Two atoms in an anisotropic harmonic trap

0.21 MB·English

by  Z. Idziaszek

Two atoms in an anisotropic harmonic trap
Z. Idziaszek1,2 and T. Calarco1,3
1Istituto Nazionale per la Fisica della Materia, BEC-INFM Trento, I-38050 Povo (TN), Italy
2Centrum Fizyki Teoretycznej, Polska Akademia Nauk, 02-668 Warsaw, Poland
3European Centre for Theoretical Studies in Nuclear Physics and Related Areas, I-38050 Villazzano (TN), Italy
We consider the system of two interacting atoms confined in axially symmetric harmonic trap.
Within the pseudopotential approximation, we solve the Schr¨odinger equation exactly, discussing
thelimits ofquasi-oneandquasi-two-dimensionalgeometries. Finally,wediscusstheapplication of
an energy-dependentpseudopotential, which allows to extendthevalidity ofourresults tothecase
of tight traps and large scattering lengths.
PACSnumbers: 34.50.-s,32.80.Pj
5
0
0 Atomic interactions at ultralow temperatures are of resonances.
2 central importance for recent research on quantum de- We considertwointeractingatomsofmassmconfined
n generate gases [1]. A typical feature of experiments on in an axially symmetric harmonic trap with frequencies
a ultracoldmatteristhepresenceofaweaktrappingpoten- ω⊥ and ωz. In the following we use dimensionless vari-
J tial, whichmodifies the propertiesof the cloud ofatoms, ables, in which all lengths are expressed in units of har-

4 while it does not affect the collisions of individual parti- monic oscillator length a = ~/mω , and all energies
z z
2 cles. Developmentofopticallattice technology,however, are expressed in units of ~ωzp. In these units the trap-
2 hascreatedsystemswheretheatomsaretightlyconfined pingpotentialisVT(r)= 21(η2ρ2+z2),whereη =ω⊥/ωz
in the wells of optical potential [2]. In addition, the ex- and ρ2 = x2+y2. We assume that the range of the in-
v
3 perimental achievement of the Mott insulator phase [3] teratomic potential is much smaller than the oscillator
6 has allowedfor a precise controlover a number of atoms lengths az and a⊥ = ~/mω⊥, which guarentees that
1 storedin a single well. This has opened a way for exper- theinteratomicpotentipalisnotdistortedbytheharmonic
0 imentalstudies of interactionsofindividualatoms inthe trap. Forsufficientlylowenergies,thescatteringispurely
1 presence of trapping potential and, together with other of s-wave type and we model the atom-atom interaction
4 approaches to micromanipulation of neutral atoms like by a Fermi pseudopotential V(r) = 4πaδ(r) ∂ r with s-
0 ∂r
atom chips [4, 5] or tight dipole traps [6], it represents wavescatteringlengtha[12]. Fortheharmonicconfining
/
h a major candidate for the implementation of quantum potential, the total Hamiltonian
p information processing. A theoretical understanding of
- 1 1
t the dynamics of few atoms in deformed tight-confining Hˆ = 2 2+V (r )+V (r )+V(r r ), (1)
n geometries would be of great help in all these contexts. −2∇1− 2∇2 T 1 T 2 1− 2
a
u From the theoretical side, the analytical solution for can be splitted into center of mass part: Hˆ =
CM
q two atoms interacting in a harmonic trap is known only 1 2 +V (R), and the relative motion part: Hˆ =
v: forthesphericallysymmetriccase[7,8]. Thecorrespond- −21∇R2 +VT(r)+V(√2r), where r = (r r )/√2realnd
ing problem for axially symmetric trap was studied nu- −2∇r T 1 − 2
Xi merically in [9]. However, there the authors considered R = (r1 +r2)/√2. To solve the Schr¨odinger equation
for the relative motion, we decompose the wave function
r only the limiting regimes of quasi-one and quasi-two-
a dimensional traps. In this letter we present the exact in the basis of eigenstates of the noninteracting prob-
lem, substitute this decomposition into the Schr¨odinger
solution for the axially symmetric harmonic trap of ar-
equation, and then extract the expansion coefficients by
bitrary geometry. In particular, when the ratio of axial
projectingonto noninteracting states [7]. This yields the
to radial trapping frequency is an integer, or the inverse
wave function of m = 0 states, with vanishing angular
of an integer, we give the explicit analytic form of the z
momentum along z-axis
exact solution. In the other cases, we derive an efficient
recurrencerelationthatallowsforevaluatingit. Further- ∞ exp tE z2 cotht ηρ2 coth(ηt)
more,we study the asymptotic behaviorofeigenenergies Ψ(r)= η dt h − 2 − 2 i.
and eigenfunctions in the limit of quasi-one and quasi- (2π)23 Z0 sinh(t)sinh(ηt)
two-dimensional traps.
p (2)
Astandardtreatmentofultracoldatominteractionsis
based on the replacement of a real physical potential by The harmonic oscillator states with m = 0 vanish at
z
ans-wavedelta-functionpseudopotential. Toextendthe r = 0, and they are not influenced by 6the pseudopo-
validityofthismodelinteractiontothecaseoftighttraps tential. Eq. (2) represents the wave function which is
and large scattering lengths, one can utilize the concept not normalized, and is related with the single particle
of an effective, energy-dependent scattering length [10]. Green function of the anisotropic harmonic oscillator by
We discuss this idea and show how our results can be Ψ(r)= 2G(r,0). We note that the integral representa-
−
generalizedto the case ofmagneticallytunable Feshbach tion(2)isvalidforenergiessmallerthanthegroundstate
2
energyoftheharmonicoscillator: E =1/2+η. Theva- too large. To determine the energy levels in the limit
0
lidity of Eq. (2), however, can be extended for E E ofquasi-one-andquasi-two-dimensionaltraps,we derive
0
≥
by means of the analytic continuation. the asymptotic form of (x,η) for η 1 and η 1.
F ≫ ≪
The presenceofthe trappingpotentialimplies the dis- Let us first focus on the case of η 1. Performing an
≫
crete character of the energy spectrum. The allowed expansion in the integral (4) for large η and making use
values of energy E has to be determined from equa- of the recurrence formula (7) we arrive at
tion: 1/ √2πa = [(∂/∂r)rΨ(r)] , which results
fromde−riva(cid:0)tionof(cid:1)Eq.(2),andexpressre=s0aboundarycon- (x,η)η≫1√πη[ζ(1,1+x/η)+√ηΓ(x)/Γ(x+1)], (8)
dition imposed by zero-range interaction. Investigation F ≈ 2 2
of the integral in Eq. (2) for small values of r, shows where ζ(s,a) denotes the Hurwitz zeta function. This
that Ψ(r) behaves like 1/(2πr) as r 0. This diver- asymptotic formula is valid for x > η, which corre-
→ −
gence is removed by the regularizationoperator (∂/∂r)r sponds to the range of energies E < E +2η. For the
0
inthe Fermipseudopotential. Subtractingfromthe inte- lowest excited states: 0<E E 2η we approximate
0
− ≪
gral (2), the part which gives rise to the 1/r singularity, ζ(1/2,1+x/η) by ζ(1/2,1) in Eq. (8), and match the
the condition for the eigenenergies can be rewritten as resulting energy spectrum with the energy spectrum of
twoatomsinaone-dimensionaltrap. Thelatterisdeter-
−√2π/a=F(−(E−E0)/2,η), (3) mined by √2a1D = Γ((E0−E)/2)/Γ((E0+1−E)/2)
[7]. The two spectra are identical, provided that the
where
one-dimensional scattering length is a = 1/ηa
1D
∞ ηe−xt 1 ζ(1/2,1)/√2η, which agrees with the value o−f the r−e-
(x,η) dt . (4)
F ≡Z (cid:20)√1 e−t(1 e−ηt) − t3/2(cid:21) normalized scattering length derived for a quasi-one-
0
− − dimensional waveguide [13]. On other hand, for energies
For particular values of the anisotropy parameter η, the E < E0, we can use (x,η) √πηζ(1/2,x/η), which
F ≈
function (x,η) can be calculated analytically. In the follows from Eqs. (7) and (8). This approximation, sub-
F
case of cigar shaped traps with η =n, where n is a posi- stituted into (3), leads to the condition determining the
tive integer, we obtain energy of a bound state:
n−1 √2/a+√ηζ(1/2,(E E)/(2η))=0, (9)
√πΓ(x) F 1,x;x+ 1;ei2πnm 0−
2
(x,n)= mX=1 (cid:16) (cid:17) 2√πΓ(x), which is identical to the known result derived for the
F Γ(x+ 1) −Γ(x 1) quasi-one-dimensionalwaveguide [13, 14].
2 − 2
(5) In the case of quasi-two-dimensionaltraps: η 1, we
≪
where F(a,b;c;x) denotes the hypergeometric function obtain the following approximate formula for (x,η):
F
and Γ(x) is the Euler gamma function. It can be easily
verified that the sum in Eq. (5) involving complex roots η≪1
(x,η) Φ(x) log(η) ψ(x/η), (10)
of unity is a real number for x R. On the other hand, F ≈ − − −
∈
forpancakeshapedtrapswith anisotropyparameterη = where
1/n, the following result holds
Φ(x)= 2 log(1+x) (11)
n−1 −
2√π Γ(x+m/n) ∞
F(x,1/n)=− n Γ(x 1/2+m/n). (6) +2 (2k)! (k+ 1)log x+k +1 ,
mX=0 − Xk=1(2kk!)2 (cid:20) 2 x+k+1 (cid:21)
For n = 1, we recover obviously the well known re-
and ψ(z) = (d/dz)logΓ(z) denotes the digamma func-
sult for the spherically symmetric trap: (x,1) =
F tion. This result is valid for x > 1, which corre-
2√π Γ(x)/Γ(x 1/2) [7]. We note that Eqs. (5) and −
(−6) are derived fr−om the integral representation (4) ap- sponds to energies E < E0 + 2. For the lowest ex-
cited states: 0 < E E 2, we approximate Φ(x) by
plicable for x > 0, however, their validity for x < 0 is − 0 ≪
Φ(0)inEq.(10), andcomparethe resultingenergyspec-
extended by virtue of the analytic continuation.
trum to that of the two-dimensionalsystem. In the two-
In the general case, when η does not meet the con-
dimensional trap, the eigenenergies of two interacting
ditions of Eqs. (5) and (6), the energy spectrum can be
atoms are given by log(2a2 η) = ψ((E E)/(2η))
determinednumerically. ForE <E0thefunctionF(x,η) [18]. Inthiswaywefi−ndtheva2lDueofthetwo0-d−imensional
is given by by Eq. (4), while for E >E , one can utilize
0 scattering length a for which both spectra are the
the following recurrence relation 2D
same: a = exp[1( √2π/a )]/√2, where =
2D 2 D − 3D D
(x,η) (x+η,η)=η√πΓ(x)/Γ(x+1/2), (7) Φ(0) 1.938. This result agrees with the value of a2D
F −F derive≃dfor a quasi-two-dimensionalsystem without con-
whichcanbeeasilyderivedfromthedefinitionof (x,η). finement in the radial direction [15]. In the range of en-
F
From the practical point of view, the use of the exact ergiescorrespondingto a bound state, we use anasymp-
results of Eqs. (5) and (6) is efficient as long as n is not toticexpansionofψ(x/η)forx/η 1inEq.(10),which
≫
3
be determined from the following expansions
120
0.6 Ψ(r)= ηe−ηρ2/2 ∞ 2ηmΓ(2ηm−E)L (ηρ2)
80z 2π3/22E/2 mX=0h 2 m
E/ 0.4 DE−2ηm(z √2) , (13)
× | | i
40 e−(ηρ2+z2)/2 ∞ ( 1)k
0.2 Ψ(r)= 2π3/2 Xk=0(cid:20)2−2kk!H2k(z)Γ(cid:16)kη − 2Eη(cid:17)
0 U k E ,1,ηρ2 . (14)
(a) (b) 0.0 × (cid:16)η − 2η (cid:17)i
-2 -1 0 1 2 -20 -10 0 10 20
Here = E E , L (x) and H (z) are respectively
a/az a/az E − 0 m k
the Laguerre and Hermite polynomials, D (x) is the
ν
FIG. 1: Energy spectrum of two atoms interacting via parabolic cylinder function and U(a,b,z) denotes the
regularized delta potential in a three-dimensional trap with confluent hypergeometric function. As it can be easily
η=ω⊥/ωz =100 (a) and η=0.01 (b). Panel (a): Theexact observed, the first expansion involves the harmonic os-
energylevels(solidlines)arecomparedwiththeenergyspec-
cillator wave functions in the radial direction and the
trum of the one-dimensional system with renormalized scat-
one-dimensionalsolutionfortwointeractingatomsinthe
teringlength(dashedlines),andwiththeenergiesofabound
axialdirection. Wehaveverifiedthatforelongatedtraps
statecalculated from Eq.(9). (dottedlines -almost indistin-
(η 1), the first term of this series provides already a
guishable from the solid ones). Panel (b): The exact energy
≫
levels (solid lines) are compared with the energy spectrum quite good approximation for the wave function of the
of the two-dimensional system with renormalized scattering lowest excited states. A similar feature is observed for
length (dashed lines), and with theenergies of abound state the second series in Eq. (14) in the traps with η 1.
≪
calculatedfromEq.(12)(dottedlines). Thescatteringlength Conversely, for energies E < E the two series involve
0
a is scaled in h.o. unitsaz = ~/mωz. generally several terms. In this regime, we can analyze
p the behavior of the wave functions on the basis of the
integralrepresentation(2). Due to the complicatedform
yields (x,η) Φ(x) logx. Substitutingthisapproxi- of the latter integral, we focus here only on the limiting
F ≈− −
mationinto(3),weobtaintheequationwhichdetermines caseofquasi-one-orquasi-two-dimensionaltraps,andin-
the energy of a bound-state in quasi-two-dimensional vestigate only the behavior of the axial (ρ = 0) and the
traps: radial (z =0) profiles of the wave functions.
ExpandingtheintegralinEq.(2)forη 1,weobtain
√2π/a=Φ((E0 E)/2)+log((E0 E)/2). (12) the axial (Ψz(z) Ψ(zˆz)) and radial (Ψ≫⊥(ρ) Ψ(ρρˆ))
− − ≡ ≡
profiles of the wave function, applicable for E <E
0
For a shallow bound-state (E E 1) one can ap-
0
bpirnodxiinmgateeneΦr(g(yEi0s−giEve)/n2b)ybyEΦ(0−)Ean=d≪0in.2t8h8isexrpe(g√im2eπ/thae) Ψz(z)η≫1 η ∞ exp(cid:16)−2|z|pmη−E/2(cid:17), (15)
[15]. 0 − ≈ 2π mX=0 mη−E/2
Fig. 1 shows the energy spectrum of two interact- Ψ⊥(ρ)η≫1e−ηρ2/2 ρ−1+p√ηζ 1, E /(2π). (16)
ing atoms calculated for η = 100 (a) and η = 0.01 ≈ h (cid:16)2 −2η(cid:17)i
(b). Fig. 1(a) compares the exact energy levels given For z √ 1, the main contribution to the sum in
by Eqs. (3) and (5), with the energy spectrum of Eq.(|1|5)−coEm≫esfromthe firstterm. Inthis casethe wave
the one-dimensional system with renormalized scatter- function exhibits the exponential decay, which is simi-
inglengtha1D, andwithbound-state energiescalculated lar to the behavior of one-dimensional bound-state in a
from Eq. (9). Fig. 1(b) presents the exact result of Eqs. free space: Ψ(z) exp( √ 2 z ). On the other hand,
(3) and (6), the energy spectrum of the two-dimensional the wave function∼in the−rad−ialEd|ir|ection has a Gaussian
system with renormalized scattering length a2D, and profile, characteristic for the ground-state of harmonic
bound-state energy calculated from Eq. (12). We have oscillator,whereasthe divergentterm1/(2πρ)arisesdue
not included the energy levels calculated from approxi- to the interaction potential.
mations (8) and (10), which for η = 100 and η = 0.01 In quasi-two-dimensional traps, for energies E < E ,
0
are indistinguishable from the exact result. We observe we found the following radial and axial profiles of the
that for E >E0 the one- and the two-dimensional spec- wave-functions
trafitverywelltheexacteigenenergies,whereastheyare
∞
ilnatctoerrr,ehctowweivther,reasrpeewcteltlodtehsceribboeudnbdy-sEtaqtse.e(9n)eragnieds.(1T2)h.e Ψ⊥(ρ)η≪≈1π−32 (2(m2mm)!!)2K0(2ρ m−E/2), (17)
mX=0 p
We now turn to the calculation of wave functions. η≪1e−z2/2 1 Φ( /2)+log( /2)
While for E < E0 they can be evaluated from the in- Ψz(z) −E −E , (18)
tegral representation (2), in the general case, they can ≈ 2π (cid:20) z − √π (cid:21)
| |
4

Finally we would like to stress that our derivation can
0.8 be easily supplemented to include an energy-dependent
0.20 scatteringlength[10,11,16]. Thisextendsthevalidityof
1/2 a0z.6 the pseudopotential approximation to scattering lengths
0.15
= 0 much larger than the trap size, and allows to properly
r z = 0
0.4 describe the entire molecular spectrum. The energy-
0.10
dependent effective scattering length is defined through
0.2 z = 0 = 0 0.05 t~hkeiss-wthaveerpehlaatsieveshmiftomδ0e:natueffm(E[1)9=].−Ttahneδa0p(kp)li/cka,tiwohneroef
(a) (b)
0.0 0.00 this model interaction in our derivations leads to substi-
0.0 0.1 0.2 0.3 0 1 2 3 4 5 6
r/az r/az tutionofabyaeff(E)inEq.(3)determiningtheeigenen-
ergies,andrequiresaself-consistentsolvingforthe value
FIG. 2: The axial (ρ=0) and the radial (z =0) profiles of
ofE. FormagneticallytunableFeshbachresonances,the
theground-statewave function for two atoms interacting via
a regularized delta potential with a = ±∞. The atoms are s-wavephase shift is knownanalytically [17], and in this
case one can derive an explicit formula for a (E) [10].
confined in a harmonic trap with η =ω⊥/ωz =100 (a), and eff
η = 0.01 (b). The exact profiles (solid lines) are compared
Insummary,wesolvedanalyticallytheproblemoftwo
withtheapproximateresultsofEqs. (15)-(17)(dashedlines).
atomsinteractinginanaxiallysymmetricharmonictrap
All lengths are scaled toaz = ~/mωz.
with arbitrary trap anisotropy. For integer ratios of the
p
trapping frequencies we gave closed formulas for the so-
lutions. Furthermore, by introducing an effective energy
where K (x) is a modified Bessel function. The asymp-
0 dependence in the scattering length [10, 11], we can find
toticbehaviorofK (x)forx 1isgovernedbyK (x)
0 ≫ 0 ∼ the solutions for any value of the latter. Therefore our
π/2xe−x. Hence, for ρ√ 1, the sum in (17) is
−E ≫ result allows for a direct exact evaluation of the dynam-
pdominated by the first term, and the asymptotic decay
ics of a pair of interacting neutral atoms in very tight
of the wave function in the radial direction is similar to
traps, possibly in reduced dimensionality and under an
the one observed for a bound state in two dimensions:
arbitrary external magnetic field, even in the presence
Ψ(ρ) K (√ 2 ρ). Along the tightly confined, ax-
∼ 0 − E of Feshbach resonances. Applications include a signifi-
ial direction, the wave function has a Gaussian profile,
cantrangeofsituationsinvolvingquantumcontrolatthe
which is modified at short distances by the interaction
atomiclevel,fromsingle-atominterferometrytoquantum
potential.
information processing.
The behavior ofthe ground-statewavefunction in the
unitarity limit (a = ) in the quasi-one-dimensional WethankL.P.Pitaevskii,M.Holland,G.OrsoandM.
±∞
(η = 100) and quasi-two-dimensional traps (η = 0.01) Wouters for valuable discussions. We are grateful to E.
is presented in Fig. 2. The figure compares the exact Bolda for making available his numerical data reported
profiles evaluated from Eqs. (13) and (14) with the ap- in [9]. T. Calarco acknowledges support from the EC
proximate results of Eqs. (15)-(17). We observe that all undercontractIST-2001-38863(ACQP)andfromMIUR
approximate curves fit quite well the exact functions. (COFIN 2002).
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