Table Of ContentJosephson dynamics of a spin-orbit coupled Bose-Einstein condensate
in a double well potential
Dan-Wei Zhang,1 Li-Bin Fu,2 Z. D. Wang,3 and Shi-Liang Zhu1,4,
∗
1Laboratory of Quantum Information Technology and SPTE,
South China Normal University, Guangzhou, China
2Science and Technology Computation Physics Laboratory,
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
2 3Department of Physics and Center of Theoretical and Computational Physics,
1 The University of Hong Kong, Pokfulam Road, Hong Kong, China
0 4 Center for Quantum Information, IIIS, Tsinghua University
2
We investigate the quantum dynamics of an experimentally realized spin-orbit coupled Bose-
n Einstein condensate in a double well potential. The spin-orbit coupling can significantly enhance
a
the atomic inter-well tunneling. We find the coexistence of internal and external Josephson effects
J
in thesystem, which are moreoverinherently coupled in a complicated form even in theabsence of
0 interatomic interactions. Moreover, we show that the spin-dependent tunneling between two wells
2 can inducea net atomic spin current referred as spin Josephson effects. Such novel spin Josephson
effects can be observable for realistically experimentalconditions.
]
s
a PACSnumbers: 03.75.Lm,67.85.Hj
g
-
t
n I. INTRODUCTION
a V(x) J
u (a) (b) L(cid:112) (cid:112)(cid:112) R(cid:112)
Based on the Berry phase effect [1, 2] and its non-
q
at. Afiebldelsiainngneenuetrraalilzaattioomns[3b],ythceoncrteroaltliionngoaftsoymnt-hliegthitcginatuegre- L R (cid:58) J(cid:110)(cid:112) (cid:58)
m action has attracted great interest in recent theoretical 2 J 2
- studies [4–18], and has been realized in spinor Bose- (cid:112)(cid:110)
nd EmiennsttseionftchoendNeInSsTatgesro(uBpE[1C9s,)2i0n]atnhdeaplisoonienersienvgereaxlpseurbi-- (cid:16)b 0 (cid:14)b x L(cid:110) J R(cid:110)
o
sequent experiments of other groups [21–23]. The neu- (cid:110)(cid:110)
c
[ tral atoms in the generated effective Abelian and non-
FIG.1: (Coloronline)Aschematicrepresentationof(a)aSO
Abelian gauge fields behave like electrons in an electro-
1 coupledBECinadoublewelltrap. (b)Thedynamicprocess
magnetic field [19, 21, 22] or electrons with spin-orbit
v of the system, where the blue solid and dashed lines repre-
(SO) coupling [20]. Different from electrons that are
6 sent the inter-well tunneling without and with spin-flipping,
0 fermions, the atoms with the synthetic SO coupling can respectively, the green circles represent the Raman coupling.
3 be bosonsandtypicallyBECs. Thisbosoniccounterpart The inter-well spin-flippingtunneling induced by theRaman
4 oftheSOcoupledmaterialshasnodirectanaloginsolid- coupling is negligible under current experiment conditions
. state systems and thus has received increasing attention [20, 50]. The atomic interaction terms are not shown in the
1
0 [24–44]fordifferenttypes ofSOcoupling,differentinter- figure.
2 nalatomic structures(pseudospin-1/2,spin-1 andspin-2
1 bosons, etc.), and different external conditions (homoge-
: nous, trapped and rotated). These theoretical investiga- particular, the coherent atomic tunneling between two
v
i tionsfocusmainlyonthestaticpropertiesofSOcoupled wells results in oscillatory exchange of the BEC, which
X BECsandhaverevealrichphasediagramsoftheground- is analogous to the Josephson effects (JEs) for neutral
r states [25–28, 30, 31] and exotic vortex structures [33– atoms [45–48]. The weakly interacting BECs provide a
a
35, 39–41]. However, to our knowledge, their dynamics further context [46–48] for JEs in superconductor sys-
has been less studied [24, 42–44], where the SO coupled tems because they display a nonlinear generalization of
BECsaredemonstratedtoexhibitinterestingrelativistic typical d.c. and a.c. JEs and macroscopic quantum self-
dynamics, such as analogs of self-localization [42], Zit- trapping,allofwhichhavebeenobservedinexperiments
terbewegung [43] and Klein tunneling [44] under certain [49–51]. Apartfromtheconventionalsingle-speciesBECs
conditions. [45–48],theJosephsondynamicsoftwo-speciesBECs[52]
On the other hand, quantum dynamics of a BEC in andspinorBECswithoutSOcoupling[53]havealsobeen
a double well potential has been widely investigated. In studied [54–57]; however, the dynamics of SO coupled
BECs is yet to be explored.
Inthispaper,weinvestigatethedynamicsofaspecific
SO coupled BEC, which was realized in the experiment
∗Electronicaddress: [email protected] of the NIST group, in a double well trapping potential.
2
We find that the SO coupling in the system contributes is E (k ) = h¯2kx2 4α2k2+Ω2/4, which exhibits a
to and increases the atomic tunneling to a large extent, struc±turxe of tw2mo b±ranpches. xWe are interested in the
whichcansignificantlyenhancethe atomicJEs. The full lower energy one E (k ). There is only one minimum
x
dynamicsofthesystemcontainsbothinternalandexter- in k = 0 for large−Raman coupling Ω > 4E , where
x r
nalJEs,whicharemoreoverinherentlycoupledinacom- the atoms of both atomic levels condense. However,
plicated form even in the absence of interatomic interac- the lower branch for Ω < 4E presents two minima for
r
tions. We further demonstrate that the spin-dependent condensation of dressed pseudo-spin-up (left one) atoms
Josephsontunnelingcanleadtoanetatomicspincurrent anddressedpseudo-spin-down(rightone)atoms,respec-
by varying conditions, which we refer to as spin Joseph- tively. The Raman coupling and a small δ modulate the
son effects. The predicted spin-Josephson currents are populationof atoms inthese two states [20]. Here we fo-
robustagainsttheparameteradjustmentandvaryingini- cus onthe laterregime, i.e. Ω<4E , because suchBEC
r
tial conditions, and can be observable in the SO coupled with spin-separated and non-zero central momentum is
BECs under realistic experimental conditions. more interesting in contrast to a regular BEC with zero
The paper is organized as follows. In Sec. II we con- central momentum.
struct a model that can be used to study the quantum
To be clearer, we can rewrite the Hamiltonian (2) as
dynamics of a SO coupled BEC in a double well poten-
tial. Then, in Sec. III, the Josephson dynamics of the
H Ω/2
ceodnlystcrouucpteleddsiynstteermnailsainndveesxtitgearnteadl,JwEisthbetihneg caodmdrpelsicseadt-. hˆ0 =(cid:18)Ω/↑2 H (cid:19), (3)
↓
In Sec. IV we demonstrate that the spin JEs exhibit in
the system under realistic conditions. Brief discussions where H = h¯2(kˆ2 +2k kˆ )+ δ and H = h¯2(kˆ2
and a short conclusion are given in Sec. V. 2k kˆ ) ↑δ. S2inmce ixt is moLrexstrai2ghtforwar↓d to2dmescrxib−e
L x − 2
the system in terms of dressed pseudo-spin states com-
pared with using bare ones, we will work in the dressed
II. MODEL
pseudo-spin space and simply refer to dressed pseudo-
spinas spinfor conveniencehereafter. We alsonote that
In a very recent experiment, the NIST group realized
the parameters k , Ω and δ in the single-particle Hamil-
a synthetic SO coupling in the 87Rb BEC, in which a L
tonian can be tuned independently in a wide range [20],
pairofRamanlasersgenerateamomentum-sensitivecou-
making the SO coupled BEC a suitable platform for in-
plingbetweentwointernalatomicstates[20]. Inthebare
vestigatingtheJosephsondynamicsinthepresenceofSO
pseudo-spin basis = m = 0 and = m =
| ↑ib | F i | ↓ib | F coupling.
1 , the SO coupling is described by the single particle
− i Now we turn to consider such a SO coupled BEC in a
Hamiltonian given by [20]
double well potential denoted by V(x) as shown in Fig.
ˆh= 2pm2 Iˆ+ 21(cid:18)Ωe−δ2ikLx Ωe2iδkLx (cid:19), (1) 1to(ab).eNspoitne-itnhdaetptehneddeonut.blTeoweinllvpesottiegnattiealthheeredyisnaasmsuicmseodf
− the system, we adopt the two-mode approximation [45–
where p is the atomic momentum in the xy plane, m is 48] with the field operator
the atomic mass, δ is the tunable detuning behaved as a
Zeemanfiled, kL isthewavenumberoftheRamanlaser, Ψˆ (x) aˆ ψ (x)+aˆ ψ (x), (4)
σ Lσ Lσ Rσ Rσ
and Ω is the Raman coupling strength. Such kind of ≃
SO coupling is equivalentto that of anelectronic system
where ψ (x) is the ground state wave function of the j
withequalcontributionfromRashbaandDresselhausSO jσ
well (j =L,R) with spin σ (σ = , ), and aˆ is the an-
coupling, and thus it is effective just in one-dimension jσ
↑ ↓
nihilationoperatorforspinσ inthej well,satisfyingthe
(1D). So we restrict our discussions in 1D and focus on
the motion of atoms along x axis by freezing their y and bosonic commutation relationship [aˆjσ,aˆ†kσ′] = δjkδσσ′.
z degrees of freedom. Thevalidityofthe two-modeapproximationholdsunder
To proceed further, we introduce the dressed pseudo- twoconditions: theweakatomicinteractionandsmallef-
spins =e−ikLx b and =eikLx b[20,41],then fectiveZeemansplitting,astheatomscannotbepumped
the sin|g↑lie-particle|H↑aimilton|ia↓ni in 1D (a|l↓oing x axis) can outoftheloweststateofeachwellinthiscase. Inthesec-
be written as ondquantizationformalism,the totalHamiltonianreads
¯h2kˆ2 Ω δ
hˆ0 = 2mx +2αkˆxσz + 2σx+ 2σz, (2) H=Z dxΨˆ†(x)hhˆ0+V(x)+hˆintiΨˆ(x), (5)
where kˆ is the atomic wave vector operator, and α =
x
E /k is the SO coupling strength with E = h¯2k2/2m where the two-component field operator Ψˆ =(Ψˆ ,Ψˆ )T,
beringLthe single-photonrecoil energy. TherdispersioLnre- and the interaction Hamiltonian hˆ will be s↑peci↓fied
int
lation of the single particle Hamiltonian (2) with δ = 0 below. SubstitutingEq. (4)intoEq. (5),onecanrewrite
3
the total Hamiltonian as
0.8 0.8
H=Xj,σ εjσaˆ†jσaˆjσ +Xσσ′ (cid:16)Jσσ′aˆ†LσaˆRσ′ +h.c.(cid:17) 0.6 (a) I0 = 1 µm 0.6 (b) I0 = 2 µm
+Ω2 Xj (cid:16)aˆ†j↑aˆj↓+h.c.(cid:17) (6) η()J / Er 00..24 00..24
0
δ
+2Xj (cid:16)aˆ†j↑aˆj↑−aˆ†j↓aˆj↓(cid:17)+Hint, −−00..42 110000JJ((TV)) −0.02
1 2 3 |J(SO)|4 1 2 3 4
where ε = dxψ (x)[H + V(x)]ψ (x) 1¯hω 0.8 100J(R) 0.8
Er is thjσe sinRgle-paj∗rσticle gσround statjeσener≈gy2in jth−e 0.6 (c) I0 = 3 µm 0.6 (d) I0 = 4 µm
j well with ω being the harmonic frequency of this
j Er 0.4 0.4
wedlxl,ψJσσ(x=)ΩRψdxψ(xL∗)σ((wx)it[Hhσσ+anVd(σ¯x)r]eψfRerσr(ixn)gatonddiJffσeσr¯en=t η()J / 0.2 0.2
L∗σ 2 Rσ¯
Rspins) are the tunneling terms shown in Fig. 1(b). In 0 0
addition, the interaction Hamiltonian is given by
Hint = 21Xj (cid:16)g↑(j↑)aˆ†j↑aˆ†j↑aˆj↑aˆj↑+g↓(j↓)aˆ†j↓aˆ†j↓aˆj↓aˆj↓ FIG. 2−:0.21(Color o2nbl i/n µem)3The ene4rg−y0.21 scales o2fbt u/ µnmne3ling ter4ms
+2g(j)aˆ†j aˆ†j aˆj aˆj , J(η) as a function of b for (a) l0 = 1 µm, (b) l0 = 2 µm, (c)
↑↓ ↑ ↑ ↓ ↓(cid:17) l =3 µm, and (d) l =4 µm, respectively. In (a)-(d) we set
(7) 0 0
where gσ(jσ)′ = 2h¯m2al2⊥σσ′ dx|ψjσ(x)|2|ψjσ′(x)|2 is the effec- Ω/¯h=Er/¯h=22.5 kHz.
ofitsicsnvaceeitmltl1eaeDrtniontri.gnltNleeenrnoagtgcteththitnhabgasesttsowthRcreeieaernntegetwsdhpeitnwohiaσtavhaehnaaidgσrnσmσ′o′orbnaeedniicdntgvhl⊥eetrhtibniecetaisenl-rgw-cwaothevnlee-l w√hl01e√reπeϕ−(0(Lx)−(dx))2/2=l20 w√ithl01√l0π=e−p(x+¯hd/)m2/ω2l20b,eianngdthϕe(0oRs)c(ixll)ato=r
atomic interactionsbecause the s-wavescattering length length. Substituting Eq. (9)intotheexpressionsofJσσ′,
(which is on the order of nanometers) is much smaller one can obtain
than the inter-well distance (which is on the order of
J =J(T)+J(SO)+J(V)+J(Z),
micrometers). We have also dropped the inter-well cou- J↑↑ =J(T)+J(SO)+J(V) J(Z), (10)
pling since its strength is exponentially smaller than the J↓↓ =J =J(R), −
intra-well counterpart. The Hamiltonian (6) describes
↑↓ ↓↑
the dynamic process of the system schematically shown where the terms J(T) = h¯2 dxϕ(L)ϕ′′(R), J(SO) =
in Fig. 1(b). −2m 0 0
For simplicity, we assume the double well potential to −Er dxϕ(0L)ϕ(0R), J(V) = dRxϕ(0L)V(x)ϕ(0R), J(Z) =
be symmetric as shown in Fig. 1(a), with each well hav- 2δ dRxϕ(0L)ϕ(0R), and J(R) = Ω2R dxϕ(0L)e−2ikLxϕ(0R).
ing the same harmonic trapping frequency ω. Thus we RComparedwiththeatomictuRnnelingofaregularBEC,
have ε =ε and g(L) =g(R). Such kind of double well the SO coupled BEC in this system exhibits two addi-
L R σσ′ σσ′
potential can be generated in experiments [51] with the tional tunneling channels, the SO coupling induced tun-
form neling term J(SO) and the Raman coupling induced one
J(R). Toclarifytheeffectsofthesetermsinthetunneling
V(x)=a(x2 b2)2, (8)
− processes,weneedtoworkoutandtocomparetheenergy
where the parameters a and b are both tunable in the scales of all the terms J(η), where η = T,V,Z,SO,R .
{ }
experiments [51]. Expanding V(x) near x= b, one ob- Substituting Eqs. (9) and (8) into Eq. (10), we can ob-
tains its harmonic form as V(2)(x) =. 1mω2(±x b)2 and tain the following analytical solutions
2 ±
tthhuesBaEC=inmeωa2c/h8bw2e.llTphoetegnrtoiaulnwditshtaetaechwaspvienfucnacntibonesapo-f J(η) =ξηe−b2/l20, (11)
proximately represented by its corresponding lowest en- where ξ = h¯2 (1 b2), ξ = h¯2 (3l4 b2l2+b4),
ergy single-particle wavefunction, which can be worked T 2ml20 2 − l20 V 8mb2l40 4 0 − 0
oεutψby s(ohlevrieng thaereeqfouratjio=nsL[H,Rσ,+re12spmeωct2i(vxel±y)b.)2T]ψhejσre=- ξmZan=fiδl/e2d, ξδSiOs =ind−eEpern,daenndtlξyRt=unΩabel−ekL2tol20.thSeindcoeutbhlee-Zweeel-l
jσ jσ
sults are [24] ± structure and should be small, we here further assume
δ E and thus we focus on the comparison among
r
ψL =ϕ(0L)(x)e−ikLx, J(S≪O,R,T,V). TheeffectsofZeeman-splittinginducedtun-
ψψLR↑↓ ==ϕϕ(0(0LR))((xx))eei−kLikxL,x, (9) nelFinorglw0il∼l b√e2spb,ecwifieehdaivnetξhTe S∼ec0. IaVn.d ξV ∼ 4mh¯2b2, and
ψR↓↑ =ϕ(0R)(x)eikLx, thus ||ξξSVO|| = 8π2(λbL)2 >∼ 100. Here we have assumed
4
the same wavelength λ = 2π/k = 0.8 µm and recoil = 0 in Eq. (6), in which the single-particle Hamil-
L L int
H
frequency E /¯h = 22.5 kHz as those in the experiments tonian is given by
r
[20], b and l to be on the order of micrometers [50, 51].
0
In fact, in the regime of b2/l02 ∼[0.5,2], we find that H0 ≃J↑↑(cid:16)aˆ†L↑aˆR↑+aˆL↑aˆ†R↑(cid:17)+J↓↓(cid:16)aˆ†L↓aˆR↓+aˆL↓aˆ†R↓(cid:17)
|ξSO|>∼100max{|ξT|,|ξV|}. (12) +Ω2 (cid:16)aˆ†L↑aˆL↓+aˆL↑aˆ†L↓+aˆ†R↑aˆR↓+aˆR↑aˆ†R↓(cid:17)
aBnedsidλes,=on0e.8µcamn,cahnedckthtuhsatthξeRR∼amΩaen−-c6o4ufpolrinlg0i∼ndu1cµemd +2δ (cid:16)aˆ†L↑aˆL↑−aˆ†L↓aˆL↓+aˆ†R↑aˆR↑−aˆ†R↓aˆR↓(cid:17). (14)
L
tunneling is negligible in this system. The comparisons Here we have dropped the tunneling terms Jσσ¯ since
amongJ(η) forsometypicalparametersareshowninFig. these spin-flipping tunneling progressescanbe negligible
2. Inotherwords,wefindthatunderrealisticexperiment in the current experiment conditions [20, 50]. In order
conditions [20, 50], the spin-flipping tunneling induced to study the dynamic properties of the system, we need
by Raman coupling is negligible but the SO coupling in- to workwith the equationofmotion. The corresponding
duced tunneling term J(SO) dominates and moreover it Heisenberg equations read
greatly enhances atomic tunneling in this system. Thus
we may rewrite the tunneling terms as i¯hddtaˆjσ =[aˆjσ,H0]=Jσσaˆ¯jσ+ Ω2aˆjσ¯ +(−1)p2δaˆjσ,
(15)
J =J 0, where σ and σ¯ refer to different spin, while j and ¯j
J↑↓ J↓↑ ≈J(SO) = γE , (13) to different wells, and p = 0,1 are for σ = , , re-
↑↑ ≈ ↓↓ ≈ − r spectively. Using the mean-field approximation,↑on↓e has
nwohteerethγat=thexepn(−ewb2/tul02n)n∼elin[0g.1t,e0r.m6]s. JIt(SisO)woarntdhwJh(Rile) tino acˆajσn≃rewhaˆrijtσeit≡heaejqσuawtiitohnsajoσf mbeoitnigoncansumbers. Thus we
this SO coupled system are both tunable, enabling us to
studytheinterestingeffectsofSOcouplingintheatomic i¯ha˙jσ =Jσσa¯jσ+ Ω2ajσ¯ +(−1)p2δajσ. (16)
inter-well tunneling. For instance, one can decrease the
By defining a four-component wavefunction Φ =
effective wavenumber in x axis to the scale k 1/l so
tthuantneJl(inRg) ∼ca0n.3r7evΩivaen.dTthhiesnctahnebReamacahnie-cvoeudpbliyLnga∼dinjudsut0cinedg (HaML↑Φ,a,Lw↓h,earRe↑,thaeR↓H)Tam, Eiltqo.ni(a1n6)ofisthreewsryisttteemn aissgii¯hvdedtnΦby=
the angle between the applying Raman lasers and the
tlarragpeprinwgavpeolteenngttiha.l Ionraadltdeirtnioanti,vienlythbeysuamsinegwlaayseornsewciathn Ω2δ Ω2δ J0↑↑ J0
tune the recoil energy E to identify the enhancement HM = J2 −02 δ Ω↓↓ . (17)
of atomic tunneling duerto the SO coupling (i.e. the 0↑↑ J Ω2 2δ
effect of J(SO)) in experiments. As a first step to inves- ↓↓ 2 −2
tigate the system under current experiment conditions, We now look into the JEs in this system. Let us fur-
wehereconcentrateonthetunnelingregimegovernedby ther express ajσ as ajσ = Njσeiθjσ, where the particle
Eq. (13). numbers Njσ and phasespθjσ are all time-dependent in
general. According to Eq. (16), we can obtain
III. FULL DYNAMICS OF THE SYSTEM i¯hN˙2jσ −¯hNjσθ˙jσ =Jσσ NjσN¯jσei(θ¯jσ−θjσ) (18)
+Ω2 NjσNjσ¯epi(θjσ¯−θjσ)+(−1)pδ2Njσ.
We are now in the position to investigate the quan- p
tum dynamics of the system constructed in the previous Separating the image and real parts of Eq. (18) yields
section. We first address the non-interacting case, i.e. two groups of equations as
N˙jσ = 2Jh¯σσ NjσN¯jσsin(θ¯jσ−θjσ)+ Ωh¯ NjσNjσ¯sin(θjσ¯ −θjσ), (19)
θ˙jσ = Jhσ¯σqpNN¯jjσσ cos(θ¯jσ−θjσ)+ 2Ωh¯qNNjjσσp¯ cos(θjσ¯ −θjσ)+(−1)p2δh¯.
Eq. (19)consistsactuallyofeightcoupledequations. To φ = θ θ and ρ = N N for the phase and
j j j j j j
simplify these equations, we introduce φ = θ θ particle↓nu−mbe↑rdifferencesbe↓tw−eent↑wospinsinthesame
σ Rσ Lσ
−
and ρ =N N for the phase and particle number
σ Rσ Lσ
−
differences between two wells with the same spin σ, and
5
well j, respectively. Thus we can obtain ergyandtheJosephsondynamicsinthepresenceofweak
interactions will be presented in the next section.
1
ρ˙ = sinφ + ( 1)q sinφ ,
σ 1 σ 2 j
L 2 − L
Xj
1
ρ˙j = ( 1)p 1sinφσ+ 2sinφj, (20) IV. JOSEPHSON EFFECTS IN WEAK RAMAN
2 − L L
Xσ COUPLING REGIMES
where = 4Jσσ√N N , = 2Ω N N , and
q = 0,L11for j−=h¯R,L, RreσspeLcσtivLel2y (p−=h¯0p,1 foj↑r σj↓= , , In the preceding section, we have shown that the SO
respectively). From the above Eq. (20), we find↑th↓e coupledBECinadoublewellpotentialexhibitsthecom-
coexistence of internal JE related to ρ˙ (φ ) and external plicatedcoupledexternalandinternalJosephsondynam-
j j
JErelatedtoρ˙ (φ ). Moreover,theinternalandexternal ics. We, in this section, consider a specific dynamic pro-
σ σ
JEs are inherently coupled in a more complicated form. cess of the system in the weak Raman coupling regime
Before ending this section, we briefly discuss the (i.e. Ω/Er 1), where the external Josephson dynamic
≪
weakly interacting cases, which have been assumed to dominates. Infact,themanipulationanddetectionofthe
meet the requirement of two-mode approximation. In SO coupled BECs in this regime havebeen performedin
this regime, the mean-field analysis still works well, and experiments [20].
the dropped term can be taken into count within For the weak Raman coupling, we find that the ra-
int
H
the previous discussions. This leads to two additional tio ν J /Ω can reach several hundreds from Eq.
σσ
≡ | |
terms related to interactions into Eq. (16), and now the (13). Thus within the time scale τ ¯h/Ω 45 ms
∼ ≃
equations of motion are given by for Ω = 0.001E , one can ignore the effects of the spin-
r
flippingtunneling,whichleadstotwoexternalJosephson
i¯ha˙jσ =Jσσa¯jσ+ Ω2ajσ¯ +(−1)pδ2ajσ (21) tunneling processes for different spins. The spins in this
+gσσ ajσ 2ajσ +gσσ¯ ajσ¯ 2ajσ, regime are conservedand then the total particle number
| | | |
of spin σ N = N +N are time-independent con-
σt Lσ Rσ
wgσhσe′re=t√h√e2πh¯m2ianl2⊥σtσle0′r.acTtinhge essttriemnagttihongσoσf′thceanintbeerafcotuionndeans- setqaunattsi.onWseofamssoutmioenN(2↑1t)=inNt↓htis=caNset cfoarnsbime rpeliwcirtiytt.enThaes
d a ( 1)pδ +g a 2+g a 2 J a
i¯h Lσ = − 2 σσ| Lσ| σσ¯| Lσ¯| σσ Lσ . (22)
dt(cid:18)aRσ (cid:19) (cid:18) Jσσ (−1)p2δ +gσσ|aRσ|2+gσσ¯|aRσ¯|2 (cid:19)(cid:18)aRσ (cid:19)
Bydefiningthenormalizedinter-wellparticlenumberdif- berealizedbyFeshbachresonance[58]. Underthiscondi-
ference for spin σ as =[N N ]/N ( 1 tion, the two external Josephson tunneling processes for
σ Rσ Lσ t σ
Z − − ≤Z ≤
1), the equations of motion (19) become rather simple different spins are decoupled. We numerically calculate
in this case (similar to those for the regular two species Eqs. (23), with some typical results of the time evolu-
BECs [54]), which are given by tion of for different initial conditions being shown in
σ
Z
Fig. 3. In the calculations, we have assumed the zero
Z˙σ =−2Jh¯σσ√1−Zσsinφσ, Zeeman filed δ = 0 in Fig. 3(a) and (b), and small Zee-
φ˙σ = Jh¯σσ √1Zσ 2 cosφσ+ Uh¯σσZσ + Uh¯σσ¯Zσ¯ +(−1)p2δh¯. man field δ = 0.01Er in (c) and (d). Compared with
−Zσ zero Zeeman field cases, a small Zeeman field results in
(23)
a deviation in Josephson tunneling strengths J and in
The spin-dependent atomic density current is given by σσ
time-cumulative phases δ/2h¯ for different spins. Here
I =N ˙ . (24) (t) and (t) demonstrate the oscillatory Josephson
σ t·Zσ Ztu↑nnelings Zw↓hich are similar to the early results in Ref.
From Eq.(24), we can define the net spin current as [47]. As shown in Fig. 3(a-d), they are spin-dependent
and the dynamic evolution of each one depends on its
I =I I , (25)
s owntunnelingstrength,phaseandinitialconditions. We
↑− ↓
also calculate I , I and I in this regime with typical
and the total atomic current as σ s a
results being shown in Fig. 4. It is interesting to see
I =I +I . (26) that the spin-dependent atomic density currents due to
a
↑ ↓ the spin-related Josephson tunnelings give rise to a net
WefirstconsidertheJEsofthesysteminthenoninter- spin current (cf. Fig. 4), and moreover in some certain
acting limit, i.e. U = U = 0 in Eq. (23), which can
σσ σσ¯
6
1 (a) 1 (b) 1 (a) 1 (b)
0.5 0.5 0.5 0.5
Zσ 0 0 Zσ 0 0
−0.5 −0.5 −0.5 −0.5
−110 2 4 6 8 1 0ZZ−↑↓110 2 4 6 8 10 −110 2 4 6 8 10 −ZZ1↑↓10 2 4 6 8 10
(c) (d) (c) (d)
0.5 0.5 0.5 0.5
Zσ 0 0 Zσ 0 0
−0.5 −0.5 −0.5 −0.5
−10 2 4 6 8 10−10 2 4 6 8 10 −10 2 4 6 8 10 −10 2 4 6 8 10
t / ms t / ms t / ms t / ms
FIG. 3: (Color online) The time evolution of Zσ in nonin- FIG.5: (Coloronline)ThetimeevolutionofZσ intheweakly
teracting limitation. In (a) and (b) we have δ = 0 and interacting regime with Uσσ = 0.01Er and Uσσ′ = 0.011Er.
J↑↑ = J↓↓ = −0.1Er; In (c) and (d) we have δ = 0.01Er, Other parameters and initial conditions in (a)-(d) are the
J↑↑ =−0.905Er,andJ↓↓ =−0.105Er. Theinitialconditions same with those in Fig. 3(a)-(d),respectively.
are Z↑(0) = −Z↓(0) = 0.3, φ↑(0) = 0.5φ↓(0) = π/4 in (a)
and (c); and Z↑(0)=Z↓(0)=0, φ↑(0)=φ↓(0)=π/4 in (b)
and (d).
4000 the interaction energy U and U , which should be
(a) σσ σσ¯
Nt 2000 Uσσ,Uσσ¯ ¯hω due to the two-mode approximation.
(I, I, I) / σsa−20000 I Ttaehffrheiesicsctroseruegopqifmluetie≪dhr,eevmtihinaeetneattrtwaorocmetsisiuocplnitin,sn-wtJieneorsaseUchptσohiσwosno≈isnn. tFUTuiσongσ¯n.ue5n≪lidsnoegr0ms.pt1eraEotnyrcd.epsitcshIeanesl
↑
−40000 2 4I↓ 6 8 10 results of the time evolution of σ for the same initial
Z
4000 I conditions and parameters in Fig. 3. It clearly demon-
s
Nt 2000 (b) Ia strate that the modification of Zσ(t) due to atomic in-
(I, I, I) / σsa−20000 tc3wea(irbstaeh)cs,tt5(iho(sebun)cst)huissnainnnsecoltetihntseghigecenaniinsfieetrcegafroyna.rtctTZaionhσnde(r0eee)nvfoe=errnge0yvtehairsenysdsmmpδiainnl=loJcroo0simneinppshaoFsrmioegdne.
−4000 dynamics stillexhibit a similar oscillatoryfeature in this
0 2 4 6 8 10
t / ms regime.
FIG. 4: (color online) Josephson currents in the noninter-
acting limit. The time evolution of spin-dependent atomic To see more clearly the oscillatory properties of the
currents Iσ, a net spin current Is and a total atomic current spin JEs, we have numerically calculated the frequency
Ia in (a) for the same conditions in Fig. 3(a); and in (b) for spectra of the net spin currents I for various conditions
s
thesome conditions in Fig. 3(b).
(such as those in Fig. 3 and Fig. 5). We find that the
spectra for different cases exhibit a single peak centered
at the slightly shifted frequency, as seen in Fig. 6. The
initial conditions the total atomic current can be zero, single-peakfeature shownin Fig. 6 implies that the spin
which leads to a new interesting pure spin currents (cf. current I (t) can well be described by a sin-function,
s
Fig. 4(b)). We call such new JEs as spin Josephson whiletheweakinteratomicinteractionsorthesmallZee-
effects, which can be observable in experiments by mea- man field can merely modify the period and amplitude
suring the time-evolution of spin-dependent population of the current slightly. Thus we conclude that the spin
imbalance of the atomic gas [59]. JEs in this system are robust against the parameter ad-
For weakly interacting cases, we have to estimate justment and initial conditions.
7
atomicinter-welltunnelingandconsideronlytheinternal
dynamics in each single well. The atomic tunneling be-
tween two spins refers to spin-flipping is induced by the
Raman coupling, and thus such Josephson tunneling is
I 3(a) in the spin space. Considering the atomic gas condenses
s
of 3(c) with a finite but opposite momentum for different spins,
m 5(a) the internal JEs connect interesting quantum tunneling
u 5(c) in the momentum space.
r
ct In summary, we have investigated the quantum dy-
e
p namics of a SO coupled BEC in a symmetric double
S
well potential. The SO coupling contributes to atomic
tunneling between wells and significantly enhances JEs
for realistic conditions. We have predicted a novel spin
Josephsoneffectswhichcanbeobservedinapracticalex-
0.0 0.2 0.4 0.6 0.8 1.0 perimentsince allthe requiredingredients,including the
Frequency (kHz) SOcoupledBECsandthe tunable double wellpotential,
have already been achieved in the previous experiments.
FIG.6: (Color online) Spectraof thenetspin currentsIs for
the cases in Fig. 3(a,c) and in Fig. 5 (a,c). A single peak in
the spectrum of each case implies that the spin current Is(t)
is well described bya sin-function.
Acknowledgments
V. DISCUSSION AND CONCLUSION
This work was supported by the NBRPC
Before concluding, we briefly discuss another specific (No.2011CBA00302),the SKPBRC (Nos.2011CB922104
dynamic process of the system in the relatively strong and 2011CB921503), the NSF of China (Grant Nos.
Raman coupling regime, Ω J , which can be real- 10974059, 11075020 and 11125417), and the GRF
σσ
≫ | |
izedsuchasbytuningJ 0.1E andΩ E . Inthis (HKU7058/11P) and CRF (HKU8/CRF/11G) of the
σσ r r
∼− ∼
regime, within the time scale ¯h/J one can ignore the RGC of Hong Kong.
σσ
| |
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