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Dynamically generating arbitrary spin-orbit couplings for neutral atoms Z. F. Xu1 and L. You1 1State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China (Dated: January 23, 2012) Spin-orbit coupling (SOC) can give rise to interesting physics, from spin Hall to topological insulators, normally in condensed matter systems. Recently, this topical area has extended into 2 atomicquantumgasesinsearchingforartificial/syntheticgaugepotentials. Theprospectsoftunable 1 interaction and quantum state control promote neutral atoms as nature’s quantum emulators for 0 2 SOC. Y.-J. Lin et al. recently demonstrated a special form of theSOC kxσy: which they interpret as an equal superposition of Rashba and Dresselhaus couplings, in bose condensed atoms [Nature n (London) 471, 83 (2011)]. This work reports an idea capable of implementing arbitrary forms of a SOC by switching between two pairs of Raman laser pulses like that used by Lin et al.. While one 0 J fpraoimr adffieffcetrsenktxσdyirefcotriosnosmaendtimaes,uabsseeqcuoenndt pspaiinr croretaatteiosnkiynσtyoo±vkeyrσoxt.heWrittihmseusffiwciitehntRmamanaynppuullsseess, 2 theeffectiveactionsfromdifferentdurationsaresmallandaccumulateinthesameexponentdespite that kxσy and ±kyσx do not commute. Our scheme involves no added complication, and can be ] demonstrated within current experiments. It applies equally to bosonic or fermionic atoms. s a g PACSnumbers: 03.75.Mn,67.85.Fg,67.85.Jk - t n Introduction. Atomic quantum gases are increasingly Several existing theoretical proposals are capable of a u viewedasfavoredmodelsystemsforemulatingcondensed implementingSOCwithrotationsymmetryinlaseratom q matter physics. Optical lattices resulting from ac Stack coupled models. For instance, in a tripod scheme [7], . t shifts to atomic levels, are easily implemented with co- when one-photon resonant couplings between the three a herentlaserbeams,whichconfine atomslike electronsin lower-energystates and a higher-energyone are allowed, m solidstates. Aninterestingtopicconcernsstrongcorrela- twodarkstatesemerge,althoughspontaneousemissionis - d tionsasininteger/fractionalquantumHalleffectandthe alwaysacauseofconcerninthiscase. D.L.Campbellet n analogous spin Hall effect. The standard description for al. [10]proposedanalternativeschemebycyclicallycou- o the former involvesU(1) Abelian gaugefields, which can pling three or four ground or metastable internal states. c be simulated in neutral atoms through rotation [1, 2] or Withsufficientlaserintensities,the aboveinducedSOCs [ adiabatictranslationsinfar-off-resonantlaserfields[3–6]. can possess a continuous rotation symmetry. Another 2 Non-Abelian gauge fields, e.g., as in spin-orbit coupling scheme by Jay D. Sau et al. [11] employs an effective v (SOC) [7–11], enable richer possibilities like fractional two-dimensionalperiodicpotentialcreatedfromtwolaser 5 quantum Hall states. As a result, active researches are beamsandtheir reflectedlights propagatingalongxˆ and 0 7 targetingtheimplementationsof(SOC)insimpleatomic yˆ directions in 40K atoms. In the limit of small Raman 5 systems. coupling, their corresponding effective SOC is of a pure 0. For atoms with multiple internal states, or (pseudo- Rashba type in the first Brillouin zone. 1 spin)spinordegreesoffreedom,SOCchangessinglepar- InthisLetter,wedescribeadynamicapproachforim- 1 ticle spectra and competes with density-density or spin- plementingrotationalsymmetricSOCofarbitraryforms 1 dependent interactions, (i.e., spin-exchange and singlet- within a pseudo-spin 1/2 atomic system. We adopt the : v pairing interactions). Strong correlations often lead to JQImodelandstartfromthesimpleSOCtheyproposed i exotic ground states [12–21], such as the plane-wave and recently demonstrated [9]. The key to our idea is X phaseandthestripedphasediscoveredrecentlyinpseudo optimalcontroltheoryappliedwithrepeatedlaserpulses r a spin-1/2[13–15]orspin-1condensates[13]. Otherexam- to rotate atomic pseudo-spins. Our idea works for both ples offer the triangular-latticedphase or square-latticed atomic fermions and bosons, and can be easily adopted phase in spin-2 condensates with axisymmetric SOC to other atomic models. Thus it constitutes a power- [16, 17]. In a recent experiment, the JQI group of Spiel- ful new direction for engineering synthetic atomic gauge manobservedbothAbelian[6]andnon-Abelian[9]gauge potentials. fields in a pseudo spin-1/2 atomic Bose gas, albeit in The equally weighted sum of Rashba and Dresselhaus a special form k σ of SOC, which is an equally types SOC of k σ [9], can be rotated into a form x y x y ∝ ∝ weighted sum of Rashba ( k σ k σ ) and Dressel- k σ , by performing single atomspin rotationthrough x y y x y x ∝ − ± haus ( k σ +k σ ) couplings [9]. More generally, a a Rabi pulse. Such a coherent control idea when re- x y y x ∝ SOC form of continuous rotation symmetry, or an arbi- peatedovertime,canrealizek σ and k σ typesSOC x y y x ± traryweightedsumofRashbaandDresselhauscouplings, in subsequent time intervals of duration δt. The result- exists in solid-state materials. ing dynamics is then described respectively by an effec- 2 tive Hamiltonian with pure Rashba or Dresselhaus SOC two laser beams affect two photon resonant Raman cou- with the first order approximation for small δt. The ac- pling (Ω ) between nearby ground Zeeman states, far R companiedchangeofatomicmomentum,canbenullified detuned from the excited states. Effectively, such a through a variety of means as we describe below step by coupling scheme produces an artificial magnetic field step. We start with a review of the experiment by Y.-J. alongthex-axisdirectionwiththeresultingHamiltonian Lin et al. [9], which helps to introduce our idea. Hˆ = Ω F cos(2k yˆ+∆ω t), where F are 3 3 R R x L L x,y,z × The JQI protocol. Consider a F = 1 atomic Bose- spin-1 matrices, k = √2π/λ with λ is the laser wave- L Einstein condensate (BEC) under a bias magnetic field length, and E = ¯h2k2/2m, the unit of photon recoil L L along zˆ located at the intersection of two Raman laser energy. In explicit forms, after adiabatically eliminating beams propagating along yˆ+ zˆ and yˆ+ zˆ, with an- excited states, the total Hamiltonian becomes − gular frequencies ω and ω +∆ω , respectively. The L L L ¯h2kˆ2 E+ 0 0 Ω 0 cos(2kLyˆ+∆ωLt) 0 Hˆ = + 0 E 0 + R cos(2k yˆ+∆ω t) 0 cos(2k yˆ+∆ω t) , (1) 3 0 L L L L 2m   √2   0 0 E− 0 cos(2kLyˆ+∆ωLt) 0     where E+, E0 and E− are Zeeman (eigen-) energies of wherethesecondlineshowsanexplicitSOCterm kˆyσz ∝ M =1,0, 1spin states,respectively. Under the rotat- when viewed after a unitary transformation. F − ing wave approximation, it turns into ¯h2kˆ2 E+ 0 0 Hˆ = + 0 E 0 3 0 2m   0 0 E− Ω   R + F cos(2k yˆ+∆ω t) x L L 2 Ω R F sin(2k yˆ+∆ω t). (2) y L L − 2 Further introduce a frame transformation: ψ˜ = e−iFz∆ωLtψ, where ψ and ψ˜ are the wave functions in the laboratory and transformed frames, respectively, we arrive at the Hamiltonian ¯h2kˆ2 2h¯ωq+3δ/2 0 0 Hˆ = + 0 δ/2 0 +E δ/2 3 0 2m   − 0 0 δ/2 − Ω  Ω  R R + F cos(2k yˆ) F sin(2k yˆ), (3) x L y L 2 − 2 where h¯ωZ = E− −E0, ¯h∆ωL = ¯hωZ +δ, E0 −E+ = FIG. 1: (Color online). (a) A schematic illustration of the ¯hω 2h¯ω , δ is detuning and ¯hω is the quadratic Zee- Z q q JQIimplementationforSOC[9],whereaF =1atomicBose- − man shift. When ¯hωq is sufficiently large and the Ra- Einsteincondensateinteractswithabiasmagneticfieldalong man coupling Ω = Ω /√2 is small, we neglect the state zˆ and two Raman laser beams propagating along yˆ+zˆ and R MF = 1 and a constant term E0 δ/2. The effec- −yˆ+zˆ, with angular frequencies ωL and ωL+∆ωL, respec- t|iveHamilitonianfortheremainingtwo−nearlydegenerate tively. (b) LEFT: Linear Zeeman shifts of the three hyper- fine spin states. MIDDLE: Zeeman shifts of the three hy- states becomes perfinespinstatesincludingbothlinearandquadraticterms. ¯hk2 δ Ω Ω RIGHT:Zeemanshifts intherotatingframe(with frequency Hˆ2 = 2m + 2σz + 2σxcos(2kLyˆ)− 2σysin(2kLyˆ) ∆ωLFz) of thepseudo-spin pointing along zˆ. ¯h2k2 δ Ω ¯h2k kˆ = eikLyˆσz + σz + σx+2 L yσz +EL Dynamically generating arbitrary SOC. Our protocol 2m 2 2 2m ! forimplementing the Rashbatype SOC is illustratedbe- e−ikLyˆσz, (4) low in Fig. 2. It relies on our ability of being able to × 3 switch atomic pseudo-spin from along z- to along y-axis (andvice versa)usingRamanpulses. Inthefirsthalfpe- riod, Raman lasers L and L are turned on. In the sec- 1 2 ondhalf,L andL areturnedoninstead. L isthesame 3 2 3 as L except it propagates along opposite direction. At 1 themiddlepoint,wepulseonanextraπ/2pulsetorotate the pseudo-spinfromy-to z-axis,describedbythe operator exp[ i(σ /2)π/2] in the transformed frame, or the x − operatorexp[iF ∆ω t]exp[ i(σ /2)π/2]exp[ iF ∆ω t] z L x z L − − inthelabframe;intheendofeachperiod,wepulseonan π/2 pulse for the reverse rotation. Both spin rotation − pulses can be accomplished with either Raman coupling from appropriately detuned lasers or rf plus microwave coupling between the two remaining internal states. In the first half, the system is then governedby Hˆ of 3 the Eq. (3) as in the JQI experiment [9]. In the second half,theHamiltonianinthetransformedframefollowing that in the Eq. (3) becomes ¯h2kˆ2 2h¯ωq+3δ/2 0 0 Hˆ′ = + 0 δ/2 0 +E δ/2 3 2m   0− 0 0 δ/2 − Ω  Ω  R R + F cos(2k zˆ)+ F sin(2k zˆ), (5) x L y L 2 2 FIG. 2: (Color online). (a) A schematic diagram for dynam- which completes one period of our prescribed protocol. ically generating arbitrary spin-orbitcoupling. (b)Thepulse For large¯hω and smallΩ, the same conditionas in Ref. q sequenceusedtoimplementRashbatypeSOC.Theblueand [9], the effective Hamiltonian for the reduced two-state cyan ones are suitable momentum impulses used to compen- model becomes sate for the unwanted photon recoils, which can be accomplished with artificial or real inhomogeneous magnetic fields ¯hk2 δ Ω Ω Hˆ′ = + σ + σ cos(2k zˆ)+ σ sin(2k zˆ) or suitably arranged state dependent Bragg pulses. The red 2 2m 2 z 2 x L 2 y L ±π/2 pulserotates atomic spin. ¯h2k2 δ Ω ¯h2k kˆ = e−ikLzˆσz + σ + σ 2 L zσ +E z x z L 2m 2 2 − 2m ! eikLzˆσz. (6) direction and with a spatial gradient. Thus they can × be nullified by real magnetic field gradients or synthetic The pair of π/2 pulse (before) and π/2 pulse (after) magnetic field gradients generated from spatial depen- affects a unitary transformation − dentacStarkshifts. Forinstance, U1 iscompensatedfor by a magnetic field pointing along z-axis and a spatial ei(σx/2)π/2Hˆ′e−i(σx/2)π/2 gradient (B′) along y-axis, with an adjustable impulse 2 ′ ′ ′ ¯hk2 δ Ω Ω overδt whereE µB isthe appropriateZeemanen- = + σy + σxcos(2kLzˆ)+ σzsin(2kLzˆ) ergy gradient. Aft≡er−the control pulse δt, the sign of B′ 2m 2 2 2 is changed to affect a second impulse, which then leads ¯h2k2 δ Ω ¯h2k kˆ = eikLzˆσy + σ + σ +2 L zσ +E to the following y x y L 2m 2 2 2m ! ×e−ikLzˆσy. (7) e−iE′yˆFzδt′/h¯e−iHˆ3δt/h¯eiE′yˆFzδt′/h¯ ¯h2k2 e−IinkLsyûσiztaabnldyUtr2an=sfeoirkmLzêσdy,frEaqmse(s4,)raesnpde(c7ti)vreelyvewalitehxpUl1ic=it = exp(cid:26)−i(cid:18) 2m +(h¯ωq+δ)Fz +¯hωqFz2+E0 SOC terms kˆyσz and kˆzσy. They cannot, however, be ΩR ¯h22kLkˆy + F +2 F +E δt/¯h , (8) simply added together in the forms above. To combine x z L 2 2m ! ) theabovetwohalvesintoasingleRashbaorDresselhaus type SOC, we have to eliminate these unitary transfor- ′ ′ ′ mations. BothU andU correspondsto spindependent provided E δt = 2h¯k , where we assume E is strong 1 2 L phase shifts, they can be viewed as from the impulse of enough so that we can neglect the contribution from Hˆ 3 ′ an artificial or real small magnetic field along a suitable duringtheshortpulseδt ( δt). Theeffectivetwo-state ≪ 4 dynamics is then approximately governby and its corresponding two-state approximation, exp(−i ¯h22mk2 + 2δσz + Ω2σx+2¯h22kmLkˆyσz!δt/¯h),(9) exp(−i ¯h22mk2 + 2δσz + Ω2σx−2¯h22kmLkˆzσz!δt/¯h). (11) apart from a overallphase term involving a constant en- For the special case of Rashba SOC, the suggested ergy in the exponent. Similarly, U2 is nullified as well, pulsesequenceareillustratedinFig. 2(b),wheretheblue resulting in andcyanonesaresuitablemomentum impulsesfor com- pensating the unwanted momentum recoils in the first eiE′zˆFzδt′/h¯e−iHˆ3′δt/h¯e−iE′zˆFzδt′/h¯ and second half cycles respectively. The red pairs are ¯h2k2 π/2 pulses for rotating the pseudo-spin. If the π/2 one = exp −i 2m +(h¯ωq+δ)Fz +¯hωqFz2+E0 ±precedes the π/2 pulse, we find in one period T =2δt, (cid:26) (cid:18) − Ω ¯h22k kˆ the total evolution operator under two-state approxima- R L z + F 2 F +E δt/¯h , (10) tion is given by x z L 2 − 2m ! ) U(T,0) = ei(σx/2)π/2 eiE′zˆFzδt′/h¯e−iHˆ3′δt/h¯e−iE′zˆFzδt′/h¯ e−i(σx/2)π/2 e−iE′yˆFzδt′/h¯e−iHˆ3δt/h¯eiE′yˆFzδt′/h¯ (cid:16)¯h2k2 δ Ω ¯h2k kˆ (cid:17) (cid:16) ¯h2k2 δ Ω ¯h2k kˆ(cid:17) L z L y exp i + σ + σ 2 σ δt/¯h exp i + σ + σ +2 σ δt/¯h y x y z x z ≃ (− 2m 2 2 − 2m ! ) (− 2m 2 2 2m ! ) ¯h2k2 δ Ω ¯h2k exp i + (σ +σ )+ σ + L(kˆ σ kˆ σ ) 2δt/¯h . (12) y z x y z z y ≃ − 2m 4 2 2m − (cid:26) (cid:18) (cid:19) (cid:27) According to the Floquet theorem, the quasienergy ǫ B~ = byeˆ , which is equivalent to B~ = byeˆ upon an y z − − of time-periodic system is derived from det[U(T,0) axis rotation, A from precisely needed for implementing e−iǫT]=0. Then from Eq. (12), we can easily infer tha−t U above. 1 under first order of T approximation, the quasienergy Likewise, the above B-field gradient can be simu- of our system is the same as the spectra of that with lated using ac stark shifts from position dependent laser Rashba SOC. Reversing the two red π/2 pulses intro- fields far off resonant coupled to the two states form- ± duces a minus sign ” ”, the Rashba SOC then changes ing an atomic pseudo-spin. Assuming a one-photon − into Dresselhaus SOC. By adjusting the timing constant resonant coupling Rabi frequency Ω (~r) and a detun- L δt, we can extend the above discussion to SOC of arbi- ing ∆ = Ω Ω , the ac Stark shift takes the form trary form β(kˆyσz kˆzσy)+ 1 β 2 (kˆyσz +kˆzσy). σ ΩL2(~r)/L(2−∆ )0 σ I (~r)/∆ . A linear spatial gra- The steady state of−the effective s−ys|tem| Hamiltonian is ∝ z| L | L ∝ z L L dient can thus be affected with a laser intensity gradi- p reacheddue to elastic atomic collisions. Although in the ent, which can be implemented using many methods, simplest case, one period of the control protocol is often including the use of a gradient neutral density filter. sufficient,the actualimplementationcanaimatahigher Stronger gradients arise from interfering several waves precisionofthe effective SOC Hamiltonian by increasing formingastandingwave,e.g.,withI (~r) cos2(qy/2)= L the number of cycles, or simple reducing δt. ∼ (1+cosqy)/2,lineargradientis qy aroundthenodal ∝± points of cosqy. A magnetic field gradient was first used in Ref. [6] for More generally, the state dependent gradients can be implementing Abelian gauge fields with neutral atoms. engineeredtocouplestatesinthesameZeemanmanifold. However, since real static B-field is subjected to the Maxwell’sequations B~ =0and B~ =0,onecannot For example, the above ac Stark shifts from one-photon ∇· ∇× couplingcanbesubstitutedwithtwo-photonRamancou- simply obtain a linear gradient along one direction, e.g. a B-field like B~ = (B by)eˆ is illegitimate because of pling with suitable differential detuning, like in Bragg 0 y − scattering, which then implements impulses yˆσ , its non-vanishing divergence. The simplest linear gradi- z ∝ ± zˆσ or zˆσ . entB-field, thereforeneeds to havetwocomponents, like y z ∝± ∝± that of the commonly used two dimensional quadruple Summarizing We present a coherent control protocol field, B~ =bxeˆ byeˆ . When the system is of a reduced capable of realizing the Rashba type SOC in a pseudo- x y − dimension, not including x-direction as in Ref. [6], one spin 1/2 atomic quantum gas [9]. For most systems, is then equippedwith a one-dimensionalB-fieldgradient our protocol can be implemented in one cycle, involving 5 two separate resonant Raman coupling. More elaborate Phys. Rev. A 71, 053614 (2005); G. Juzeliu¯nas, J. forms are possible with multiple control pulses. When Ruseckas, P. O¨hberg, and M. Fleischhauer, Phys. Rev. more than one control cycle is implemented, we can fur- A 73, 025602 (2006). ther enhance the precision and strength of the SOC, or [5] KennethJ.Gu¨nter,MarcCheneau,TarikYefsah,Steffen P. Rath, and Jean Dalibard, Phys. Rev. A 79, 011604 thecorrespondingartificiallycreatedgaugepotentials. In (2009) addition, the scheme we suggest is independent of quan- [6] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, tumstatisticsofatoms,thuscanbeadoptedtofermionic J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102, atoms as well. Our idea thus opens the door for dy- 130401 (2009); Y.-J. Lin, R. L. Compton, K. Jimenez- namicallyimplementingartificialgaugepotentialsincold Garcia,J.V.Porto,andI.B.Spielman,Nature(London) atomic systems based on coherent control theory. 462, 628 (2009). Finally, we compare our idea with two previous [7] J. Ruseckas, G. Juzeliu¯nas, P. O¨hberg, and M. Fleis- chhauer, Phys. Rev.Lett.95, 010404 (2005). schemes [10, 11]. In Ref. [10], three and four laser fields [8] Gediminas Juzeliu¯nas, Julius Ruseckas, and Jean Dal- are needed, cyclically coupled to three or four internal ibard, Phys. Rev.A 81, 053403 (2010). states. Nearly pure Rashba or Dresselhaus SOC then [9] Y.-J.Lin,K.Jiménez-Garc´ıa,andI.B.Spielman,Nature results respectively in the limit of large intensity laser (London) 471, 83 (2011). fields. It remains open to find a suitable experimental [10] D.L.Campbell,G.Juzeliu¯nas,andI.B.Spielman,Phys. system. In Ref. [11], along each axis of x- and y- two Rev. A 84, 025602 (2011). lasers with different frequency and their respective re- [11] Jay D. Sau, Rajdeep Sensarma, Stephen Powell, I. B. flections are needed. Only in the far-detuned and small Spielman,andS.DasSarma,Phys.Rev.B83,140510(R) (2011). Raman coupling Ω limit, Rashba or Dresselhaus SOC R [12] Tudor D. Stanescu, Brandon Anderson, and Victor Gal- can be implemented, which results in a relatively small itski, Phys.Rev.A 78, 023616 (2008). SOC, proportional to Ω . Our scheme, however, takes R [13] ChunjiWang,ChaoGao,Chao-MingJian,andHuiZhai, the full advantage of the JQI protocol [9]. By simply Phys. Rev.Lett. 105, 160403 (2010). turning on several pulses, and making use of the coher- [14] Tin-Lun Ho and ShizhongZhang, Phys. Rev.Lett.107, ent control, we can dynamically generate arbitrary SOC 150403 (2011). for neutral atoms. [15] S.-K. Yip,Phys. Rev.A 83, 043616 (2011). [16] Z. F. Xu, R. Lu¨, and L. You, Phys. Rev. A 83, 053602 This work is supported by the NSFC (Contracts (2011). No. 91121005 and No. 11004116). 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