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 in 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=mc{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.

L 0 /2 Single δ 2 –1 –1 minimum ωz L

δ/2 E(q)/E Energy, –2 0 0 b + 3 /2 –3 ωq δ –2 –1 0 1 2 Detuning, δ / E

+1 q/k Equal population Quasimomentum, L cd –2 0 k = 0 = 2 L qq L 1

Ω 2 k = –2 L = 0 –4 3 0 1 2 3 4 5 Ω E k Raman coupling, / L 4 = 0 = 2 L b 5 q 0 –1 Raman coupling, /E = –2k = 0 10 6 L

7 L 10–2 –1.0 –0.5 0.0 0.5 1.0 –1.0 0.0 1.0 Phase Metastable 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, δ / E –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. d,Spin–momentum 0.0 0.19 decomposition. Data for sudden laser turn-off: d < 0, V 5 2EL (top image pair), Ω = 0.1E Ω = 0.3E Ω = 0.6E and V 5 6EL (bottom image pair). For V 5 2EL, | "9æ consists of :, x<0 and L L L { ;, x< 2kL ,and | #9æ consists of :, x<2kL and ;, x<0 . effectively describes spinless bosons with a tunable dispersion rela- tion16 with which we engineered synthetic electric17 and magnetic fields18 for neutral atoms. In the absence of Raman coupling, atoms with spins j"æ and j#æ spatially mixed perfectly in a BEC. By increasing V we observed an abrupt quantum phase transition to a new state where the two dressed Figure 2 | Phases of a SO-coupled BEC. a, b, Mean field phase diagrams for infinite homogeneous SO-coupled 87Rb BECs (1.5-kHz chemical potential). spins spatially separated, resulting from a modified effective inter- The background colours indicate atom fraction in | "æ and | #æ. Between the action between the dressed spins. dashed lines there are two dressed spin states, "9æ and #9æ. a, Single-particle 87 5 | | We studied SO coupling in oblate Rb BECs with about 1.8 3 10 phase diagram in the V2d plane. b, Phase diagram (enlargement of the grey atoms in a l 5 1,064-nm crossed dipole trap with frequencies (fx, fy, rectangle in a), as modified by interactions. The dots represent a metastable fz) < (50, 50, 140) Hz. The bias magnetic field B0^y generated a vZ/ region where the fraction of atoms f"9,#9 remains largely unchanged for th 5 3s. 2p < 4.81 MHz Zeeman shift between j"æ and j#æ. The Raman beams c, Miscible-to-immiscible transition. Phase line for mixtures of dressed spins and images after TOF (with populations N < N ), mapped from | "9æ and | #9æ propagated along ^y+^x and had a constant frequency difference DvL/ " # 2p < 4.81 MHz. The small detuning from the Raman resonance showing the transition from phase-mixed to phase-separated within the ‘metastable window’ of detuning. d 5 B(DvL 2 vZ) was set by B0, and the state jmF 511æ was decoupled owing to the quadratic Zeeman effect (see Methods). the existence of the SO coupling associated with the Raman dressing. We prepared BECs with an equal population of j"æ and j#æ at V, We kept d < 0 when turning on V by maintaining equal populations in d 5 0, then we adiabatically increased V to a final value up to 7EL in bare spins j"æ, j#æ (see Fig. 1d). 70 ms, and finally we allowed the system to equilibrate for a holding We experimentally studied the low-temperature phases of these time th 5 70 ms. We abruptly (toff , 1 ms) turned off the Raman lasers interacting SO-coupled bosons as a function of V and d. The zero- and the dipole trap—thus projecting the dressed states onto their temperature mean-field phase diagram (Fig. 2a, b) includes phases constituent bare spin and momentum states—and absorption-imaged composed of a single dressed spin state, a spatial mixture of both them after a 30.1-ms time of flight (TOF). For V . 4EL (Fig. 1d), the dressed spin states, and coexisting but spatially phase-separated BEC was located at the single minimum q0 of E2(q) with a single dressed spins. momentum component in each spin state corresponding to the pair This phase diagram can largely be understood as the result of non- {j", q0 1 kLæ, j#, q0 2 kLæ}. However, for V , 4EL we observed two interacting bosons condensing into the lowest-energy single particle momentum components in each spin state, corresponding to the state, and can be divided into three regimes (Fig. 2a). In the region of two minima of E2(q)atq" and q#. The agreement between the data positive detuning marked j#9æ, there are double minima at q 5 q", q# in (symbols), and the expected minima locations (curves), demonstrates E2(q)withE2(q#) , E2(q") and the bosons condense at q#. In the

region marked j"9æ the reverse holds. The energy difference between states, which are always spin-orthogonal.) Because c’:; and c2 have the two minima is D(V, d) 5 E2(q") 2 E2(q#) < d for small d (see opposite sign here, the dressed BEC can go from miscible to immiscible Methods). In the third ‘single minimum’ regime, the atoms condense at the miscibility pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi threshold19 for a two-component BEC z z ~ z at the single minimum q0. These dressed spins act as free particles with c0 c2 c’:; 2 c0ðÞc0 c2 , when V 5 Vc (this result is in agree- group velocity BKx/m (with an effective mass m* < m, for small V), ment with an independent theory presented in ref. 23). where Kx 5 q 2 q",#,0 for the different minima. Figure 2b depicts the mean field phase diagram including interac- We investigated the phase diagram using BECs with initially equal tions, computed by minimizing the interaction energy HI plus the spin populations prepared as described previously, but with d ? 0and single particle detuning D(V, d) < d. This phase diagram adds two t up to 3 s. We probed the atoms after abruptly removing the dipole new phases, mixed (hashed) and phase-separated (bold line), to those h . V R 2{ 2 ˆ trap, and then ramping 0 in 1.5 ms. This approximately mapped present in the non-interacting case. The c2 r^; r^: 2terminHI j"9æ and j#9æ back to their undressed counterparts j"æ and j#æ (see implies that the energy difference between a j"æ BEC and a j#æ BEC Methods). We absorption-imaged the atoms after a 30-ms TOF, during 2 is proportional to N c2. The detuning required to compensate for this the last 20 ms of which a Stern–Gerlach magnetic field gradient along ^y difference slightly displaces the symmetry point of the phase diagram separated the spin components. downwards. As evidenced by the width of the metastable window 2wd Figure 3a shows the condensate fraction f#9 5 N#9/(N#9 1 N"9)inj#9æ in Fig. 2b, for jdj , wd the spin-population does not have time to relax at V 5 0.6EL as a function of d,atth 5 0.1 s, 1 s and 3 s, where N"9 and to equilibrium. The miscibility condition does not depend on atom N#9 denote the number of condensed atoms in j"9æ and j#9æ, respec- number, so the phase line in Fig. 2c shows the system’s phases for tively. The BEC is all j"9æ for d=0 and all j#9æ for d>0, but both jdj , wd: phase-mixedpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for V , Vc and phase-separated for V . Vc dressed spin populations substantially coexisted for detunings within where Vc< {8c2=c0EL<0:19EL. 6wd (obtained by fitting f#9 to the error function where d 56wd We measured the miscibility of the dressed spin components from corresponds to f#9 5 0.50 6 0.16). Figure 3b shows wd versus V for their spatial profiles after TOF, for V 5 0to2EL and d < 0 such that hold times t . wd decreases with t ; even by our longest t of 3 s it h h h NT"9 < NT#9, where NT"9,#9 is the total atom number including both the has not reached equilibrium. 9æ 9æ 23 condensed and thermal components in j" , j# . For each TOF image, Conventional F 5 1 spinor BECs have been studied in Na and we numerically re-centred the Stern–Gerlach-separated spin distribu- 87Rb without Raman coupling19–21. For our j"æ and j#æ states, the tions (Fig. 2c, and see Methods), giving condensate densities n"9(x, y) interaction energy depends on the local density in each spin state, and n#9(x, y). Given that the self-similar expansion of BECs released and is described by: ð from harmonic traps essentially magnifies the in situ spatial spin dis- hi 24 DEDE 1 c2 2 c2 tribution, these reflect the in situ densities . 1 ^ 3 2 2 2 HI~ d rc0z r^:zr^; z r^ {r^ z c2zc’:; r^:r^; ~ { 2 2 2 2 2 ; : A dimensionless metric s 1 n:’n;’ = n:’ n;’ quantifies where r^: and r^; are density operators for j"æ and j#æ, and normal the degree of phase separation (where Æ...æ is the spatial average over a ordering is implied. In the 87Rb F 5 1 manifold, the spin-independent single image). s 5 0 for any perfect mixture n"9(x, y) / n#9(x, y), and 212 3 s 5 1 for complete phase separation. Figure 4 displays s versus Raman interaction is c0 5 7.79 3 10 Hz cm , the spin-dependent inter- 22 214 3 coupling V with a hold time t 5 3 s, showing that s < 0 for small V (as action is c2 523.61 3 10 Hz cm , and c’:;~0. Because h ? expected given our miscible bare spins) and s abruptly increases above jjc0 jjc2 , the interaction is almost spin-independent, but c2 , 0, so the two-component mixture of j"æ and j#æ has a spatially mixed ground a critical Vc. The inset to Fig. 4 plots s as a function of time, showing ˆ that s reaches steady state in 0.14(3) s, which is much less than th.To state (is miscible). WhenHI is re-expressed in terms of the dressed spin obtain V , we fitted the data in Fig. 4 to a slowly increasing function states, c’

Phase separations, s 3210 1.0 103 Hold time, t (s) 3 s 0.5 s 0.1 s 0.2 h 0.8

0.6 / h (Hz) 0.1 s δ 0.5 s 102 0.0 0.4 234567 234567 2 0.01 0.1 1 0.2 3 s Width, w Ω E Raman coupling, / L

Fraction of condensate 0.0 101 –1,000 –500 0 500 1,000 0.0 0.2 0.4 0.6 Figure 4 | Miscible to immiscible phase transition. Phase separation s versus h Ω E V 5 Detuning, δ/ (Hz) Raman coupling, / L with th 3 s; the solid curve is a fit to the function described in the text. The power-law component of the fit has an exponent a 5 0.75 6 0.07; this is not a Figure 3 | Population relaxation. a, Condensate fraction f#9 in | #9æ at critical exponent, but instead results from the decreasing size of the domain wall V 5 0.6EL versus detuning d at th 5 0.1, 0.5 and 3 s showing wd decrease with between the regions of | "9æ and | #9æ as V increases. Each point represents an increasing th. The solid curves are fits to the error function from which we average over 15 to 50 realizations and the uncertainties are the standard obtained the width wd. b, Metastable detuning width. Width wd versus V at deviation. Inset, phase separation s versus th with V 5 0.6EL fitted to an 22 th 5 0.1, 0.5 and 3 s; the data fits well to a[b 1 (V/EL) ](dashedcurves). exponential showing the rapid 0.14(3)-s timescale for phase separation.

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Observation of spinor dynamics in optically trapped 87Rb Bose- Einstein condensates. Phys. Rev. Lett. 92, 140403 (2004). METHODS SUMMARY 21. Ho, T.-L. Spinor Bose condensates in optical traps. Phys. Rev. Lett. 81, 742–745 System preparation. Our experiments began with nearly pure 87Rb BECs of (1998). 5 30 22. Widera, A. et al. Precision measurement of spin-dependent interaction strengths approximately 1.8 3 10 atoms in the jF 5 1, mF 521æ state confined in a 87 crossed optical dipole trap. The trap consisted of a pair of 1,064-nm laser beams for spin-1 and spin-2 Rb atoms. N. J. Phys. 8, 152 (2006). { 2 { { 23. Ho, T.-L. & Zhang, S. Bose–Einstein condensates in non-abelian gauge fields. propagating along ^x ^y (1/e radii of wx^z^y<120 mm and w^z<50 mm) and ^x ^y Preprint at Æhttp://arxiv.org/abs/1007.0650æ (2010). 2 (1/e radii of w^x{^ysemiconductors: search for mean-square field stability gmBBRMS=h=80 Hz. The field drifted slowly on longer topological Majorana particles in solid-state systems. Phys. Rev. B 82, 214509 timescales (but changed abruptly when unwary colleagues entered through our (2010). laboratory’s ferromagnetic doors). We compensated for the drift by tracking the 30. Lin, Y.-J., Perry, A. R., Compton, R. L., Spielman, I. B. & Porto, J. V. Rapid production of 87Rb Bose-Einstein condensates in a combined magnetic and optical potential. radio-frequency and Raman resonance conditions. Phys. Rev. A 79, 063631 (2009). The small energy scales involved in the experiment meant that it was crucial to minimize magnetic field gradients. We detected stray gradients by monitoring the Acknowledgements We thank E. Demler, T.-L. Ho and H. Zhai for conceptual input; and 52 æ 5 æ we appreciate conversations with J. V. Porto and W. D. Phillips. This work was partially spatial distribution of jmF 1 –jmF 0 spin mixtures after TOF. Small magnetic supported by ONR, ARO with funds from the DARPA OLE programme, and the NSF field gradients caused this otherwise miscible mixture to phase-separate along the through the Physics Frontier Center at the Joint Quantum Institute. K.J.-G. direction of the gradient. We cancelled the gradients in the x^{^y plane with two pairs acknowledges CONACYT. of anti-Helmholtz coils, aligned along ^xz^y and ^x{^y,togm B’=h =0.7 Hz mm21. B Author Contributions All authors contributed to writing of the manuscript. Y.-J. L. led Full Methods and any associated references are available in the online version of the data-taking effort in which K.J.-G. participated. I.B.S. conceived the experiment; the paper at www.nature.com/nature. performed numerical and analytic calculations; and supervised this work. Author Information Reprints and permissions information is available at Received 27 November 2010; accepted 27 January 2011. www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of this article at 1. Kato, Y. K., Myers, R. C., Gossard, A. C. & Awschalom, D. D. Observation of the spin www.nature.com/nature. Correspondence and requests for materials should be Hall effect in semiconductors. Science 306, 1910–1913 (2004). addressed to I.B.S. ([email protected]).

METHODS V 2 a a V 2 87 aðÞ2 ~ < System preparation. Our experiments began with nearly pure Rb BECs of 2 z 2 256 E 5 30 4EL Bvq L approximately 1.8 3 10 atoms in the jF 5 1, mF 521æ state confined in a crossed optical dipole trap. The trap consisted of a pair of 1,064-nm laser beams In these expressions, we have retained only the largest term in a 1/vq expansion. In 2 propagating along ^x{^y (1/e radii of wx^z^y<120 mm and w^z<50 mm) and {^x{^y our experiment, where Bvq 5 3.8EL, d is substantially changed at our largest coup- 2 (1/e radii of w^x{^y

? is, approximately sgn(d/EL)log10jd/ELj for jjd dmin, to linear, that is, approxi- To analyse the miscibility from the TOF images where a Stern–Gerlach gradient = mately d for jjd dmin, where dmin/EL 5 0.001EL 5 1.5 Hz. separated individual spin states, we re-centred the distributions to obtain n"9(x, y) In our measurement of the dressed spin fraction f#9 (see Fig. 3a), d 5 0is and n#9(x, y). This took into account the displacement due to the Stern–Gerlach determined from the NT"9 5 NT#9 condition. We identify this condition as gradient and the non-zero velocities B x=m of each spin state (after the adiabatic d 5 d0 and apply it for all hold times th. Because jd0j < 3 Hz is below our appro- mapping). The two origins were determined in the following way: we loaded the ximately 80-Hz root-mean-square field noise, we are unable to distinguish d0 dressed states at a desired coupling V but with detuning d chosen to put all atoms ~ { 2 2 from 0. in either j#9æ or j"9æ. Because q:,; + 1 V 32EL kL (see Fig. 1c), these velo- ~ z { Recombining TOF images of dressed spins. To probe the dressed spin states cities B x=m B q: kL m, B q; kL m depend slightly on V, and our tech- (equation (3)), each of which is a spin and momentum superposition, we adiabatically nique to determine the origin of the distributions accounts for this effect. ~ z ~ { mapped them into bare spins, :, x q: kL and ;, x q; kL , respectively. Calibration of Raman coupling. Both Raman lasers were derived from the same Then,ineachimageoutsidean,90-mm radius disk containing the condensate for Ti:sapphire laser at l < 804.1 nm, and were offset from each other by a pair of each spin distribution, we fitted nT"9,T#9(x, y) to a gaussian modelling the thermal acousto-optic modulators driven by two phase-locked frequency synthesizers near background and subtracted that fit from nT"9,T#9(x, y) to obtain the condensate two- 80 MHz. We calibrated the Raman coupling strength V by fitting the three-level dimensional density n"9,#9(x, y). Thus, for each dressed spin we readily obtained the Rabi oscillations between the mF 521, 0 and 1 1 states driven by the Raman temperature, total number NT"9,T#9, and condensate densities n"9,#9(x, y). coupling to the expected behaviour.