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 Schrödinger 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 Schrödinger 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 problem, substitute this decomposition into the Schrödinger 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 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 excited 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 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 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 solutions. 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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Two atoms in an anisotropic harmonic trap

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