A unitary quantum gas is a gas of quantum particles with a binary interaction
of infinite scattering length and negligible range. It has been produced in
recent experiments with gases of fermionic atoms by means of a Feshbach
resonance. Using the Fermi pseudo-potential model for the atomic interaction,
we show that the time evolution of such a gas in an isotropic three-dimensional
time dependent harmonic trap is exactly given by a gauge and scaling transform.Comment: submitted 23 March 200

Bourdel

Carlson

Castin

Greiner

Jochim

Kagan

Madison

Menotti

Nozières

O'Hara

Olshanii

Petrov

Pitaevskii

Randeria

Vogels

Yvan Castin

Zwierlein

English

arXiv

Comptes Rendus Physique

gligibleg theic three-uement etmoyennens un gaztrons quemporellemosteractionsnanyctivee BroglieC. R. Physique 5 (2004) 407–410Physics/Statistical physics, thermodynamicsExact scaling transform for a unitary quantum gasin a time dependent harmonic potentialYvan CastinLaboratoire Kastler Brossel, École normale supérieure, 24, rue Lhomond, 75231 Paris, FranceReceived 23 March 2004; accepted 30 March 2004Presented by Jean DalibardAbstractA unitary quantum gas is a gas of quantum particles with a binary interaction of infinite scattering length and nerange. It has been produced in recent experiments with gases of frmionic atoms by means of a Feshbach resonance. UsinFermi pseudo-potential model for the atomic interaction, we show that the time evolution of such a gas in an isotropdimensional time dependent harmonic trap is exactly given by a gauge and scaling transform.To cite this article: Y. Castin,C. R. Physique 5 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméÉvolution d’un gaz quantique unitaire dans un potentiel harmonique variable : solution par changement d’échelle.Nous entendons par « gaz quantique unitaire » une assemblée de particules dont le mouvement est décrit quantiqqui interagissent par un potentiel de longueur de diffusion infinie et de portée négligeable devant leur distanceet leur longueur d’onde thermique. Un tel gaz a été produit récemment à l’aide d’une résonance de Feshbach dad’atomes fermioniques. En modélisant les interactions entre particules par le pseudo-potentiel de Fermi, nous monl’évolution d’un gaz unitaire dans un potentiel de piégeage harmonique isotrope tridimensionnel de dépendence tequelconque est décrite exactement par la composition d’un chagement d’échelle et d’une transformation de jauge.Pour citercet article : Y. Castin, C. R. Physique 5 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords: Unitary quantum gas; Scaling transform; Harmonic trapMots-clés : Gaz quantique unitaire ; Changement d’échelle ; Potentiel harmonique1. IntroductionExperiments with quantum gases of spin 1/2 fermionic atoms are currently making rapid progresses. One of thefascinating properties of these fermionic gases is the possibility to freely tune the sign and the strength of the atomic intwithout reducing the lifetime of the sample: the value of thes-wave scattering lengtha of two particles with opposite spicomponents can virtually be adjusted from−∞ to +∞ with a Feshbach resonance technique [1], without inducinginstability of the gas even in the unitary limita = ±∞ [2]. These systems are still gases, in the sense that the efferange of the interaction potential is negligible as compared to the mean interparticle separation and to the thermal dE-mail address: yvan.castin@lkb.ens.fr (Y. Castin).1631-0705/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crhy.2004.03.017408 Y. Castin / C. R. Physique 5 (2004) 407–410d Fermiring throughnd on therfpand andedalretical–Einsteincalingtit of anuse of aiation ofosonsncouplingy. Thethe non-e,errm ofll inwavelength. We shall take advantage of this crucial property and model the true interaction potential by the so-callepseudo-potential [3].Such stability of the strongly interacting Fermi gases opens up fascinating possibilit es, e.g. the study of the crossovebetween a Bose–Einstein condensate of dimers (already observed, see [4–7]) and a BCS condensate of pairs, by passthe strongly interacting regimekF |a|  1, wherekF is the Fermi momentum [8,9]. In the unitary limitkF |a| → +∞, thethermodynamic properties of the spatially homogeneous gas are universal: they depend only on the Fermi energy atemperature. At zero temperature, the chemical potential of the homogeneous gas is thenµ = ηµ0 whereη is a pure numbeandµ0 is the chemical potential of the ideal Fermi gas [2,10]. An accurate measurement ofη would provide a crucial test omany-body theories [2,5,7].The standard imaging technique used with quantum gases is to switch off the trapping potential, to let the gas exto perform a light absorption imaging of the atomic cloud. Such a ballistic expansion acts as a magnifying lens: it was usto reveal the vortex lattice in arotating Bose–Einstein condensate [11], and very r cently to obtain the value of the universnumberη for the unitary Fermi gas [7]. Clearly the interpretation of the time of flight images strongly relies on a theounderstanding of the time evolution of the gas in a time dependent harmonic potential. In the case of a pure Bosecondensate in the regimekF |a|  1, this was achieved starting from the Gross–Pitaevskii equation by a gauge plus stransform [12,13]. When the Bose or Fermi gas enters the strongly interacting regimekF |a| > 1, no solution starting from firsprinciples is available1 and one relies on the hydrodynamic approximation [14].Here we consider the idealized case of an isotropic and harmonic three-dimensional trapping potential. In the liminfinite scattering length, we show that the Fermi pseudo-potential has a scaling invariance that rigorously allows thegauge plus scaling transform similar to the one of [15] to describe the time evolution of the gas due to an arbitrary varthe trapping frequency.2. The model based on the Fermi pseudo-potentialConsider an assembly ofN non-relativistic particles, with an arbitrary spin. These particles may be indistinguishable bor fermions, or even be distinguishable. All the particles have the same massm nd interactvia the same binary interactiopotential independent of the spin degrees of freedom. The interaction potential is the Fermi pseudo-potential withconstantg related to thes-wave scattering lengtha by g = 4πh̄2a/m. At this stage, 0< |a| < +∞, we shall take the unitarlimit |a| → +∞ later.Let ψ(r1, . . . , rN) be the wavefunction of the gas corresponding to a given (but arbitrary) spin configurationwavefunctionψ then evolves according to the Schrödinger equation:ih̄∂tψ =N∑i=1[− h̄22mri + U(ri )]ψ +∑1i<jNgδ(ri − rj )ψ regij . (1)Hereri is the three-dimensional Laplacian with respect to the spatial coordinatesri of particle numberi, U is the trappingpotential seen by each particle and eachψ regij, the so-called regular part ofψ in ri = rj , is the following function ofN − 1vectors of coordinates:ψregij({rk, k = i, j},Rij) = limrij →0∂∂rij(rij ψ), (2)whererij is the norm ofrij ≡ ri − rj and where the limit and the partial derivative are taken for fixed positionsrk of theN − 2particles other than particlesi andj , and for a fixed position of the center of mass of the particlesi andj, Rij ≡ (ri + rj )/2.The domain of the Hamiltonian corresponding to the Fermi pseudo-potential is therefore not the Hilbert space ofinteracting gas, but a functional space withspecific boundary conditions for the wavefunctionψ . More precisely, as we now sethe model amounts to replacing the true interaction potential bycontact conditions, i.e. by boundary conditions onψ when thedistancerij between two particles tends to zero, the wavefunctionψ otherwise evolving with the interaction free Schrödingequation [16]. As the wavefunctionψ does not contain any delta singularity, each delta singularity in the interaction teEq. (1) has indeed to be compensated by a delta singularity in the kinetic energy term. In 3D this implies thatψ can diverge as1/rij whenrij → 0, a divergence which is still square integrable. Two cases can occur:1 It was expected that a gauge plus scaling transform would provide an exact solution to theN -body problem in a strictly two-dimensionaBose gas in an isotropic harmonic trap [15], for an interaction potential modeled by a delta distribution. Actually a delta interaction potentia2D does not lead to a mathematically well defined problem and therequired regularisation [17] breaks the scaling invariance.Y. Castin / C. R. Physique 5 (2004) 407–410 409y fromnet:o that weth onlyeonisnsform (seenshe– case (i): limrij →0 ψ = 0: no delta singularity occurs from the kinetic energy operator, and there is no delta singularitthe interaction term asψ regijvanishes. For instance, this is the case when particlesi andj are indistinguishable fermions ithe same spin state.– case (ii):ψ has a 1/rij singularity:ψ = Arij+ B + O(rij ), (3)whereA andB are still functions of therk ’s with k = i, j , and ofRij. The regular part ofψ is thenψ regij = B. Writing thekinetic energy operator for the pair of particlesi, j asri + rj = 12Rij + 2rij and using(1/r) = −4πδ(r), we findthat the total coefficient ofδ(ri − rj ) in the right-hand side of Eq. (1) vanishes provided thatA + aB = 0.A way of summarizing the two cases is thensimply to impose the boundary conditions:ψ(r1, . . . , rN) = A({rk, k = i, j},Rij)[r−1ij − a−1] + O(rij ) (4)the first case corresponding toA = 0 and the second one toA = 0. Having ensured that all the delta singularities cancel in thSchrödinger equation, we can now restrict it to the manifold where the positions of the particles are two by two distincih̄∂tψ =N∑i=1[− h̄22mri + U(ri )]ψ for rij = 0, ∀i = j. (5)Eqs. (4) and (5) constitute the basis of our model.3. Scaling transform in the unitary limitWe specialize the previous section to the case of the unitary quantum gases: the scattering length is now infinite, sset to zero the 1/a term in the boundary conditions (4), to obtainψ(r1, . . . , rN) =A({rk, k = i, j},Rij)rij+ O(rij ). (6)Note that the unitary limit does not look in any way singular in our formulation of the model, as the scattering lengappears through its inverse in Eq. (4). Also we restrict to the case of an isotropic harmonic potential:U(r) = 12mω2(t)r2, (7)where the oscillation frequencyω(t) of a particle in a trap is constant and equal toω0 for t < 0 and has an arbitrary timdependence fort  0. The ballistic expansion of the gas mentio ed in the introduction corresponds to settingω to zero fort  0.We assume that the state vector of the gas fort < 0 is a steady state of Schrödinger’s equation with the energyE. Thecorresponding wavefunction for the considered spin configuration isψ0, which in particular obeys the boundary conditiequation (6). Our ansatz for the time-dependent wavefunction, inspired from [12,13,15], is thenψ(r1, . . . , rN, t) = N (t)ei∑Nj=1 mr2j λ̇(t )/2h̄λ(t)ψ0(r1/λ(t), . . . , rN/λ(t)), (8)where the time dependent normalisation factorN (t) and the time dependent scaling factorλ(t) need to be determined. Thansatz is the combination of a gauge transformation (see the Gaussian phase factor) and a time dependent scaling trathe rescaling of the coordinates byλ(t) in ψ0).The first step is to check that the ansatz equation (8) is in the right Hilbert space,i.e. that it satisfies the boundary conditioequation (6). The scaling transform indeed preserves th boundary conditions: it rescales and multiplies the functionA by λbut does not lead to the appearance of a O(1) term in Eq. (6). Note that for a non-zero value of 1/a the conclusion would bedifferent if A = 0. The gauge transform also preserves the boundary conditions: for fixedk ’s we writer2i + r2j = 2R2ij + r2ij /2so that for a fixedRij, the gauge transform involves only as a non-constant factoreimr2ij λ̇/4h̄λ = 1+ imr2ij λ̇/4h̄λ + O(r4ij). (9)As r2ij × A/rij = O(rij ), the boundary conditions are preserved by the gaugetransform, a conclusion that extends to tnon-zero 1/a case.410 Y. Castin / C. R. Physique 5 (2004) 407–410choice offnormrbitrarythe zerothe limitts. LKBool,1..What is left is to check that the free particle Schrödinger equation (5) is satisfied by the ansatz for an appropriateλ(t) andN (t). One calculates the time derivative and the Laplacian of the ansatz. One uses the fact thatψ0 is an eigenstate oenergyE to express the action of the kinetic energy operator onψ0 in terms ofr2j ψ0 terms andEψ0. Equating the termsr2j ψ0on both sides of Eq. (5) leads toλ̈(t) = ω20λ3(t)− ω2(t)λ(t) (10)to be solved with the initial conditionsλ(t < 0) = 1, λ̇(t < 0) = 0. For a ballistic expansion, one findsλ(t) = (1 + ω20t2)1/2.Equating the terms proportional toψ0 gives Ṅ /N = −3Nλ̇/(2λ) − iE/(h̄λ2) which is readily integrated inN (t) =λ−3N/2(t)exp[−iE ∫ t0 dτ/λ2(τ)h̄]. The first factor in the right-hand side of this equation ensures the conservation of theof the wavefunction.The ansatz equation (8) therefore gives, in the unitary limit, the exact time evolution of the initial state vector for an atime dependence of the isotropic harmonic potential. It rigorously confirms the scaling law that one would obtain fromtemperature hydrodynamic approximation. It also allows to use the symmetry considerations developed in [15]: e.g.of linear response to a sudden change in the frequencyω away fromω0 gives rise to an undamped oscillation at frequency 2ω0which reveals the existence ofN -body stationary states of energyE ± 2h̄ω0 coupled toψ0 by the excitation operator∑Ni=1 r2i .AcknowledgementsWe thank L. Pricoupenko for discussions, A. Sinatra, I. Carusotto, J. Dalibard and C. Cohen-Tannoudji for commenis a research unit of École normale supérieure and Université Pierre et Marie, associated to CNRS.References[1] J.M. Vogels, C.C. Tsai, R.S. Freeland, S. Kokkelmans, B.J. Verhaar, D.J. Heinzen, Phys. Rev. A 56 (1997) R1067.[2] K.M. O’Hara, S.L. Hemmer, M.E. Gehm, S.R. Granade, J.E. Thomas, Science 298 (2002) 2179.[3] Y. Castin, in: R. Kaiser, C. Westbrook, F. David (Eds.), Coherent Atomic Matter Waves, Lecture Notes of Les Houches Summer SchEDP Sciences and Springer-Verlag, 2001, pp. 1–136.[4] M. Greiner, C.A. Regal, D.S. Jin, Nature 426 (2003) 537.[5] S. Jochim, M. Bartenstein, A. Altmeyer, S. Riedl, C. hin, J.H. Denschlag, R.Grimm, Science 302 (2003) 2101.[6] M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, S. Gupta, Z. Hadzibabic, W. Ketterle, Phys. Rev. Lett. 91 (2003) 250401.[7] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M.Teichmann, L. Tarruell, S. Kokkelmans, C. Salomon, cond-mat/040309[8] P. Nozières, S. Schmitt-Rink, J. Low Temp. Phys. 59 (1985) 195.[9] M. Randeria, in: A. Griffin, D. Snoke, S. Stringari (Eds.), Bose–Einstein Condensation,Cambridge University Press, Cambridge, 1995[10] J. Carlson, S.-Y. Chang, V.R. Pandharipande, K.E. Schmidt, Phys. Rev. Lett. 91 (2003) 050401.[11] K.W. Madison, F. Chevy, W. Wohlleben, J. Dalibard, Phys. Rev. Lett. 84 (2000) 806.[12] Y. Castin, R. Dum, Phys. Rev. Lett. 77 (1996) 5315.[13] Y. Kagan, E.L. Surkov, G. Shlyapnikov, Phys. Rev. A 54 (1996) R1753.[14] C. Menotti, P. Pedri, S. Stringari, Phys. Rev. Lett. 89 (2002) 250402.[15] L.P. Pitaevskii, A. Rosch, Phys. Rev. A 55 (1997) R853.[16] D.S. Petrov, C. Salomon,G.V. Shlyapnikov, cond-mat/0309010.[17] M. Olshanii, L. Pricoupenko, Phys. Rev. Lett. 88 (2002) 010402.

Exact scaling transform for a unitary quantum gas in a time dependent harmonic potential

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Exact scaling transform for a unitary quantum gas in a time dependent
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Exact scaling transform for a unitary quantum gas in a time dependent harmonic potential

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