Generation of nonground-state Bose-Einstein condensates by modulating atomic interactions

0.25 MB·English

by  E. R. F. Ramos

Generation of nonground-state Bose-Einstein condensates by modulating atomic
interactions
E. R. F. Ramos,1,∗ E. A. L. Henn,1 J. A. Seman,1 M. A. Caracanhas,1 K.
M. F. Magalha˜es,1 K. Helmerson,2 V. I. Yukalov,3 and V. S. Bagnato1
1Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo,
Caixa Postal 369, 13560-970, S˜ao Carlos-SP Brazil
2Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
3Bogolubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research Dubna 141980, Russia
9 A technique is proposed for creating nonground-state Bose-Einstein condensates in a trapping
0 potential bymeans of the temporal modulation of atomic interactions. Applyinga time-dependent
0 spatially homogeneous magnetic field modifies the atomic scattering length. A modulation of the
2 scattering length excites the condensate, which, under special conditions, can be transferred to an
n excited nonlinear coherent mode. It is shown that a phase-transition-like behavior occurs in the
a time-averagedpopulation imbalancebetweenthegroundandexcitedstates. Theapplication ofthe
J technique is analyzed and it is shown that the considered effect can be realized for experimentally
9 available condensates.
2
PACSnumbers: 03.75.Kk,03.75.Lm,03.75.Nt

]
h
p I. INTRODUCTION for available atomic systems is demonstrated.
- ThecoherentstatesofatrappedBECaredescribedby
t
n thesolutionstotheGross-Pitaevskiiequation(GPE).To
a Bose-Einsteincondensation(BEC)isaneffectofavery transfer the BEC fromthe groundto a nongroundstate,
u broad interest, touching on a variety of physical topics
it is necessary to apply a time-dependent perturbation,
q and creating, sometimes unexpected, links among differ-
at a frequency close to the considered transition. As a
[
ent physical subjects. Many exciting possibilities have
result[10,11]theresonantlyexcitedcondensatebecomes

3 been investigated in recent years and are discussed in an effective two-levelsystem. The external fields consid-
v books[1]andreviewarticles[2,3,4]. TheFeshbachreso-
eredin the previous works[2, 10, 11, 14] were formedby
3 nancetechnique,whichenablesavariationofthescatter-
spatiallyinhomogeneousalternatingtrappingpotentials.
8 inglengthviaamagneticfield,isoneofthemostpromis-
Now,weconsideraverydifferentsituationrepresentedby
0
ing toolsfor manipulatingthe propertiesofquantumde-
3 a spatially homogeneous time-oscillating magnetic field,
generategases. Thistechnique,e.g.,hasmadeitpossible
. whichcanbeeasilyimplementedwithpresentexperimen-
5 totuneanunstablesystemintoastableone[5,6],toform
tal techniques.
0
molecular condensates [7], and to investigate the nonlin-
8
eardynamicsofaBECwithatime-dependentscattering
0
length [8, 9].
: II. MODULATION OF SCATTERING LENGTH
v
i In the present letter, we show that the temporal mod-
X
ulation of the scattering length can also be used for gen- TheGPE,describingazero-temperatureweaklyinter-
r erating nonground-state condensates of trapped atoms. acting Bose gas, is given by
a
Suchstatesaredescribedbynonlineartopologicalcoher-
ent modes and can be excited by a resonant modulation ∂Φ ℏ2
iℏ = − ∇2+U (r)+A |Φ|2 Φ, (1)
of the trapping potential [10, 11]. We advance an alter- trap s
∂t (cid:20) 2m (cid:21)
0
native way for exciting the coherent modes of a trapped
BEC by including an oscillatory component in the scat- where U (r) is the trapping potential, the interaction
trap
tering length. The main idea is to superimpose onto the strengthA =4π(N−1)ℏ2a /m , a the s-wavescatter-
s s 0 s
BEC a uniform magnetic field with a small amplitude ing length, m the atomic mass and N is the number of
0
time variation. Due to the Feshbach resonance effect, condensed atoms. In the presence of a spatially uniform
suchan oscillatoryfield createsan externalperturbation magnetic field, a near a Feshbach resonance is given by
s
in the system, coherently transferring atoms from the the well known relation
groundtoachosenexcitedcoherentstate. Thefeasibility
of the experimental implementation of this phenomenon ∆
a =a 1− , (2)
s nr
(cid:18) B−B (cid:19)
res
where a is a non-resonant scattering length, B is
nr res
∗Electronicaddress: [email protected] the value ofthe magnetic fieldwhere the resonancein as
2
occurs,and∆isthe correspondingresonancewidth. Let Eq.(10) in Eq.(6), one obtains a set of differential equa-
us consider the time-dependent magnetic field tions for the coefficients c (t) and c (t),
0 p
dc
B(t)=B0+bcos(ωt), (3) iℏ 0 = A0|cp|2c0(2I0,p,0−I0,0,0)
dt
with |b| ≪ |B0 −Bres|. In such a case, Eq.(2) can be + Aei∆ωt |c |2c I +2|c |2c I
p p 0,p,p 0 p 0,0,p
expanded to first order as 2 (cid:16) (cid:17)
A
+ e−i∆ωtc∗c2I , (11a)
as(t)≃a0+acos(ωt), (4) 2 p 0 p,0,0
where iℏdcp = A |c |2c (2I −I )
0 0 p p,0,p p,p,p
dt
∆ a b∆
a0 =anr(cid:18)1− B0−Bres(cid:19) a= (B0n−rBres)2 . (5) + A2e−i∆ωt(cid:16)|c0|2c0Ip,0,0+2|cp|2c0Ip,p,0(cid:17)
A
The scattering length then possesses an oscillatory com- + ei∆ωtc∗c2I , (11b)
2 0 p 0,p,p
ponent around the average value.
Combining Eq.(4) and Eq.(1), one gets the GPE with where the integral Ij,k,l is defined as
the additional oscillatory term V = V (r,t). With the
I = φ∗|φ |2φ dr. (12)
notation H = H0+V, one has j,k,l Z j k l
HΦ=iℏ∂Φ/∂t, (6) Inderiving the latter equations,two assumptions,whose
mathematical basis has been described in detail in
Refs. [2, 10, 11], are made. First, the time variation
ℏ2
H0 =− ∇2+Utrap(r)+A0|Φ|2, (7) of c0(t) and cp(t) are to be much slower than the the
2m0 exponential oscillations with the transition frequency
ω = (E − E )/ℏ. This condition is fulfilled, when
2 p0 p 0
V =Acos(ωt)|Φ| , (8) theamplitudesA I andAI aresmallerthanℏω .
0 j,k,l j,k,l p0
Thesecondistheresonancecondition,whentheexternal
4π(N −1)ℏ2 alternating field connects only the two chosen nonlinear
A = a , states. Another point concerns damping due to colli-
0 0
m
0 sionsbetween particlesin the desiredmodes orcollisions
withthethermalcloud. Althoughtheoscillationtimefor
4π(N −1)ℏ2 populations takes tens of trapperiods, this time is much
A= a. smaller than the lifetime of a typical BEC or a vortex
m
0 state [12, 12]. So, we expect that damping occurs but
SolvingEq.(6),wekeepinmindthatthefrequencyωis not as a dominate process. Thus, we have left out the
chosentobeclosetothetransitionfrequencybetweenthe damping effect for this model.
groundandanexcited mode. We startingconsideringas Another important aspect is that the total number of
thetotalwavefunctionalinearcombinationofacomplete atomsdo notvaryontime,butthe numberineachstate
set of modes as follows does. This variation is taking into account in Eqs. (11)
since these equations depend on the population of each
Φ(r,t)= cj(t)φj(r)e−iEjt/ℏ, (9) state,representedby|c0|2 and|cp|2. However,themodes
Xj φj in equation (9) are stationary solutions of Equation
(7) when all atoms are in state j. Thus, if there is a
where φ (r) are stationary solutions for the equation variation in the atom number of some state, there is a
j
H0φj = Ejφj, with eigenenergies Ej. Then, was proved variation in the wave function that represents this state.
before[2,10]theonlyrelevanttermsthatsurvivearetwo So,thetotalwavefunctionshouldbewrittenintheform
modesconnectedbythemodulatingperturbation(8). In
Φ(r,t)= d (t)φ′(r,t), (13)
this case, the total wavefunction (9) can be represented, j j
in a good approximation, by Xj
wherethe number dependence is insertedinthe time de-
Φ(r,t)=c0(t)φ0(r)e−iE0t/ℏ+cp(t)φp(r)e−iEpt/ℏ, (10) pendence. In this way, the population of a state j would
be given by |d (t)|2 and not by |c (t)|2, since the expan-
j j
where the label 0 refers to ground state and p to an
sions (9) and (13) are different. However, in the case of
excited state. We investigate the time evolution of the
our study, the system is in a weak-coupling regime, i.e.,
BEC, with the initial condition, where all atoms are in
g is small, so the variation of the wave function can be
the ground state, i.e., c (0) = 1 and c (0) = 0. Our
0 p neglectedandthepopulationofastatej canbegivenby
aim is to study the transfer between the fractionalmode
populations of the ground and the excited states. Using nj(t)≈|cj(t)|2.
3
III. APPLICATION TO A CYLINDRICALLY
SYMMETRIC TRAP
1.00
We consider a cylindrically symmetric harmonic trap
0.75
m
Utrap = 20(ωr2r2+ωz2z2), (14) 0.50 (a)
0.25
and use the optimized perturbation theory, as discussed
in Ref. [2, 10, 11], for finding the modes φ and φ . It is 0.00
0 p
convenient to define the dimensionless variables 1.00
H(φ) 0.75
ψ(x)=l3/2φ(r), H(ψ)= , n
r ℏω o
r ati 0.50 (b)
x = r, x = z, pul 0.25
r l z l o
r r p
of 0.00
n 1.00
g0 =4π(N −1)alr0 , λ= ωωzr , δ = (ω−ωrωp0) , (15) actio 0.75
r
F 0.50 (c)
where l = ℏ/m ω . Using the fourth-order Runge-
r 0 r
Kutta methopd [15], we calculate the time evolution of 0.25
the coefficients c0(t) and cp(t) for different values of the 0.00
detuning δ and external-field amplitude. The functions
1.00
c (t) and c (t) define the fractional mode populations.
0 p
For an excited mode, we take the radial dipole state 0.75
{100},whichisthelowestenergystateabovetheground 0.50 (d)
state {000}. Here {nmk} implies the notation for the
0.25
threequantumnumbers,wheren,mandkrefertoradial,
azimuthal and axial mode numbers respectively. Fig.1, 0.00
where λ = 0.2 and g0 = 70, shows the time evolution of 0 40 80 120 160 200
themodepopulationsn0andnpfordifferentvaluesofthe -1
detuningδ anda/a ,whichisgivenbyEq. (5). Thecho- t (units of r )
0
senparameterscorrespondtotypicalexperimentalsetups
and are easily controlled in a laboratory. The solutions
demonstratedifferentbehaviorsofthe statepopulations. FIG. 1: (Color online) Populations of the ground state n0
For a/a = 0.7 and δ = 0, Fig.1(a), the populations (solidline)andexcitedstate{100}np (dashedline)asafunc-
0 tion of time for λ=0.2 and g0 =70 with (a) a/a0 =0.7 and
display small oscillation amplitudes, with a considerably
δ = 0; (b) a/a0 = 0.7 and δ = 0.04; (c) a/a0 = 0.72 and
larger population in the ground state. Increasing the
δ=0.04; (d)a/a0 =1 and δ=0.04
detuning to δ = 0.04 results in the behavior shown in
Fig.1(b). Although on average atoms stay longer in the
groundstate, for some intervals of time n is largerthan
p
Here, the average of each population is performed over
n . Changing the amplitude to a/a = 0.72 and main-
0 0
the full cycle of an oscillation.
taining δ = 0.04, as in Fig.1(c), yields a very different
temporal behavior. Atoms now stay longer in the ex- The above order parameter η displays a nontrivialbe-
citedstateratherthaninthegroundstate. Theshapeof havior when the ratio a/a0 is modified. For different de-
the functions shows the inherent nonlinearity of the sys- tunings,the variationofη asa functionofthe ratioa/a0
tem. If the amplitude is increased further to a/a0 = 1, is presented in Fig.2. The variation of η vs. a/a0 can
with δ = 0.04, as in Fig. 1(d), the system shows a full be smooth, when δ ≤0.03, or can show sudden changes,
population inversion. For certain times, when the mode whenδ >0.03. Smallervaluesofa/a0keepatomsprefer-
population fully migrates from the ground {000} to the ablyinthe groundstate. Forsomecriticalvalueofa/a0,
excited {100}state, it is possible to havea pure conden- η becomes negative,which means that BEC stayslonger
sate in the coherent topological excited mode. with a larger population in the excited state than in the
A convenient way to quantify the population behav- ground state. This situation is ideal for detecting the
ior is through the introduction of an order parameter η, formation of topological modes. Step like behavior ob-
definedasthe difference betweenthe time-averagedpop- served on η is not surprising for a nonlinear system. For
ulations for both states [14, 16], example, the classical driven anharmonic oscillator [17]
exhibits a shift onits resonancecurve,which we alsoob-
η =n¯ −n¯ . (16) serve in our system [18]. This shift for a large driven
0 p
4
Atom Bres(G) ∆(G) anr B0(G) b(G) a0 N(×104)
85Rb 155.0 10.7 -443 164.4 -0.9 63 0.2
1.0 87Rb 1007.34 0.17 100 1007.53 0.02 11 0.9
0.8 7Li 735 -113 -27.5 636 10 3.9 9.3
39K 403.4 -52 -23 357.9 4.55 3.3 4.7
0.6 = 0
= 0.03 TABLE I: Amplitudes B0 and b of the magnetic field for
0.4
= 0.04 four different species of atoms . We set g0 = 70, λ = 0.2,
0.2 = 0.06 a/a0 = 0.8 and b = 0.1(Bres −B0). Scattering length is
= 0.10 expressed in units of the Bohr radius. Data for 85Rb were
0.0 obtainedinRef.[21];for87Rb,Ref.[22];for7Li,Ref.[19];for
39K,Ref. [20].
-0.2
-0.4
A final point to be addressed concerns losses intro-
0.0 0.2 0.4 0.6 0.8 1.0
duced by collisions, specially near a Feshbach resonance.
a/a
0 With an off-resonant magnetic field, the dominant loss
mechanism is a three body collision, whereas close to
FIG.2: (Coloronline)Orderparameterηasafunctionofthe the resonance, the molecular formation dominates the
ratioa/a0,fordifferentdetuningsshowingaphase-transition atom loss mechanism [23]. Although the fields consid-
like behavior. ered in Table I and in the 7Li example above are within
the resonance linewidth, they are far enough to the res-
onance, then we consider three body collision as the
strengthcan resultin a bistable behavior. In our system main loss mechanism. In the case of 7Li, that the res-
that appears as a step like response with η. This bista- onance linewidth is large, as we are in the border of
bility appears only for positive detunings, as observed this linewidth, the loss rate is very small, as shown in
in our calculation. Negative detunings follows a smooth Ref. [19]. So, we can neglect the atom loss, specially be-
curve without step occurrence. cause we apply the magnetic field in a short period of
An important question is the feasibility of the exper- time.
imental creation of such coherent modes. The main pa-
rameter here is a/a . It shows us that this value is
0
strongly dependent on the Feshbach resonance width ∆, IV. DISCUSSION
characteristic of each type of atoms. For systems with
small ∆, the required value of a/a0 for obtaining the In conclusion, we have shown that using a magnetic
transition in η will occur only for B0 ≈ Bres. In this modulation field, applied to a trapped Bose-condensed
case, the necessary condition b ≪ Bres − B0 can only gas, it is possible to transfer the atomic population
be fulfilled for very small values of b, which creates an from the ground state to an excited state, producing a
extra difficulty with the present techniques of magnetic nonground-state condensate. The time averaged popu-
field control [19, 20]. As an experimentally realistic ex- lation imbalance between the ground and excited states
ample,let us considerthe caseofb=0.1(Bres−B0) and represents an order parameter, which demonstrates an
a/a0 =0.8, which corresponds to the atomic parameters interestingbehavioras a function ofthe modulationam-
listed in Table I. Setting g0 = 70, ωr = 2π ×120Hz, plitude. Depending on the detuning, the behavior of η
we obtain a value of b of 0.9G for 85Rb, and of 0.02G canbeeithersmoothorratherabrupt. Thisistheconse-
for 87Rb, which would be difficult to control. On the quenceofthe strongnonlinearityoftheinteractions. For
other hand, 10G, for 7Li, and 4.55G, for 39K, are val- somerangeofdetunings,ηbecomesnegativeaboveacrit-
ues that can be realized with present technical capabili- ical value of the modulation amplitude. This occurs be-
ties [19, 20]. causeofthepopulationinversionrealizedduringthepro-
Let us consider now a condensate containing 105 7Li cessofthemodeexcitation. Largerdetuningsandout-of-
atoms in a trap with radial frequency ω =2π×120Hz, resonancemodulationskeepthepopulationintheground
r
g = 70, and λ = 0.2. With these conditions, the tran- stateandnopopulationinversionisobserved. Numerical
0
sition frequency is ω = ω = 2π×209.6Hz. Also, calculations, accomplished for 7Li atoms, show that the
p0 100,0
togetherwiththeinformationfromTableI,Eqs. (5)and valuesforthe amplitude andmodulationofthe biasfield
(15), we obtain the bias magnetic field B = 632.5G are within realistic experimental conditions.
0
and the amplitude b = 11.68a/a . Then, we observe The justification of the approach for generating non-
0
the critical values for b ranging from 4G for a detuning linear coherent modes by the resonant modulation of an
ω−ω =2π×21Hzto10.5G,forω−ω =2π×6.3Hz. alternating field has been explained in detail in our pre-
p0 p0
Such oscillating amplitudes correspond to less than 1% viouspublications[10,11,14,24,25]. Itwouldbeunrea-
of the total bias field B . sonable to repeat here all this justification in full. How-
0
5
ever, for the convenience of interested readers, we recall employed for deriving the equations for the fractional
in brief some important points. mode coefficients c (t), is a well justified method based
n
When a system is subject to the action of a time- on rigorous mathematical theorems [29]. Therefore, this
dependent external field V(t), alternating with a fre- method does not require some other justifications.
quencyω,thenthereexisttwoprincipallydifferentsitua- Moreover, the results, obtained by means of the av-
tions,dependingonthetemporalalternationbeingeither eraging techniques, when treating the generation of the
slow or fast [26]. Respectively, there occur two different nonlinearcoherentmodesbyaresonantalternatingfield,
physical cases. have been thoroughly compared with the results of the
Ifthetemporalvariationisslow,suchthatthealterna- directnumericalsimulationoftheGross-Pitaevskiiequa-
tion frequency ω is much smaller than the characteristic tion in Refs. [33,34]. Both ways of treating the problem
system energy E, then one says that the perturbation is havebeenfoundto be ina verygoodquantitativeagree-
adiabatic [26]. For a quantum-mechanical system, the ment.
energy E is an eigenvalue of the system Hamiltonian H. Inarealexperiment,ofcourse,amodulation,evenbe-
When the latter varies slowly in time, so that the varia- ing perfectly resonant, might, nevertheless, lead to heat-
tion is adiabatic, then the adiabatic picture is in order, ing,when the neighboringnonresonantlevels become es-
withtheeigenvaluesE(t)andrelatedeigenfunctionsϕ(t) sentially involved in the process. This means that such
slowly varying in time. For such an adiabatic variation, a resonant mode generation can be effective only during
the notion of nonlinear coherent modes is not of much a finite time. This problem has been analyzed in detail
use. earlier [11,35], where it has been found that, taking the
modulation amplitude sufficiently small, it is feasible to
Aprincipallydifferentsituationoccurs,whenthetem-
avoid heating during the lifetime of atoms inside a trap.
poral variation is fast, so that ω is in resonance with
one of the transition frequencies ω ≡ (E − E )/~. For instance, the heating time was estimated [11,35] to
ij i j
be of order 10−100 s.
It is exactly this case that is considered in the present
Thenonground-statenonlinearcoherentmodesareof-
paper. Then the solutions of the stationary equation
ten termed topological, since the wavefunctions of dif-
H ϕ = E ϕ define the nonlinear coherent modes ϕ .
0 j j j j
ferent modes possess drastically different spatial shapes,
Constructingaclosedlinearenvelope,spanningthetotal
with a different number and location of zeros. As an
set{ϕ },onegetstheHilbertspaceH≡Span{ϕ },with
j j
example, we can recall the wavefunctions of the first ex-
thetypicalproperty[27]thatanyfunctionfromHcanbe
citedmodes foratomsinaharmonictrap,foundinRefs.
expanded over the set {ϕ }. The latter set forms a total
j
[11,24]. These functions ψ (x ,ϕ,x ), represented in
basis [11, 28], over which the solution of Eq. (6) can be nmk r z
dimensionlessunitsofEq. (14),arelabelledbythequan-
expanded as in Eq. (9). Looking for the solution of Eq.
tum numbers n, m, and k and are the functions of the
(6) in the form of expansion (9), we employ the method
dimensionless cylindrical variables x ,ϕ, and x . The
of the parameter variation. Separating in expansion (9) r z
ground-state wavefunction is
the time-dependent coefficient function into two factors,
the fastly oscillating exponential exp(−iE t/~), and the
tselocwhnlyiqvuaersy[i2n9g]aenndvetlhoepesccajl(ets)e,pwaeractaionnuaspepjrtohaecahv[e3r0a,g3i1n]g. ψ000 =(cid:18)u200π0v3000(cid:19)1/4 e−(u000x2r+v000x2z)/2 .
A similar technique in optics is called the slowly-varying
The radial dipole mode has the form
amplitude approximation [32]. Substituting expansion
(9) into Eq. (6) and involving the averaging techniques
t[2h9e–s3a1m],ewaesohbatsaibneEenqsd.o(n1e1)infoRrecfns.(t[)1.0T,1h1i]s. procedure is ψ100 =(cid:18)u210π0v3100(cid:19)1/4 u100x2r −1 e−(u000x2r+v000x2z)/2 .
(cid:0) (cid:1)
In this way, one should not confuse the adiabatically
The vortex mode is also a possible stationary solution
slowvariationofanexternalfield,whenthemodeprofiles
and energies are certainly changing in time, and the fast
resonantfield oscillation, when the fractional mode pop- ψ010 =u010 vπ0130 1/4xr eiϕ e−(u000x2r+v000x2z)/2 .
ulations varybetweenthe stationarycoherentmodes. In (cid:16) (cid:17)
thelattercase,thesolutiontoEq. (6)canberepresented And the axial dipole mode is
in form (9) and the averaging techniques are applicable,
yuiaetlidoinnghEerqes.is(1e1q)uifvoarletnhte tcooetffihceiernetsofnuanncttioenxsc.itaTthioensiot-f ψ001 =(cid:18)4u200π13v0301(cid:19)1/4xz e−(u000x2r+v000x2z)/2 .
an atom, as is discussed in Refs. [24,25]. In the same
way as for an atom, the external resonant field should The quantities u and v here are defined by the
nmk nmk
notbetoostronginordernottodisturbtheenergy-level optimization conditions and depend on all system pa-
classification. For this, it is sufficient to take the modu- rameters (see calculational details in Refs. [11,24]). As
lation amplitude b in Eq. (3) appropriately small, which is evident, the spatialshapes of atomic clouds,described
is always possible. by |ψ |2, are principally different for different quan-
nmk
It is important to stress that the averagingtechnique, tum numbers.
6
If the scattering length is modulated according to Eq. set {n,m,k}, vanish for the vortex mode ψ =ψ and
p 010
(4), then the resulting Eq. (6) acquires the additional for the axial dipole mode ψ = ψ . Hence, in these
p 001
term(8)playingtheroleofamodulatingfield. Therefore cases, the vortex and the axial dipole modes cannot be
the mechanism of generating nonground-state modes is excited by modulating the scattering length. However,
thesameforboththe cases,whetherthe additionalterm the radial dipole mode ψ can be generated by this
100
is caused by the trap modulation or by the scattering- method,sincetherelatedintegrals(12)arenonzero. This
length modulation. In any case, the general structure of is why, we haveconsideredhere exactly this case. Away
Eqs. (6) to (8) is similar for both these setups. outoftheproblem,whichwouldallowforthe generation
The main difference between the methods of generat- of other modes, including the vortex mode, could be by
ing nonlinear coherent modes by the trap modulation or employing a nonsymmetric trapping potential.
by the scattering-length modulation is as follows. In the
method of the trap modulation, by choosing the appro-
priatespatialdependenceofthemodulatingfieldV(r,t),
Acknowledgments
itispossibletogenerateanytypeofmodes. While,inthe
method of the scattering-length modulation, only those
modescanbe excited,forwhichtheintegralsinEq. (12) Authors thank R. G. Hulet and G. Roati for provid-
are nonzero. For example, in the case of cylindrically ing us with experimental data on Feshbach resonances,
symmetric harmonic trap with potential (13), the inte- and L. Tomio for fruitful suggestions. This work was
grals I , I , I , and I , in which p implies a supported by CAPES, CNPq and FAPESP.
0,p,p 0,0,p p,0,0 p,p,0
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