Phase-imprint induced domain formations and spin dynamics in spinor condensates

0.28 MB·English

by  Chengjun Tao

Phase-imprint induced domain formations and spin dynamics in spinor condensates
∗
Chengjun Tao and Qiang Gu
Department of Physics, University of Science and Technology Beijing, Beijing 100083, P.R. China
(Dated: April1, 2009)
We demonstrate that certain domain structures can be created both in ferro- and antiferro-
magnetic spinor condensates if the initial phase is spatially modulated. Meanwhile, spin dynamics
ofthecondensatewithmodulatedphasesexhibitsexoticfeaturesincomparisonwiththoseofacon-
9 densate with a uniform phase. Weexpect that these phenomena could be observed experimentally
0 using a phase-imprintingmethod.
0
2 PACSnumbers: 03.75.Mn,03.75.Kk,75.45.+j

n
a
Spinor Bose-Einstein condensate (BEC) has been at- nations for the spin dynamics are mainly based on the
J
tracting growing attentions in the last decade since it assumptionthateachcomponentsharesthesamespatial
0
displays a variety of exotic phenomena associated with wavefunction[12,13,14],calledthesingle-modeapprox-
2
its spin degree of freedom [1, 2, 3]. Spin-domain forma- imation(SMA).ItseemsthattheSMAworkswellforthe

] tion and spin dynamics are definitely among such topics AFM BEC [4], but becomes invalid for the FM conden-
s
of particular interest. sateduetoitsdomainstructures. Wehaveeverproposed
a
g Early in 1998, soon after the experimental realiza- a two-domain model to account for the later case and
- tion of the spinor BEC [1], the MIT group investigated suggest that domain formations inside FM BECs bring
t
n the miscibility of different spin domains in the spinor about significant influence on spin dynamics [15].
a 23Na condensate[3]. The spin-dependantinteractionbe- Inthispaper,weproposeaschemeforgeneratingspin
u tween 23Na atoms is antiferromagnetic (AFM) and do- domains both in FM and AFM condensates, and dis-
q
main structures were pre-created by applying a gradient cuss the exotic spin dynamics caused by domain forma-
.
at magnetic field. However, the |mF = ±1i domains be- tions. According to this scheme, domain structures can
m come almost-completely miscible as the gradient field is be created by the spatially modulated phases, which is
turned down, indicating that spin domains is hard to be expected to be realized via phase-imprinting method in
-
d formed spontaneously in the AFM spinor condensate. A experiments[16]. Phase-imprinting,asaversatiletoolto
n new experimental result further rules out spontaneous manipulate BECs, has already been used to create dark
o domain formation in 23Na condensate [4]. Very recently, solitons [17], vortices [18, 19] and vortex rings [20] in
c a theoretical work demonstrates that the homogeneous scalar or two component condensates. One can further
[
magneticfieldcanleadtospatialmodulationalinstability expect that it applies to spinor-1 BECs as well.

1 in AFM condensates, followed by the generation of spin We start with the mean-field energy functional for a
v domains[5]. Incontrast,thespontaneousdomainforma- spinor-1 Bose condensate, which is expressed as [2]
2
tion has been observed in the ferromagnetic (FM) 87Rb
5 ~2
condensate using an in-situ phase-contrast imaging [6]. ∗ ∗
1 E = dr ∇ψ ∇ψ +V (r)ψ ψ
3 This is a pioneering approach in exploring spin domains Z (cid:20)2m i i ext i i
. in spinor bosons. Theoretically, the domain formation 1 1
1 ∗ ∗ ∗ ∗
0 is attributed to the FM interaction between 87Rb atoms + 2c0ψiψjψjψi +2c2ψiψkFijFklψlψj(cid:21) , (1)
9 which leads to spontaneous polarization in the ground
0 state [7, 8, 9, 10]. where ψ denotes the condensate wave function for the
α
: Spin dynamics in spinor BECs, usually referring to atomic BEC in the α-th internal state |m =αi and re-
v F
i evolutions of spin populations in different spin compo- peatedindicesareassumedtobesummed. Thec0 andc2
X
nents [11, 12, 13, 14], arises directly from spin exchange termsdescribecontributionsofthespin-independentand
r collisions. Taking the F = 1 manifold for example, the spin-dependent interactions between atoms respectively.
a
collision process can be expressed as |m = 1i+|m = The spin-dependent interaction could be FM if c < 0
F F 2
−1i↔2|m =0i,whichnaturallyholdtheconservation or AFM if c > 0. F is the vector of spin matrices and
F 2
oftotalspins. OwingtothequantumnatureofBECs,the V (r) is the external trap potential. Then equations of
ext
collisionprocessiscoherentsothatitleadstooscillations motion for the spinor condensate, derived from Eqn. (1)
of spin populations. Such coherent behaviors have been via variational principles, are given by [2]
observedexperimentallyin87Rbcondensatesofboththe
tFhe=AF1M[132]3Naandco2nd[1e4n]samtean[4if]o.ldSso,faarn,dthveeorryetrieccaelnetxlyplain- i~∂∂tψ+1 =[H+c2(n+1+n0−n−1)]ψ+1+c2ψ02ψ−∗1 ,
∂
i~∂tψ0 =[H+c2(n+1+n−1)]ψ0+2c2ψ+1ψ−1ψ0∗ ,
∂
∗Electronicaddress: [email protected] i~∂tψ−1 =[H+c2(n−1+n0−n+1)]ψ−1+c2ψ02ψ+∗1 ,(2)
2
where H=−2~m2 ∇2+Vext+c0(n+1+n0+n−1) and nα 3 3
represents density of the mF = α atoms. Equations (2) (a) t=0 +1 (b) t=0.002
0
are just the Gross-Pitaevskii (GP) equations for spinor- 2 -1 2
1 condensates, with the condensate wave function ex-
pressed as ψ (r,t)= n (r,t)eiθα(r,t). 1 1
α α
The GP equationsphave been intensively employed to
describespindynamicsanddomaininstabilitiesofspinor 00.0 0.2 0.4 0.6 0.8 1.0 00.0 0.2 0.4 0.6 0.8 1.0

BECs, wherein the phase θ acts as a crucial factor since 3 3
istatreefl[9e,ct1s0,th1e1,q1u2a,n1tu3m]. Hchoawreavceterr,isθticcosuoldf pthlaeycaonmdoenre- 2 (t,r)| 2 (c) t=0.01 2 (d) t=0.07
important role than recognized previously. In previous
|
works, θ is usually considered to be spatially invariant. 1 1
Hereinafter, we look at the case that the phase θ can
0 0
be spatially modulated. Such an extension is by no 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

meanstrivial. Wewillshowthatintriguingdomainstruc- 3 3
turesandspindynamicscouldbeinducedbyappropriate (e) t=0.14 (f) t=0.21
2 2
phase-modulations.

We consider a one-dimensional (1D) BEC as an ex-
1 1
ample, supposing that it is confined in the infinitely
deep square well potential, approximatelycorresponding
0 0
to the elongated cigar-shaped BEC in experiments [21]. 0.0 0.2 0.4 r 0.6 0.8 1.0 0.0 0.2 0.4 r 0.6 0.8 1.0
The initial condensate density profile is set to be
FIG. 1: Spin domain formation in the ferromagnetic spin-1
homogeneous-likeexceptatboundarieswhereittends to
condensate.
zero[22],asplottedinFig. 1(a). Wenotethattheinitial
phase is not homogeneous, but has been modulated as
ofthe spinorcondensatecanbe simulatedusingEqs. (2)
θ (r,0)=α(1−2θ(r−L/2)) , (3)
α and (3), and the results are shown in Fig. 1. It can be
seenthatthedensityprofilesofthethreecomponentsare
where
changingin different mannersafter the evolutionbegins,
1 x<0 althoughthey arethoroughlyinthesameshapeinitially.
θ(x)=(cid:26)0 x>0 (4) Theformationprocessofα=±1domainsareapparently
demonstrated from Fig. 1(b) to 1(d). On the right half-
is the step function and L is the width of the well. sideofthefigures,the+1componentoverwhelmsthe−1
Theaboveinitialconditionsareverysimilartothecase component,i.e.,the+1domainisformed. Meanwhilethe
of generating dark solitons in a one-dimensional scalar −1 domain appears on the left. The two symmetrically-
condensate, where the phase modulation is realized via located domains are in accord with the specific choice of
thephase-imprintingmethod[17]. Thescalarcondensate initialphasesasgivenby Eq. (3),whichimplies thatthe
contains only one component, corresponding to the case domains are closely related to the modulated phases.
of α = 1 in our model. For spin-1 condensate, there are Obviously,boththeshapeandlocationofdomainsare
three individual components with probably different ini- varying with the time. A very interesting phenomenon
tial phases, as described by Eq. (3) with α=±1,0 [23]. occurs during the period from Fig. 1(d) to (f), where
For the 1D condensate, the coupling constants c and the ±1 domains have been exchanging their positions.
0
c can be estimate from the 3D scattering length [21]. It is worth noting that the two domains exchange their
2
As a simple approximation, c1D/c3D ≈ l/V, where l is positions periodically with time. This kind of dynamical
0 0
the elongatedlengthofthe condensate,equivalenttothe
width of the well in our model, and V is the condensate
volume. In following calculations, l is set to be the unit
of the length. And the unit of time is defined as t = 0.1
2ml2/~. The effective interactioncoefficients are alusnoitre- 2 00..35
scaled, e.g., c′ =2ml2c /~2 where c is the 1D coupling
0 0 0
′
constant. Wenotethatbotht andc areproportional
unit 0 M0
to the square of the length.
First, we investigate a FM condensate, such as 87Rb.
According to the experiment data in Ref. [6], the 87Rb -2
condensate contains 2.1×106 atoms and the length of
the condensate is about 334µm. So tunit ≈300s and the 0.0 0.1 0.2 0.3 0.4 0.5
′ ′ t
re-scaled coefficients c and c are approximately equal
to300and−1.4respec0tively. T2hendynamicalproperties FIG. 2: Local magnetization density at r=0.1,0.3 and 0.5.
3
3 3
3.0 +1 (a) t=0 +1 (b) t=0.0002
n 0 0
atio -T1otal 2 -1 2
ul2.8
op 1 1
P
1.0
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0.8 3 3
0.60.0 0.1 t 0.2 0.3 2 (t,r)| 2 (c) t=0.002 2 (d) t=0.007
|
FIG.3: Evolutionofpopulationsofdifferentspincomponents 1 1
and total spins in a ferromagnetic condensate.
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

3 4
(e) t=0.01 00..13 (f)
behavior of the condensate is somewhat similar to the 2 0.5
2
soliton-like dynamics in FM spinor condensates studied 0 M
byZhangetal. recently[24]. InRef. [24],thesoliton-like
1
behaviors are attributed to the exchange interaction c , -2
2
while here they are resulted from the phase modulation. 0 -4
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2
Astoonedomain,theinsidemagnetizationdensityM = r t
n+ −n− exhibits perfect oscillation. Figure 2 plots the FIG.4: Domain formation in aferromagnetic spinorconden-
local magnetization density M at three different points
sate with phases are modulated according to Eq. (5). (a-e)
in the left side of the condensate. M at r = 0.1 and 0.3 show the distribution of each component at different times.
exhibits a good periodicity with almost the same period (f) shows the evolution of the local magnetization density at
withinthe time ofoursimulation. The oscillationperiod different positions.
isdependentonthedimensionalityofthecondensateand
theparameterc . Nevertheless,attheboundarybetween
2
two domains r =0.5, the magnetization density is equal imprinting induced domain structure is strongly depen-
to 0 and remains unchanging, owing to the symmetry of dant on the modulation of initial phases. Regular initial
initialdensityprofilesandthesymmetricchoiceofinitial phases can induce regular domain structures. As indi-
phases. cated above, one can even produce a simple two-domain
structure so as to facilitate experimentalprobing. More-
Figure 3 illustrates the evolution of total population
over, if the initial phases are not that regular as above,
of each spin component (spin population) and the to-
anirregularmulti-domainstructurecanbeproduced. To
tal spins. The spin population smears out detailed in-
demonstratethispoint,wesupposethattheinitialphases
formation of inner structures inside the condensate, but
are modulated in a more complicated manner, such as
it is an important factor concerning spin dynamics of
spinor condensatesandhas been intensivelystudied pre-
θ (r,0)=sin((4+α)πr) (5)
α
viously [11, 12, 13, 14]. It is already discovered that the
spin population exhibits a quantum oscillating feature; where different components are imprinted with different
thispointisalsoconfirmedinourcalculations. Moreover, phases. Figs. 4(a-e) show evolution of the condensate.
the total spin conserves during the evolution, although A multi-domain structure emerges and the condensate
the local magnetization is allowed. Comparing Figs. 2 displays very complicated dynamical behaviors. For ex-
and 3, one can get an interesting point that the period ample,thelocalmagnetizationdensityshowninFig. 4(f)
of population oscillations is different from the oscillation is still oscillating, but not periodically.
period of the local magnetization density. This reveals Another striking difference between the two cases is
that the spin population is not sufficient to characterize the time scale of producing domains. The characteris-
the whole feature of spin dynamics when the condensate tic time for the spontaneous domain formation strongly
has certain inner structures, e.g., domain structures. depends on the spin-dependant interaction (t =
unit
It is important to point out that domain structures ~/|c |n), whereas that for phase-imprint induced do-
2
can appear spontaneously in the FM condensate even if mains is mainly determined by the size of the conden-
the initial phases are not modulated at all. This kind of sate (t = 2ml2/~). The spontaneous domain forma-
unit
domain formation is attributed to the spontaneous sym- tion takes a period of relaxation time of nearly 100ms
metrybreakinginthespinspacecausebytheFMinterac- in 87Rb condensates [6]. In a similar condensate with
tion between bosons [7, 8, 9, 10] and has been confirmed l=334µm,ittakesabout0.12t ≈36stoformthetwo
unit
experimentally in the 87Rb condensate [6]. The spon- domainstructureresultedfromthephasemodulation,as
taneous domain formation usually leads to some multi- can be seen from Fig. 2. In this case, domains appear
domain structures and domains seem distributed ran- much more slowly. However, the domain formation time
domly in the condensate [6, 25]. However, the phase- can be significantly reduced by shortening the conden-
4
spin dynamics measurements show good agreementwith
3 3
(a) t=0 +1 (b) t=0.002 predictions made on the base of the SMA. Nevertheless,
2 -01 2 spin domains may be induced by applying some driving
factors. For example, a recent theory predicts that spin
1 1 domains can be generated by applying an external ho-
mogeneous magnetic field to the AFM condensates [5].
00.0 0.2 0.4 0.6 0.8 1.0 00.0 0.2 0.4 0.6 0.8 1.0 So a question arises: whether domain formation could

3 4 be induced if modulating initial phases? To answer this
(c) t=0.01 (d) t=0.06 question, we need to simulate dynamic behaviors of the
3
2 AFMcondensateaccordingtoEqs. (2)and(3),withthe
2 (t,r)| 1 12 cp′araanmdect′ertoc2bseet1t0o0baendpo1s0itriveesp.eWcteivcehloyoasnedthtehepaorbatmaienteedr
| 0 2
results are shown in Fig. 5. The evolution process of
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 AFMcondensatesisquitesimilartotheFMcase,except

3 3
that the density profile of each spin component is differ-
(e) t=0.18 (f) t=0.30
entfrom thatin the FM condensate. This indicates that
2 2
one can really create spin domains in AFM condensates
1 1 by the phase imprinting method, as does in FM con-
densates. These simulation results await experimental
0 0 validation, for example, in the AFM 23Na condensate.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
r r
FIG. 5: Evolution for domain formation in a anti- In summary, we have established that the phase-
ferromagnetic condensate. Rather similar to the ferromag- imprinting could induce spin domains both in FM and
netic case, the domain structure is mainly dependent on the AFMspinorcondensates. Domainscomeintobeingafter
initial phase. theinitialphasesareimprintedandthedomainstructure
is strongly related to the spatial modulation of phases.
These characters make the phase-imprint induced do-
sate length. When the length is reduced to about 10µm, mainformationdiffersfromthespontaneousdomainfor-
the formation time can be decreased to within 100ms. mation in the FM condensate. Even more interestingly,
Analternativewayto reducethe domainformationtime phase-imprintingofferstheopportunitytostudyspindo-
is to enrich the phase modulation pattern. As Fig. 4(f) mainsandrelatedspindynamicsintheAFMcondensate
shows,domainsarewelldevelopedatt≈0.01tunit,much wheredomainstructurecannotformspontaneously. Our
faster than the two-domain case where t ≈ 0.12tunit. In results demonstrate that the phase engineering can play
some sense, enriching the phase modulation pattern is more important roles in manipulating the quantum fea-
equivalent to reducing characteristic length of the con- ture of spinor Bose condensates than previously done.
densate.
IncomparisonwiththecaseofFMcondensate,domain This work is supported by the National Natural Sci-
formations in AFM spinor condensates, such as 23Na, is ence Foundation of China (Grant No. 10504002), the
morefascinating. Averyrecentexperimentsuggeststhat Fok Yin-Tung Education Foundation, China (Grant No.
the spatial domain structure could not be formed spon- 101008),andtheMinistryofEducationofChina(NCET-
taneously in the 23Na condensate [4]. Meanwhile, the 05-0098).
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