Macroscopic dynamics of a trapped Bose-Einstein condensate in the presence of 1D and 2D optical lattices

4 Pages·0.14 MB·English

by  M. Kraemer

Macroscopic dynamics of a trapped Bose-Einstein condensate in the presence of 1D
and 2D optical lattices
M. Kr¨amera, L. Pitaevskiia,b and S. Stringaria
aDipartimento di Fisica, Universit`a di Trento,
and Istituto Nazionale per la Fisica della Materia,
I-38050 Povo, Italy
b Kapitza Institute for Physical Problems,
ul. Kosygina 2, 117334 Moscow, Russia
(February 1, 2008)
the so called recoil energy and s is a dimensionless pa-
2
Thehydrodynamicequationsofsuperfluidsforaweaklyin-
0 rameter providing the intensity of the laser beam. The
teractingBosegasaregeneralizedtoincludetheeffectsofpe-
0 opticalpotentialhasperiodicityd=π/q =λ/2alongthe
riodic optical potentials produced by stationary laser beams.
2 z-axis. Thecaseofa2Dlatticewillbediscussedlater. In
Thenewequationsarecharacterizedbyarenormalizedinter-
n the following we will assume that the laser intensity be
action couplingconstantandbyaneffectivemassaccounting
a large enough to create many separated wells giving rise
for the inertia of the system along the laser direction. For
J
large laser intensities the effective mass is directly related to to an array of several condensates. Still, due to quan-
2 the tunneling rate between two consecutive wells. The pre- tum tunneling, the overlap between the wave functions
2
dictions for the frequencies of the collective modes of a con- of two consecutive wells can be sufficient to ensure full

densate confined by a magnetic harmonic trap are discussed coherence. In this case one is allowed to use the Gross-
1
v forboth1Dand2Dopticallatticesandcomparedwithrecent Pitaevskii (GP) theory for the order parameter to study
8 experimental data. both the equilibrium and the dynamic behaviour of the
9 system at zero temperature [8]. Eventually, if the tun-
3 The experimentalrealizationofopticallattices[1–6]is
nellingbecomestoosmall,thefluctuationsoftherelative
1 stimulating new perpectives in the study of coherence
phasebetweenthecondensateswilldestroythecoherence
0 phenomena in trapped Bose-Einstein condensates. A
2 of the sample giving rise to new quantum configurations
firstdirectmeasurementofthecriticalJosephsoncurrent
0 associated with the transition to a Mott insulator phase
has been recently obtained in [3] by studying the center
t/ [2,6].
a ofmassmotionofamagneticallytrappedgasinthepres-
In the presence of coherence it is natural to make the
m enceofa1Dperiodicopticalpotential. Underthesecon-
ansatz
- ditions the propagation of collective modes is a genuine
d quantum effect produced by the tunneling through the Ψ(r)= Ψ (x,y)f (z)eiSk(x,y) (2)
n k k
barriersand by the superfluid behaviour associatedwith
o Xk
the coherence of the order parameter between different
c
: wells. Theeffectoftheopticalpotentialistoincreasethe for the order parameter in terms of a sum of many
v
inertia of the gas along the direction of the laser giving condensate wave-functions relative to each well. Here
i
X rise to a reduction of the frequency of the oscillation. Sk(x,y) is the phase of the k-component of the order
r The purpose of the present work is to investigate the parameter, while Ψk and fk are real functions. We
a
collectiveoscillationsofamagneticallytrappedgasinthe will make the further periodicity assumption fk(z) =
presenceof1Dand2Dopticallatticestakingintoaccount f0(z kd) where f0 is localized at the origin. The above
−
the effect of tunneling, the role of the mean field inter- assumptionsforΨandfk arejustifiedforrelativelylarge
action and the 3D nature of the sample. Under suitable values of s where the interwell barriers are significantly
conditionstheseeffectscanbedescribedbyproperlygen- higher than the chemical potential. In this case the con-
eralizing the hydrodynamic equations of superfluids [7]. densate wave functions of different sites are well sepa-
Letusassumethatthegas,atT =0,betrappedbyan rated (tight binding approximation).
externalpotentialgivenbythesumofaharmonictrapof Using the ansatz (2) for the order parameterone finds
magnetic originV andof a stationaryopticalpotential thefollowingresultforthemeanfieldexpectationvalueof
ho
Vopt modulatedalongthez-axis. The resultingpotential the effective HamiltonianH = j p2j/2m+Vext(rj) +
is given by g j<kδ(rj −rk): P (cid:0) (cid:1)
P
1
Vext = 2m(cid:0)ωx2x2+ωy2y2+ωz2z2(cid:1)+sERsin2qz (1) E =hHi=(cid:20)Z dz 2¯hm2 (∂zf0)2+f02Vopt(cid:21) ZdxdyΨ2k
whereω ,ω ,ω arethefrequenciesoftheharmonictrap, Xk
x y z
g
q =2π/λisfixedbythewavelengthofthelaserlightcre- + dzf4 dxdyΨ4
ating the stationary 1D lattice wave, ER = h¯2q2/2m is 2 (cid:20)Z 0(cid:21)Xk Z k
1
¯h2 g˜n ¯h2
+(cid:20)Z dzf02(cid:21) Zdxdy(cid:20)2m(∂r⊥Ψk)2 E =Z dVnM(cid:20) 2M +Vho+ 2m(∂r⊥S)2−δcos[d∂zS](cid:21) , (7)
Xk
¯h2 wherewehaveintroducedtherenormalizedcouplingcon-
+ Ψ2kVho(r⊥,kd)+ 2mΨ2k(∂r⊥Sk)2(cid:21) stantg˜=gd f04dz,wehaveneglectedquantumpressure
terms originaRting from the radial term in the kinetic en-
−δ ZdxdyΨkΨk+1cos[Sk−Sk+1] , (3) ergy and we have set ΨkΨk+1 ∼ Ψ2k = dnM. We have
Xk also omitted some constant terms (first two terms in eq.
(3)) which do not depend on n or on S.
M
where in the two-body and in the magnetic interaction
Withrespecttothefunctionalcharacterizingatrapped
terms as well as in the radial kinetic energy we have
Bose gas in the absence of optical confinement, one no-
ignored the overlap contributions arising from different
tices twoimportantdifferences: firstthe interactioncou-
wells. Intheevaluationoftheaxialkineticenergyandof
pling constantis renormalizeddue to the presence ofthe
the optical potential term we have instead kept also the
optical lattice. This is the result of the local compres-
overlap terms originating from consecutive wells. These
sionofthegasproducedbythetightopticalconfinement
are proportionalto the quantity
which increases the repulsive effect of the interactions.
Secondthe kinetic energyterm alongthe z-directionhas
¯h2
δ = 2 dz ∂ f (z)∂ f (z d)+f (z)f (z d)V , no longer the classical quadratic form as in the radial
z 0 z 0 0 0 opt
− Z 2m − −
direction,butexhibits aperiodicdependence onthe gra-
(4) dient of the phase. By expanding this term for small
gradients, which is the case in the study of small am-
related to the tunneling rate and responsible for the oc- plitude oscillations, one derives a quadratic term of the
currence of Josephson effects. form(h¯2/2m∗) dVn (∂ S)2characterizedbytheeffec-
M z
By setting Sk = 0 (groundstate configuration), the tive mass R
variation of E with respect to f0 yields the differential m mδd2 δ π2
equation = = (8)
m∗ ¯h2 ER 2
¯h2 ∂2 where δ is defined by eq. (4). Notice that within the
(cid:20)−2m∂z2 +sERsin2qz(cid:21)f0(z)=ε0f0(z) (5) employedapproximationthevalueofδ,andhenceofm∗,
does not depend on the number of atoms, nor on the
where ε0 is introduced to ensure the normalization con- mean field interaction.
dition d/2 dzf2 = 1 which implies that the functions The equilibrium density profile, obtained by minimiz-
−d/2 0
Ψ areRnormalized to the number of atoms N occu- ing eq.(7) with S =0 has the typicalformof aninverted
k k
pying each site: Ψ2dxdy = N . In eq. (5) we have parabola [10]
k k
ignored the contrRibution arising from the two-body in- 1
teraction. Estimates of [9] show that this is a good ap- n0M =(cid:18)µ− 2m(ωx2x2+ωy2y2+ωzz2)(cid:19)/g˜, (9)
proximationalreadyatmoderatelylarges. Wehavealso
which conserves the aspect ratio of the original mag-
neglected the external magnetic potential which is justi-
netic trapping. The size of the condensate has instead
fied if √sE >>¯hω ,¯hω . Since in the following we are
R x y
increasedsinceg˜>g. Forlargestheincreaseofthecou-
interestedin the lowenergy excitationsof the systemwe
plingconstantcanbelarge(g˜ s1/4 [9]). However,since
willalwayskeepthefunctionf0 equaltothegroundstate ∼
the radius of the sample scales like the 1/5-th power of
solution of (5).
g˜ the resulting increase in the size of the system is not
In order to discuss the macroscopic properties of the
very spectacular (for s = 15 we find an increase of the
system, including its low energy dynamics, it is conve-
size by 20% for the experimental setting of [3]).
nient to transform the discretized formalism described ∼
The functional (7) can be used to carry out dy-
above into the one of continuum variables. This is ob-
namic calculations. In this case one needs the action
tained through the replacement (1/d) dz in the
k → A= dt H i¯h ∂ ,with the secondterm givenby
various terms of the energy. ThrPough such aRprocedure h i− h∂ti
one naturally introduces a smoothed or ”macroscopic” i¯h (∂R/∂t)(cid:0) = dV¯hn(cid:1)MS˙. The resulting equations of
h i −
density defined by motion are obtaiRned by imposing the stationarity condi-
tion on the actionwith respect to arbitraryvariations of
n (x,y,z)=(1/d)Ψ2(x,y) (6) the density n and of the phase S. The equations take
M k M
the form
with z dk, and a smoothed phase S (Sk S(x,y,z)). ¯h δd
By ap≃plying the smoothing procedureto→eq.(3)we ob- n˙M + m∂r⊥(nM∂r⊥S)+ ¯h ∂z(nMsin[d∂zS]) =0, (10)
tain the following macroscopic expression for the energy ¯h2
functional ¯hS˙ +g˜nM +Vmag + 2m(∂r⊥S)2−δcos[d∂zS] =0. (11)
2
In particular, at equilibrium these equations reproduce and (13) show that in the case of center-of-mass oscilla-
result (9) for the equilibrium density. Furthermore, tions, the linearity condition is achieved for initial dis-
Josephson-typeoscillationsareamongthosecapturedby placements ∆x of the trap satisfying ∆x<< 2δ/mω2,
z
eqs. (10) and (11). To see this consider the case of a a condition that becomes more and more sevpere as the
uniform gradient of the phase along z, ∂ S = P (t)/¯h, laser intensity increases. For larger initial displacements
z Z
whereP isatime-dependentparameter. Fromeqs. (10) the oscillation is described by the pendulum equations.
Z
and(11)one canthen deriveequations ofmotionfor the For verylargeamplitudes the motionis howeverdynam-
center of mass Z(t)= dVzn (t)/N and for the conju- ically unstable [11,13].
m
gate momentum variabRle PZ [3,11] From the previous discussion it emerges that the ef-
fective mass is the crucial parameter needed to pre-
P
¯hZ˙ δdsin d Z =0, (12) dict the value of the small amplitude collective frequen-
− (cid:20) ¯h (cid:21) cies. An estimate of m/m∗ can be made by neglecting
P˙ +mω2Z =0, (13) the magnetic trapping as well as the role of the mean
Z z
field interaction. Within this approximation the effec-
which have the typical Josephson form. tive mass is easily obtained from the excitation spec-
In the limit of small oscillations the solutions of eqs. trum of the Schr¨odinger equation for the 1D Hamilto-
(10)and (11) havethe form n=n0 +δn(r)eiωt with δn nian H = (h¯2/2m)∂2/∂z2+sE sin2qz, avoiding the
M R
−
obeying the hydrodynamic equations: explicit determination of the tunneling parameter (4).
Onelooksforsolutionsoftheformeipz/h¯f (z)wherepis
µ V µ V p
−ω2δn=∂r⊥(cid:20) −m ho∂r⊥δn(cid:21)+∂z(cid:20) −m∗ho∂zδn(cid:21) , (14) the quasi-momentumof the atomand fp(z)is a periodic
function of period d. The resulting dispersion law ε(p)
whereµ=g˜n0 (0)isthechemicalpotentialofthesample provides, for small p, the effective mass according to the
and n0 (0) isMthe equilibrium density (9) evaluated at identification ε(p) ε0+p2/2m∗. The value of m/m∗,
M ≃
which turns out to be a universal function of the inten-
the center. The solutions of (14) provide the low energy
sity parameter s, has been evaluated for a wide range of
excitations of the system. In the absence of magnetic
∗
values of s (see fig.1). These results for m can be used
trapping one finds phonons propagating at the velocity
c = g˜n0 /m∗, in agreement with the result obtained to estimate the actualvalue of the collective frequencies.
M ∗
The method described here to calculate m is expected
in [1p2] for a 1D array of Josephson junctions. In the
to be reliable not only for very large laser intensities s
presenceofharmonictrappingthediscretizedfrequencies
when the tight binding approximation applies and the
of the time-dependent solutions of (14) do not depend
effectivemasscanbe expressedintermsofthe tunneling
on the value of the coupling constant. By applying the
transformation z m∗/mz, one actually finds that rate(seeeqs. (8),(4)),butalsoforsmallervaluesofs. Of
→ course for very small laser intensities, as in the experi-
the new frequencies arpe simply obtainedfromthe results
∗
ment[14],thedeterminationofm requirestheinclusion
of [7] by replacing
of the mean field interaction and of the magnetic trap-
ω ω m/m∗. (15) ping through the explicit solution of the GP-equation.
z z
→
p Infig. 2wecompareourpredictionsforthefrequencies
For an elongated trap (ωx = ωy = ω⊥ ωz) the of the center-of-mass motion with the recent experimen-
≫
lowest solutions are given by the center-of-mass mo- tal data obtained in [3]. The comparison reveals good
tion ωD = m/m∗ωz and by the quadrupole mode agreement with the experiments. Our results also agree
ω = 5/2pm/m∗ω . The center-of-mass frequency well with those obtained from the numerical solution of
Q z
coincideps witph the value obtained from eqs. (12) and the time-dependent GP-equation [11,13].
(13) in the limit of small oscillations. Concerning the The above formalism is naturally generalized to in-
quadrupolefrequency we notethat the occurrenceofthe clude a 2D optical lattice where the optical potential is
factor 5/2 is a non-trivial consequence of the mean V = sE sin2qx + sE sin2qy. The actual potential
opt R R
field inpteraction predicted by the hydrodynamic theory now generates an array of 1D condensates which has al-
of superfluids in the presence of harmonic trapping [7]. ready been the object of experimental studies [4]. For a
In addition to the low-lying axial motion the system ex- 2D-lattice the ansatz for the order parameter is [15]
hibitsradialoscillationsathighfrequency,oftheorderof
ω⊥. The mostimportantones are the transversebreath- Φ(r)= Ψkx,ky(z)fkx,ky(x,y)eiSkx,ky(z). (16)
ingandquadrupoleoscillationsoccuringatω =2ω⊥ and kXx,ky
ω =√2ω⊥ respectively. Forelongatedtrapsthefrequen-
Inthe TF-limit the groundstatesmootheddensity n =
ciesofthesemodesshouldnotbeaffectedbythepresence M
Ψ2 /d2 still has the familiar form n0 = (µ V )/g˜
of the optical potential. Different scenarios are obtained kx,ky M − ho
fordisc-shapedtraps(ωz >>ω⊥). Theaboveresultsap- with the redefined coupling constant g˜ = g d dxf04 2,
ply to the linear regime of small oscillations. Eqs. (12) where f0 is still given by the solution of eq.(cid:0)(5R) and (cid:1)we
3
have used the same approximations as in the 1D case. [12] J. Javanainen, Phys. Rev.A 60, 4902 (1999).
Also with regard to dynamics, one can proceed as for [13] A. Trombettoni, A. Smerzi, unpublished.
the 1D lattice. One finds that the equations of motions, [14] S.Burger,F.S.Cataliotti,C.Fort,F.Minardi,M.Ingus-
after linearization, take the form cio, M.L. Chiofalo and M.P. Tosi, Phys. Rev. Lett. 86,
4447 (2001).
δ¨n=∂z(cid:20)µ−mVho∂zδn(cid:21)+∂r⊥(cid:20)µ−mV∗ho∂r⊥δn(cid:21) . (17) [15] Ilinbraiunmalo,gtyhewiqthuatnhteityresΨul2ktxs,koyf [i9s]gwiveefinndbythaant,iantveerqtueid-
parabola as a function of z. The number of particles
The frequencies of the low energy collective modes are N = dzΨ2 occupying the corresponding site
kx,ky kx,ky
t[7h]enbyobstimainpelyd rfreopmlactinhgoseωin the ambs/emnc∗eωofatnhde lωattice is given byRNkx,ky =N0,0 1−kx2/Kx2−kx2/Ky2 3/2 with
m/m∗ωy. For large lasexr i→ntenpsities thexvalue oyf m→∗ Kx,y = ¯hω¯/mωx2,yd2 1(cid:0)5Na d dxf04 2/aho(cid:1)1/5 and
pcoincides with the one calculated for the 1D array. If N0,0 = 25pπN/KxKy. (cid:16) (cid:0) R (cid:1) (cid:17)
ω >> ω m/m∗,ω m/m∗, the lowest energy solu- [16] S. Stringari, Phys.Rev.A 58, 2385 (1998).
z x y
tions involvpe the motiopn in the x y plane. The oscil- [17] T.-L.HoandM.Ma,J.Low.Temp.Phys.115,61(1999).
lations in the z-direction are inste−ad fixed by the value [18] C. Menotti and S. Stringari, cond-mat/0201158.
ofω . Theseinclude the center-of-massmotion(ω =ω )
z z
andthe lowestcompressionmode (ω =√3ω )[7,8]. The
z
frequencyω =√3ωz coincideswiththevalueobtainedby

directlyapplyingthehydrodynamictheoryto1Dsystems
1000
[16,17]and reveals the 1D nature of the tubes generated
bythe2Dlattice. Iftheradialtrappinggeneratedbythe

lattice becomes too strong the motion along the tubes 100
caanndnoonelojunmgeprsbientdoemscorirbeecdorbryeltahteedm1eDanrefigeimldeseq[u18a]t.ions m*/m

10
StimulatingdiscussionswithF.Cataliotti,C.Fort,M.
Inguscio, A. Smerzi and A. Trombettoni are acknowl-

edged. This researchis supported by the Ministero della
1
Ricerca Scientifica e Tecnologica (MURST). 0 5 10 15 20 2s5 30 35 40 45 50
FIG.1. Effective mass as a function of thelaser intensity
s (see eq.(1)) calculated neglecting the effects of interaction
and harmonic trapping.
10
[1] B.P. Anderson and M.A. Kasevich, Science 282, 1686 9
(1998). 8
[2] C.Orzel,A.K.Tuchman,M.L.Fensclau,M.Yasuda,and 7
M.A. Kasevich, Science 291, 2386 (2001).
[3] Fn2a9.Sr3d.,iC,8a4At3a.l(Ti2or0to0tmi1,)bS.e.tBtounrig,eAr,.CS.mFeorrzti,,PM..MIandgduascloion,i,SFci.eMncie- frequency [Hz] 456
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[5] O. Morsch, J.H. Mu¨ller, M. Cristiani, D. Ciampini,
1
E. Arimondo, Phys.Rev. Lett. 87, 140402 (2001).
0
[6] M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ansch, 0 1 2 3 4 5 6 7 8 9 10
s
I.Bloch, Nature415, 39 (2002). FIG.2. Frequency of the center-of-mass motion for a con-
[7] S.Stringari, Phys.Rev. Lett. 77, 2360 (1996). densate trapped by the combined magnetic and optical po-
[8] F.Dalfovo,S.Giorgini, L.P.Pitaevskii,andS.Stringari, tential(1)asafunctionofthelaserintensity. Thecirclesand
Rev.Mod. Phys. 71, 463 (1999). triangles are, respectively, the experimental and theoretical
[9] P. Pedri, L. Pitaevskii, S. Stringari, C. Fort, S. Burger, data of [3]. The triangles have been obtained by evaluating
F.S. Cataliotti, P. Maddaloni, F. Minardi, M. Inguscio, the tunneling rate within a Gaussian approximation for the
Phys.Rev.Lett. 87, 220401 (2001). order parameter in each well [3]. The solid line refers to our
[10] The profile (9) can also be obtained by applying the theoretical prediction.
smoothing procedure (6) to the equilibrium solution for
Ψ2 given byeq.(8) in [9].
k
[11] A.Trombettoni, PhD-thesis,SISSA,Trieste (2001).
4