Nonlinear Supression of Tunneling and Strong Localization in Bose-Einstein Condensates with Spatially Inhomogeneous Interactions

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by  Victor M. Perez-Garcia

Nonlinear Supression of Tunneling and Strong Localization in Bose-Einstein
Condensates with Spatially Inhomogeneous Interactions
V´ıctor M. P´erez-Garc´ıa∗
Departamento de Matem´aticas, E. T. S. de Ingenieros Industriales,
and Instituto de Matem´atica Aplicada a la Ciencia y la Ingenier´ıa (IMACI),
Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain.
(Dated: February 8, 2008)
Westudythepropertiesofthegroundstateof Bose-Einstein condensateswith spatially inhomo-
geneousinteractionsandshowthattheatomdensityexperiencesastronglocalization atthespatial
region wherethescattering length isclose tozero whiletunnelingtoregions with positivevaluesof
thescattering length is strongly supressed by thenonlinear interactions.
7
PACSnumbers: 03.75.Lm,05.45.Yv,42.65.Tg
0
0
2
The experimentalgenerationofBose-Einsteinconden- The model and its Thomas-Fermi limit.- The ground
n
sates (BEC) with ultracold dilute atomic vapors [1] has stateofaBECinthemeanfieldlimitisthereal,positive
a
J turned out to be of exceptional importance for physics. solution of the Gross-Pitaevskiiequation
The formation of a condensate occurs when the temper-
3
1
ature is low enough and most of the atoms occupy the λφ= ∆φ+V(x)φ+g(x)φ2φ, (1)

] groundstateofthesystem. Thisprocessisvisiblebothin −2 | |
S
momentum and in real space due to the spatial inhomo-
P which minimizes the energy
geneities exhibited by the order parameter on a macro-
.
n scopic scale because of the trapping potentials. 1 1
i E(φ)= φ2+V(x) φ2+ g(x)φ4 , (2)
nl The properties of the ground state of trapped BECs ZR3(cid:20)2|∇ | | | 2 | | (cid:21)
are well known. In the mean field limit simple analyt-
[
ical expressions are available in the Thomas-Fermi ap- under the constraint of a fixed number of particles N =
2 proximation [2] and beyond [3]. These approximations φ2. Eq. (1)iswritteninnondimensionalunitswhere
8v and direct numerical simulations describe accurately the tRhRe3|co|ordinates x and time t are measured in units of
2 properties of the experimentally found groundstates [4]. a0 = ~/mω and 1/ω, respectively, while the energies
0 Nonlinear interactions between atoms in a Bose- and frpequencies are measured in units of ~ω and ω re-
2 Einstein condensate are dominated by the two-body col- spectively, ω being a characteristic frequency of the po-
1 lisions that can be controlled by the so-called Feschbach tential. Finally, g(x)=4πa(x)/a0 is proportionalto the
6
resonance (FR) management [5]. The control in time of local value of the s-wave scattering length a.
0
/ the scattering length has been used to generate bright The Thomas-Fermi approximation proceeds by ne-
n
solitons [6] and induce collapse [7] and has been the ba- glecting the kinetic energy or equivalently the term pro-
i
l sis for theoretical proposals to obtain different nonlinear portional to ∆φ in Eq. (1). In the case of spatially ho-
n
: waves[8,9,10]. Moreover,interactionscanbemadespa- mogeneousinteractionsg(x)=g0thisleadstoφTF(x)=
Xiv ttihaellyladseerpeinntdeennstitbyy(iancttihnegocanseeitohfeorptthiecaml acognntertoicl fiofelFdRosr i[n(λho−mVog(xen))eo/ugs0]t1h/2e.sWamheenforthmealinmtearnaicptuiolantsioanrelesapdastitaolly
r [11]) acting on the Feschbach resonances. This possibil-
a ityhasmotivatedmanytheoreticalstudiesonthebehav- φ (x)= (λ V(x))/g(x), (3)
TF
−
iorof solitonsin Bose-Einsteincondensates(BECs)with p
spatially inhomogeneous interactions [12, 13, 14, 15, 16]. which diverges on the set G = x R3 : g(x) = 0 .
{ ∈ }
Inthispaperwestudytheeffectofspecificspatiallyin- Obviously, in the vicinity of G the Thomas-Fermi ap-
homogeneous interactions on the ground state of a BEC proximationis not correctbut anyway this divergence is
in the mean field limit. We will show that when the a first indication of a phenomenon to be studied in this
scattering length is non-negative, a striking localization paper: the tendency of the ground state in Bose-Einstein
phenomenon of the atom density occurs at the regions condensates with spatially inhomogeneous interactions to
where the scattering length vanishes. By tuning appro- localizestronglyontheregionswherethescatteringlength
priately the control(magnetic or optical) fields this phe- is close to zero provided the system is sufficiently nonlin-
nomenoncanbeusedtodesignregionswithlargeparticle ear, i.e. for sufficiently large values of gN.
densities and prescribedgeometries. Another interesting A simple example.- Let us first consider an exactly
phenomenon to be studied in this letter is the nonlinear solvable “toy” example which displays the main features
limitation of tunneling of atoms to the regions in which of the phenomena to be studied in this paper: a quasi-
the interactions are stronger. one dimensional BEC in a box, i.e. setting V(x) = 0,
2
FIG. 1: [Color online] Spatial distribution of the density
n(x) = |φ(x)|2 for the ground state solutions of Eq. (4)
with L = 2,a = 1 and different values of the chemical po-
tential λ corresponding to differentvalues of N (a) From the
lower to the upper curve g0N = 2,3.55 (corresponding to
λ∗) and g0N = 6. (b) From the lower to the upper curve
g0N =25,50,100,200. FIG.2: [Color online] (a) Ratiobetween themaximumatom
density and the atom density at x = 0 as a function of the
scaled scattering length g∗ on the spatial region |x| < a (for
φ(x= L)=0withscatteringlengthgivenbyg(x)=g0, g0 =1) (b-c) Scaled atom density profiles for (b) g∗ =0 and
for x ±<a and g(x)=0, for x >a. (c)g∗ =0.2inbothcasesforatotalscalednumberofparticles
| | | | N =1000. The insets show theprofile of g(x).
In this simple case, Eq. (1) becomes
φxx+2λφ= 2g0φ3, x a, (4a)
| |≤ a certain critical density is achieved. This region corre-
φ +2λφ= 0, a< x <L, (4b)
xx sponds to the nonlinear analogue of the classically for-
| |
bidden region in ordinary potentials and is energetically
and its positive, even solution, satisfying the boundary
lessfavourableduetotheextrarepulsiveenergyprovided
conditions φ( L)=0,φ′(0)=0 is given by
± by the nonlinear interactions. However,the tunneling of
atoms in this region is essentially limited to a constant
φ(x)=Csin √2λ(x L) ,a< x <L, (5) value, independently of the number of atoms, which dif-
h − i | |
fers essentially from ordinary tunneling.
while for x <a the solutions are given by The supression of tunneling depends strongly on the
| |
value of the scattering length in the inner region, that
αλ sn x√λα+δ;k2 , λ<λ we have taken to be zero up to now. If instead we set
g0 ∗
q (cid:16) (cid:17) g(x)=g when x <a and study the dependence of the
φ(x)=π/ 2√2g0|a−L| , λ=λ∗, (6) ratio bet∗ween th|e|maximum atom density and the atom
qαg(cid:0)0λ dc(cid:16)x√λα;k(cid:1)2(cid:17), λ>λ∗ dofentshietytuantnxel=ing0),(wwehicfihndisaasmtreoansgurdeepoefntdheencaemopnlittuhdies
where α(k)=2/(1+k2), sn and dc are two of the stan- parameter as shown in Fig. 2(a).
dard Jacobi elliptic functions and k is the elliptic mod- This effect is also seen in the atom density profiles
ulus. Both the elliptic modulus and amplitude C can when comparing the cases with g = 0 [Fig. 2(b)] and
∗
be obtained from the matching conditions for φ(a) and g = 0.2 [Fig. 2(c)] for N = 1000. Larger values of N
∗
φ′(a). These conditions also give λ = π2/ 8(a L)2 . lead to a stronger effect.
∗
−
Finally, the cutoff value of the chemical pote(cid:2)ntial λc ca(cid:3)n Numerical results.- Let us now consider Eq. (1) in
be obtained from the condition of maximum slope at morerealisticquasione-dimensionalscenariosby includ-
x = a which leads to λ = π2/[2(a L)2]. In that case ing a potential V(x) = 0.02x2. We have computed
c
−
thenumberofparticlesinthecondensateisinfinite,since numerically the ground state for scattering lengths of
the amplitude in the outer region C . the form g0(x) = 1,g1(x) = exp( x2/200),g2(x) =
The spatialprofilesofthe grounds→tat∞e density fordif- exp( x2/50). These choices allow us−to study different
−
ferent values of g0N shown in Fig. 1 support our con- degrees of localization of the interactions starting from
jecture based on the Thomas-Fermi solution, i.e. the the case of no localization. Our results are summarized
existence of a strong localization of the atom density in in Fig. 3.
the region where the interactions vanish. In Fig. 3(a) we observe how the maximum density
It is also remarkable that the atom density in the in- (n(x)= φ(x)2) increasesdrasticallyfor spatially decay-
| |
ner part of the domain, i.e. the region where there are ing nonlinearities (blue and red curves) as a function of
nonlinearinteractions,remainsalmostconstantindepen- the number of effective number of particles in the quasi-
dently onthe numberofparticlesinthe condensateonce onedimensionalcondensateN. Thisamplitudegrowthis
3
FIG.4: [Coloronline]Groundstatemaximumparticledensity
(a)andwidthW =R x2n(x)dx/N (b)forV(x)=0.02x2 and
g0(x)=1(redlines) andg4(x)=(1−0.001x2)+ (bluelines).
FIG. 3: [Color online] Ground states of Eq. (1) for V(x) =
0.02x2 and g0(x) = 1,g1(x) = exp(−x2/200), and g2(x) =
exp(−x2/50),fordifferentvaluesofthescalednumberofpar-
ticles N. (a) Maximum particle density maxxn(x) and (b)
width W2 = R x2n(x)dx/N for g0 (green), g1(x) (red) and
g2(x)(blue). (c)Spatialprofilesofpn(x)forN =10(blue),
N = 100 (green), N = 1000 (red) for g(x) = g1(x). The FIG. 5: [Color online] Ground states for V(x) = 0.02x2 and
dashed black line is the ground state with homogeneous in- g(x)=1−exp(−x2/50) (shown in panel (c)). (a) pn(x) for
teractionsandN =1000. (d)Sameas(c)butforN =10000 N =10 (blue line), N =100 (green line) and N =1000 (red
(blue), and N = 40000 (green), in comparison with the case line). (b) Condensate width as a function of N.
of spatially homogeneous interactions (dashed black lines).
morefavourablethe localizationcloseto the pointwhere
due to a strong localization effect near the region where the interactions vanish.
g(x) vanishes as it is seen in Fig. 3(c,d). In contrast, Finally, in Fig. 5 we get again the localization phe-
the condensate density for spatially homogeneous inter- nomenon but now near x = 0 for localized interactions
actionsgrowsslowlyaccordingtotheThomas-Fermipre- givenbyg(x)=1 exp( x2/50)andV(x)=0.02x2. Itis
diction maxn(x) N2/3. When the number of particles interesting how lo−calized−the density becomes to “avoid”
∝
is small, the size of the atomic cloud is smaller than the penetrating into the regions with appreciable values of
localization region of g(x). For larger values of N the the scattering length.
ground state extends beyond the localization region of Rigorous results.- In Ref. [17] the equation
g(x) and the atom density becomes more and more lo-
calized near its edge. This effect is more clear for larger ∆u=λu a(x)(ur+f(x,u))u, (7)
− −
number of particles and is accompanied by a saturation
in the amplitude growth in the region where g(x) is far was considered on a bounded domain Ω RN with
from zero [Fig. 3(d)]. C1,1 boundary under the conditions u∂Ω ⊂= 0. r >
In the case of spatially homogeneous interactions the 0, a(x) C(Ω¯) is nonnegative and| the open set
width grows according to the law W N1/3 [Fig. 3(b)]. D = x∈ Ω:a(x)>0 satisfies D¯ Ω and pos-
Wheninteractionsarespatiallydepen∝dentandduetothe sesses a{ fi∈nite number }of C1 connect⊂ed components
localizationoftheamplitude closetothezeroofg(x)the D ,1 j l such that D¯ D¯ = if i = j.
j i j
width growth saturates for large values of N to a value The fu≤nction≤f : Ω¯ [0,+ ) ∩ R, satis∅fies f,f6 =
u
depending on the size of g(x). ∂f/∂u C Ω¯ [0, ×);R ∞and→the growth conditions
These effects are even more clear when the nonlinear- limu↓0f∈(,u)(cid:0)=0×,lim∞u↑∞fu(cid:1)(,u)/ur−1 =0, uniformly in
itydecaystozerofaster. Forinstance,takinganonlinear Ω¯. Inthe·casewhenf =0,Eq·. (7)isthelogisticmodelof
coefficient given by g4(x) = (1 0.001x2)+ (i.e. an in- populationdynamics[18]whereΩistheregioninhabited
−
vertedparabolawithmaximumamplitudeg =1atx=0 bythespeciesu,λmeasuresits birthrateanda(x)mea-
and zero values for x > 31.6) as shown in Fig. 4, we suresthecapacityofΩtosupportthespeciesu. Givenan
| |
see that the maximum densities are even higher than for open subset Ω1 Ω with a finite number of components
g1(x),g2(x). In this case, in comparison with our first let us denote by⊂σ1Ω1[ ∆] the minimum of the principal
−
simple example, we cansee aneven strongerlocalization eigenvalues of the operator ∆ on each of the compo-
−
since the existence of the potential makes energetically nents of Ω1 subject to homogeneous Dirichlet boundary
4
conditions on∂Ω1 it was proven[17]that Eq. (7) has an
unique positive solution for λ (σΩ[ ∆],σΩ0[ ∆]) and
1 1
that no such solution exists for∈λ σ−Ω0[ ∆]. M−oreover,
itgrowstoinfinitywhenλ σΩ0[ ≥∆]o1nc−ompactsubsets ∗ Electronic address: [email protected];
↑ 1 − URL:http://matematicas.uclm.es/nlwaves
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