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LETTER Doi:10.1038/Nature09887 LETTER doi:10.1038/nature09887 Spin–orbit-coupled Bose–Einstein condensates Y.-J. Lin1, K. Jime´nez-Garcı´a1,2 & I. B. Spielman1 Spin–orbit (SO) coupling—the interaction between a quantum a parametrizes the SO-coupling strength; V 52gmBBz and d 52gmBBy particle’s spin and its momentum—is ubiquitous in physical sys- result from the Zeeman fields along ^z and ^y, respectively; and sx,y,z are tems. In condensed matter systems, SO coupling is crucial for the the 2 3 2 Pauli matrices. Without SO coupling, electrons have group 1,2 3–5 spin-Hall effect and topological insulators ; it contributes to velocity vx 5 Bkx/m, independent of their spin. With SO coupling, their 2 the electronic properties of materials such as GaAs, and is import- velocity becomes spin-dependent, vx 5 B(kx 6 2am/B )/m for spin j"æ ant for spintronic devices6. Quantum many-body systems of ultra- and j#æ electrons (quantized along ^y). In two recent experiments, this cold atoms can be precisely controlled experimentally, and would form of SO coupling was engineered in GaAs heterostructures where therefore seem to provide an ideal platform on which to study SO confinement into two-dimensional planes linearized the native cubic SO coupling. Although an atom’s intrinsic SO coupling affects its coupling of GaAs to produce a Dresselhaus term, and asymmetries in the electronic structure, it does not lead to coupling between the spin confining potential gave rise to Rashba coupling. In one experiment a and the centre-of-mass motion of the atom. Here, we engineer SO persistent spin helix was found6, and in another the SO coupling was coupling (with equal Rashba7 and Dresselhaus8 strengths) in a only revealed by adding a Zeeman field10. neutral atomic Bose–Einstein condensate by dressing two atomic SO coupling for neutral atoms enables a range of exciting experi- spin states with a pair of lasers9. Such coupling has not been rea- ments, and importantly, it is essential in the realization of neutral atom lized previously for ultracold atomic gases, or indeed any bosonic topological insulators. Topological insulators are novel fermionic band system. Furthermore, in the presence of the laser coupling, the insulators including integer quantum Hall states and now spin interactions between the two dressed atomic spin states are modi- quantum Hall states that insulate in the bulk, but conduct in topo- fied, driving a quantum phase transition from a spatially spin- logically protected quantized edge channels. The first-known topo- mixed state (lasers off) to a phase-separated state (above a critical logical insulators—integer quantum Hall states11—require large laser intensity). We develop a many-body theory that provides magnetic fields that explicitly break time-reversal symmetry. In a quantitative agreement with the observed location of the trans- seminal paper3, Kane and Mele showed that in some cases SO coupling ition. The engineered SO coupling—equally applicable for bosons leads to zero-magnetic-field topological insulators that preserve time- and fermions—sets the stage for the realization of topological insu- reversal symmetry. In the absence of the bulk conductance that plagues lators in fermionic neutral atom systems. current materials, cold atoms can potentially realize such an insulator Quantum particles have an internal ‘spin’ angular momentum; this in its most pristine form, perhaps revealing its quantized edge (in two can be intrinsic for fundamental particles like electrons, or a combina- dimensions) or surface (in three dimensions) states. To go beyond the tion of intrinsic (from nucleons and electrons) and orbital for composite form of SO coupling we created, almost any SO coupling, including particles like atoms. SO coupling links a particle’s spin to its motion, and that needed for topological insulators, is possible with additional generally occurs for particles moving in static electric fields, such as the lasers12–14. nuclear field of an atom or the crystal field in a material. The coupling To create SO coupling, we select two internal ‘spin’ states from 87 results from the Zeeman interaction {m:B between a particle’s mag- within the Rb 5S1/2, F 5 1 ground electronic manifold, and label netic moment m, parallel to the spin s, and a magnetic field B present in them pseudo-spin-up and pseudo-spin-down in analogy with an elec- the frame moving with the particle. For example, Maxwell’s equations tron’s two spin states: j"æ 5 jF 5 1, mF 5 0æ and j#æ 5 jF 5 1, dictate that a static electric field E 5 E0^z inÀÁ the laboratory frame (at rest) mF 521æ. A pair of l 5 804.1 nm Raman lasers, intersecting at 2 h 5 d gives a magnetic field BSO 5 E0ðÞB=mcÀÁ{ky,kx,0 in the frame of an 90u and detuned by from Ramanpffiffiffi resonance (Fig. 1a), couple ~ 2 2 object moving with momentum Bk~B kx,ky,kz ,wherec is the speed of these states with strength V;hereBkL 2pB l and EL 5 B kL/2m are light in vacuum and m is the particle’s mass. The resulting momentum- the natural units of momentum and energy. In this configuration, the dependent Zeeman interaction 2m?BSO(k)!sxky{sykx is known as atomic Hamiltonian is given by equation (1), with kx replaced by a the Rashba7 SO coupling. In combination with the Dresselhaus8 coupling quasimomentum q and an overall EL energy offset. V and d give rise ^ ^ / 2sxky 2 sykx, these describe two-dimensional SO coupling in solids to effective Zeeman fields along z and y, respectively. The SO-coupling to first order. term 2ELqsy kL results from the laser geometry, and a 5 EL/kL is set by In materials, the SO coupling strengths are generally intrinsic l and h, independent of V (see Methods). In contrast with the electronic ~ z properties, which are largely determined by the specific material and case, the atomic Hamiltonian couples bare atomic states ji:, x q kL ~ { ~ the details of its growth, and are thus only slightly adjustable in the and ji;, x q kL with different velocities, B x=m BðÞq+kL =m. laboratory. We demonstrate SO coupling in an 87Rb Bose–Einstein The spectrum, a new energy–quasimomentum dispersion of the SO- condensate (BEC) where a pair of Raman lasers create a momentum- coupled Hamiltonian, is displayed in Fig. 1b at d 5 0 and for a range of sensitive coupling between two internal atomic states. This SO coupling couplings V. The dispersion is divided into upper and lower branches is equivalent to that of an electronic system with equal contributions of E6(q), and we focus on E2(q). For V , 4EL and small d (see Fig. 2a), 15 Rashba and Dresselhaus9 couplings, and with a uniform magnetic field B E2(q) consists of a double well in quasi-momentum , where the group in the ^y{^z plane, which is described by the single-particle Hamiltonian: velocity hE2(q)/hBq is zero. States near the two minima are dressed hi spin states, labelled as j"9æ and j#9æ.AsV increases, the two dressed B2k^2 B2k^2 V d spin states merge into a single minimum and the simple picture of H^ ~ 1 { BzB k^ :m~ 1 z s z s z2a^k s ð1Þ 2m SO 2m 2 z 2 y x y two dressed spins is inapplicable. Instead, that strong coupling limit 1Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, Maryland, 20899, USA. 2Departamento de Fı´sica, Centro de Investigacio´n y Estudios Avanzados del Instituto Polite´cnico Nacional, Me´xico DF, 07360, Me´xico. 3 MARCH 2011 | VOL 471 | NATURE | 83 ©2011 Macmillan Publishers Limited. All rights reserved RESEARCH LETTER ab a 1 4 L E )/ 0 q ( /2 E Single δ 2 –1 –1 minimum ωz L E / δ/2 Energy, –2 δ 0 0 b + 3 /2 –3 ωq δ –2 –1 0 1 2 Detuning, +1 q/k Equal population Quasimomentum, L cd –2 0 k = 0 = 2 L qq L 1 E Ω 2 k = –2 L = 0 –4 3 0 1 2 3 4 5 pling, / Ω E k Raman coupling, / L 4 = 0 = 2 L b 5 q 0 –1 Raman cou = –2k = 0 10 6 L 7 L 10–2 E –1.0 –0.5 0.0 0.5 1.0 –1.0 0.0 1.0 / Phase Metastable δ Minima location (k ) Quasimomentum, q/k separated window L L 0 2ωδ Figure 1 | Scheme for creating SO coupling. a, Level diagram. Two Mixed l 5 804.1 nm lasers (thick lines) coupled states | F 5 1, mF 5 0æ 5 | "æ and Detuning, –10–2 | F 5 1, mF 521æ 5 | #æ, differing in energy by a BvZ Zeeman shift. The lasers, B with frequency difference DvL/2p 5 (vZ 1 d/ )/2p, were detuned d from the –10–1 Raman resonance. | mF 5 0æ and | mF 511æ had a B(vZ 2 vq)energy difference; because Bvq 5 3.8EL is large, | mF 511æ can be neglected. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 b, Computed dispersion. Eigenenergies at d 5 0forV 5 0 (grey) to 5EL. When Ω E Raman coupling, / L V , 4EL the two minima correspond to the dressed spin states | "9æ and | #9æ. 9 9 V d 5 c c, Measured minima. Quasimomentum q",# of | " , # æ versus at 0, Phase mixed Phase separated Raman corresponding to the minima of E2(q). Each point is averaged over about ten coupling experiments; the uncertainties are their standard deviation.
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