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Preview Resonant control of spin dynamics in ultracold quantum gases by microwave dressing

Resonant control of spin dynamics in ultracold quantum gases by microwave dressing Fabrice Gerbier,∗ Artur Widera, Simon F¨olling, Olaf Mandel, and Immanuel Bloch Institut fu¨r Physik, Johannes Gutenberg-Universit¨at, 55099 Mainz, Germany. (Dated: February 6, 2008) We study experimentally interaction-driven spin oscillations in optical lattices in thepresence of an off-resonant microwave field. We show that the energy shift induced by this microwave field 6 can be used to control the spin oscillations by tuning the system either into resonance to achieve 0 near-unity contrast or far away from resonance to suppress the oscillations. Finally, we propose a 0 schemebased on thistechniqueto createa flat sample with eithersingly- or doubly-occupiedsites, 2 starting from an inhomogeneous Mott insulator, where singly- and doubly-occupied sites coexist. n PACSnumbers: 03.75.Mn,03.75.Lm,03.75.Qk a J 8 The fact that coupling an atomic system to a strong m=-1 m=0 m=+1 ] radiation field can modify its level structure is well- a b er known,leadingtosucheffectsastheAutler-Townesdou- F=2 h blet or the Bloch-Siegert shift (see, e.g., [1]). Today, (1) z t this phenomenon is often explained in a dressed-atom H o G t. picture, where the bare atomic states are replaced by 6.8 wµ a eigenstates of the total atom-plus-field system. For cold ~ ω (2) m atoms, this picture helps to understand how magnetic F=1 m=-1 - trapping potentials can be engineered with off-resonant m=0 m=+1 (3) d radio-frequency or microwave radiation [2], allowing for n example to create nearly two-dimensional [3] or double- o c well trapping potentials [4, 5]. [ Here, we show how an off-resonant microwave field FIG. 1: Anoff-resonant microwave field with approximately 1 (hereafter called dressing field) can be used for coher- π-polarizationandfrequencyωµw ((a))isusedtocontrolthe detuningforcoherentspinoscillationsinopticallattices. Case v ent control of the dynamics of a spinor quantum gas. (b1) corresponds to zero or weak microwave intensity. With 1 Ultracold gases with internal (spin) degrees of freedom properly chosen microwave parameters, one can compensate 5 [6] are novel quantum systems where atom-atom inter- thisdetuningandachievefullresonance(b2)orovercompen- 1 actions play a key role. In an optically trapped gas of sate to suppress thespin-changing collisions (b3).) 1 0 spin-f = 1 atoms, spin-dependent interactions can pro- 6 mote anatompair fromaninitial two-atomstate ,where 0 both atoms are initially in the m=0 magnetic sublevel, the topic of spin oscillations in optical lattices. Control- / t to a final state with one atom in m = +1 and the other ling the spin oscillations enables one to use them as a a inm= 1[7,8,9,10,11]. Thecoherenceofthisprocess tool for quantum state detection, preparation or manip- m − was demonstrated in [12, 13] for atom pairs in optical ulation. Ina separatepaper,we have describedhow this - d lattices and in [14, 15] for a seeded Bose-Einstein con- technique can be used to probe atom number statistics n densate. In the former case, the dynamics is controlled across the superfluid to Mott insulator (MI) transition o by the energy mismatch, or detuning, between the ini- [16]. We propose here to use it in order to create an c tial and final spin states of the colliding atom pair. This homogeneous MI with singly-occupied sites out of a in- : v detuningoriginatesnotonlyfromdifferentquadraticZee- homogeneous system where singly- and doubly-occupied Xi manshiftsintheinitialorfinaltwo-atomstates,butalso sites coexist. fromdifferentinteractionenergies. Ifpositive,asin87Rb In our case, the microwave is relatively weak, so that r a [10,13,15],this“residual”interactiondetuningprevents the dressed states almost coincide with the bare atomic to reach the resonance for spin oscillations by magnetic states and the level shifts can be calculated perturba- means. tively [1]. We consider a 87Rb atom in a constant and Inthispaper,weshowhowtotunethespinoscillations homogeneous magnetic field B, plus an arbitrary po- inandoutofresonanceusingtheenergyshiftinducedby larized microwave field with amplitudes B(µw) , with q q an off-resonant dressing field (see Fig.1). We apply this q = 0, 1 for π and σ± polarizations, re{spectiv}ely. Far ± technique to an ensemble of atom pairs in a deep optical from any hyperfine resonance, we find the level shift for lattice and initially polarized in the m=0 sublevel. We the state F =1,m as | i showthattheconversiontoanensembleofm= 1pairs canbe achievedwith nearunit efficiency, evenin±a finite ~Ω2 µ B ∆E πf x= B . (1) magneticfield. Thesignificanceofthisworkgoesbeyond m ≈ 4∆ m(cid:18) ~∆ (cid:19) 2 1 . After these pulses, the magnetic field is eventu- y − i sit0,15 ally ramped down to a final value (typically B = 0.4G n a bde0,10 for the experiments described here). Subsequently, colli- al0,05 sional spin oscillations take place at each individual lat- c pti0,00 tice site where an atom pair is present. As discussed in O -600 -200 0 200 600 [12,13],theatompairpreparedwithbothatomsinitially Position(µm) in m = 0 behaves as a two-level system, and undergoes Rabi-likeoscillationsat aneffective oscillationfrequency FIG. 2: (a) Absorption image of an expanded cloud after tosc = 0 ms (upper image) and tosc = 15 ms (lower image). Thedashedlineindicatestheboxused tofitthem=0com- ~Ωeff = (~Ω0)2+(∆ǫ+Us)2. (3) ponent. (b) Horizontal cut through the lower image in (a) p (black line). Subtracting a Gaussian fit (red dashed line) to Here the coupling strength ~Ω0 = 2√2Us and the in- the m=0 component yields a new image (blue line) used to teraction part of the detuning are both fixed by a spin- count separately thepopulation in m=±1by integration. dependent interaction energy U = U˜(a a )/3, with s 2 0 U˜ an on-site interaction energy per Bo−hr radius a B [13] and a the scattering length characterizing scatter- In Eq. (1), ∆ = ω ω denotes the microwave de- F µw − hf ing in the total spin-F channel [6, 12]. The quantity tuning,whereω isthemicrowavefrequencyandwhere ωhf isthehyperfiµwnefrequencysplittinginthe5S1/2 man- ∆enǫce=inǫ+Z1ee+maǫ−n1e−ner2gǫi0es∝ǫ Bb2ectowrereensptohnedisnittoialthaenddififfnear-l ifold of Rubidium 87. We have introduced the Rabi m states. Without dressing field, ∆ǫ = (µ B)2/2~ω is frequency Ω = µ B(µw)/~ for the π-polarized transi- B hf π B π determined by the magnetic field. In the presence of the tion, with µ the Bohr magneton, and assumed that B dressing field, the Zeeman sublevels shift according to x=µ B/~∆ 1. The function B ≪ Eq. (1), so that one obtains 3C I 1 fm(x)=Xq 4m,q Iπq 1−(m+ q2)x, (2) ∆ǫ+Us =~×α(Ω2π −Ω2res), (4) accounts for different microwave polarizations and the whereα=(f+1(x)+f−1(x)−2f0(x))/4∆andwherethe resonance Rabi frequency Ω solves multi-level structure. Here C = F = 1,m;1,q F = res m,q |h | 2,m + q 2 are square moduli of Clebsch-Gordan coef- (µ B)2 cfircoiewnatvse, cia|onmdpIoqneisntt.hWe einctaenlisbirtayteofthtehemiqc−ropwoalavreizinedtenmsii-- ~αΩ2res+ 2~Bωhf +Us =0. (5) tiesforeachpolarizationq bydrivingfast,single-particle When normalized to the total atom number, the oscilla- Rabi flopping between the hyperfine states F = 1,m = tion amplitude for the lattice as a whole is reduced by a | 1 and F = 2,m = 1+q . From this, we obtain factor n , corresponding to the fraction of atom pairs in − i | − i 2 Iσ−/Iπ ≈ 0.33 and Iσ+/Iπ ≈ 0.02 for our experimental the lattice [12, 13]. Forourtrappingparametersandour parameters. The peak Rabi frequency observed in our atom number, approximately half of the atoms are in a experiment is Ωπ =2π×82(1) kHz. central region with two atoms per site (n2 ≈0.5) [12]. To observe the microwave-induced shift, a MI of We have investigated the resonantly dressed spin dy- around 2 105 Rb atoms is first created in the F = namics at a magnetic field of 0.4 G, by varying the mi- × | 1,m= 1 Zeemansublevel. This MI is heldina purely crowave power (or equivalently the dressing frequency), − i optical lattice potential at a magnetic field of 1.2 G. In andrecordingseveralperiodsofspinoscillationsforeach this work, we use a typical lattice depth of 40Er, with power. The microwavefieldwas detuned by severalhun- Er = h2/2Mλ2 and λ = 840nm the lattice laser wave- dred MHz to the red of the hyperfine resonances to sup- length. Afteranexperimentalphasedescribedbelowand press population transfer to F = 2. Three representa- a 12-ms-long free expansion including a magnetic gradi- tive examples are shown in Fig. 3a, corresponding to ent pulse, we detect the population in each Zeeman sub- the cases illustrated in Fig. 1: the unperturbed case level as shown in Fig. 2 [12, 13]. We find the population (Ω Ω ,squares),thenear-resonantcase(Ω Ω , π res π res in the m=0 sublevelby fitting a Gaussianfunction to a circle≪s) where the microwave dressing compensa≈tes ex- restricted area, by substracting it from the data, and by actlythe“bare”detuning,andtheovercompensatedcase integrating over the cloud areas to find the populations (Ω Ω , triangles), where the dynamics is blocked π res in m= 1 (see Fig. 2). due ≫to the large detuning induced by the microwave ± The main topic of this paper is the study of spin dy- dressing. We have found empirically that the data were namics in a dressing field. In our experiments, the spin well described by a fitting function of the form dynamics is initialized as described in [12, 13] by first transferring the sample to F = 1,m = 0 by apply- N t | i 0 =A 1 e−γofft +Bcos 2π e−γosct. (6) ing two microwave π pulses starting from F = 1,m = N − (cid:18) T (cid:19) − | (cid:0) (cid:1) osc 3 1 0.5 0.8 s N 0.4 a ne N)/-1 0.3 2 1 ationsianstat 0.6 + 0.2 ulm 0.4 1 pe + oe N PZ 0.2 ( 0.1 0 3 0 0 100 200 300 Time(ms) 0 10 20 30 40 50 60 70 80 Time(ms) FIG. 4: Spin oscillations recorded for a magnetic field B = 30 ) 1.2G.FilledcirclesdenotethepopulationintheZeemansub- ms 25 b statem=0,andhollowcirclesthesumofthepopulationsin 2 ( substates m=±1. d 20 3 o eri 15 1 P 10 expected behavior. The fit parameters include the same 0 5 10 15 20 25 30 35 40 parameters α,Ω as in the previous case, plus an ad- RabiFrequencyΩπ (kHz) ditional parametreesr n corresponding to the fraction of 2 atoms participating in the oscillation. The values for α 0.4 ne andΩ areconsistentwiththosededucedfromthefitto od 0.3 c res cillatimplitu 0.2 itshselimghetalsyusrmedaplleerritohda.nTthheeofrvaecrtailolnamn2pl=itu0d.5eonf2a≈tom0.4p0a(i3rs) s Oa 0.1 expectedintheMottphase. Wehaveperformedanother measurementwhere the atomic sample is held at a mag- 0 0 5 10 15 20 25 30 35 40 netic field of 1.2G using the dressing field on resonance RabiFrequencyΩπ (kHz) accordingtoEq.(5). Thespinoscillationcurveshownin Fig. 4 yields an amplitude of n 0.47(2), much closer 2 ≈ to the expectation than for the B = 0.4G case. We at- FIG. 3: Spin oscillations at a magnetic field B = 0.4 G. (a) tribute part of this 15 % reduction in contrast to a Spin oscillations without microwave dressing (squares), for a finite ramp time of t∼he magnetic field in the latter case, resonantdressing(circles)andforanovercompensateddress- inducing a small amount of adiabatic population trans- ing (triangles). (b) and (c), period and amplitude of the fer at the beginning of the oscillation [13]. From the oscillation (circles), together with a fit to the expected Rabi datashowninFig.4,wehavealsodeterminedthedamp- dependance. ing times γ ,γ and the offset value A, as defined in osc off Eq. 6. Within our experimental accuracy, we find that The second term describes the spin oscillations with pe- they are independent of microwave power and magnetic riod T , including a damping term with rate γ . The field. The average offset value is A 0.6, and the av- osc osc first term describes a slowly varying offset approaching erage damping times γ 140s−1 a≈nd γ 160s−1. osc off the value A with a time constant γ−1. Although the origin of the≈damping is not enti≈rely clear, off The measured period of the oscillation is plotted in the fact that γ γ indicates a correlation between osc off ≈ Fig. 3b as a function of the Rabi frequency Ω of the the damping of the oscillations and the rise of the offset. π dressing field. The period is fitted to the expected Possible damping mechanisms can be either local (i.e. form of the Rabi frequency given in Eq. (3,4) treat- involving only on-site processes) or non-local (involving ing U , Ω and α as free parameters. We find good tunneling to a neighboring site, particularly effective if s res agreement between the expected 2πα = 0.069s, Ω = atoms are transferredto excited Bloch bands by heating res 2π 23.4kHzandthe measuredvalues 2πα=0.066(1)s, processes). On-site damping mechanisms could possibly × Ω = 2π 22.8(1)kHz. The “bare” Rabi frequency be off-resonant Raman transitions between the m = 1 res × ± U /h = 10.5(1)Hz is in excellent agreement with an sublevels induced by the lattice beams. We believe the s independent measurement of the spin-dependent colli- limitations of the present experiment to be of technical sionalcoupling,whichweredonewithoutadressingfield origin. In principle, even longer coherence times could [12, 13]. be achieved. In Fig. 3c, we show the oscillation amplitude versus Because of these long coherence times, the technique microwave Rabi frequency Ω , together with a fit to the presented here could find applications beyond the study π 4 singly- doubly- tem with only singly-occupied sites in the ground state. occupied occupied This would help to remove local defects with two atoms F=2 onasite,thusprovidinganovelcoolingtechniqueunique F=1 to the MI state. We acknowledgesupportfromthe DFG,fromAFOSR and from the EU under the OLAQUI program. ∗ email: [email protected] remove [1] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg (Wiley VCH, 1997). [2] R.J.C.Spreeuw,C.Gerz,L.S.Goldner,W.D.Phillips, S. L. Rolston, C. I. Westbrook, M. W. Reynolds, and I. F. Silvera, Phys. Rev.Lett. 71, 3162 (1994). time m =-1 0 +1 [3] Y. Colombe, E. Knyazchyan, O. Morizot, B. Mercier, F V.Lorent,andH.Perrin,Europhys.Lett.67,593(2004). [4] P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. FIG. 5: Illustration of the proposed scheme to create a ho- H¨ansch, and J. Reichel, Phys. Rev. Lett. 92, 203005 mogeneousMIwithonlysingly-occupiedsitesstartingfroma (2004). sample with singly- and doubly-occupied Mott plateaus. Af- [5] T. Schumm, S. Hofferberth, L. M. Andersson, S. Wil- ter half a spin oscillation period, atoms in singly-occupied dermuth,S.Groth,I.Bar-Joseph,J.Schmiedmayer,and sites are transferred into the upper hyperfine level. An- P. 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Fortier, W. Zhang, L. You, and M. S. Chapman, Phys. Rev.Lett. 92, 140403 (2004). of spin oscillations. For example, Refs. [17, 18, 19] dis- [11] T. Kuwamoto, K. Araki, T. Eno, and T. Hirano, Phys. cuss how spin squeezing experiments could benefit from Rev. A 69, 063604 (2004). such a control. In Fig.5, we show how a homogeneous [12] A.Widera,F.Gerbier,S.F¨olling,O.Mandel,T.Gericke, MI with one atom per site only can be created, starting and I.Bloch, Phys. Rev.Lett. 95, 190405 (2005). from of an inhomogeneous system where Mott plateaus [13] A.Widera,F.Gerbier,S.F¨olling,O.Mandel,T.Gericke, with n = 1,2 coexist. This could be important for the and I.Bloch, in preparation (2005). implementationofparallelcollisionalquantumgatesina [14] J. Kronj¨ager, C. Becker, M. Brinkmann, R. Walser, P. Navez, K. Bongs, and K. Sengstock, cond- lattice [20, 21], where an inhomogeneous filling degrades mat/0509083 (2005). the fidelity of the process (see also [22]). Similarly, one [15] M.-S.Chang,Q.Qin,W.Zhang,L.You,andM.S.Chap- canproduceanarrayofdoubly-occupiedsitesinm= 1, man, Nature Physics 1, 111 (2005). − as first proposed in [23] (see also [12, 24]). [16] F. Gerbier, A. Widera, S. F¨olling, O. Mandel, and In conclusion, we have demonstrated resonant tuning I. Bloch, cond-mat/0511080 (2005). [17] H.PuandP.Meystre,Phys.Rev.Lett.85,3987(2000). ofspinoscillationsinopticallatticesusinganoff-resonant [18] L.-M.Duan,A.Sørensen,J.I.Cirac,andP.Zoller,Phys. microwave field. This provides dynamical control over Rev. Lett.85, 3991 (2000). the spinoscillations,whichtogetherwiththe longcoher- [19] A. Sorensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Na- ence times enables to use them as a tool for detection, ture 409, 63 (2001). preparation and manipulation of the quantum state of [20] D.Jaksch,H.J.Briegel,J.I.Cirac,C.W.Gardiner,and the many-atom system. As an illustration, we propose a P. Zoller, Phys. Rev.Lett. 82, 1975 (1999). filteringtechniqueproducingaflatMIwithasingleatom [21] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. H¨ansch, and I. Bloch, Nature425, 937 (2003). per site, starting fromasystem withsingly- anddoubly- [22] P. Rabl, A. J. Daley, P. O. Fedichev, J. I. Cirac, and occupied sites. From a thermodynamical point of view, P. Zoller, Phys. Rev.Lett. 91, 110403 (2003). this flat MI would be metastable with a lifetime on the [23] A. Widera, O. Mandel, M. Greiner, S. Kreim, T. W. order of the tunneling time in the system. This method H¨ansch,andI.Bloch,Phys.Rev.Lett.92,160406(2004). couldalsobe appliedtoathermodynamicallystablesys- [24] G. Thalhammer, K. Winkler, F. Lang, S. Schmid, 5 R. Grimm, and J. HeckerDenschlag, cond-mat/0510755 (2005).

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