RESEARCH ARTICLE◥

QUANTUM SIMULATION

Realization of two-dimensionalspin-orbit coupling forBose-Einstein condensatesZhan Wu,1,2,3 Long Zhang,1,4,5 Wei Sun,1,2,3 Xiao-Tian Xu,1,2,3 Bao-Zong Wang,1,4,5

Si-Cong Ji,1,2 Youjin Deng,1,2,3 Shuai Chen,1,2,3* Xiong-Jun Liu,4,5* Jian-Wei Pan1,2,3*

Cold atoms with laser-induced spin-orbit (SO) interactions provide a platform toexplore quantum physics beyond natural conditions of solids. Here we propose andexperimentally realize two-dimensional (2D) SO coupling and topological bands for arubidium-87 degenerate gas through an optical Raman lattice, without phase-lockingor fine-tuning of optical potentials. A controllable crossover between 2D and 1D SOcouplings is studied, and the SO effects and nontrivial band topology are observed bymeasuring the atomic cloud distribution and spin texture in momentum space. Ourrealization of 2D SO coupling with advantages of small heating and topologicalstability opens a broad avenue in cold atoms to study exotic quantum phases,including topological superfluids.

The spin-orbit (SO) interaction of an elec-tron is a relativistic quantum mechanicseffect that characterizes the coupling be-tween themotion and spin of the electronwhen moving in an electric field. In the

rest frame, the electron experiences a magneticfield proportional to the electron velocity andcouples to its spin by the magnetic dipole in-teraction, rendering the SO coupling. The SO in-teraction plays an essential role in topologicalinsulators, which have been predicted and ex-perimentally discovered in two-dimensional (2D)and 3D materials (1, 2), and topological super-conductors (3, 4), which host exotic zero-energystates called Majorana fermions (5) and stillnecessitate rigorous experimental verification.For topological insulators, the strong SO inter-action leads to band inversion, which drivestopological phase transitions in such systems.In superconductors, triplet p-wave pairing mayoccur when SO coupling is present and resultsin topologically nontrivial superconductivityunder proper conditions (6).Recently, there has been considerable interest

in emulating SO effects and topological phases

with cold atoms, driven by the fact that cold atomscan offer extremely clean platforms with full con-trollability to explore such exotic physics. In coldatoms, the synthetic SO interaction can be gen-erated by Raman coupling schemes that flip atomspins and transfer momentum simultaneously(7, 8); 1D SO interaction (9) has been successfullydemonstrated in experiment for both cold boson(10, 11) and fermion degenerate gases (12, 13).With the 1D SO coupling, which corresponds toan Abelian gauge potential (14), one can studyeffects such as the magnetized or stripe groundstates for bosons (15–18), spin dynamics (19, 20),and 1D insulating topological states for fermions(21). Realizing higher dimensional SO couplings,which correspond to non-Abelian gauge poten-tials (7, 8), can enable the study of a broader rangeof nontrivial quantum states such as topologicalinsulators driven by 2D and 3D SO interactions(1, 2). Furthermore, a 2D SO interaction is theminimal requirement to reach a gapped topolog-ical superfluid phase through a conventional s-wave superfluid state (22, 23).Several schemes have been proposed for gen-

erating 2D and 3D SO couplings (7, 8, 24–26).Notable progress was recently achieved when 2DSO couplingswere demonstrated for pseudospinsdefined by two dark states in an empty tripodsystem (27). However, realizing 2D SO couplingfor quantum degenerate atom gases remainschallenging. Very recently, it was proposed that2D SO coupling can be realized by a simple opticalRaman lattice scheme that applies two pairs oflight beams to create the lattice and Raman po-tentials simultaneously (28). However, this schemerequires the challenging realization of two Ramantransitions with a locked relative phase. Here wepropose a minimal scheme that overcomes these

challenges and realizes 2D SO coupling with 87RbBose-Einstein condensates (BECs).

Theoretical proposal

We aim to realize 2D SO coupling and topologicalband for ultracold atoms on a square lattice withthe Hamiltonian

H ¼ ℏ2k2

2mþ Vlattðx; zÞ

� �� 1þ

Mxðx; zÞsx þMyðx; zÞsy þmzsz ð1Þ

where ħ is Planck’s constant h divided by 2p, k isthewave vector that represents themomentumofthe atoms, 1 is the 2-by-2 unit matrix, sx;y;z arePauli matrices acting on the spins,m is the massof an atom, Vlatt denotes the lattice potential inthe xz plane, Mx;y are periodic Raman couplingpotentials, and mz is a tunable Zeeman constant.The lattice potential Vlatt is spin-independent andcan induce nearest-neighbor hopping that con-serves the atomspin,whereasMx;y inducehoppingthat flips atom spin. This is different from the laser-assisted tunneling schemewithout spin-flip (29–31).The overall effect of hopping along x^ and z^ direc-tions results in 2D SO coupling, which can lead tonontrivial topological bands for the square lattice.Here we propose to realize the Hamiltonian

(Eq. 1) through aminimal scheme,which is genericand applicable to both boson and fermion atoms.Figure 1 illustrates the realization in a 87Rb Bosegas, with j↑i ≡ j1;−1i and j↓i ≡ j1;0i; the hyper-fine state j1;þ1i can be removed by a sufficientlylarge two-photon detuning. The minimal ingre-dients of the realization include a blue-detunedsquare lattice created with two light componentsdenoted by the blue lines, and the periodic Ramanpotentials generated together with additionallight components denoted by the red lines (Fig.1A). Both ingredients can be achieved with a sin-gle in-plane (xz) linearly polarized laser source.The initial phases of the light beams have noeffect on this optical Raman lattice scheme (32),and we neglect them in the following discussion.The optical lattice is generated by E1x and E1z

(blue lines), which are incident from horizontal(x) and vertical (z) directions, respectively (Fig. 1A),and can be created from a single light of frequencyw1 by a beam splitter. The beams are reflectedby two mirrors (M1 andM2) and form standingwaves in the intersecting area described byE1x ¼ z^ E1xeiϕL =2cosðk0x −ϕL=2Þ and E1z ¼x^ E1zeiϕL=2 cosðk0z − ϕL=2Þ, where E1x=1z areamplitudes and the phase ϕL ¼ k0L is acquiredthrough the optical path L from the intersectingpoint to mirror M1, then to M2, and back to theintersecting point, with k0 ¼ w1=c. For alkaliatoms, we can show that the optical potentialgenerated by linearly polarized lights is spin-independent when the detuning D is muchlarger than the hyperfine structure splittings(32). The lattice potential then takes the form

Vlattðx; zÞ ¼ V0xcos2ðk0x − ϕL=2Þþ

V0zcos2ðk0z − ϕL=2Þ ð2Þ

whereV0x=0z ¼ ℏjWx=z j2=D.TheRabi frequencyam-plitudes of the standing wavesWx=z≡deff ⋅E1x=1z=ℏ,

RESEARCH

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1Shanghai Branch, National Laboratory for Physical Sciencesat Microscale and Department of Modern Physics, Universityof Science and Technology of China, Shanghai 201315,China. 2Chinese Academy of Sciences (CAS) Center forExcellence and Synergetic Innovation Center of QuantumInformation and Quantum Physics, University of Science andTechnology of China, Hefei, Anhui 230026, China. 3CAS-Alibaba Quantum Computing Laboratory, Shanghai 201315,China. 4International Center for Quantum Materials, Schoolof Physics, Peking University, Beijing 100871, China.5Collaborative Innovation Center of Quantum Matter, Beijing100871, China.*Corresponding author. Email: shuai@ustc.edu.cn (S.C.);xiongjunliu@pku.edu.cn (X.-J.L.); pan@ustc.edu.cn (J.-W.P.)

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where E1x=1z ¼ E1x=1zz^=x^, and the effective dipole

matrix deff takes into account the transitionsfrom a ground state ðg↑;↓Þ to all relevant excitedstates in D1 and D2 lines (Fig. 1B). The latticepotential induces spin-conserved hopping, asillustrated in Fig. 1D.TheRaman couplings are inducedwhenanother

beam E2z of frequency w2 is incident from the zdirection. The light components E1z and E2z

can be generated from a single laser source viaan acoustic-optic modulator (AOM), which con-trols their frequency difference dw ¼ w1−w2 andamplitude ratio E2z=E1z . The light E2z generatesplane-wave fields E2z ¼ x^E2zeik0z and E2x ¼z^E 2xeið−k0xþϕL−dϕLÞ, where the irrelevant initialphase is neglected (32). The relative phase dϕL ¼Ldw=c, acquired by E2x , is a crucial parameter,which can be preciselymanipulated by changing theoptical path L or dw, and it controls the di-mensionality of the realized SO coupling. Thestanding-wave and plane-wave beams form adouble-L type configuration (Fig. 1C), with E1x

and E2z generating one Raman potential via jF ;0iin the form M0xcosðk0x −ϕL=2Þeiðk0z−ϕL=2Þ, andE1z and E2x producing another one via jF ;−1i asM0ycosðk0z −ϕL=2Þe−iðk0x−ϕL=2Þ−idϕL . Note thatlattice sites of a blue-detuned lattice are locatedaround zeros of optical fields. It follows that termssuch as cosðk0x −ϕL=2Þcosðk0z−ϕL=2Þ, which isantisymmetric with respect to each site in both xand z directions, have small contribution to low-band physics. Neglecting such terms yields

Mxðx; zÞ ¼ Mx−MycosdϕL ð3Þ

Myðx; zÞ ¼ MysindϕL ð4ÞHere Mx=y ¼ M0x=ycosðk0x=z − ϕL=2Þsinðk0z=x −ϕL=2Þ, with M0x=M0y ¼ E1xE2z=ðE1zE2xÞ(32). Together with an effective Zeeman termðmz ¼ ℏd=2Þ, which is controlled by tuning thetwo-photon detuning d (Fig. 1C), we reach theeffective Hamiltonian (Eq. 1). Note thatMx ðMyÞis antisymmetric with respect to each latticesite along the x^ðz^Þ direction (Fig. 1, E and F). Thisfeature has an important consequence that Mx

ðMyÞ leads to spin-flipped hopping only along x(z) direction. Moreover, the phase difference dϕL

governs the relative strength of the sx and syterms and thus determines the dimensionality ofthe SOcoupling. For example, setting dw =50MHzyields dϕL ¼ p=2 for L = 1.5 m, resulting in op-timal 2D SO coupling. Further increasing theoptical path to L = 3.0 m gives dϕL ¼ p, and theSO coupling becomes 1D. This enables a fullycontrollable study of the crossover between 2Dand 1D SO couplings by tuning dϕL and providesa comparison measurement to confirm the real-ization of 2D SO interaction.

2D SO coupling and topological band

The Hamiltonian (Eq. 1) has an inversion sym-metry defined by ðsz � R2DÞHðsz � R2DÞ−1 ¼ H ,where the 2D spatial operator R2D transformsthe Bravais lattice vector R→ −R. For the s-band, the Bloch Hamiltonian is given byHðqÞ ¼½mz − 2tx0ðcosqxaþ cosqzaÞ�sz þ 2tsosysinqxaþ2tso sxsinqza, which, around the G point, takesthe formHðqÞ ¼ ½mz − 4t0 þ t0a2ðq2x þ q2zÞ�szþ

lsoqxsy þ lsoqzsx , with the SO coefficient lso ¼2atso being tunable by varying Raman couplingstrength (32), unlike in the previous schemes(10–13, 27). Here, t0 and tso denote the spin-conserved and spin-flip hopping coefficients,respectively, and a is the lattice constant. Thisis a quantum anomalousHall model driven by SOcoupling (28), which cannot be exactly realizedin solid-state materials. It was shown (33) thatthe topology of inversion symmetric Chern bandscan be determined by the product of the spin-polarizations PðLjÞ at four highly symmetricmo-menta Q ¼ P4

j¼1sgn½PðLjÞ�, with the momentafLjg¼ fGð0; 0Þ;X1ð0; pÞ;X2ðp;0Þ;Mðp; pÞg. Thetopological (or trivial) phase corresponds to Q ¼−1 (or +1). Two typical examples are shown inFig. 2 by exactly diagonalizingH, withV0x;z ¼ 5Er,M0x;y ¼ 1:2Er, dϕL ¼ p=2, and mz ¼ 0:1Er (Fig.2, A, B, E, and F) ormz ¼ 0:4Er (Fig. 2, C and D),where the recoil energy is Er ¼ ℏ2k20=2m. For thechosen parameters, the lowest two subbands aregapped (Fig. 2, A toD).Whenmz ¼ 0:1Er, the spinpolarizations at the G andM points are opposite(Fig. 2, B, E, and F), implying that the band istopologically nontrivial. In contrast, the polar-izations are the same for mz ¼ 0:4Er (Fig. 2D),and the band is trivial.The present scheme displays several essential

advantages: (i) Fluctuations, such as those causedby the mirror oscillations, have a tiny effect on Land, thus, do not affect dϕL appreciably. Theinitial phases of light beams globally shift theoptical Raman lattice but cannot affect the rel-ative configuration between Vlatt and the Raman

84 7 OCTOBER 2016 • VOL 354 ISSUE 6308 sciencemag.org SCIENCE

∆∆ + ∆

1,

, , + 1, − 1

latt

, 2 ,

, 1 ,

, 1 , , 2 ,

1

2

1,− 11,0

21 1

,− 1 ,02

3/ 2

21/ 2

21/ 2

∆ + ∆∆

Fig. 1. Proposal of the optical Raman lattice scheme. (A) Sketch of thesetup for realization. The light components E1x;1z (blue lines) form a spin-independent square optical lattice in the intersecting area and generate twoperiodic Raman potentials, together with the light components E2x;2z (red lines).(B)Optical transitions to generate latticepotentials byE1x;1z for the states j1;mFiðmF ¼ 0;−1Þ, with j↑i ¼ j1;−1i and j↓i ¼ j1;0i, including all relevant D1 and D2

transitions. Here, F is the quantum number of hyperfine states, mF is the

quantum number of magnetic substates, and Ds denotes the fine-structuresplitting. (C) Two periodic Raman potentials are generated through a double-Ltype configuration by E1x;2z and E1z;2x, respectively. (D to F) Profiles of latticepotential for V0x ¼ V0z ¼ 5Er (D) and Raman potentials Mx (E) andMy (F) forM0x ¼ M0y ¼ 1:2Er. The color intensity characterizes the relative height of thepotentials.The Raman potentialsMx andMy are antisymmetric (or symmetric)with respect to the lattice site along the x (or z) and z (or x) directions, respectively.

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potentials (32). Thus, the present scheme is in-trinsically immune to any phase fluctuations inthe setting, which avoids the need for phase-locking, a challenging task inpractical realizations.(ii) As long as E1x ¼ E1z and E2x ¼ E2z , whichare easily accomplished, the system becomesuniform in the x and z directions: V0x ¼ V0z andM0x ¼ M0y . No fine-tuning of optical potentialsis required. (iii) All of the coupling beams can becreated from only a single laser source, simplify-ing the experimental layout. (iv) Compared withthe generation of 1D SO coupling, the presentdouble-L Raman configuration (Fig. 1C) does notsuffer additional instability or heating in therealization. These advantages render the presentscheme immediately feasible in ultracold atomexperiments with the current technology.

Experimental setup

In our experiment, a BEC of about 1.5 × 105 87Rbatoms in the state j1;−1i is prepared in a crossedoptical dipole trap with trapping frequencies offwx;wy;wzg ¼ 2p� f45; 45; 55gHz, which cansuppress the antitrapping effect of blue detunedlights. A bias magnetic field of 49.6 G is appliedalong the z^ direction to generate the Zeemansplitting and determine the quantization axis. Asshown in Fig. 3A, three laser beams (wavelength =767 nm) in the xz plane illuminate the atoms forthe generation of the Hamiltonian (Eq. 1). Amongthese laser beams, a pair of counterpropagatinglasers with the same frequency w1 (the blue linesin Fig. 3A labeled as “lattice lasers”) produces thetwo-dimensional optical lattice. These two lasers

are incident along the x^ and z^ directions, re-spectively, and are reflected by two mirrors (M1

and M2) to form the standing waves in bothdirections. The polarizations are set in the xzplane so that the interference between the x^and z^

directions is automatically avoided. The thirdlaser with frequency w2 (the red line in Fig. 3Alabeled as “Raman laser”) is a running wave,which is incoming along the z^ direction with thesame polarization as the lattice lasers. All threelaser beams are generated from the same Ti:sapphire laser, and the frequencies and amplitudesof these beams are controlled by two phase-lockedAOMs. Thus, the phase coherence is automa-tically kept, and no additional phase-locking isneeded. The Raman and lattice lasers are alsocoupled into the same optical fiber and then leadto the science chamber, which helps to avoid thephase noise due to the imperfect overlap in prop-agation. The frequency difference w1−w2 is setto 35 MHz to match the Zeeman splitting be-tween the j1;−1i and j1; 0i states. The j1; 1i stateis effectively suppressed because of a large qua-dratic Zeeman splitting, and the system can betreated as a two-level system. The detuning mz

can be adjusted by tuning the bias magneticfield. By controlling the intensities of the latticeand Raman lights, we set the lattice depth asV0x ¼ V0z and the Raman coupling strength asM0x ¼ M0y .In the experiment, the BEC is first prepared

in the dipole trap with the bias magnetic fieldbeing switched on. Then, the intensities of thelattice and the Raman beams are simultaneously

ramped up to the setting value in 40 ms. As aconsequence, the BEC atoms are adiabaticallyloaded in the local minimum of the lowest bandat the G point. The phase difference dϕL in theHamiltonian can be achieved by setting the prop-agating length between the two mirrors M1 andM2. The detection is performed in the same wayas described in (11, 18, 34). The spin-resolvedtime-of-flight (TOF) imaging is taken after alllaser beams and the bias magnetic field aresuddenly turned off and the gas has expandedfreely for 24 ms within a gradient magnetic fieldto resolve both the momentum and spin.

Experimental results

Todemonstrate the realization of 2DSO coupling,we study the crossover effect in the BEC regime bytuning dϕL. Atmz = 0 and by preparing the atomsin the spin-up state, we can adiabatically load the87Rb condensate into the G point. Then, we per-form the spin-resolved TOF expansion, whichprojects Bloch states onto free momentum stateswith fixed spin polarizations. Figure 3B showsthe TOF images for various values of dϕL. For thespin-up ðj↑iÞ state, five atom clouds are observed:The major portion of the BEC cloud remains atmomentum ðkx; kzÞ ¼ ð0;0Þ, whereas four smallfractions are transferred to momenta ðT2k0; 0Þand ð0; T2k0Þ by the first-order transitions due tothe lattice potentialVlatt. Depending on dϕL, twoor four small BEC clouds are formed in the j↓istate at the diagonal corners with momentaðTk0; Tk0Þ. This is a consequence of SO coupling;the atom clouds are generated by the Raman

SCIENCE sciencemag.org 7 OCTOBER 2016 • VOL 354 ISSUE 6308 85

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3.5

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4.5

m = 0.4E

3.4

3.8

4.2

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Topological

m = 0.1E

3.4

3.6

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Plowest subband

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second subband P

q /k0x q /k0z q /k0z

q/k

0z

q /k0x

q/k

0z

q /k0x

E(q

)/E

r

E(q

)/E

r

E(q

)/E

r

E(q

)/E

r

q /k0x

Fig. 2. Band structure and spin texture with 2D SO interaction.This figure shows an example of gapped band structure with nontrivial band topology (A), spintexture along the loop G-X-M-G (B), and spin polarization distributions hszi of the lowest band (E) and the second band (F) formz ¼ 0:1Er. (C and D) Example of atrivial bandwith gapped band structure (C) and spin texture along the loop G-X-M-G (D) formz ¼ 0:4Er. For all panels,we takeV0x ¼ V0z ¼ 5Er,M0x ¼ M0y ¼ 1:2Er,and dϕL = p/2.

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transitions, which flip spin and transfer momentaofmagnitude

ffiffiffi2

pk0 along the diagonal directions.

As given in Eqs. 3 and 4, the Raman terms Mx

and My depend on dϕL. For dϕL ¼ p=2, foursmall clouds in the j↓i statewith TOFmomentum

k→ ¼ ðTk0; Tk0Þ are observed (Fig. 3B), reflectingthe 2D SO coupling. On the other hand, by tuningthe relative phase to dϕL ¼ 3p=4, the populationof atom clouds in the two diagonal directionsbecomes imbalanced. Furthermore, the system

reduces to 1D SO couplings when dϕL ¼ p and2p, with Mx ¼ MxTMy and My ¼ 0. In thiscase, the Raman pumping generates only asingle diagonal pair of BEC clouds, as shown inFig. 3B for dϕL ¼ p; 2p. This is similar to the1D SO coupling in the free space in (11), wherethe Raman coupling flips the atom spin and gen-erates a pair of atom clouds with opposite mo-menta. Figure 3B also shows that there is adifference of distribution between the lowerleft and upper right BEC clouds at j↓i, which isdue to non–tight-binding correction. A simpleanalysis reveals that although the fully anti-symmetricRaman terms cosðk0x þ aÞcosðk0z þ bÞhave negligible effects in the tight-binding limitof the lattice, they give finite contributions inthe moderate lattice regime and are responsiblefor such difference of distribution (32). To quantifythe crossover effect,wedefineW ¼ ðN x^ − z^−N x^þ z^Þ=ðN x^ − z^ þN x^þ z^Þ to characterize the imbalance ofthe Raman coupling–induced atom clouds, withN x^ T z^ denoting the atom number of the two BECclouds along the diagonal x^Tz^ direction. W canbe fitted by a simple cosine curve cosdϕL (Fig.3C), reflecting the crossover between the 2D and1D SO couplings realized in the present BECregime.Next, we focus on the 2D isotropic SO coupling

with dϕL ¼ p=2, measure the spin distribution inthe first Brillouin zone, and detect the topologyof the bands by varying the biasmagnetic field totune d, which governs mz. For this purpose, weneed a cloud of atoms with a temperature suchthat the lowest band is occupied by a sufficientnumber of atoms, whereas the population ofatoms in the higher bands is small. A similarprocedure used in the above BEC measurementis followed, except that the atoms are cooled torelatively higher temperatures,which aremeasureda posteriori using the momentum distributionof hot atoms. After a TOF expansion, we obtainthe atom distributions of both spin-up and spin-down states in themomentumspace and thenmapthem back to the Bloch momentum space accord-ing to the plane-wave expansion of eigenfunctions.We define the spin polarization PðqÞ ¼ ½n↑ðqÞ−n↓ðqÞ�=½n↑ðqÞ þ n↓ðqÞ�, with n↑;↓ðqÞ being thedensity of atoms of the corresponding spin statein the first Brillouin zone. Figure 4, A andB, showsthe numerical results and experimentally mea-sured spin polarizations at different temperatures,respectively, for mz ¼ 0, V0x ¼ V0z ¼ 4:16Er, andM0x ¼ M0y ¼ 1:32Er. In performing numericalsimulation, the finite temperature effect is takeninto account based on the Bose-Einstein statisticsf ðEÞ ¼ 1=½eðEq −m=kBT Þ−1�, with kB the Boltzmannconstant and Eq given by band energies of theHamiltonian (Eq. 1), plus the kinetic energyℏ2q2y=2m due to the motion in the out-of-latticeplane (y) direction. The average atom density istaken as n = 3 × 1019 m–3, which determines thechemical potential m. There is good agreementbetween the theoretical and experimental results,which demonstrates the feasibility and reliabilityof the spin polarization measurement. Further-more, the results in Fig. 4 suggest that a tem-perature around T = 100 nK is optimal to extract

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Lattice Lasers,E1

Raman Lasers,E2

Probe lasers

Optical DipoleTrap Lasers

B

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1

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y

x

k /k0x

k/k

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0

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1

WO

ptical Density

Fig. 3. Experimental realization of 2D SO interaction and 1D-2D crossover. (A) Experimental setup.B is the biased magnetic field, which generates the Zeeman splitting and gives the quantum axis of theatoms. (B) Spin-resolved TOF images of BEC atoms for dϕL ¼ p/2, dϕL ¼ 3p/4, dϕL ¼ p, and dϕL ¼ 2p.Theother parameters are measured as V0x ¼ V0z ¼ 4:16Er, M0x ¼ M0y ¼ 1:32Er, and mz ¼ 0. (C) MeasuredimbalanceW between the Raman coupling–induced atoms in the two diagonal directions as a function of therelative phase dϕL, compared to a cosine curve cosdϕL. The results are averaged over ~30 TOF images.

0-0.5-1 0.5 1 0-0.5 0.5 1 0-0.5 0.5 1 0-0.5 0.5 1

-0.5

0

0.5

1

T=75nK T=125nK T=175nK T=250nK P-1

T=84 16nK T=118 13nK T=169 15nK T=263 12nK

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0

0.5

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-1

0-0.5-1 0.5 1 0-0.5 0.5 1 0-0.5 0.5 1 0-0.5 0.5 1

P

q /k0x

q/k

0z

q/k

0z

q /k0x

+1

-1

+1

-1

Fig. 4. Spin texture at different temperatures with dϕL = p/2. (A) Numerical calculations for temper-atures T = 250, 175, 125, and 75 nK. P is the polarization of the atomic spin, where +1 means spin up and–1 means spin down. (B) Experimental measurements of spin polarization at different measured tem-peratures. Parameters for the numerical calculation are the same as the experimental parameters:V0x ¼ V0z ¼ 4:16Er, M0x ¼ M0y ¼ 1:32Er, and mz ¼ 0.

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the spin-texture information of the lowest band.In comparison, if the temperature is too high,atoms are distributed over several bands and thevisibility of the spin polarization will be greatlyreduced, whereas too low a temperature can alsoreduce the experimental resolution because theatoms will be mostly condensed at the bandbottom.We then measure the spin polarization as a

function of detuningmz to reveal the topology ofthe lowest-energy band, with V0x ¼ V0z ¼ 4:16Er

and M0x ¼ M0y ¼ 1:32Er. The numerical calcu-lations and TOF-measured images of PðqÞ are inagreement (Fig. 5A and Fig. 5, B toD, respectively).In Fig. 5E, we plot the values of polarization PðLjÞfor the four highly symmetric momenta G, X1,M,and X2. PðX1Þ and PðX2Þ always have the samesign, whereas the signs of PðGÞ and PðMÞ areopposite for small jmz j and the same for largejmz j, with a transition occurring at the criticalvalue of jmc

z j that is a bit larger than 0:4Er . Attransition points, the spin polarization PðX1Þ

or PðX2Þ vanishes due to the gap closing andthermal equilibrium. From the measured spinpolarizations, the productQ and the correspondingChern number, given by Ch1 ¼ − 1−Q

4

X4

j¼1sgn

½PðLjÞ� (32, 33), can be read off (Fig. 5F). A nu-merical calculation using exact diagonalization(32) in the present experimental parameter regimecan show two transitions between the topolog-ically trivial and nontrivial bands near mc

z ¼T0:44Er, according to the theory in (33), whichagrees with the experimental observation. Notethat aroundmz = 0, the spin polarizations at X1,2

change sign through zero, implying the gap closingat X1,2 and a change of Chern number by 2. Thisconfirms that for the 2D SO-coupled system re-alized in the present experiment, the energy bandis topologically nontrivial when 0 < jmz j < jmc

z j,whereas it is trivial for jmz j > jmc

z j.Estimation of heating

The heating rate of the dipole trap ismeasured tobe 18 nK/s, mainly owing to the photon scattering

and the intensity noise. This results in a BEClifetime of about 10 s. The heating rate of thelattice and the Raman lights caused by photonscattering is about four times that of the dipoletrap, in the regime for V0 ¼ 4:16Er and M0 ¼1:32Er (32). Nevertheless, in the current exper-iment, residual heating is induced by the fluc-tuation of the bias magnetic field, which drivesadditional spin-flip dynamics in the presenceof resonant Raman couplings. This contribu-tion to the heating is about one order higherthat of the dipole trap, reducing the lifetime ofthe SO-coupled BEC to just above 300 ms. Thislifetime is sufficient to explore both single-particleand interacting physics for the 87Rb BEC system.Moreover, stabilizing the biasmagnetic fieldmayresult in an even longer lifetime of seconds inappropriate parameter regimes.

Discussion and outlook

The 2D SO coupling we realized here is for realspins (hyperfine eigenstates) of atoms, which can

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Fig. 5. Spin texture and band topology with dϕL = p/2. (A and B) Spin texture at different mz by tuning the two-photon detuning. Experimentalmeasurements (B) are compared to numerical calculations at T = 100 nK (A). (C and D) Measured spin texture in topologically trivial bands atmz ¼ −0:6Er (C)and mz ¼ 0:6Er (D). (E and F) Measured spin polarization PðLjÞ at the four symmetric momenta fLjg ¼ fG;X1;X2;Mg as a function of mz (E), and the productQ ¼ P4

j¼1sgn½PðLjÞ� (F), which determines the Chern number Ch1 and characterizes the topology of the band. In all the cases, we set V0x ¼ V0z ¼ 4:16Er andM0x ¼ M0y ¼ 1:32Er.

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be precisely measured and engineered experi-mentally. In comparison, a 2D SO coupling via atripod system (7, 24, 27) or ring-coupling scheme(25) corresponds to pseudospins defined by super-positions of multiple hyperfine levels with super-position coefficients being spatially dependent.This conceptual difference manifests the advan-tages of the present realization for future broadstudies of SO effects and interacting physics.Furthermore, owing to the realization in theoptical lattice, the present 2D SO coupling canbring about much richer physics than a pure2DRashba correspondence. In the s-band regime,the present Bloch Hamiltonian describes a quan-tum anomalous Hall model driven by SO coupling,which cannot be exactly realized in solid-statematerials. Thus, even in the single-particle regime,our realization leads tonontrivial topological bands,whereas a single-particle 2D Rashba system istopologically trivial. Moreover, even richer physicscan be obtained if considering the higher-band(e.g., p-band) regimes.Many experimental studies—including themea-

surement of topological Hall effects, Berry phasemechanism, and k-space monopole—can be per-formed on the basis of the present realization. Onthe other hand, with the high controllability of thepresent realization, the SO interaction can bereadily switched on and off and can be adjustedbetween 1D and 2D limits. This may lead to richquench spin dynamics in the optical lattice withnontrivial band topology. Moreover, with thepresent SO coupling in the optical lattice, onemay explore states of matter [such as SO-coupledMott insulators with interacting bosons (35, 36)]that have no analog in solids.Furthermore, the present optical Raman lattice

scheme is generic and can be immediately appliedto fermion systems (e.g., 40K), in which case, thequantum anomalous Hall effect in the single-particle regime and topological superfluid (28)or novel magnetic phases (37) in the interactingregimes will be especially noteworthy. In partic-ular, the topological superfluid phase is highlysought after because it hosts Majorana quasi-particles, which obey non-Abelian statistics (38)and have attracted attention in both condensedmatter and cold atom physics (5). Finally, al-though the present study is focused on a 2D latticesystem, generalizing our scheme to 3D opticallattices may lead to the realization of topologicalphases in 3D systems, including the Weyl topo-logical semimetals (39, 40).

REFERENCES AND NOTES

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ACKNOWLEDGMENTS

We thank J. Ho, T.-F. J. Poon, and G.-B. Jo for helpful discussions.This work has been supported by the Ministry of Science andTechnology of China (under grants 2016YFA0301601 and2016YFA0301604), National Natural Science Foundation of China, theCAS, the National Fundamental Research Program (under grant2013CB922001), Fundamental Research Funds for the CentralUniversities (under grants 2030020028 and 2340000034), andPeking University Initiative Scientific Research Program. X.-J.L.acknowledges support from the National Natural Science Foundation ofChina (grant 11574008). X.-J.L. and L.Z. are also supported by theThousand-Young-Talent Program of China.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/354/6308/83/suppl/DC1Supplementary TextFigs. S1 to S6References (41, 42)

11 March 2016; accepted 12 September 201610.1126/science.aaf6689

REPORTS◥

GEOPHYSICS

Localized seismic deformation in theupper mantle revealed by denseseismic arraysAsaf Inbal,* Jean Paul Ampuero, Robert W. Clayton

Seismicity along continental transform faults is usually confined to the upper half of thecrust, but the Newport-Inglewood fault (NIF), a major fault traversing the Los Angelesbasin, is seismically active down to the upper mantle. We use seismic array analysis toilluminate the seismogenic root of the NIF beneath Long Beach, California, and identifyseismicity in an actively deforming localized zone penetrating the lithospheric mantle.Deep earthquakes, which are spatially correlated with geochemical evidence of a fluidpathway from the mantle, as well as with a sharp vertical offset in the lithosphere-asthenosphere boundary, exhibit narrow size distribution and weak temporal clustering.We attribute these characteristics to a transition from strong to weak interaction regimesin a system of seismic asperities embedded in a ductile fault zone matrix.

Earthquakes occurring along transform plateboundaries are generally confined to the up-per portions of the crust, with uppermantledeformation being predominantly aseismic(1). Seismological investigationsof active fault-

ing at lower crustal depths are limited by highlyattenuated signals whose level barely exceeds thenoise at Earth’s surface, and by the sparseness ofregional seismic networks. Consequently, importantphysical parameters characterizing the transition

from brittle fracture to ductile flow at the base ofthe seismogenic zone are generally very poorlydetermined (2).Because seismic tomography usually cannot re-

solve features whose spatial extent is less thanabout 10 km in the mid-lower crust (3–5), the

88 7 OCTOBER 2016 • VOL 354 ISSUE 6308 sciencemag.org SCIENCE

Seismological Laboratory, California Institute of Technology,Pasadena, CA 91125, USA.*Corresponding author. Email: ainbal@gps.caltech.edu

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(6308), 83-88. [doi: 10.1126/science.aaf6689]354Science Pan (October 6, 2016) Si-Cong Ji, Youjin Deng, Shuai Chen, Xiong-Jun Liu and Jian-Wei Zhan Wu, Long Zhang, Wei Sun, Xiao-Tian Xu, Bao-Zong Wang,Bose-Einstein condensatesRealization of two-dimensional spin-orbit coupling for

 Editor's Summary

   

, this issue p. 83; see also p. 35Scienceatoms, it is expected that the setup would also work for fermions.spin-orbit coupling (see the Perspective by Aidelsburger). Although this experiment used bosonicinvolves only a single laser source and can be continuously tuned between one- and two-dimensional

conceived and experimentally demonstrated a simple scheme thatet al.be tricky to engineer. Wu purity and controllability of this experimental setting. However, the necessary spin-orbit coupling can

Studying topological matter in cold-atom systems may bring fresh insights, thanks to the intrinsicSpin-orbit coupling in an optical lattice

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