Table Of ContentTunnelling of condensate magnetization in a double-well potential
O¨. E. Mu¨stecaplıog˘lu,1 M. Zhang,2 and L. You3
1Koc¸ University, Department of Physics, Rumelifeneri Yolu, 34450 Sarıyer, Istanbul, Turkey
2Center for Advanced Study, Tsinghua University, Beijing, 100084, China and
3School of Physics, Georgia Institute of Technology, Atlanta GA 30332, USA
(Dated: February 1, 2008)
Westudyquantumdynamicalpropertiesofaspin-1atomicBose-Einsteincondensateinadouble-
wellpotential. Adoptingameanfieldtheoryandsinglespatialmodeapproximation,wecharacterize
our model system as two coupled spins. For certain initial states, we find full magnetization oscil-
lations betweenwells not accompanied bymass (oratom numbers)exchange. Weidentify dynamic
regimes of collective spin variables arising from nonlinear self-interactions that are different from
theusualJosephsonoscillations. Wealsodiscussmagnetizationbeatsandincompleteoscillationsof
5 collectivespinvariablesotherthanthemagnetization. Ourstudypointstoanalternativeapproach
0 toobserve coherent tunnellingof a condensate through a (spatial) potential barrier.
0
2 PACSnumbers: 03.75.Lm,73.43.Jn,75.45.+j,75.40.Gb
n
a
Indifferentbranchesofphysics,varioussystemslarger interactionstrengthinsideeachwell[18]. Whilethisphe-
J
than atomic size have been examined to have signatures nomenon is well understood for scalar condensates, it is
7
ofpurelyquantumeffectsparticularlyinquantumcoher- so far unexplored for spin-1 condensates [19, 20, 21]. In
2
ence and tunnelling [1, 2, 3]. The macroscopic quantum thisarticle,weproposeadifferentandmoredirectmech-
1 tunnelling of magnetization (MQTM) [4, 5, 6] is such anism for observing coherent condensate Josephson dy-
v an effect that has been vigorously sought after and has namicsbasedonmagnetizationoscillationsduetoanex-
5 beenclaimedtobeobservedinmolecularnanomagnetsof ternal(spatial)Josephsoncouplingofaspin-1condensate
6 Mn acetate and Fe compounds [7, 8]. The surpris- in a double well potential, building upon previous stud-
12 8
1 −
ing event of MQTM was also believed to occur in iron ies of a spin 1/2 condensate in a double well potential
1
storage protein ferritin [9]. Apart from its fundamen- [22, 23, 24].
0
5 tal significance at the interface of quantum and classical Our model system consists of a spin-1 condensate in
0 realms,MQTMinalargespinsystemisexpectedtoplay a double-well potential. When tunnelling through the
/ a key rolein realizing technologicaladvances for future’s potential barrier is weak, the wave function effectively
h
quantum computers and for the newly emerging field of behaves as a superposition of those localized in the left
p
- spintronics. Being macroscopic quantum objects them- (L) and right (R) wells. In the mean field approxima-
nt selves, optically trapped atomic Bose-Einstein conden- tion, we write the three component order parameter as
a sates (BECs) [10, 11] are also proposed to be promising ψi(~r,t) = φLi(~r)ξi(t)+φRi(~r)ηi(t) with i = ,0,+ re-
−
u candidatesforMQTM[12],thoughstrictlyspeaking,one spectivelyforeachZeemanstate M . φ istheground
F νi
q may classify such systems as mesoscopic as well. statespatialwavefunction ofthe|i-thispincomponentin
:
v theν-thwell(ν=L,R).Validityofthemeanfieldapprox-
In optical traps hyperfine spin degrees of freedom be-
i imation is discussed in the conclusion part of the paper.
X comeaccessibleandforacondensateofatomsintheF =
As has been studied in great detail [25, 26], both spin-1
r 1 hyperfine ground state, its order parameter is a three condensates of 87Rb and 23Na atoms are dominated by
a component (M = 1,0,+1) spinor. The tunnelling as-
F − spinsymmetricinteraction. Thisleadstothewideuseof
sociatedwiththeinternal(spin)degreeoffreedomcanbe
single spatial mode approximation (SMA) [26], whereby
induced by a transverse magnetic field when it exceeds
the mode function is taken to be the same of all three
a critical value depending on the strength of magnetic
components, andis itself determined only fromthe sym-
spindipole-dipole interaction[12]. SuchJosephsonoscil-
metric interactions. We will adopt this approximation
lationswithinternaldegreesoffreedomcanbecompared
as recentstudies havefully delineatedits validity regime
to the well-known superfluid 3He Josephson oscillations
[26, 27, 28]. Denote atomic density as n (~r), hence we
observed in recent years [13]. The tunnelling associated ν
take φ = n (~r), which leads to
with the external (spatial) center of mass degree of free- νi ν
dom has been demonstratedwith cold atoms [14, 15, 16] p
Ψ(~r,t)= n (~r)ξ~(t)+ n (~r)~η(t), (1)
as well as atomic condensates in an optical lattice [17]. L R
The underlying strong correlations in spin and motional
p p
degrees of freedom in such systems may eventually lead where f~ν (f~L = ξ~ and f~R = ~η) is the spin-1 spinor asso-
to novel applications in spintronics. Beyond the semi- ciated with the condensate in the ν-th well, and should
classical macroscopic Josephson oscillation physics, two be interpreted as a column vector, while f~† a row vector
ν
tunnelling coupled condensates can also display many in our notation.
particle quantum correlations when the strength of tun- We follow the conventionof Ref. [26] to split the total
nellingcouplingistunedrelativetothetypicalmeanfield Hamiltonianforaninteractingspin-1condensateintotwo
2
parts H = H +H . After integrating over the spatial
S A
variablesandadoptingtheSMAintroducedaboveweget
1
H = ǫ f~†f~ + λ(S)f~†f~ f~†f~
S ν ν ν 2 ν ν ν ν ν 0.5
ν=L,R(cid:18) (cid:19)
X
+J(ξ~†~η+~η†ξ~), (2)
I
0
1
H = λ(A) f~†F f~ f~†F f~ , (3) 0
A 2 ν ν j ν ν j ν 1
ν=L,R j=x,y,z
X X
where F are spin-1 matrices. We have defined λ(S/A) = −0.5 0
j ν
c d~rn2(~r), the single well ground state energy 0.5 M
S/A ν 0
R ~2 R0 −0.5 −1
ǫ = d~r [ √n ]2+√n V√n , (4)
ν ν ν ν
2M ∇
Z (cid:18) (cid:19) FIG.1: TheJosephsonphasediagramforJ =0.02andλA=
and a positive tunnelling coefficient −0.01. ThetrajectoriesofEq. (9)stayonclosedpaths. λA<
0 (> 0) refers to a ferromagnetic (antiferromagnetic) spin-1
~2 condensate as for 87Rb(23Na) atoms and |λA|≪|λS|.
J = d~r √n √n +√n V√n . (5)
L R L R
2M∇ ·∇
Z (cid:18) (cid:19)
Thespinsymmetricandasymmetriccoefficientsarec = oscillationdynamics that might be useful to guide ongo-
S
4π~2(a +2a )/3M andc =4π~2(a a )/3M interms ing experimental efforts. For these solutions to be phys-
0 2 A 2 0
of the scattering length a (a ) in the−total spin channel ically meaningful, we have also studied their stabilities
0 2
of0(2)ofthetwocollidingspin-1atoms. Thetunnelling by a standard linearization analysis, and only stable so-
coupling is the same for all spin components as the po- lutions are considered.
tential is assumed spin independent, i.e., V = Vδ for We now try to formulate the evolution of the most
ij ij
simplicity. Thetermsinvolvingsmallspatialoverlapsbe- easilymeasurablequantity: magnetizationineachofthe
tween the wave functions in each well are neglected [19]. symmetric double well. Since the total magnetization
The time dependent Schrodinger equations for the of our system is a conserved quantity, we therefore only
spinorsareobtainedfromi~dξ~/dt=δH(ξ~,~η,ξ~†,~η†)/δξ~†, needtoconsiderthemagnetizationintheoneofthewells,
e.g. the left well. We further consider a special initial
which gives in a compact form,
state η = ξ , η = ξ , and η = ξ . The density
+ − 0 0 − +
dξ~ matrix of the system then becomes ρ = ξ~∗ ξ~T with
i~dt = ǫL+(λ(LS)+λ(LA))|ξ~|2−λ(LA)hL ξ~+J~η, ρij = ξi∗ξj for i,j = +,0,−. It may be not⊗ed that ρ
d~η (cid:16) (cid:17) stands for single-particle density matrix, as the many-
i~dt = Jξ~+ ǫR+(λ(RS)+λ(RA))|~η|2−λ(RA)hR ~η, (6) body density matrix is beyond the scope of a mean field
analysis. The dynamical variables contributing to the
(cid:16) (cid:17)
the equivalent coupled Gross-Pitaevskiiequation for the evolution of magnetization M are given by
twospinors. Wehavedefinedashorthandnotationh
ν
f~′∗ν ⊗f~′Tν with f~′T = (f−,−f0,f+). Both ξ~ and ~η ar≡e M = ρ1++−ρ−−; n0 =ρ00; 1
normalizedtounity,soweattempttofindthestationary R = (ρ ρ +c.c); R = (ρ +ρ );
± +0 0− 0 +− −+
states satisfying 2 ± 2
1 1
I = (ρ ρ c.c); I = (ρ ρ ). (8)
i~ d ξ~ =µ ξ~ . (7) ± 2i +0± 0−− 0 2i +−− −+
dt ~η ~η
(cid:18) (cid:19) (cid:18) (cid:19) We note that R2 + I2 =
0 0
[(1 n )2+M)][(1 n )2 M]/4, which comes di-
In this short article, we will focus on the tunnelling − 0 − 0 −
rectly from the definitions of R , I , M, and the
dynamics in a symmetric double well with ǫ = ǫ = 0 0
ǫ,λ(S) = λ(S) = λ ,λ(A) = λ(A) = λ . InLthis cRase, normalization condition for f~. Quite generally, there
L R S L R A is a conserved quantity, R , in this system. Further
inter-well tunnelling couples the same spin components, +
restricting the initial state to ξ = 0 we find a simple
while intra-well interactions cause spin mixing, coupling 0
situation analogous to the standard Josephson junction
components + and to 0 . We have performed ex-
| i |−i | i for a scalar condensate in a double well [19, 29]
tensivesimulationsanddiscoveredavarietyofinteresting
solutions: self-trapping for each spinor component, for M˙ = 4JI ,
combinations of different spinor components, as well as 0
general nonlinear Josephson type complete oscillations. R˙0 = 2λAI0M,
−
Toillustratetheseresults,wewillselectivelydisplaysome I˙ = 2λ R M JM. (9)
0 A 0
−
3
1 1500
0.5
1000
M 0 τ
500
−0.5
0
−−1100 −50 θ0 50 100 0 1 2J/|λA| 3 4
+−
FIG. 2: The phase portrait for J =0.0051 and λA=−0.01. FIG. 3: Dependenceof theperiod to the2J/|λA|.
1 1
This set of equations can be interpreted as the familiar
two state optical Bloch equations, albeit nonlinear ones,
with R and I being the analogous real and imaginary
0 0 0.8 0.5
parts of the atomic dipole moment. M then acts as the
population inversion. The fixed point I = M = 0 is a
0
center, which is not an attractor. When J λ , there
A
≤| | M 0.6 0
exists another fixed point I =2λ R J =0, which is
0 A 0
−
unstable since one of its three characteristic frequencies
is positive. Since M is physically bounded between 1
and 1, we anticipate there exists a limit cycle in this 0.4 −0.5
−
system. For ξ =1, we find R =(1 M2)λ /4J. The
+ 0 A
−
phasediagramsfor the caseoffullJosephsonoscillations
is shown in Fig. 1. Alternatively, we can construct two 0.2 −1
0 500 1000 0 500 1000
dimensional phase portraits by introducing an auxiliary t t
variable θ =tan−1(I /R ) as illustrated in Fig. 2.
+− 0 0
Equation(9)canbeanalyticallysolvedintermsofthe FIG. 4: Initial state is ξ~(0) = (1,0,0) and ~η(0) = (0,0,1).
Jacobi elliptic functions [29] with the oscillation period
ǫ = 1.0, λS = 1.0, and λA = −0.01. Left figure shows self
given by
trappingwhen2J ≤|λA|,solidlineforJ =0.001,dashedline
for J =0.0049. Right figure shows magnetization oscillation,
τ = 2K(|λ2JA|)/|λA|, 2J <|λA|, (10) solid line for J =0.0051, dashed line for J =0.01.
( 4K(|λ2JA|)/(2J), 2J >|λA|,
where K(.) is the complete elliptic integral of the first interactioncausedself-trappingdisappears,whilethefull
kind. The behavior of the oscillation period with re- magnetizationoscillationreappearsinthepreviouslyself-
spect to 2J/λ is shown in Fig. 3. As we can see, trapped regime. Yet, surprisingly, the presence of the
A
| |
for small values of 2J/λ the gradually increasing pe- nonlinearityduetoatomicself-interactionremainsvisible
A
| |
riod resembles that of a nonlinear pendulum oscillation, through dynamic phenomena analogous to self-trapping
which exhibits self-trapping. The sel-trapping condition in the collective variables of other combinations of the
is therefore 2J < λ . 2J/λ = 1 is the critical value single particle density matrix elements. In particular if
A A
| | | |
corresponding to the homoclinic orbit of the equivalent we examine the collective variable ρ ρ , we find
++ 00
−
pendulum being completely in the top position. Beyond that it does not always exhibit full oscillations. As an
thattheperioddecreaseswithincreasing2J/λ ,asthe example, we demonstrate this effect, unique to a spinor
A
| |
equivalent pendulum assumes a librator rotation, i.e. a condensate, in Fig. 5 for ρ ρ . With the given ini-
++ 00
−
full oscillation of the magnetization. Figure 4 clearly tial conditions full (complete) oscillations between 1 and
illustrates these different behaviors for the initial state 0 occur when the spin dependent atomic self-interaction
ξ~(0)=(1,0,0) and ~η(0)=(0,0,1). is turned off, as shown by the dashed-line. Inclusion of
When the population in the Zeeman state M = 0 the self-interaction, on the other hand, inhibits the full
F
| i
becomes non-zero, even very small, for instance with oscillation, leading to a similar self-trapping effect as in
ξ = η = 0.9962, ξ = η = 0.0872, and ξ = η = 0, thenon-linearJosephsondynamicsofascalarcondensate
+ − 0 0 − +
4
have simply taken the standard deviation σ(N) of the
1
atom number fluctuation to be √N consistent with
∼
0.8 the meanfieldtheory. We canalsoestimate the absolute
phase diffusion times of our system due to atom number
0.6 fluctuations. The spin symmetric interaction in a spin-
(S)
1 condensate, gives rise to a diffusion time of τ 20
00 0.4 c ∼
ρ (seconds)fortheoverallcondensatephase;whilethespin
−
++ 0.2 asymmetric interaction, gives rise to a diffusion time of
ρ
τ(A) 5000(seconds)fortherelativephasesbetweendif-
c
0 ∼
ferent condensate components. In the above estimates,
−0.2 we have simply taken τc(S) N/[σ(N)cS n ] as for a
≈ h i
scalarcondensate[18]andτ(A) N/[σ(N)c n ]forthe
c A
−0.4 ≈ h i
0 0.5 1 1.5 2 2.5 spin mixing dynamics [26]. We have also assumed an
t x 104 average condensate density of n = 1.7 1013(cm)−3
for N = 107 87Rb atoms, whichh ciorrespon×ds to a typi-
FIG. 5: Oscillation of the collective variable ρ++ −ρ00 for cal Josephson oscillation period of 1 (second). Thus
an initial state ξ~(0) = (0.9962,0.0872,0). Other parameters we conclude that macroscopic magn∼etization tunnelling
used are ǫ = 1.0, λS = 1.0, λA = −0.01 (solid line), and in a spinor condensate as modelled here can be exam-
λA=0(dashedlineforthecaseofnospindependentatomic ined faithfully with the mean field theory. We further
interaction) , and J =0.001.
note that in a recent experiment with a smaller number
of atoms N 1000, the nonlinear Josephson oscillations
∼
observed were in complete agreement with the predic-
in a double well. As a general rule, we find oscillations
tions of the mean field theory [32].
now become incomplete over any regular interval. The
values ofρ ρ canevenbecome negativeas a result
++ 00 In conclusion, we have studied the double well system
−
of including the self-interaction. We can summarize the
of a spin-1 atomic condensate using the mean field the-
above observations based on the phase space nonlinear
oryandSMA.Wehavecharacterizedthevariousregimes
dynamics. When λ is non-zero and in the self trapping
A of the nonlinear dynamics for the resulting macroscopic
regime when 2J < λ , our model system possesses one
A quantum tunnelling phenomena. In addition to features
| |
attractor, which attracts trajectories of different collec-
commonlyattachedtothescalarJosephsondynamicsofa
tive variables depending on the initial conditions.
singlecomponentcondensateinadoublewell,e.g. coher-
Intheoppositeregimetoself-trappingwhen2J > λ ,
A ent atomic population oscillation and macroscopic self-
| |
the system possesses two distinct attractors, leading to
trapping [19, 29], we havefound interestingeffects solely
a complete oscillation of magnetization between 1 and
due to the spinor nature of the condensate such as the
−
1. We can adjust the system parameters to shrink the
macroscopic oscillation and self-trapping of condensate
distance between the two attractors in the phase space,
magnetization without net changes of total atom num-
and generate beatings in the magnetization oscillation
bers within each well. The physics of these correlated
with appropriately chosen initial conditions. The math-
tunnelling dynamics is essentially the same as what was
ematical reason is simple. Due to the presence of the
found before in a double well system of a two compo-
self-interaction in the limit of 2J > λ , the orbits of
A nent (or spin 1/2) condensate [23], but can become sig-
| |
magnetization in the phase space now become open and
nificantly richer due to the additional freedom in atomic
continuously circulate around two closely spaced focuses
internal states. For instance, amplitude modulations in
(of the two attractors), leading to beating patterns.
themagnetizationoscillationbecomepossibleinaspin-1
Finally,weprovidesomesupportingargumentsforthe
condensate. We have illustrated the tunnelling induced
meanfieldtreatmentofthetunnellingdynamics. Theva-
macroscopic magnetization oscillations with several ex-
lidityofthemeanfieldtheorycanbeintuitivelyexpected
amples for a spin-1 condensate of ferromagnetic interac-
due to the large numbers of atom in our model. At N
tions. Our results highlight the coherent Josephson dy-
107 atoms,oursystemis beyondthe microscopicregim∼e,
namics of a spinor condensate in a double well, and lay
andbecomesatleastmesoscopicifnotcompletelymacro-
the ground work for future studies of the quantum state
scopic. Wenotethattheperiodofmagnetizationoscilla-
of atoms in such a system.
tion, given in Eq. (10), has a simple dependence on the
number of atoms τ 1/N. Following the same analysis We thank Dr. A. Smerzi for helpful comments
∼
asinRef. [30],wecanestimatetheeffectofatomnumber and discussions. O.E.M. acknowledges support from a
fluctuations by [τ(N √N) τ(N)]/τ(N) 1/√N, TU¨BA/GEBI˙P award. The work of L. Y. is supported
| ± − | ∼
which is less than 0.003% for the present system. We by the NSF and NASA.
5
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