Effect of Induced Spin-Orbit Coupling for Atoms via Laser Fields

Xiong-Jun Liu, Mario F. Borunda, Xin Liu, and Jairo Sinova

Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA(Received 5 September 2008; published 26 January 2009)

We propose an experimental scheme to observe spin-orbit coupling effects of a two-dimensional Fermi

atomic gas cloud by coupling its internal electronic states (pseudospins) to radiation in a Lambda

configuration. The induced spin-orbit coupling can be of the Dresselhaus and Rashba type with a

Zeeman term. We show that the optically induced spin-orbit coupling can lead to a spin-dependent

effective mass under appropriate conditions with one of them able to be tuned between positive and

negative effective masses. As a direct observable we show that in the expansion dynamics of the atomic

cloud the initial atomic cloud splits into two clouds for the positive effective mass case regime, and into

four clouds for the negative effective mass regime.

DOI: 10.1103/PhysRevLett.102.046402 PACS numbers: 71.70.Ej

Spin-orbit (SO) coupling effects in semiconductors haveemerged in the solid-state community as a very active fieldof research, fueled in part by the field of spintronics [1],e.g., the engineering of devices where the spin degree offreedom of the electron is exploited for improved function-ality. This has led to new developments in the anomalousHall effect [2] and the spin Hall effect (SHE) [3,4]. Incorrespondence to the spin of an electron, the internaldegree of freedom of an atom (pseudospin) is representedby the superposition of its electronic states (hyperfinelevels). SO coupling can be equivalently depicted as theinteraction between an effective non-Abelian gauge poten-tial and a particle with (pseudo)spin. In quantum systems,the idea generating a gauge field adiabatically was pro-posed by Wilczek and Zee more than 20 years ago [5].Recently, such an idea was applied to atomic systems,where the motion of atoms in a position-dependent laserconfiguration gives rise to an effective non-Abelian gaugepotential [6–10], which can lead to an effective SO inter-action in an ultracold atomic gas [11–13].

Realization of SO interaction in atomic gases opens newpossibilities of studying spintronic effects, e.g., spin re-laxation [11], Zitterbewegung [12], and SHE, in atomicsystems which provide an extremely clean environment,allowing in a controllable fashion unique access to thestudy of complex physics. However, experimental detec-tion of such SO effects in atoms requires one to measurethe pseudospins (not just hyperfine levels) that are usuallynot directly observable for atomic systems. In this Letter,we propose an experimental scheme to study SO couplingeffects, based on a trapped two-dimensional (2D) Fermiatomic gas with a simple internal three-level -type setup.We demonstrate that an effective SO interaction, e.g.,Rashba and linear Dresselhaus terms, can be obtained bycoupling atoms with a three-level configuration to spatiallyvarying laser fields. The optically induced SO coupling canlead to a spin-dependent effective mass under proper con-dition. A direct observable of this effect is in the expansiondynamics for each of the effective mass cases after the

external trap is switched off, and we predict that the initialatomic cloud splits into two or four clouds.We consider a cloud of quasi-2D (y-z plane) Fermi

atomic gas with internal three-level -type configuration[see Fig. 1(a)] coupled to radiation. The transition jbi !jai is coupled by the laser field with Rabi frequency1 ¼10 exp½i1ðrÞ, and the transition jci ! jai is coupled byanother laser field 2 ¼ 20 exp½i2ðrÞ, where 1;2ðrÞare position-dependent phases. The Hamiltonian of asingle particle reads as H ¼ H0 þ VtrapðrÞ þHI, where

VtrapðrÞ is the 2D trap, and

HI ¼ @jaihaj ð@1jaihbj þ @2jaihcj þ H:c:Þ: (1)

Diagonalizing the interacting Hamiltonian HI yields threeeigenstates: jDi ¼ ðsinÞjbi ðcosÞeijci, jB1

i ¼ðcosÞðcosÞjbi þ ðcosÞðsinÞeijci þ ðsinÞei1 jai,and jB2

i ¼ ðsinÞðcosÞjbi þ ðsinÞðsinÞeijci ðcosÞei1 jai. Here ¼ 1 2, the mixing angles

tan ¼ 20=10 and tan ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

10 þ220

q= 0=.

The corresponding eigenvalues are ED ¼ 0 and EB1;2¼

@ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ 42

0

qÞ=2. Since spatially varying lasers are

employed, diagonalization of the interacting HamiltonianHI leads to a SU(3) gauge potential [6–9]. We consider thelarge detuning case, 2 2

0, where jED EB1j 0

|a⟩

|b⟩ |c⟩

Ω2∝ exp(iφ

2(r))Ω

1∝ exp(iφ

1(r))

∆

z

y

(a) (b)

(1/2)Ω0ei(k

1y+k

2z)

(1/2)Ω0ei(k

1y−k

2z)

FIG. 1 (color online). (a) Three-level -type system coupledto position-dependent laser fields with large detuning. (b) Laserconfiguration for 1.

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and jDi; jB1i spans a near-degenerate subspace, with

their eigenvalues far separated from that of EB2. We can

then apply the adiabatic condition by neglecting the statejB2

i, which leads to a U(2) non-Abelian adiabatic gauge

potential based on the near-degenerate subspace spannedby jD;B1

i [14]. This situation is different from the cases in

Refs. [9,11] where the adiabatic condition is also assumedbetween the states jDi and jB1

i, and thus the spin-

dependent gauge potential is still Abelian in their case[9]. The adiabatic non-Ablelian gauge potential for thepresent near-degenerate subspace is obtained via

A ðrÞ ¼ i@c

ehDj hB1

jrjB1i jDi: (2)

For our purpose, we shall set the parameters 1 ¼ 2 ¼k1y and ¼ k2z, which means 1;2 are standing waves in

the z direction but plane waves in the y direction. Suchconfiguration can be achieved by applying two laser fieldsfor each atomic transition. For1, for instance, we can settwo laser fields with the same strength, where one travelswith the wave vector k1ey þ k2ez and the other travels with

k1ey k2ez [see Fig. 1(b)]. The Rabi frequencies are then

given by 1ðrÞ ¼ 0 cosðk2zÞeik1y and 2ðrÞ ¼0 sinðk2zÞeik1y. For simplicity, in what follows, we usethe spin language and denote by j"i ¼ jDi, j#i ¼ jB1

i.Under this configuration the gauge potential (2) can berecast into A ¼ m1zey m2yez m1Iey with the

coefficients 1 ¼ @k120=ð2m2

0Þ and 2 ¼ @k2=m. In ad-

dition, the scalar potentials are given by ’"" ¼ @2

2m

20

2 k22,

’## ¼ @2

2m

20

2 k21, and ’#" ¼ 0.

To this step we can obtain the effective Hamiltonian forthe near-degenerate subspace with SO coupling in the form(neglecting constant terms):

H ¼ H0 þHSO þHz þ Vtrap; (3)

where H0 ¼ P2y

2m þ P2z

2m , HSO ¼ 1zPy þ 2yPz, Hz ¼M0z with M0 ¼ @

2

4m

20

2 k2 þ @2

0

2 , and the 2D harmonic

trap Vtrap ¼ 12m!2ðy2 þ z2Þ. A Hamiltonian of this form is

predicted to give SHE [3] for M0 ¼ 0 and planar Halleffect [15] for M0 0 in solid-state systems.

The term HSO þHz can be readily diagonalized

in the momentum space, ðHSO þHzÞdiag ¼@

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ðM0=@Þ 1ky2 þ 2

2k2z

qz, where ky;z are wave vec-

tors of atoms in the y and z directions. The associatedeigenstates are given by jþi ¼ ½cos#=2; i sin#=2T ,ji ¼ ½i sin#=2; cos#=2T with tan# ¼ 2@kz=ðM0 1@kyÞ. However, even in this situation, the full

Hamiltonian H is not diagonalizable. Practically, we canconsider the case where M2

0 22@

2ðkzFÞ2 21@

2ðkyFÞ2(the validity in realizable experimental setups will be dis-cussed below), ky;zF are the y=z components of the Fermimomenta, and expand the term ðHSO þHzÞdiag to the k2

order: ðHSO þHzÞdiag ðM0 þ 22P2z

2M0 1PyÞz. Substi-

tuting this result into Eq. (3) yields

Hð1Þ ¼ 1

2mðPy1mzÞ2þ 1

2 ~mP2zþM0zþVtrap; (4)

where 1~m ¼ 1

m þ 22

M0z. Equation (4) shows that the SO

coupling leads to a spin-dependent effective mass of atomsmoving in the z direction, with ~m ¼ ð1 m=mÞ1mand m ¼ @

2k22=M0. The eigenfunction of Eq. (4) reads

ny;nzðy; zÞ ¼ c nyð!y; yÞ ~c

nzð!z ; zÞei1my=@ji; (5)

with ji ¼ ji. Here c ny ;~c nz are harmonic oscillator

wave functions. The eigenvalues are given by Eny;nz ¼

ðny þ 12Þ@!y þ ðnz þ 1

2Þ@!z þ M0 1

2m21, where !y ¼

!, !z ¼ !

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ m=m

pare spin-dependent effective

trap frequencies due to SO coupling, and ny; nz are

integers.The results (4) and (5) are valid for the situation m<

m, where the effective mass of particles in state ji ispositive, i.e., ~m > 0. In that case the expansion of theHamiltonian up to k2 order suffices and its validity can beverified by comparing with the exact expression. On theother hand, the negative effective mass regime for the jistate can be reached when m>m. In this case one can

check the dispersion relation "k ¼ H0 þHðÞSO þHðÞ

z (for

atoms in ji) represents a double-well potential in the kzaxis; therefore a higher-order expansion with momentumkz is required to derive the effective Hamiltonian.Similarly, we consider that 4

2@4ðkzFÞ4 M4

0 and can

then expand "k up to the k4z order, which gives a double

well4-type potential form in kz momentum space ( ~m <

0), i.e., "kz ¼42

8M30

P4z þ 1

2 ~mP2z [16]. Higher-order terms in

the expansion can equivalently lead to small corrections tothe coefficients of k2z and k4z terms. Finally we get aneffective Hamiltonian for atoms at state ji:

Hð2Þ ¼ 1

2mðPy þ1mÞ2 þ 4

2

8M30

ð1þ1ÞP4z

þ 1

2 ~mð1þ2ÞP2

z M0 þ 1

2m!2r2 1

2m2

1: (6)

Here the coefficient correction 1 ¼ 4ðm=mÞ½1þðm=mÞ2 1,

2 ¼ 21þðm=mÞ 1 are small under the condition

42@

4ðkzFÞ4 M40, say,

21;2 1 in the present case.

Unlike the positive mass regime, calculation of theFermi energy in the negative mass case cannot be doneanalytically. Nevertheless, we are interested in systemswith a large number of atoms, where the Thomas-Fermi(TF) approximation is suitable [17–19]. Then the Fermienergy can be calculated by solving the equation

N ¼ X¼þ;

Zk<k

F;r<R

F

d2rd2kð2Þ ðr;k; T ¼ 0Þ; (7)

where the atomic distribution function in phase space

is given by ð2Þ ¼ ð2Þ2ðeðHð2Þ

ðr;kÞ FÞ þ 1Þ1 with

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¼ 1=kBT, and the initial size of the atomic cloud is given

by RF ¼ ð2"F 2M0 2 minf"kzgÞ1=2=ðm1=2!Þ. From

Eq. (7) the relation of Fermi energy to the number of atomsN, trap frequency, and SO coupling strength can be ob-tained numerically as shown in Fig. 2.

There are specific observables of this spin-dependenteffective mass induced by the spin-orbit coupling. First,the anisotropy of the effective mass can lead to the aniso-tropic momentum distribution. Considering the Thomas-Fermi approximation, at zero temperature and for thepositive mass case, the momentum distribution is given

by nðkÞ ¼ 12m!2 ½"F M0 @

2

2m ðk2y þ m~m

k2zÞ. This con-trasts with the result for the usual isotropic mass, wherethe atomic distribution in the momentum space is alwaysisotropic irrespective of the shape of the trap [17]. Forthe negative effective mass case, nþðkÞ has the same

form, while nðkÞ ¼ 12m!2 f"F þM0 @

2

2m ½k2y þ m~m

ð1þ2Þk2z þ m4

2@2

4M30

ð1þ 1Þk4zg. Also, a fully polarized Fermi

gas is obtained when "F <M0, and nþðkÞ ¼ 0. The an-isotropy of the momentum distribution can lead to anisot-ropy of the Fermi velocity in the y-z plane. For instance,when m=m ¼ 3=4 and "F <M0, we have vz

F vyF=2.

The anisotropy of Fermi velocity can be directly detectedby time-of-flight absorption [18,19].

However, the dramatic signature of SO effects in thepresent Fermi atomic gas resides in the expansion dynam-ics of the atomic cloud after the 2D external trap and thelaser fields are switched off. The evolution of the atomicdistribution in phase space ðr;k; T; tÞ can be calculatedby the Boltzmann transport equation assuming that at t ¼0, Vtrap ! 0 and M0 ! 0. For the present cold dilute non-

interacting Fermi gas, the evolution of b;cðr;k; T; tÞ is

followed by the ballistic law b;cðr;k; T; t > 0Þ ¼b;cðr @kt=m;k; T; 0Þ [here jk; i; ð ¼ b; cÞ are the ei-genstates after the trap and laser fields are turned off]. Thetemporal atomic spatial density can be calculated bynb;cðr; T; tÞ ¼

Rd2kb;cðr @kt=m;k; T; 0Þ. The eigen-

states jk; i; ð ¼ b; cÞ are related to the initial pseudospinbasis ji by jk; bi 1

2 jkþ k2ez;þi þ 12 jk k2ez;þi þ

12i jkþ k2ez;i 1

2i jk k2ez;i and jk; ci

12 jk þ k2ez;i þ 1

2 jk k2ez;i 12i jk þ k2ez;þi þ

12i jk k2ez;þi. We find that the atomic density is

nbðr;T;tÞ¼X¼

Z d2k

4

ðjÞ

r@t

mðkþk2ezÞ;k;T;0

þðjÞ

r@t

mðkk2ezÞ;k;T;0

; (8)

where ðjÞ ðr;k; T; 0Þ denote the distribution functions for

atoms in the state ji for the positive mass (j ¼ 1) andnegative mass (j ¼ 2) cases. The function ncðr; T; tÞ can becalculated in the same way. Practically, we can assume thatbefore the expansion begins "F <M0, and then

ðjÞþ ðr;k; T; 0Þ ¼ 0. For the positive mass case, the evolu-

tion of the atomic density can be calculated exactly:

nbðr;T;tÞ¼ffiffiffiffiffiffiffiffiffiffiffim ~m

p8ð1þ!2t2Þ@2

1þ!2t2

1þ ~mm !2t2

1=2

ln1þe ~Eþðr;tÞ

e ~Eþðr;tÞþ ln

1þe ~Eðr;tÞ

e ~Eðr;tÞ

; (9)

where ~Eðr; tÞ ¼ F þM0 m!2y2

2ð1þ!2t2Þ m!2ðz@k2t=mÞ22½1þð ~m=mÞ!2t2 .

One can verify that at t ¼ 0 the maximum point of nb isobtained at r ¼ 0, while after a sufficiently long time thereare two maximum points at y ¼ 0, z ¼ @k2t=m. As aresult, Eq. (9) represents an initial atomic cloud that splitsinto two clouds each moving in opposite directions withgroup velocities vg ¼ @k2ez=m. Using typical parame-

ters,M0 ¼ 106@ s1,m 0:963 1026 kg (6Li atoms),and k2 ¼ 0:87 107 m2, one finds ~m 4 m and vg 9:0 cm=s.The evolution of the atomic density in the negative

effective mass regime is more complicated. Again, assum-ing "F <M0, we obtain from Eq. (8) the temporal atomicdensity as (denoted by ~z ¼ z @k2t=m):

nbðr;T;tÞ¼X

¼þ;

Z d2k

162

1þexp

1þ!2t2

2m@2k2y

þ

2m@2

@kzþ

~m!2t~z

1þ ~mm !2t2

4

þmþ ~m!2t2

2m ~m@2k2zþ m!2y2

2þ2!2t2

þ m2!2~z2

2mþ2 ~m!2t2M0 F

1; (10)

where ¼ m42@

2ð1þ 1Þ=ð4M30Þ. To make a qualitative

analysis, we first calculate the time independent mo-

mentum distribution: nbðk; T; tÞ ¼ 18m!2 ðln1þe~EþðkÞ

e~EþðkÞ þln1þe~EðkÞ

e~EðkÞ Þ. Here ~EðkÞ¼ FþM0þ @22

2m @2

2mfk2yþ½ðkzk2Þ22g with ¼mð1þ2Þ=ð2 ~mÞ.Note that if k2

ffiffiffiffi

p, nbðk; T; tÞ has four distinct maxi-

mums at kz ¼ k2 ffiffiffiffi

p, k2 þ

ffiffiffiffi

p, k2

ffiffiffiffi

p, k2 þffiffiffiffi

p

, which indicates that the initial atomic cloud is com-

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

N

(×104)

εF+M0 (×104h⋅ s−1)

δm/m=1.07δm/m=1.13δm/m=1.20

FIG. 2 (color online). Number of atoms versus Fermi energy atm=m ¼ 1:07; 1:13; 1:20, corresponding to the effective mass~m ¼ 15m ;7:7m ;5m.

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posed of four overlapping clouds that will travel in the zaxis at different speeds. Using parameters similar to theprevious system,m 0:963 1026 kg (6Li atoms), k2 ¼1:0 107 m1, and M0 ¼ 0:90 106@ s1, we find that 4:43 1015 m2 and ¼ 8:9 1012 m2. Thus thefour maximums correspond to vg ¼ 13:4;7:4 cm=s.

Figure 3 displays numerical estimates of the splitting.The time at which such splitting can be observed is foundby estimating the initial size of the atomic cloud RF. For asystem with N 104–5 atoms and a trap frequency !35 Hz, RF 230 m in the positive effective mass caseand RF 127:8 m when the effective mass is negative,assuming T < TF. As a result, for the positive effectivemass regime, one can verify that the two atomic clouds willbe fully separated after the system evolves by t 5 ms.For the negative mass regime, the time is aboutt 9 ms.It should be emphasized that measurement of the presentexpansion dynamics needs to detect the density of atoms inthe hyperfine level jbi (or jci), rather than to detect thepseudospin states ji that are not differentiable for theatomic system, thus the SO effect obtained here is directlyobservable in experiments by direct imaging of the sepa-rated atomic clouds.

Now we confirm the validity of the approximationM2

0 22@

2ðkzFÞ2 21@

2ðkyFÞ2. The second inequality is

always valid, since 2 1 in the large detuning

case. Note that kzF can be calculated by kzF ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ 2mð~"F þ @

22=2mÞ=ð@2Þpfor ~m < 0 and kzF ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 ~m~"F=@2p

for ~m > 0. For the system composed ofN 104 atoms and a trap frequency! ¼ 35 Hz, and usingthe previously employed parameters, we find kzF 1:41106 m1 with 2

2@2ðkzFÞ2=M2

0 0:019 1 for ~m > 0,and kzF 3:55 106 m1 with 4

2@4ðkzFÞ4=M4

0

0:028 1 for ~m < 0 (note for the case ~m < 0 theapproximation is up to the k4z order). Thus, the first inequal-ity is also valid. Finally we estimate the TF approximationthat is considered in the calculation. TF approximationfails in a small periphery region RF R < r < RF ofthe atomic cloud [17]. For the 2D Fermi atom gas, onecan find that the ratio of R to the atomic cloud size RF

satisfies R=RF N1=2. Therefore such a small regioncan be safely neglected for the case with a large number ofatoms, say N > 104. Note that this model can be readilyextended to Bose-Einstein condensate systems [20], where,together with the atom-atom interaction, the SO couplingmay lead to intriguing new physics.In conclusion, we have proposed an experimental

scheme to study SO coupling effects for a cloud of a 2Dtrapped Fermi gas. Under certain conditions, the opticallyinduced SO coupling in atoms leads to a spin-dependenteffective mass which can be positive or negative. In theexpansion dynamics of the atomic cloud after switching offthe trap, it is shown that the initial atomic cloud splits intotwo or four clouds moving oppositely depending on tuna-ble spin-orbit coupling parameters. The present schemeprovides an applicable way to directly observe the SOcoupling in cold Fermi atoms.We gratefully acknowledge discussions with V. Galastki.

This work was supported by ONR under Grant No. ONR-N000140610122, by NSF under Grant No. DMR-0547875,and by SWAN-NRI. J. S. is a Cottrell Scholar of theResearch Corporation.

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FIG. 3 (color online). Splitting of atomic cloud for the positiveeffective mass case (a)–(c) and negative mass case (d)–(f) afterturning off the trap and laser fields. The n label represents thevalue each panel has been multiplied by to keep figure to scale.

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