N e s the number of Rydberg atoms and denotes the average over the probablty dstrbuton. The Q parameter measures the devaton of a probablty dstrbuton from a Possonan, for whch t s zero, whereas for a super- sub- Possonan t s postve negatve. The experment showed a quenchng of the Rydberg number dstrbuton,.e., a decrease of Q, for ncreasng nteracton strength as theoretcally confrmed 11,12. The dfferences between the theoretcal calculatons Q, for all n and the measured values n FIG. 8. Color onlne Comparson of the Q parameter n the blockade squares and antblockade crcles confguraton as a functon of the prncpal quantum number n for a sample wth a homogeneous atomc densty =8 1 9 cm 3 and for an exctaton pulse length =2 s. Q was determned by 1 5 successve measurements of N Ry and N 2 Ry. The Rab frequences, are 4.,.24 MHz squares and 22.1,.8 MHz crcles. Q can be attrbuted to shot-to-shot fluctuatons of the number of ground state atoms n the experment 25. The exctaton parameters n 1 were n the blockade regme, where the sngle-atom exctaton lne exhbts a sngle peak at =. Therefore, there s a volume correlaton hole around each Rydberg atom, where the exctaton of addtonal atoms s strongly suppressed. On the other hand, n the parameter regme of the antblockade, where the exctaton lne shows a double-peak structure, there s n addton a shell around each Rydberg atom, n whch addtonal exctatons are strongly enhanced. Thus, the statstcs of the Rydberg exctatons should depend on the structure of the sngle-atom exctaton lne and the antblockade can be detected ndrectly even n the gas phase by measurng the atom countng statstcs. Fgure 8 shows the calculated Q parameter as a functon of the prncpal quantum number n for the blockade and antblockade regme. In the blockade confguraton squares one observes a monotonc decrease of Q wth n n accordance wth the measurements n 1. In the antblockade regme crcles, however, Q s nonmonotonc,.e., the dstrbuton s slghtly broadened, and the quenchng starts at much hgher n. Although the broadenng of the dstrbuton may be dffcult to observe expermentally, the dfference n the functonal form of Q n provdes a clear expermental sgnature n a mesoscopc regon of the MOT, where the atomc densty s approxmately homogeneous. V. CONCLUSIONS We have developed a smple approach, whch allows one to descrbe the dynamcs n ultracold gases n whch Rydberg atoms are excted va a resonant two-step transton. Startng from a quantum master equaton, whch ncorporates the full dynamcs of an nteractng gas of three-level atoms, we have derved a many-body rate equaton. It covers the correlated dynamcs of the system, yet, t can easly be solved by Monte Carlo samplng for a realstcally large number of atoms. Our approach, vald under well defned condtons typcal for experments, s based upon two approxmatons: an

10 ATES et al. adadabtc approxmaton on the sngle-atom level to elmnate the atomc coherences and the neglgence of multphoton transtons n the nteractng system. Solvng the problem of two nteractng atoms exactly wth a quantum master equaton we could show that the approxmate soluton based on the rate equaton s n very good agreement wth the exact result. The present approach s capable of reproducng the partal exctaton blockade observed n 8 qualtatvely. Qualtatvely n accordance wth our calculatons regardng the exctaton lne shape and the so-called Q parameter are also the expermental results of 1. Fnally, the careful analyss of the two-step exctaton scheme has led to the predcton of an antblockade effect due to an Autler-Townes splttng of the ntermedate level probed by the Rydberg transton n the approprate parameter regme. Ths antblockade should be drectly observable for a lattce gas, realzed, e.g., wth an optcal lattce. As we have demonstrated, t could also be observed ndrectly n the gas phase through the atom countng statstcs, whch dffers qualtatvely from ts counterpart n the blockade regme. ACKNOWLEDGMENTS We thank M. Wedemüller and hs group for provdng us wth expermental data and for useful dscussons. T. Pohl acknowledges fnancal support by NSF through a grant for the Insttute of Theoretcal Atomc, Molecular and Optcal Physcs ITAMP at Harvard Unversty and Smthsonan Astrophyscal Observatory. APPENDIX: EXPRESSIONS FOR ee and _ The steady-state soluton of the OBE 6 for the Rydberg populaton s ee = The exctaton rate n Eq. 11 can be wrtten as where PHYSICAL REVIEW A 76, A1 = a + a a 4 4, A2 a = , a 2 = , a 4 = A3a A3b A3c 1 T. F. Gallagher, Rydberg Atoms Cambrdge Unversty Press, Cambrdge, England, W. R. Anderson, J. R. Veale, and T. F. Gallagher, Phys. Rev. Lett. 8, I. Mourachko, D. Comparat, F. de Tomas, A. Forett, P. Nosbaum, V. M. Akuln, and P. Pllet, Phys. Rev. Lett. 8, M. D. Lukn, M. Fleschhauer, R. Côté, L. M. Duan, D. Jaksch, J. I. Crac, and P. Zoller, Phys. Rev. Lett. 87, I. Bouchoule and K. Mølmer, Phys. Rev. A 65, 4183 R M. Saffman and T. G. Walker, Phys. Rev. A 66, D. Tong, S. M. Farooq, J. Stanojevc, S. Krshnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, Phys. Rev. Lett. 93, K. Snger, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Wedemüller, Phys. Rev. Lett. 93, T. Vogt, M. Vteau, J. Zhao, A. Chota, D. Comparat, and P. Pllet, Phys. Rev. Lett. 97, T. Cubel Lebsch, A. Renhard, P. R. Berman, and G. Rathel, Phys. Rev. Lett. 95, C. Ates, T. Pohl, T. Pattard, and J. M. Rost, J. Phys. B 39, L J. V. Hernández and F. Robcheaux, J. Phys. B 39, F. Robcheaux and J. V. Hernández, Phys. Rev. A 72, C. Cohen-Tannoudj, J. Dupont-Roc, and G. Grynberg, Atom- Photon Interactons: Basc Processes and Applcatons John Wley & Sons, New York, C. Ates, T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. Lett. 98, K. Snger, J. Stanojevc, M. Wedemüller, and R. Côté, J. Phys. B 38, S A. Renhard, T. C. Lebsch, B. Knuffman, and G. Rathel, Phys. Rev. A 75, A. Schwettmann, J. Crawford, K. R. Overstreet, and J. P. Shaffer, Phys. Rev. A 74, 271 R W. L, P. J. Tanner, and T. F. Gallagher, Phys. Rev. Lett. 94, D. A. Steck, Rubdum 87 D Lne Data 23 ; URL steck.us/alkaldata 21 M. Wedemüller prvate communcaton. 22 T. Amthor, M. Reetz-Lamour, S. Westermann, J. Denskat, and M. Wedemüller, Phys. Rev. Lett. 98, S. Frebel, C. D Andrea, J. Walz, M. Wetz, and T. W. Hänsch, Phys. Rev. A 57, R M. F. Sykes, D. S. Gaunt, and M. Glen, J. Phys. A 9, T. Cubel Lebsch, A. Renhard, P. R. Berman, and G. Rathel, Phys. Rev. Lett. 98, 1993 E