The pseudopotential in resonant regimes Ludovic Pricoupenko
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Ludovic Pricoupenko. The pseudopotential in resonant regimes. 2005. hal-00007776v1
HAL Id: hal-00007776 https://hal.archives-ouvertes.fr/hal-00007776v1 Preprint submitted on 3 Aug 2005 (v1), last revised 10 Jan 2006 (v3)
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ccsd-00007776, version 1 - 3 Aug 2005 r odto ntewv ucini the in function wave the on by bound- condition suitable Deuteron a ary to with the force possible on neutron-proton was the it properties replacing that scattering shown numerous [1] Peierls get R. and First, Bethe Physics. H. Nuclear in interaction neutron-proton the suooeta 4.Tefc htterneo h nu- the of range the ( that Fermi force so-called fact the clear The of formulation [4]. final Breit the pseudopotential G. to by led further and developed [3] between was Equivalence approaches proton. two a these and be- neutron slow cross-section scattering a tween of a computation of the idea approxi- in Born the mation a introduced for [2] allowing Fermi potential effective E. zero-range after, year One nel. eto-rtntiltsatrn egh( length scattering triplet neutron-proton aiyo qiaetzr-ag suooetas the introducing pseudopotentials: by zero-range generalized equivalent λ been of has family idea a for pseudopo- This Fermi [6] the BCS using tential. or implemented a bosons be can As for is fermions [5] this approaches potential. 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resonances. broad -wave ≃ s 4 wv chan- -wave . 10 3 λ Dtd uut4 2005) 4, August (Dated: uoi Pricoupenko Ludovic doesn’t − s -wave 15 m) r eido h eoatrgm nthe in studies regime These resonant 17]. the 15] [16, on 14, gas 13, relied quantum [12, are unitary problem obtain- the body for for few tool the and a in as results understood exact be ing scattering to energy has low a but two-body only properties appears not the it is case, of pseudopotential this parameterization zero-range In the fruitful. that ap- be clearly im- zero-range to another the proven which is has in proach problem research body of direction few portant The quasi discovered. in have been interaction systems two-dimensional tunable by quasi of is and concept This one-dimensional the function. that wave way 3D this the Bethe-Peierls on the of condition from properties deduced boundary dimensional scattering be low can where particles also 11] confined 10, concerns [9, gases configurations pseudopoten- ultra-cold powerful and in A approaches tials [8]. zero-range [7] traps of two-dimensional systems application quasi dimensional in Hartree-Fock- also three the and in using formalism condensate the Bogolubov the of action on back the excitations self-consistently account into take hl h onapoiaincnb dutdt the energy. to colliding adjusted arbitrary be for problem, The solution can two-body zero-range approximation the exact Born of the treatment while exact the change tegho orltoscnb ue rirrl while arbitrarily the tuned that fact be the Major can from correlations hand. comes of at systems strength now these channels are in in momentum interest crossover angular BCS-BEC higher the of of studies in scattering possible resonant to 25]. p enlarges [24, ques- problem domain body accurately the many studying Actually, quantum itself for the in sit- in model interesting such raised a tions only of as example not serves is also spectacular but system a the is where 23] field uation 22, gases of 21, the Fermi achievement 20, two-components in [19, experimental in directions crossover The BCS-BEC challenging the more atoms. ultra-cold the actually of of appears one resonances as scattering through tained of use magnetic by external systems an by field. ultra-cold tuned in [18] achieved resonances Feshbach be can which wv 2,2,2,2,3]and 30] 29, 28, 27, [26, -wave oegnrly td fhg orltosrgmsob- regimes correlations high of study generally, More n euaie clrpoutaatdto adapted product scalar regularized a ing λ ptnilhsbe ple o lr-odbsn to bosons ultra-cold for applied been has -potential hnteFripedptnilwhich pseudopotential Fermi the then s n hr w-oysatrn sres- is scattering two-body where ons rneshm ecp o h Fermi the for (except scheme -range ihafml fpseudopotentials. of family a with i ee 5 France. 05, Cedex ris ee, d wv hnes[1,and [31], channels -wave sdescribed is s wv channel -wave 2 the mean inter-particle distance remains large with re- However, it is of interest when dealing with wave func- spect to the potential radius R (for example, the s-wave tions of arbitrary symmetry (for example in presence of scattering length can be adjusted to several order of mag- an anisotropic external potential) to have the full expres- nitude larger than R). As a consequence, one can ex- sion of the pseudopotential, that is an expression which pect that short range physics is not directly involved in contains implicitly the projection operator over the in- a large class of many body properties and can be de- teracting channel (noted Πl for the channel of angular scribed only in terms of the low energy two-body behav- quantum number l). ior. It is worth pointing out that this fundamental feature has been verified in the BCS-BEC crossover of the two- component Fermi gas. For this reason, the zero-range C. Organization of the Paper approach which is parameterized only by the two-body low energy physics is a very appealing tool for study- ing these regimes. Actually, the state of the art is as The paper is organized as follow: in the first part, the follow: broad s-wave resonances (like the one in 6Li at regime of interest (two-body resonant scattering) is intro- 835 Gauss [21]) can be accurately parameterized with the duced and characterized in each partial wave channel by scattering length by use of the Fermi pseudopotential, a set of two parameters in the phase-shift. In the second and for narrow s-wave resonance (for example in 23Na part, the different tools of the zero-range approach in this at 907 Gauss [32]) where the effective range is large and regime are defined. We show that the problem can be de- negative, the method has been generalized by modifying fined in terms of boundary conditions on the wave func- the Bethe-Peierls boundary conditions in Ref.[14]. Re- tion which generalize the Bethe-Peierls approach. We cently, a pseudopotential has been introduced for p-wave express these boundary conditions in spherical coordi- resonances [33] and the aim of this paper is to extend this nates and also in Cartesian coordinates by use of the approach for resonant regimes in arbitrary partial waves. symmetric trace free tensors appearing in the usual mul- tipolar expansion. This way of defining the zero-range approach permits to include quite easily both the projec- tion operator Πl and also the l-th derivatives of the delta B. Which generalization of actual pseudopotentials distribution in the pseudopotential. It appears that in is needed ? each interacting channel there exists a family of pseu- dopotentials generated by a free parameter, thus gen- The idea of replacing a true finite range two-body po- eralizing the λ-potential approach [7, 33]. This degree tential by a zero-range pseudopotential acting on each of freedom makes possible to use the pseudopotential in partial wave has been first developed by Huang and Yang the Born approximation for mean-field approaches. The [34] in the context of the hard-sphere model in view of third and last part, illustrates the formalism. We show applications to superfluid 4He. However, due to the im- how to recover two-body scattering states by use of the portance of the short range behavior in these systems, Green’s method. Pseudopotential of Ref.[35] is recovered pseudopotentials describing three, four body and more as a particular limit of the present approach. Finally, correlations were also proposed in Ref.[34] for modeling the scattering states and also the low energy bound state the system beyond the dilute limit. The actual situa- being not mutually orthogonal in the zero-range scheme tion in ultra-cold atoms is radically different as the mean (except for the Fermi pseudopotential), we introduce a inter-particle distance is very much larger than the poten- regularized scalar product in order to solve this incon- tial radius, and two-body pseudopotentials are sufficient sistency. Applied to the low energy bound-states, we to describe the low energy behavior for the many-body show that the new scalar product leads to a normaliza- system even in high correlation regimes. Moreover as tion which coincides with the result given by the method the hard-sphere model cannot support resonances, the based on the analyticy of the scattering amplitude [36]. tools developed in Ref.[34] cannot be used for our pur- pose. Recently, a generalization of the Fermi pseudopo- tential for arbitrary partial wave and phase shift has II. ZERO-RANGE FORMALISM IN THE been proposed using delta-shell potentials in the limit RESONANT REGIME of small shell radius [35], by the way correcting a mis- take of Ref.[34]. Boundary conditions associated to the A. Regime of interest zero-range approach in Ref.[35] are defined for scattering states in free space and depend explicitly on their wave number k, so that the approach applies only for states Let us consider two particles in absence of an external of particular symmetry and in time independent situa- potential interacting via a short range isotropic potential tions. Instead, in this paper we show that in resonant U(~r ) (~r are the relative coordinates between the two par- regimes, general (that is energy independent) boundary ticles). In order to simplify the discussion and without conditions can be found. Another feature of the pseu- loss of generality, we work in the center of mass frame. dopotentials in Ref.[35] is that their action is defined on The asymptotic form of the scattering wave function ~ the radial wave function of the specified partial wave. Ψ~k(~r ) with relative wave-vector k and colliding energy 3
¯h2k2 V (r) E = defines the partial amplitudes f in each par- l 2µ l tial wave channel of angular quantum number l through the relation: E ∞ r exp(ikr) Ψ (~r ) = exp(i~k.~r )+ (2l+1) (cos γ)f , (1) ~k Pl l r Xl=0 where kr 1, γ = (~k, ~r ) and l is the Legendre poly- ≫ P 0 R r nomial of degree l. Thed partial amplitudes can be also expressed in terms of the phase shift δl with: 1 f = (exp(2iδ ) 1) . (2) l 2ik l −
Typical situations in ultra-cold atoms corresponds to bi- FIG. 1: Schematic representation of a two-body effective ra- nary collisions of particles having small enough relative ¯h2 l(l + 1) dial potential V (r) = U(r) + in a partial-wave velocities to ensure kR 1, where R is the potential ra- l 2µr2 ≪ dius. For two neutral alkali atoms in their ground state, l > 0. Er is the energy of a quasi-bound state [4, 36] which the asymptotic form of U(~r ) is a Van der Waals tail: leads to a resonant behavior in the two-body scattering for a 6 ≃ U(~r ) C6/r and the criterion for low energy scattering colliding energy E Er. processes≃ (colliding energy E E ) is fixed by [37, 38]: ≪ R phase shift in Eq.(4): 1/4 2 2µC6 ¯h R = ; ER = (3) w k2l ¯h2 2µR2 l fl(k)= 2 2l+1 , (5) −1+ wlαlk + iwlk where µ is the relative mass of the two atoms. Standard 6 values for R (and ER) range from 3.3 nm for Li (ER and follow the discussion given in Ref.[36]. The reso- 133 ∼ ≃ 7.25 mK) to 100 nm for Cs (ER 32.5 µK). In this nant regime corresponds to situations where the parame- ∼ ≃ ter w takes arbitrary large values, so that w α R2. paper we consider situations where the phase shift can | l| | l l| ≫ be parameterized in the following form: It can be shown [36] that describing the resonant regime by Eq.(5) is consistent only for αl > 0 (this assumption 2l+1 1 2 emerges also as a consequence of the regularized scalar k cot δl(k)= αlk . (4) −wl − product that we introduce in the last section). The main difference as compared to s-wave broad resonance is that Eq.(4) corresponds to the effective range approximation the resonant character of the scattering cross-section as- for a short range potential [39]: the parameter wl char- sociated to Eq.(5) depends crucially on the sign of wl as acterizes the Wigner threshold regime and αl/2 is “the can be easily seen by putting Eq.(5) in the Breit-Wigner − effective range” even if it has the dimension of a length form: only for l = 0. For neutral particles interacting with a n potential of asymptotic form U(r) Cn/r (r ) 1 Γ/2 ≃ → ∞ fl = , (6) the threshold behavior is valid for l< (n 3)/2 and the −k E Er + iΓ/2 effective range approximation for l< (n −5)/2. In ultra- − cold atoms in their ground state the power− law is given ¯h2 ¯h2k2l+1 with Er = and Γ = . For large and by n = 6 so that the expansion in Eq.(4) is rigorous only −2µαlwl µαl in the s-wave channel (w0 is the scattering length), while positive values of wl, Er < 0 and the scattering ampli- in the p-wave channel only the first term in the expan- tude has a pole corresponding to the presence of a real sion is justified. For l 2, even the Wigner threshold bound state of vanishing energy EB Er. In this regime, regime given in Eq.(4)≥ is not valid. However, we adopt f 2 doesn’t present a resonant structure.≃ However, for | l| this form as a generic way to parameterize the resonant large and negative value of wl, the bound state transform regime in an arbitrary partial wave channel. In the fol- into a long lived quasi-bound state which produces a reso- lowing, we don’t discuss the case of the broad s-wave nance in the scattering cross-section with f 2 1/ w α | l| ≃ | l l| resonance where the effective range is negligible (α0 0) in the neighborhood of E = Er see Fig.(1). It is worth and is a very particular situation with respect to what≃ pointing out that the resonant regime occurs for relative happens in other channels (this is a consequence of the wave vectors k with kR 1 which justifies the use of a absence of centrifugal barrier in the radial equation see zero-range approach. ≪ Ref.[36]). To define the resonant regime, first we write In actual experiments on ultra-cold atoms, the reso- the expression of the partial amplitude associated to the nant character in the two-body scattering is ensured by 4 use of the concept of Feshbach Resonance [40]. The in- the origin: teraction potential depends on the spin configuration of l 2 the two body system so that the energy of a short range z z /2 jl(z)= 1 + . . . (9) two-body molecular state can be tuned using an external (2l+1)!! − 1!(2l+3) magnetic field. The scattering resonance occurs when its (2l 1)!! z2/2 y (z)= − 1 + . . . , (10) energy is in the vicinity of the collisional energy of the l − zl+1 − 1!(1 2l) two scattering atoms. Feshbach resonances appear then − as a result of the coupling between a closed channel (sup- one can deduce the scattering states from the following porting the molecular state) and an open channel (associ- boundary conditions of the wave function at r = 0: ated to the scattering states). The molecular state in the (2l 1)!! k2l+1 closed channel is linked to short range properties of the lim ∂(2l+1) rl+1R (r)=0. → − r kl system and is not described by the zero-range approach. r 0 (2l)!! − tan δl(k) As a consequence, the bound state or quasi-bound state (11) appearing in the zero-range model is associated only to This expression is a first step toward the formulation of the tail of the molecular state and results from the cou- the zero range approach, but has not a general character pling between the closed and open channels. in the sense that the quantum number k appears explic- In the s-wave narrow Feshbach resonance of 23Na at itly. Instead, we search for boundary conditions which 907 Gauss [32], α0 26 nm, and R 4.5 nm. In apply on any linear combination of scattering states with higher partial wave channels,≃ scattering≃ is not isotropic different colliding energies. By this way, the formalism i.e depends also on the angular quantum number m of can be used for example in presence of any anisotropic the state. For example the p-wave resonance in 40K [27] external potential or in time dependent situations. The (R 6.8 nm) has been parameterized with the law (4) desired condition is obtained by using the expression of in the≃ states m = 0, 1 as a function of the external the phase-shift in Eq.(4) which makes possible to trans- magnetic field.| | There are two very close zero energy form Eq.(11) in the following form: resonances (Er = 0) at B 198.84 G (m = 0) and 1 B 198.31 G ( m = 1) with≃ a width characterized lim( + )rl+1R =0 , (12) → l kl respectively≃ by α| R| 2.74 and 2.81. Generalization r 0 D wl 1 ≃ of the zero range approach for taking into account this where we have introduced the differential operator: anisotropy adds technical details not essential in a first approach and in this paper, the discussion is centered on (2l 1)!! (2l+1) 2 l = − ∂ + (2l 1)αl∂ . (13) the modeling of isotropic interactions. D (2l)!! r − r We are ready now to define the zero range approach in B. Defining the zero-range approach general situations. The wave function Ψ of the interact- ing two-body system is expanded in each partial wave channel in terms of the spherical harmonics as: In this section, the interacting potential U(~r ) acting for ~r < R is replaced by a specific behavior of the wave l lm | | cΨ (r) lm function∼ associated to Eq.(4) which is imposed by bound- ~r Πl Ψ = Y (Ω) , (14) h | | i rl+1 ary conditions on the wave function as the relative coordi- mX=−l nates between the two interacting particles goes formally where Π is the projector on the subspace of angular to zero. l quantum number l, and (r, Ω) are the spherical coordi- Outside the potential radius (r > R), the scattering nates with Ω = (θ, φ). For r = 0, Ψ(~r ) is solution of wave functions Ψ (~r ) of the two-body problem are solu- ~k the free Schrdinger equation (it6 implies that the multi- tions of the free Schr¨odinger equation with the asymp- polar radial functions c lm(r) are regular functions of the totic behavior given by Eqs.(1,2,4): Ψ radius r), and for an isotropic two-body potential U(~r ) ∞ the boundary conditions on the wave function are: ~ Ψ~k(~r )= l(cos(k, ~r ) )Rkl(r) , (7) P 1 lm Xl=0 lim + c (r)=0 l m l . (15) d → l Ψ r 0 D wl − ≤ ≤ where R (r) are the radial functions which can be ex- kl Eq.(15) generalizes the Bethe-Peierls approach [1] apply- pressed in terms of the two spherical Bessel functions as ing for interaction in the s-wave channel with α = 0: [41]: 0
Rkl(r)= kl ( jl(kr) yl(kr) tan δl(k) ) . (8) ∂ (rΨ) 1 N − lim r = , (16) → r 0 (rΨ) −w0 The zero-range approach amounts to consider the formal continuation of Eq.(8) for 0 r R . As a consequence, the limit w R being associated to a zero energy broad ≤ ≤ 0 ≫ using the behavior of jl(z) and yl(z), at the vicinity of resonance. 5
Note that generalization of Eqs.(15) to the case of an [αβ... ] l S Dl anisotropic two-body interaction is straightforward by in- l 0 1 ∂ − α ∂2 troducing parameters depending on the quantum num- r 0 r α 1 3 2 1 n ∂ + α1∂ ber m: wlm, αlm . 2 r r { } α β − 1 αβ 3 5 2 2 n n 3 δ 8 ∂r + 3α2∂r α β γ − 1 α βγ 5 5 2 3 n n n 5 (n δ + permut.) 16 ∂r + 5α3∂r C. Cartesian representation of the multipolar radial functions TABLE I: Examples for the symmetric traceless tensors Sl α Eq.(17) and for the differential operators Dl in Eq.(12). n While the zero-range approach is now established in a are the components of the unit vector ~n = ~r/r. general form by Eqs.(13,14,15), it can be easier to use equivalent boundary conditions in the Cartesian coor- dinate system for applying the formalism in inhomoge- s-wave channel; for l = 1, α (0) are the components CΨ,0 neous or anisotropic situations. By the way, as will be of the dipolar momentum introduced in [33]. Quite gen- shown in the next section using these conditions leads [αβ... ] erally, Ψ,l (r) are regular functions of r analog to the to a simple construction of the zero-range pseudopoten- C lm tial (in particular, derivatives of the delta distribution coefficients cΨ (r) in Eq.(14) and characterize the behav- are easily introduced in Cartesian coordinates). In the ior of the wave function Ψ(r, Ω) in the channel l through following, we adopt the standard assumption of implicit the expansion (see Eqs.(66,69) in the Appendix): summation over repeated indexes, the three directions of (2l+1)!! ~r Π Ψ = [αβ... ](r) . (20) space x,y,z are labeled by a Greek index α 1, 2, 3 h | l| i rl+1 Sl[αβ... ]CΨ,l and [αβ{ . . . ]} denotes a set of l indexes. The∈{ idea is to} Using these notations, the generalized Bethe-Peierls use the symmetric tensors l which appear in the multi- polar expansion used in electrostaticsS [42]. They can be boundary conditions Eq.(15) can be written as: defined through their components by: 1 [αβ... ] lim( l + ) (r)=0 . (21) r→0 D w CΨ,l rl+1 1 l = ( 1)l ∂ l , (17) Sl [αβ... ] − (2l 1)!! [αβ... ] r Eqs.(19,20,21) are equivalent to Eqs.(14,15) and allow for − finding the general form of the pseudopotential. where we have introduced a short-hand notation of the l-th derivatives: D. Zero-range pseudopotential l ∂ ∂ ∂ [αβ... ] = α β ... , (18) ∂x ∂x Instead of giving the expression of the pseudopotential and xα are the three components of the radial vector directly, we prefer to detail the different steps leading to { } α ~r in the Cartesian basis ~eα : i.e. ~r = x ~eα. These ten- its achieving. Such a way of presentation can be useful if sors can be expressed in{ terms} of the component nα of one wants to derive pseudopotentials associated to oth- the unit vector ~n = ~r/r and are functions of the angles Ω ers expressions of the phase-shift in Eq.(4) or in different only. The normalization factor in Eq.(17) has been cho- contexts (low dimensions for example). In either cases, sen for later convenience and explicit expressions of a zero-range pseudopotential can be deduced from the Sl and l with l 3 are given in Tab.(I). In the following, three following constraints: (i) it cancels the delta terms we willD use some≤ useful properties of these tensors which arising from action of the Laplacian on the singularities are collected in the Appendix. By construction, are of the wave function at r = 0 in the Schr¨odinger equation, Sl traceless (δ [αβ... ] = 0) and symmetric tensors of rank (ii) it imposes the correct boundary conditions and (iii) αβSl it can be used in the Born approximation. The first con- l. They are eigenvectors of Lˆ2 (Lˆ is the angular momen- dition (item-i) can be fulfilled by introducing potential tum operator) with eigenvalueh ¯2l(l+1). Similarly to the terms in the different channels l of the form (see Eq.(74) spherical harmonics Y lm(Ω), the tensors can serve as l in the Appendix): a basis for an alternative multipolar expansionS of a wave function like in Eq.(14). To this end, we define the Carte- 2π(2l+1)¯h2 ~r V (i) Ψ = ( 1)l+1 sian representation of the multipolar radial functions of h | l | i − µ order l associated to a wave function Ψ, by a symmetric [αβ... ] l trace free tensor (r) having the components: Ψ,l (0) (∂ [αβ... ]δ)(~r ) . (22) CΨ,l C l+1 2 In order to include the boundary conditions Eqs.(21) [αβ... ] r d Ω [αβ... ] Ψ,l (r)= l Ψ(r, Ω) . (19) (item-ii), we consider the “boundary operator” l de- C l! ZΩ 4π S fined by: B For example, the monopolar (l = 0) function (r) is 1 CΨ,0 ~r Ψ = (∂ l δ)(~r ) lim( + ) [αβ... ](r). (23) l [αβ... ] → l Ψ,l just r times the projection of the wave function in the h |B | i r 0 D wl C 6
By construction wave functions belonging to the kernel of Green’s function method, in which the interacting two- l satisfy the desired boundary conditions and the family body wave function Ψ (~r ) (wavevector ~k, energy E = B ~k of pseudopotentials: ¯h2k2/2µ) is solution of the integral equation: (i) V + β l , (24) (0) 3 ′ ′ ′ l Ψ~ (~r )=Ψ (~r ) d ~r GE(~r, ~r ) ~r Vλ,l Ψ~ , (28) B k ~k −Z h | | ki where β is an arbitrary parameter, can be used to solve where the solution of the homogeneous equation is an the two-body problem in the resonant regime. Interest- (0) incident plane wave Ψ (~r ) = exp(i~k.~r ) and GE is the ingly, while exact zero-range solutions can be obtained ~k β = 0, it is not hard to be convinced that a first or- free one-body outgoing Green’s function at energy E: ∀der6 Born approximation performed for example in the ′ µ exp iku ′ computation of the two-body scattering states leads to GE(~r, ~r )= with u = ~r ~r . (29) 2 u | − | a β-dependent result. Indeed in this approximation, the 2π¯h ′ two-body wave function Ψ(0) is an eigenstate of the non- Integrating Eq.(28) over coordinates ~r gives interacting Schr¨odinger equation (a plane wave in the (0) µgλ,l [αβ... ] example chosen) and is then a regular wave function Ψ~ (~r )=Ψ~ (~r ) λ,l [Ψ~ ] k k − 2π¯h2 R k of the radius r: action of V (i) gives a zero result, but l ′l exp iku (i) (0) ∂ , (30) ~r V + β l Ψ is β-dependent. As a consequence, [αβ... ] l u r′=0 andh | we willB show| thisi property explicitly in the next sec- ′l tion, varying β offers the freedom to adjust the pseu- where ∂ [αβ... ] stands for l-th partial derivatives with re- dopotential in such a way that the first order Born ap- spect to ~r ′ coordinates. Thanks to the trace free property proximation is exact at a given energy (item-iii). Fi- (δ [αβ... ][Ψ ] = 0), contribution of the l-th derivative αβRλ,l ~k nally, the expression of the pseudopotential in Eq.(24) in Eq.(30) takes a simple form. Indeed, for a given func- although general, can be rewritten in a more convenient tion F (u), one can show that: form as in the case of the λ-potentials already introduced ′ in Refs.([7, 33]): ∂ l F (u) [αβ... ][Ψ ]= [αβ... ] ′ λ,l ~k r =0 R ~r V Ψ = ( 1)lg (∂ l δ)(~r ) [αβ... ][Ψ] . (25) 1 l h | λ,l| i − λ,l [αβ... ] Rλ,l (n n . . . ) [αβ... ][Ψ ]( r)l ∂ F (r) , (31) α β Rλ,l ~k − r r l In this expression, gλ,l = ( 1) β is the coupling constant ′ defined by: − and with F (u)= GE(~r, ~r ), Eq.(30) reads finally: (0) µgλ,l [αβ... ] 2 Ψ (~r ) = Ψ (~r ) [Ψ ] 2π(2l+1)¯h wl ~k ~k − 2 Sl[αβ... ]Rλ,l ~k gλ,l = (26) 2π¯h µ(1 λwl) l − l 1 exp(ikr) ( r) ∂r . (32) and [ . ] are symmetric trace free tensors which gener- − r r Rλ,l alize the regularizing operator introduced in Refs.[7, 33]. As r 0, one recognizes a multipolar singularity in the They act on a wave function Ψ as: two-body→ wave function: [αβ... ] [αβ... ] (2l+1)!! w − λ,l [Ψ] = lim( l + λ) Ψ,l (r) . (27) l l[αβ... ] [αβ... ] 1 l R r→0 D C Ψ~ (~r )= S [Ψ~ ]+ (r ), k − (1 λw ) rl+1 Rλ,l k O − l The parameter λ appears in Eq.(26) and Eq.(27) both (33) and by construction (like β in Eq.(24)) is a free parameter and comparison with Eq.(20) gives the multipole compo- of the theory i.e. observables are independent in a choice nents: of λ. [αβ... ] wl [αβ... ] (0) = [Ψ~ ] . (34) CΨ~k,l −1 λw Rλ,l k − l In the partial wave channels of angular quantum number III. SOME ILLUSTRATIONS AND PROPERTIES OF THE PSEUDOPOTENTIAL different from l, components of the wave function Ψ~k(~r ) coincide as expected with the partial waves of Ψ(0)(~r ). ~k A. Two-body scattering states from the Green’s All the information on the interaction are gathered in method the term [αβ... ][Ψ ] in Eq.(32). From Eq.(20), Sl[αβ... ]Rλ,l ~k one can see that it contains implicitly the projector Πl This section is both an illustration of the formalism through: and a benchmark for the proposed form of the pseudo- [αβ... ][Ψ] = potential in Eq.(25). We consider the case where two Sl[αβ... ]Rλ,l particles interact in the l-th partial wave channel without 1 l+1 lim( l + λ)r ~r Πl Ψ . (35) external potential. For this purpose we use the general (2l+1)!! r→0 D h | | i 7
A closed equation is obtained by applying the operator this is to realize that the projection operator in the par- [αβ... ][ . ] on both sides of Eq.(32). Using the tial wave (l) is implicitly present in the expression of the Sl[αβ... ]Rλ,l usual decomposition of a plane wave over spherical ones: pseudopotential Eq.(25). Indeed, assuming that Φ(~r ) is regular at r = 0, one obtains: ∞ ~ l ~ exp(ik.~r )= i (2l+1) l(cos(k, ~r ))jl(kr) , (36) [αβ... ] l ~r Πl Φ P (∂ Φ)r=0 = l! lim h | | i , (43) Xl=0 d Sl [αβ... ] r→0 rl together with Eq.(9) this leads to: moreover from Eqs.(35,43) and Eq.(70) in the Appendix, one finds: lim( + λ)rl+1 ~r Π Ψ(0) = → l l ~k r 0 D h | | i 2 d Ω Φ Πl ~r l Φ V Ψ = g lim h | | i (2l+1)!! (ik) (cos(~k, ~r )) ,(37) λ,l λ,l → l Pl h | | i Z 4π r 0 r d on the other hand: l+1 lim( l + λ)r ~r Πl Ψ . (44) r→0 l D h | | i 2l+1 1 exp(ikr) lim( l + λ) r ∂r = r→0 D r r First, we consider plane waves ~r ~k = exp(i~k.~r). The h | i ( 1)l(2l 1)!! (λ + α k2 + ik2l+1) , (38) matrix element between two such states is: − − l ′ ′ and finally, one gets: ~k V ~k = g (kk )l (cos(~k,~k ′)). (45) h | λ,l| i λ,l Pl d′ (1 λw )(ik)l (cos(~k, ~r )) For wave vectors of same modulus (k = k = √2µE/¯h) [αβ... ][Ψ ]= l l .(39) l[αβ... ] λ,l ~k − 2 P 2l+1 and for the particular value of the free parameter λ given S R 1+ wlαlk + iwlk d in Eq. (42), comparison with the expression of the partial As expected, the expression of the scattering wave func- wave amplitude Eq.(5) shows that Eq.(45) coincides with tion does not depend on the parameter λ: the on-shell T -matrix t(~k ′ ~k ) taken at energy E: ← ~ 2 (0) (2l+1)wl l(cos(k, ~r )) ′ ′ 2π¯h ′ Ψ (~r ) = Ψ (~r ) P ~ ~ ~ Born ~ ~ ~ ~k ~ 2 2l+1 t(k k )= k Vλ ,l k = f(k k ) (46) k − 1+ wlαlk + iwldk ← h | | i − µ ← l l 1 exp(ikr) ( ikr) ∂r . (40) ′ ′ − r r where f(~k ~k )= (2l+1) l(cos(~k ,~k ))fl(k) is the full scattering amplitude.← P Asymptotically for kr 1, the two-body wave function d ≫ Another interesting application of Eq.(44) is given by is: considering spherical outgoing waves: exp(ikr) ~ ~ lm Ψ~k(~r ) = exp(ik.~r )+(2l+1) l(cos(k, ~r ))fl , ~r klm = Y (Ω)j (kr) , (47) P r h | i l d (41) with the partial amplitude fl associated with the phase- with the matrix elements:: shift in Eq.(4) and given by Eq.(5). ′ l ′ ′ ′ gλ,l(kk ) It is interesting to note that the exact result can be klm Vλ,l k l m = δll′ δmm ′ . (48) obtained at the level of the first Born approximation by h | | i 4π(2l+1) choosing a particular value of the parameter λ: For λ = 0, this results coincides with the one obtained Born 2 2l+1 λ = αlk ik . (42) by Roth and Feldmeier in Ref.[43]. Here, the λ-freedom − − is a new ingredient and shows clearly that mean field cal- For example by setting λ = 0 in a first order Born approx- culations in zero-range approaches have to be developed imation gives a correct evaluation of a scattering process carefully. For example, considering a two-components with vanishing energy (Wigner threshold). Fermi gas in the unitary limit: α0 = 0 and w0 , Generalization to the case where interaction occurs in the mean field model used in [43], predicts an instability→ −∞ several partial wave channels is straightforward by simple which is not observed in experiments [26, 27, 28, 29, 30]. addition of pseudopotentials of the form of Eq.(25) for A way to understand this failure is that due to Pauli different value of the angular quantum number l. blocking, interactions concern essentially particles in the neighborhood of the Fermi surface whereas the approxi- mation made on the interacting term is valid at the Born B. Links with other formalisms level for particles of vanishing energy. By the way, it is possible to build a qualitative and simple mean field It is instructive to compute the matrix element of Vλ,l model which supports the stability of the gas at the BCS- between two states Φ and Ψ . A simplifying step for BEC crossover [44]. | i | i 8
Taking two states with angular parts characterized by identify directly the true scattering states with the states the same spherical harmonic in Eq.(44), one gets the ex- Ψ . This feature can be understood knowing that the | ~ki pression of the pseudopotential in radial coordinates: mapping between the two eigen basis is justified outside the potential radius but is not valid for r < R. Indeed, gλ,l δ(r) l+1 outside the potential radius, identification∼ between true vλ,l = lim( l + λ)(r . ) . (49) 4π rl+2 r→0 D states and states deduced from the zero-range approach r is made possible as R fixes the sphere outside which the In the particular case λ = 0, αl = 0, .h.s. of Eq.(49) potential term can be neglected in the Schr¨odinger equa- coincide with the form of the pseudopotential introduced tion. The singular boundary behavior imposed on wave in Ref.[35]: functions in Eq.(15) is the way to mimic the effect of the true 2 finite range potential for r > R but has a formal ¯h wl(2l+1)!! δ(r) 2l+1 l+1 character in itself. In order to conciliate the zero-range v0,l = lim ∂r (r . ) . (50) 2µ(2l)!! rl+2 r→0 pseudo-potential with a finite range potential of vanish- ing radius R, we consider the sphere of radius R which Note that the delta-shell regularization procedure intro- delimits the outer and inner parts of the true wave func- duced in Ref.[35] while correct appears not to be essential tions. Contribution in the scalar product of the outer- in the zero-range approach. part of the true wave functions is given by Eq.(51) with r0 = R and due to orthogonality between the true scat- tering states, scalar product of the inner-parts cancels C. The regularized scalar product this contribution. As the inner parts of the wave func- tions (region r < R) are not described in the zero-range Except for the case of the Fermi pseudopotential which approach, the idea is to modify the usual scalar prod- is a particular case of (25), two scattering states Ψ~k ′ uct for taking into account their contribution. We define ′ and Ψ (with ~k = ~k ) of expressions given by Eq.(40) then a regularized scalar product ( . . ) by subtracting ~k 6 0 are not orthogonal each others with respect to the usual the surface term (r.h.s. of Eq.(51)) from| the usual scalar scalar product. To examine this point, we exclude the product as r0 0: singularity at r = 0 in the scalar product between the → 3 ∗ two different scattering states by introducing an Ultra- (Ψ Φ)0 = Reg d ~r Ψ (~r )Φ(~r ) | r0→0 Z Violet cut-off r0: r>r0 ∞ 2l+2 2 αl r0 2 ∗ 3 ∗ r0 + lim d ΩΨ (~r ) ~r Πl Φ . (53) r0→0 2 d ~r Ψ ′ (~r )Ψ~ (~r )= ((2l 1)!!) Zr=r0 h | | i Z ~k k k2 k ′2 Xl=0 − r>r0 − 2 ∗ ∗ In general there are several possible representations of d ΩΨ ′ (~r )∂rΨ~ (~r ) Ψ~ (~r )∂rΨ ′ (~r ) (51) Z ~k k − k ~k Eq.(53) due to the various and equivalent ways to ex- r=r0 tract a regular part. We give below explicit expressions This relation is general and when applied on scattering of the regularized scalar product associated to the pseu- wave functions associated to a finite range two-body po- dopotential (25) in the l = 0 and l = 1 partial wave tential, the r.h.s. of (51) is zero for r0 = 0. However in channels: the zero-range approach, for wave functions satisfying the Particles interacting in the s-wave channel only: boundary conditions in Eq.(12), as r0 0 this term goes • → 3 ∗ to a constant value for l = 0 and α0 = 0 and even worst, (Ψ Φ)0 = lim d ~r Ψ (~r )Φ(~r ) 6 r0→0 it diverges for l> 0. Interestingly, the regular part of | Zr>r0 Eq.(51) for such wave functions can be obtained through 2 2 ∗ the following surface integration: +α0r0 d ΩΨ (~r )Φ(~r ) . (54) Zr=r0
3 ∗ This result can be written more formally by intro- Reg d ~r Ψ ′ (~r )Ψ (~r ) = ~k ~k ducing a scalar product of the form: r0→0 Zr>r0 ∞ 2l+2 α r ∗ 3 ∗ l 0 2 (Ψ Φ)0 = d ~rg(r)Ψ (~r )Φ(~r ) , (55) lim d ΩΨ~ ′ (~r ) ~r Πl Ψ~k , (52) − r0→0 ((2l 1)!!)2Z k h | | i | Z Xl=0 − r=r0 with the weight g(r)=1+ α0δ(r) . where the operator Reg . extracts the regular part at 0→ { } Particles interacting in the p-wave channel only (for r 0 • r0 = 0. The meaning of this result is that the Laplacian example in the case of polarized fermions): operator is not Hermitian with respect to the usual scalar 3 ∗ product if one considers singular functions like in Eq.(40). (Ψ Φ)0 = lim d ~r Ψ (~r )Φ(~r ) | r0→0 In another hand, the true scattering states associated to Zr>r0 the finite range potential experienced by particles, are 4 3 2 ∗ + (α1r0 r0) d ΩΨ (~r )Φ(~r ) . (56) orthogonal each others, so that it seems not possible to − Zr=r0 9
This scalar product has been first introduced in by Eq.(5), one finds at resonance for large and posi- Ref.[33], in the form of Eq.(55) with the weight tive values of wl a bound state of vanishing energy with g(r)=1+ δ(r) (α r2 r) . κ2 1/w α (obviously in the case of the s-wave broad 1 − ≃ l l resonance α0 = 0 and κ =1/w0). This is the state which plays a crucial role in the BCS-BEC crossover. In the D. Bound-states and their normalizations zero-range approach, Eq.(59) is formaly extended in the domain 0 r R and using the regularized scalar prod- ≤ ≤ By considering the normalization of bound states with uct, one finds that for (ΨB ΨB)0 = 1: | vanishing energy that appear in the zero-range scheme, 1 we show in this section that the regularized scalar prod- 2 = . (60) Nl α + ( 1)l(l + 1 )κ(2l−1) uct is consistent with the method based on the analyticy l − 2 of the scattering amplitude [36]. This last method was used in Ref.[14] in the context of s-wave narrow reso- This result is very interesting as it coincides with the one nance. In this case or also for p-wave or higher partial given by the residue of fl at the energy E = EB [36]. waves where bound states are even not square integrable To conclude this section, let us note that the scalar with respect to the usual scalar product (they don’t be- product (53) gives a constraint on the possible values of long to the 2 Hilbert space), the regularized scalar prod- the “effective range” parameter αl for a consistent de- uct appearsL then as the natural way to perform the nor- scription of the low energy bound state. Indeed, the malization in the configuration space. Moreover, this new probability of finding the molecular state outside the po- tential radius is less than one: d3~r Ψ 2 < 1. Then, scalar product allows for a simple generalization of the r>R | B| method of Ref.[36] to the case of inhomogeneous situa- using Eq.(60) and keeping theR dominant contribution as R 0, one finds that the parameter α verifies in the tions. → l If one considers solutions of the Schr¨odinger equation resonant regime for l> 0: with a finite range potential as functions of ~r and of the 2l−1 αlR > (2l 3)!!(2l 1)!! . (61) energy E: Ψ(E, ~r ), then a bound state of energy EB ∼ − − ΨB = Ψ(EB, ~r ) corresponds to a case where Ψ(E, ~r ) is In the case of a narrow s-wave resonance, in the regime square integrable. Using the Schr¨odinger equation, one of intermediate detuning (α0 w0) and w0 > R, then can show that: κ2 1/w α and the condition≫α R follows∼ directly ≃ 0 0 0 ≫ 2 2 from κR 1. By the way, Eq.(61) implies also that ¯h r ∗ d3~r Ψ 2 = 0 d2Ω Ψ (E, ~r )∂ ∂ Ψ(E, ~r ) α > 0. As≪ an example, the experimental fit performed in | B| − 2µ r E l Zr
Green’s method. While an exact treatment is invariant so that Eq.(62) can be rewritten in a change of λ, this property is in general no longer true ∞ ′ l in approximate schemes. This gives in turn a way to im- 1 1 ′ (2l 1)!! r = [αβ... ] − . (64) prove the approximation made for example in mean-field ~r ~r ′ r Sl[αβ... ]S l l! r treatments like BCS where the Born approximation is | − | Xl=0 performed implicitly. This expression can be identified with the standard ex- Scattering states being not orthogonal in the zero- pansion (for r ′ < r) [48]: range scheme (except for the Fermi pseudopotential), we ∞ have shown how the notion of regularized scalar prod- 1 1 r ′ l = (cos(~r, ~r ′)) , (65) uct emerges naturally. Normalization of bound-states by ~r ~r ′ r r Pl Xl=0 means of this new metrics gives in the free space the same | − | d result as the method based on the analyticy of the scat- where (x) is the Legendre polynomial of degree l and tering amplitude. As it applies directly in the configura- l we obtain:P tion space, this new tool gives a simple way to perform the normalization in inhomogeneous situations. (2l 1)!! ′ (cos(~r, ~r ′)) = − [αβ... ] . (66) The formalism presented in this paper can be used for Pl l! Sl[αβ... ]S l example to get scattering properties of particles confined d ′ in low dimensions in view of having informations on the For example, taking ~n = ~ez shows that: low energy behavior in 1D or 2D many-body systems. l! The few body problem is another direction where this = (cos θ) , (67) Sl(zzz...z) (2l 1)!!Pl approach should be fruitful. − Note added: while completing this paper we learn of recent related works by Z. Idziaszek and T. Calarco [45] with θ = (~r, ~ez). Eq.(66) can be used to have an expres- and also A. Derevianko [46] where the form of the pseu- sion of the projection operator Πl over the partial wave l in termsd of the tensor . For that purpose, we recall dopotential in Eq.(50) is also found. The main difference Sl of the present approach with respect to Ref.[45] (or also the addition theorem [48]: Ref.[35]) is that the pseudopotential Eq.(25) or equiv- m=l alently the generalized Bethe-Peierls conditions Eq.(15) ′ 4π lm lm ∗ ′ l(cos(~r, ~r )) = Y (Ω)Y (Ω ) , (68) have been derived in an energy independent form. P 2l+1 mX=−l d where (Ω = (θ, φ), Ω ′ = (θ′, φ′)) are the angles defining ′ ACKNOWLEDGMENTS the spherical coordinates for the two vectors (~r, ~r ). The projection of a wave function Ψ reads:
Y. Castin, F. Chevy, M. Holzmann and F. Werner are 2 ′ d Ω ′ ′ acknowledged for thorough discussions on the subject. ~r Πl Ψ = (2l+1) Ψ(r, Ω ) l(cos(~r, ~r )) . (69) h | | i Z 4π P Laboratoire de Physique Th´eorique de la Mati`ere Con- d dens´ee is Unit´eMixte de Recherche 7600 of Centre Na- together with Eq.(66) and the definition of the multipoles tional de la Recherche Scientifique. of the wave function Eq.(19), we obtain Eq.(20). In this paper, we also use an interesting property of the symmetric traceless tensors found in Ref.[47]. Let us consider two symmetric and traceless tensors and of APPENDIX: USEFUL PROPERTIES OF THE A B SYMMETRIC TRACE FREE TENSORS Sl order l, then ′ 2 ′ ′ [αβ... ] d Ω ′ ′ ′ ′ [α β ... ] This appendix collects some useful properties of the n n ...n ′ n ′ A Z 4π α β α β B symmetric trace free tensors l defined in Eq.(17) [42, S α ′ l! 47]. Let us consider the two vectors ~r = x ~eα and ~r = = [αβ... ] . (70) ′α [αβ... ] x ~eα. By definition: (2l+1)!!A B ′′ ∞ ′ ′ As an example, taking = and = allows one to 1 (2l 1)!! x αx β . . . A Sl B S l = − . (62) recover from Eqs.(66,70) the known relation: ~r ~r ′ l! Sl[αβ... ] rl+1 Xl=0 | − | 2 ′ ′′ d Ω ′ α ′ l(cos(~r, ~r )) = (2l+1) l(cos(~r, ~r )) Introducing the unit vectors ~n = ~r/r = n ~eα, ~n = P Z 4π P ′ ′ ′α ′ ~r /r = n ~eα and the symmetric trace free tensor d d′ ′′ ′ l (cos(~r , ~r )).(71) associated to ~r , we have: S Pl We close this appendix with the link betweend the sin- ′ ′ ′ n αn β = [αβ... ] , (63) gularity of the l-th partial wave of a wave function with Sl[αβ... ] · · · S l[αβ... ]Sl 11 the derivatives of the delta distribution. To this end, we This expression justifies the introduction of the potential use the so-called relation: (i) Vl in Eq.(22). Using Eq.(66), one can also write Eq.(74) in terms of Legendre Polynomials: 1 ′ ∆ = 4πδ(~r ~r ) (72) ~r ~r ~r ′ − − | − | together with Eq.(64) and the formal expansion:
∞ ′ l ′ ( r ) ′ ′ δ(~r ~r )= − (∂ l δ)(~r )n αn β ... , (73) − l! [αβ... ] Xl=0 l+1 l(cos θ) ( 1) l ∆~r P l+1 =4π − (∂z δ)(~r ) . (75) l r l! where (∂ [αβ... ]δ)(~r ) is the l-th partial derivative of the delta distribution (see Eq.(18)). Finally, we obtain:
′ ′ ∆ Sl[αβ... ] n αn β = ~r rl+1 · · · l+1 ( 1) ′ ′ 4π − (∂ l δ)(~r )n αn β ... . (74) (2l 1)!! [αβ... ] −
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