arXiv:cond-mat/0508120v3 [cond-mat.other] 10 Jan 2006 rs-iavkiadteBgluo prahs[,6] [5, approaches Bogoliubov the like the “real” schemes and the mean-field Gross-Pitaevskii Hence, using this possible potential. while not approximation two-body is Born correct treatment order the en- first approximate give small the pseudopotentials vanishly at of of class results processes large of for second a number that A ergies, is minimal collisions. property the two-body key contain modeling phase for energy, parameters scattering small ultracold correct at the which the shift of pseudopotentials reproduce the study construction respect, the by this for In gases. ) . dilute . . states, repulsion po- bound (deep two-body hard-core informations full useless the contains Consequently, tential mass). particle the ( hnthe than h ag fitrtmcfre dntdby to (denoted compared forces as interatomic large interparticle of very mean range is the gases the First, ultracold in reasons interest. distance several renewed are There this atoms. for in ultracold tools of reference physics the as appear pseudopotentials zero-range approach. this hntecaatrsi iheeg cl ¯ scale smaller high-energy much characteristic very the energies at than occur collisions two-body h ocle em suooeta 4.Tefc that of ( fact force formulation The nuclear final the the of [4]. range to pseudopotential the led Fermi and so-called [3] proton. the Breit a by and developed neutron further was slow approaches a two scatter- a these the between allowing between Equivalence of section potential computation cross effective the ing zero-range in approximation a Born of idea the the in a function wave with it the on s force on that condition neutron-proton properties showed boundary the scattering suitable [1] replacing numerous Peierls by get deuteron and to nuclear Bethe possible in was First, interaction neutron-proton the physics. on works the ≃ wv hne.Oeya fe,Fri[]introduced [2] Fermi after, year One channel. -wave oeta afacnuyatrteepoerworks, pioneer these after century a half than More h ot ftezr-ag oeta prahlein lie approach potential zero-range the of roots The 4 .Zr-ag prahi lrcl physics ultracold in approach Zero-range A. . 310 − s 15 wv eto-rtntiltsatrn length scattering triplet neutron-proton -wave hw httegoer fteeHletsae eed cruc 03.65.Nk,03.75.Ss,05.30.Fk,34.50.-s depends numbers: spaces PACS Hilbert these regular of geometry a for the introduce (except that formalism to potential f shows necessary wave zero-range is the the it in on obtained condition that contact show We a of tentials. terms in either described cteigi eoati rirr ata ae.Teform for The only used waves. be partial can arbitrary which in tential resonant is scattering )wsakyigein o h ucs of success the for ingredient key a was m) h eorneptnilapoc setne o h descr the for extended is approach potential zero-range The .INTRODUCTION I. nvri´ iree ai ui,4paeJsiu 55 Pa 75252 Jussieu, place 4 Curie, Marie et Universit´e Pierre aoaor ePyiu herqed aMati`ere Condens´ la Th´eorique de Physique de Laboratoire ≃ 210 suooeta nrsnn regimes resonant in Pseudopotential − 15 )i uhless much is m) h 2 /mR Dtd etme 0 2018) 10, September (Dated: R s wv ra eoacs nagvncanl h interaction the channel, given a In resonances. broad -wave ,s that so ), uoi Pricoupenko Ludovic 2 ( m is ry ntecs fteutaodBs a n sn the using and gas Bose en- ultracold collisional the the finite formalism, of Hartree-Fock-Bogoliubov a case for the In solution ergy. exact be the can to approximation Born adjusted the while problem, two-body rameter aiyo suooetaseuvln oteFripseu- Fermi the a the introducing to dopotential: by equivalent generalized pseudopotentials been of has family idea pseudopo- two-component Fermi This dilute the the tential. using implemented for be [7] can BCS gas or Fermi gas Bose the for oeta prah o xml,i h aeof case the in zero-range example, the For of approach. configura- application potential dimensional powerful another low are in tions properties traps Scattering (2D) quasi-two-dimensional in [9]. also three-dimensional and in [8] condensate systems the on action excitations back the the account of into take self-consistently to mits eut aebe on o he n orprilsin- particles four the and Exact in three pairs for fruitful. by found very teracting been be have to proven results potential already zero-range problem the has few-body where approach research The of area 11]. another [10, is approach discovered this been to has thanks quasi-one-dimensional systems important in quasi-two-dimensional The interaction and 12]. tunable 11, planar of the [10, in concept finding waveguides trapped atomic for particles linear way two or efficient of very amplitude a scattering is the on function wave condition 3D boundary Bethe-Peierls the scattering, e-admn-oypolm 2,2] hs studies an These is approach potential 24]. zero-range the [23, the that two-body for problems clearly show found (infinite many-body been gas and have quantum few- results exact unitary length length) the scattering scattering For two-body scat- the molecule-molecule of [14]. the terms phase of in value BEC length the the tering find the in remark- to for appear also is resonance) that and the (which molecules of bound lifetime neighborhood approach weakly the the zero-range in evaluate The large ably to to 22]. 21, possible us two-component is 20, permits the it 19, that [18, of so gas crossover Fermi will, BCS-BEC at the tuned scatter- achieve be the can experiments length these ing In [17]. resonances hbach hs tde r ffis motnei h otx of context the in importance first resonant of the are studies These h em suooeta) hsmetrics This pseudopotential). Fermi the ls eeaie h em pseudopo- Fermi the generalizes alism zdsaa rdc o aefunctions wave for product scalar ized λ nto rwt aiyo pseudopo- of family a with or unction al nteinteraction. the on ially pino iutosweetwo-body where situations of iption osntcag h xc ramn fthe of treatment exact the change not does s wv cteigrgm bandwt Fes- with obtained regime scattering -wave i ee 5 France. 05, Cedex ris λ ptnil 8.Vrigtefe pa- free the Varying [8]. -potentials ee, s wv hne 1,1,1,16]. 15, 14, [13, channel -wave λ ptnilper- -potential is s -wave 2 efficient tool beyond the two-body physics and permits developed in Ref. [36] cannot be used for our purpose. us to obtain non trivial properties in the few- and many- Recently, a generalization of the Fermi pseudopotential body problems. for arbitrary partial waves and scattering phase shifts has More generally, study of strongly correlated systems been proposed using delta-shell potentials in the limit of obtained in the vicinity of scattering resonances is actu- small shell radius [37], by the way correcting a mistake ally one of the most challenging directions in the field of of Ref. [36]. Boundary conditions associated with the ultracold atoms. These systems are not only interesting zero-range approach in Ref. [37] are defined for scattering in themselves but also permit accurate studies of issues states in free space and depend explicitly on their wave raised in the quantum many-body problem. The BCS- number k. Consequently, this approach applies only in BEC crossover in two-component Fermi gases is an exam- time-independent situations. Instead in the present pa- ple of such a situation [25, 26, 27]. The domain now en- per, we show that in resonant regimes, general (that is, larges to resonant scattering in p-wave [28, 29, 30, 31, 32] energy independent) boundary conditions can be found. and d-wave channels [33] allowing possible studies of the Another feature of the pseudopotentials in Ref. [37] is BCS-BEC crossover in channels of high angular momen- that their action is defined on the radial wave function of tum in the near future. Major interest in these systems the specified partial wave. However, it is of interest when stems from the fact that the strength of correlations can dealing with wave functions of arbitrary symmetry (for be tuned arbitrarily while the mean interparticle distance example in the presence of an anisotropic external poten- remains large with respect to the range of interatomic tial) to have the full expression of the pseudopotential, forces R (for example, the s-wave scattering length can be that is, an expression which contains implicitly the pro- adjusted to several orders of magnitude larger than R). jection operator over the interacting channel (denoted in Consequently, one can expect that short range physics the following by Πl for the channel of angular quantum is not directly involved in a large class of many-body number l). properties which can then be described only in terms of the low energy two-body behavior. It is worth point- ing out that this fundamental feature has been verified C. Organization of the paper in the BCS-BEC crossover of the two-component Fermi gas. For this reason, the zero-range approach which is The paper is organized as follows: in the first part, the parametrized only by the two-body low energy physics regime of interest (two-body resonant scattering) is in- is a very appealing tool for studying these regimes. Ac- troduced and characterized in each partial wave channel tually, the state of the art is as follows: broad s-wave by a set of two parameters in the scattering phase shift. resonances (like the one in 6Li at 835 G [20]) can be ac- In the second part, the different tools of the zero-range curately parametrized using the Fermi pseudopotential, approach in this regime are derived. We show that the and narrow s-wave resonance [34] can be studied using problem can be defined in terms of boundary conditions a generalization of the Bethe-Peierls boundary condition on the wave function which generalize the method intro- [15]. Recently, a general zero-range potential approach duced by Bethe and Peierls. We express these boundary has been introduced for p-wave resonances [35] and the conditions in spherical coordinates and also in Cartesian aim of this paper is to extend this formalism for a de- coordinates using the symmetric trace free tensors used scription of resonant scattering regime in arbitrary par- in the usual multipolar expansion. This way of defining tial waves. the zero-range approach permits us to include quite eas- ily within the expression of the pseudopotential both the projection operator Πl and the lth derivatives of the delta B. Which generalization of actual pseudopotentials distribution. We show that in each interacting channel is needed? there exists a family of pseudopotentials generated by a free parameter, thus generalizing the λ-potential ap- The idea of replacing a true finite range two-body po- proach [8, 35]. The third and last part, illustrates the tential by a zero-range pseudopotential acting on each formalism. We find the expression of the two-body scat- partial wave has been developed by Huang and Yang [36] tering states from the pseudopotential and the Green’s in the context of the hard-sphere model. However, due to function method. By choosing a specific value of the pa- the importance of the short range behavior in this system, rameter λ, this exact result is also obtained in the first pseudopotentials describing three-, four- and many-body order Born approximation for two-body processes at fi- correlations were also proposed for modeling the system nite energy. The pseudopotential of Ref. [37] is recovered beyond the dilute limit. The actual situation in ultracold as a particular limit of the present approach. Finally, atoms is radically different as the mean interparticle dis- the scattering states and also the low-energy bound states tance is very much larger than the range of interatomic are found to be not mutually orthogonal in the zero-range forces, and two-body pseudopotentials are sufficient to scheme (except for the Fermi pseudopotential). We intro- describe the low-energy behavior for the many-body sys- duce then a regularized scalar product which solves this tem even in strongly correlated regimes. Moreover as the inconsistency. We consider the low-energy bound states hard-sphere model cannot support resonances, the tools supported by the pseudopotential and show that the new 3 scalar product leads to normalization constants which co- (r → ∞) the threshold behavior is valid for l< (n − 3)/2 incide with the results given by the method based on the and the effective range approximation for l < (n − 5)/2. analyticy of the scattering amplitude [38]. In ultracold atoms in their ground state the power law is given by n = 6 so that the expansion in Eq.(4) is rigorous only in the s-wave channel (w0 is the scattering length), II. ZERO-RANGE FORMALISM IN THE while in the p-wave channel only the first term in the ex- RESONANT REGIME pansion is justified. For l ≥ 2, even the Wigner threshold regime given in Eq.(4) is not valid. However, we adopt A. Regime of interest this form as a generic way to parametrize the resonant regime in an arbitrary partial wave channel. Let us consider two particles in the absence of an ex- In broad s-wave resonances, the effective range is neg- ternal potential interacting via a short range isotropic ligible (α0 ≃ 0) and |w0| is very large as compared to potential U(~r ) (~r being the relative coordinates between the potential range R. For w0 > 0, f0 has a pole as- 2 2 the two particles). We denote their reduced mass by µ. In sociated to a shallow bound state of energy −¯h /2µw0. order to simplify the discussion and without loss of gen- Its wave function coincides (for r > R) with the molec- erality, we work in the center-of-mass frame. The asymp- ular state populated by pairs of particles in the BEC region of the BEC-BCS crossover. Remarkably, in the totic form of the scattering wave function Ψ~k(~r ) with rel- ¯h2k2 neighborhood of the resonance, the scattering cross sec- ative wave vector ~k and collisional energy E = de- tion takes large values whatever the sign of the scatter- 2µ ing length w0. As explained in Ref. [38], in the channels fines the partial amplitudes fl in each partial wave chan- nel of angular quantum number l through the relation: l > 0, this property is no longer verified, moreover the “effective range” parameter is essential for a modeling of the shape of the scattering amplitude and of the shal- ∞ exp(ikr) low bound state both. To define the resonant regime for Ψ (~r ) = exp(i~k.~r )+ (2l+1)P (~n.~n )f , (1) ~k l k l r l> 0, we follow the discussion given in Ref. [38] and use Xl=0 the expression of the partial amplitude associated with the phase shift in Eq.(4): where kr ≫ 1, ~n = ~r/r, ~nk = ~k/k and Pl is the Legendre 2l polynomial of degree l. The partial amplitudes can be wlk also expressed in terms of the phase shift δl with: fl(k)= − 2 2l+1 . (5) 1+ wlαlk + iwlk 1 fl = (exp(2iδl) − 1) . (2) The resonant regime corresponds to situations where 2ik the parameter |wl| takes arbitrary large values, so that 2 Typical situations in ultracold atoms corresponds to bi- |wlαl| ≫ R . It can be shown [38] that describing the nary collisions of particles having small enough relative resonant regime by Eq.(5) is valid only for αl > 0 (this velocities to ensure kR ≪ 1, where R is the potential assumption emerges also as a consequence of the regu- range. For two neutral alkali atoms in their ground state, larized scalar product that we introduce in the last sec- the asymptotic form of U(~r ) is a van der Waals tail: tion). The main difference as compared to s-wave broad 6 resonance is that the resonant character of the scatter- U(~r ) ≃ C6/r and the criterion for low-energy scatter- ing cross section associated with Eq.(5) depends crucially ing processes (collisional energy E ≪ ER) is fixed by [39, 40]: on the sign of wl. This property can be easily seen by writting Eq.(5) in the Breit-Wigner form: 1/4 2 2µC6 ¯h R = ; ER = (3) 1 Γ/2 ¯h2 2µR2 fl = − , (6) k E − Er + iΓ/2
Standard values for R (and ER) range from ∼ 3.3 nm 2 2 2l+1 6 133 ¯h ¯h k for Li (ER ≃ 7.25 mK) to ∼ 10 nm for Cs (ER ≃ with Er = − and Γ = . For large and 32.2 µK). In this paper we consider situations where the 2µαlwl µαl positive values of w , E < 0 and the scattering ampli- phase shift can be parametrized in the following form: l r tude has a pole which corresponds to the presence of a
2l+1 1 2 real bound state of vanishing energy EB ≃ Er. In this k cot δl(k)= − − αlk . (4) regime, |f |2 does not present a resonant structure. How- wl l ever, for large and negative value of wl, the bound state Eq.(4) corresponds to the so-called effective range ap- transforms into a long-lived quasi bound state which pro- proximation for a short range potential [41]: the param- duces a resonance in the scattering cross section with 2 eter wl characterizes the Wigner threshold regime and |fl| ≃ 1/|wlαl| in the neighborhood of E = Er. This −2αl is “the effective range” even if it has the dimension feature is a consequence of the existence of the centrifu- of a length only for l = 0. For neutral particles interact- gal barrier in the radial equation for l> 0 (see Fig.1). It n ing with a potential of asymptotic form U(r) ≃ Cn/r is worth pointing out that the resonant regime occurs for 4
Vl (r) the discussion is centered on the modeling of isotropic interactions.
E r B. Defining the zero-range approach E In this section, the interacting potential U(~r ) which can be neglected in the Schr¨odinger equation for |~r | > R is replaced by a specific behavior of the wave function 0 R r as the relative coordinates between the two interacting particles goes formally to zero. This behavior defines boundary conditions on the wave function which depend on the parametrization of the scattering phase shift in Eq.(4). Outside the potential range (r > R), the scattering
FIG. 1: Schematic representation of a two-body effective ra- wave functions {Ψ~k(~r )} of the two-body problem are so- ¯h2 l(l + 1) lutions of the free Schr¨odinger equation with the asymp- dial potential Vl(r) = U(~r) + in a partial wave 2µr2 totic behavior given by Eqs.(1,2,4): l > 0. Er is the energy of a quasi-bound state [4, 38] which leads to a resonant behavior in the two-body scattering for a ∞ ≃ collisional energy E Er. Ψ~k(~r )= Pl(~n.~nk)Rkl(r) , (7) Xl=0
where Rkl(r) are the radial functions which can be ex- relative wave vectors k verifying kR ≪ 1, which justifies pressed in terms of the two spherical Bessel functions a zero-range potential approach. as [43]: In actual experiments on ultracold atoms, the reso- nant character in the two-body scattering is ensured us- Rkl(r)= Nkl ( jl(kr) − yl(kr) tan δl(k) ) . (8) ing Feshbach resonances [42]. Interaction between atoms depends on the spin configuration of the two-body sys- The zero-range approach amounts to considering the for- tem and Feshbach resonances appear as a result of the mal continuation of Eq.(8) for 0 ≤ r ≤ R . As a con- coupling between a closed channel (supporting a short- sequence, using the behavior of jl(z) and yl(z) at the range molecular state) and an open channel (associated vicinity of the origin: with the scattering states). The energy of the short-range l 2 two-body molecular state can be tuned by use of an ex- z z /2 jl(z)= 1 − + ... (9) ternal magnetic field. The scattering resonance occurs (2l+1)!! 1!(2l+3) when the energy of the molecular state is in the vicin- (2l−1)!! z2/2 y (z)= − 1 − + ... , (10) ity of the collisional energy of the two scattering atoms. l zl+1 1!(1−2l) The molecular state in the closed channel depends on the short range properties of the system and is not de- one can deduce the scattering states from the following scribed by the zero-range approach. Consequently, the boundary conditions of the wave function at r = 0: bound state or quasi-bound state which appears in the zero-range model is associated only with the tail of the (2l−1)!! k2l+1 lim ∂(2l+1) − rl+1R (r)=0. molecular state resulting from the coupling between the → r kl r 0 (2l)!! tan δl(k) closed and open channels. (11) In the s-wave narrow Feshbach resonance (large and This expression is a first step toward the formulation of 23 negative effective range) of Na at 907 G [34], α0 ≃ the zero range approach, but does not have a general 26 nm, and R ≃ 4.5 nm. In higher partial wave chan- character in the sense that the quantum number k ap- nels, scattering is not isotropic i.e. it depends on the pears explicitly. Instead, we search for a boundary con- angular quantum number m of the state. For example dition which applies on any linear combination of scat- the p-wave resonance in 40K [29] (R ≃ 6.8 nm) has been tering states with different collisional energies. This way, parametrized with the law (4) in the states |m| =0, 1 as the formalism can be used when the collisional energy a function of the external magnetic field. There are two is not defined, as it can happen in time dependent situ- zero energy resonances (Er = 0) very close to one each ations. Using the parametrization of the phase shift in other at 198.84 G (for m = 0) and 198.31 G (for |m| = 1) Eq.(4), one obtains from Eq.(11): with a width characterized respectively by α1R ≃ 2.74 and 2.81. Generalization of the zero range approach for (2l−1)!! 1 lim ∂(2l+1) + α k2 + rl+1R (r)=0. → r l kl taking into account this anisotropy adds technical de- r 0 (2l)!! wl tails not essential in a first approach and in this paper, (12) 5
The idea is then to substitute the energy dependent term [αβ... ] l S Dl in Eq.(12) by the appropriate radial derivative. From l 0 1 ∂ − α ∂2 Eqs.(10,12) we find the desired condition: r 0 r α 1 3 2 1 n 2 ∂r + α1∂r 1 α β − 1 αβ 3 5 2 l+1 2 n n 3 δ 8 ∂r + 3α2∂r lim(Dl + )r Rkl =0 , (13) r→0 α β γ − 1 α βγ 5 7 2 wl 3 n n n 5 (n δ + permut.) 16 ∂r + 5α3∂r where we have introduced the differential operator: TABLE I: Examples for the symmetric traceless tensors Sl (2l−1)!! Eq.(18) and for the differential operators D in Eq.(13). nα D = ∂(2l+1) + (2l−1)α ∂2 . (14) l l (2l)!! r l r are the components of the unit vector ~n = ~r/r.
We are now ready to define the zero-range approach in general situations. The wave function Ψ of the interact- we adopt the standard assumption of implicit summa- ing two-body system is expanded in each partial wave tion over repeated indexes, the three directions of space channel in terms of the spherical harmonics as: {x,y,z} are labeled by a Greek index α ∈ {1, 2, 3} and [αβ . . . ] denotes a set of l indexes. The idea is to use the l c lm(r) symmetric tensors Sl which appear in the multipolar ex- h~r |Π |Ψi = Ψ Y lm(Ω) , (15) l rl+1 pansion used in electrostatics [44]. They can be defined mX=−l through their components by: where Πl is the projector on the subspace of angular l+1 l r l 1 quantum number l, and (r, Ω) are the spherical coordi- Sl [αβ... ] = (−1) ∂ [αβ... ] , (18) nates with Ω = (θ, φ). For r 6= 0, Ψ(~r ) is a solution of (2l−1)!! r the free Schr¨odinger equation (it implies that the multi- lm where we have introduced a short-hand notation for the polar radial functions cΨ (r) are regular functions of the l-th partial derivatives: interparticle distance r), and for an isotropic interaction, the potential term is replaced by the following boundary ∂ ∂ ∂ l = ..., (19) conditions on the wave function at r = 0: [αβ... ] ∂xα ∂xβ
1 lm and {xα} are the three components of the radial vector lim Dl + cΨ (r)=0 − l ≤ m ≤ l . (16) r→0 α wl ~r in the Cartesian basis {~eα}: i.e. ~r = x ~eα. These ten- sors can be expressed in terms of the component nα of Eq.(16) generalizes the Bethe-Peierls approach [1] which the unit vector ~n = ~r/r and are functions of the angles applies in the s-wave channel when the effective range Ω only. The normalization factor in Eq.(18) has been can be neglected (α0 = 0): chosen for later convenience and explicit expressions of S and D with l ≤ 3 are given in Tab.(I). In the follow- ∂ (rΨ) 1 l l lim r = − , (17) ing, we will use some useful properties of these tensors → r 0 (rΨ) w0 which are collected in the Appendix. By construction, [αβ... ] Sl are traceless (δαβS = 0) and symmetric tensors where the limit |w0| ≫ R is associated with a zero energy l 2 broad resonance. of rank l. They are eigenvectors of Lˆ (Lˆ is the angular 2 Note that the generalization of Eqs.(16) to the case momentum operator) with eigenvalue ¯h l(l + 1). Simi- lm of an anisotropic two-body interaction is straightforward larly to the spherical harmonics Y (Ω), the tensors Sl by introducing parameters that depend on the quantum can serve as a basis for an alternative multipolar expan- number m: {wlm, αlm}. sion of a wave function, like in Eq.(15). To this end, we define the Cartesian representation of the multipolar ra- dial functions of order l associated with a wave function C. Cartesian representation of the multipolar Ψ, by a symmetric trace free tensor CΨ,l(r) having the radial functions components:
l+1 2 While the zero-range approach is now established in [αβ... ] r d Ω [αβ... ] CΨ,l (r)= Sl Ψ(r, Ω) . (20) a general way by Eqs.(14,15,16), it can be easier to use l! ZΩ 4π equivalent boundary conditions in the Cartesian coordi- nate system (for example when the system is inhomoge- For example, the monopolar (l = 0) function CΨ,0(r) is just r times the projection of the wave function in neous or anisotropic). Furthermore, as will be shown in [αβ... ] the next section, using these conditions leads to a simple the s-wave channel. Quite generally, CΨ,l (r) are regu- lm construction of the zero-range pseudopotential (in par- lar functions of r analogous to the coefficients cΨ (r) in ticular, derivatives of the delta distribution are easily Eq.(15) and characterize the behavior of the wave func- introduced in Cartesian coordinates). In the following, tion Ψ(r, Ω) in the channel l through the expansion (see 6
Eqs.(69,72) in the Appendix): By construction wave functions belonging to the kernel (2l+1)!! of Bl satisfy the desired boundary conditions. Conse- h~r |Π |Ψi = S C[αβ... ](r) . (21) quently, the family of pseudopotentials defined by: l rl+1 l[αβ... ] Ψ,l Using these notations, the generalized Bethe-Peierls (i) V + β Bl , (27) boundary conditions Eq.(16) can be written as: l where β is an arbitrary parameter, can be used to solve 1 [αβ... ] lim(Dl + ) C (r)=0 . (22) the two-body problem in the resonant regime. Interest- r→0 w Ψ,l l ingly, while exact zero-range solutions can be obtained Eqs.(20,21,22) are equivalent to Eqs.(15,16). We show in ∀β 6= 0, it is not hard to be convinced that a first or- the next section that they permit to find a general form der Born approximation performed for example in the of the pseudopotential. computation of the two-body scattering states leads to Note that an alternative formulation of the condition a β-dependent result. Indeed in this approximation, the (22) in the s-wave channel amounts to imposing the fol- two-body wave function Ψ(0) is an eigenstate of the non- lowing singularity on the two-body wave function: interacting Schr¨odinger equation (a plane wave in the ex-
CΨ,0(0) r 2 3 ample chosen) and is then a regular wave function of the h~r |Π0|ψi = 1 − + A0(r +2α0r) + O(r ). (i) (0) r a radius r. Consequently, the action of Vl on Ψ gives (23) (i) (0) a zero result while h~r |Vl + β Bl|Ψ i is β-dependent. In the p-wave channel l = 1, the short-range behavior Furthermore, and we will show this property explicitly of the interacting two-body wave function takes also a in the next section, varying β offers the freedom to ad- simple form: just the pseudopotential in such a way that the first order 3 ~p.~r r 2 3 2 Born approximation is exact at a given collisional energy h~r |Π1|Ψi = 3 1 − + A1(3r − 2α1r ) + O(r ), r 3Vs (item-iii). Finally, the expression of the pseudopotential (24) in Eq.(27) although general, can be rewritten in the more where the “dipolar momentum” ~p defined by pα = convenient following form: 3Cα (0), characterizes the p-wave singularity of the Ψ,1 l l [αβ... ] wave function. In general situations the coefficients h~r |Vλ,l|Ψi = (−1) gλ,l(∂ [αβ... ]δ)(~r )Rλ,l [Ψ] . (28) (A0,A1,CΨ,0(0), ~p ) can be functions of time and of the l center of mass coordinates of the two particles. In this expression, gλ,l = (−1) β is the coupling constant defined by:
2 D. Zero-range pseudopotential 2π(2l+1)¯h wl gλ,l = , (29) µ(1 − λwl) Instead of giving a final expression of the pseudopo- tential directly, we prefer to detail the different steps in and Rλ,l[ . ] are symmetric trace free tensors which gener- its derivation. Such a way of presentation can be useful alize the regularizing operator introduced in Refs.[8, 35]. if one wants to derive pseudopotentials associated with They act on a wave function Ψ as: others expressions of the phase shift in Eq.(4) or in dif- [αβ... ] [αβ... ] R [Ψ] = lim(Dl + λ)C (r) . (30) ferent contexts (low dimensions for example). In either λ,l r→0 Ψ,l cases, a zero-range pseudopotential can be deduced from the three following constraints: (i) it cancels the delta The parameter λ appears in Eq.(29) and Eq.(30) both, terms arising from action of the Laplacian on the singu- and by construction (like β in Eq.(27)) it is a free param- larities of the wave function at r = 0 in the Schr¨odinger eter of the theory: exact wave functions obtained in the equation, (ii) it imposes the correct boundary conditions zero-range formalism do not depend on the value of λ. and (iii) it can be used in the Born approximation. The The pseudopotentials in Eq.(28) generalize for arbitrary first condition (item-i) can be fulfilled by introducing po- partial wave channels the s-wave and p-wave λ-potentials tential terms in the different channels l of the form (see already introduced in Ref. [8, 35]. Eq.(77) in the Appendix): 2π(2l+1)¯h2 h~r |V (i)|Ψi = (−1)l+1 III. SOME ILLUSTRATIONS AND l µ PROPERTIES OF THE PSEUDOPOTENTIAL C[αβ... ](0) (∂ l δ)(~r ) . (25) Ψ,l [αβ... ] A. Two-body scattering states from the Green’s In order to include the boundary conditions Eqs.(22) method (item-ii), we consider the “boundary operator” Bl de- fined by: This section is both an illustration of the formalism 1 and a way to test the proposed form of the pseudopo- h~r |B |Ψi = (∂ l δ)(~r ) lim(D + ) C[αβ... ](r). (26) l [αβ... ] → l Ψ,l r 0 wl tential given in Eq.(28). We consider the case where two 7 particles interact in the lth partial wave channel with- one can see that it contains implicitly the projector Πl out external potential and we derive the scattering states through: from the general Green’s function method. The interact- [αβ... ] ~ Sl[αβ... ]R [Ψ] = ing two-body wave function Ψ~k(~r ) (wave vector k, energy λ,l 2 2 E = ¯h k /2µ) is then the solution of the integral equa- 1 l+1 lim(Dl + λ)r h~r |Πl|Ψi . (38) tion: (2l+1)!! r→0
(0) 3 ′ ′ ′ A closed equation is obtained by applying the operator Ψ~ (~r )=Ψ (~r )− d ~r GE(~r, ~r )h~r |Vλ,l|Ψ~ i , (31) k ~k Z k [αβ... ] Sl[αβ... ]Rλ,l [ . ] on both sides of Eq.(35). Using the (0) ~ usual decomposition of a plane wave over spherical ones: where Ψ~ (~r ) = exp(ik.~r ) is the incident plane wave k ∞ and G is the one-body outgoing free Green’s function E ~ l at energy E: exp(ik.~r )= i (2l+1)Pl(~n.~nk)jl(kr) , (39) Xl=0 ′ µ exp iku ′ GE(~r, ~r )= with u = |~r − ~r |. (32) together with Eq.(9) this leads to: 2π¯h2 u ′ l+1 (0) lim(Dl + λ)r h~r |Πl|Ψ i = Integrating Eq.(31) over ~r coordinates gives: r→0 ~k l (0) µgλ,l [αβ... ] (2l+1)!! (ik) Pl(~n.~nk) , (40) Ψ~ (~r )=Ψ (~r )− R [Ψ~ ] k ~k 2π¯h2 λ,l k on the other hand: ′l exp iku ∂ [αβ... ] , (33) l u r′=0 2l+1 1 exp(ikr) lim(Dl + λ) r ∂r = r→0 r r ′l where ∂ [αβ... ] stands for the lth partial derivatives with l 2 2l+1 ′ (−1) (2l−1)!! (λ + α k + ik ) , (41) respect to ~r coordinates. Thanks to the trace free prop- l [αβ... ] erty (δαβRλ,l [Ψ~k] = 0), contribution of the lth deriva- and finally, one gets: tive in Eq.(33) takes a simple form. Indeed, for a given l function F (u), one can show that: [αβ... ] (1 − λwl)(ik) Pl(~n.~nk) Sl[αβ... ]Rλ,l [Ψ~k]= 2 2l+1 . (42) 1+ wlαlk + iwlk ′l [αβ... ] ∂ F (u) R [Ψ~ ]= [αβ... ] ′ λ,l k As expected, the expression of the scattering wave func- r =0 l tion does not depend on the parameter λ: [αβ... ] l 1 (nαnβ ... )Rλ,l [Ψ~k](−r) ∂r F (r) , (34) r (0) (2l+1)wlPl(~n.~nk) Ψ~k(~r ) = Ψ~ (~r ) − 2 2l+1 ′ k 1+ wlαlk + iwlk and with F (u)= GE (~r, ~r ), Eq.(33) reads finally: l l 1 exp(ikr) (0) µgλ,l [αβ... ] (−ikr) ∂r . (43) Ψ~ (~r ) = Ψ (~r ) − Sl[αβ... ]R [Ψ~ ] r r k ~k 2π¯h2 λ,l k 1 l exp(ikr) Asymptotically for kr ≫ 1, the two-body wave function (−r)l ∂ . (35) is given by: r r r exp(ikr) As r → 0, one recognizes a multipolar singularity in the Ψ (~r ) = exp(i~k.~r )+(2l+1)P (~n.~n )f , (44) ~k l k l r two-body wave function: where the partial amplitude fl is given by Eq.(5). (2l+1)!! wl Sl[αβ... ] [αβ... ] 1−l It is interesting to note that the exact result can be also Ψ~k(~r )= − l+1 Rλ,l [Ψ~k]+ O(r ), (1 − λwl) r obtained at the level of the first order Born approxima- (36) tion (which is a “λ-dependent” treatment) by choosing a and comparison with Eq.(21) gives the multipole compo- particular value of the parameter λ: nents: Born 2 2l+1 λ = −αlk − ik . (45) [αβ... ] wl [αβ... ] CΨ~ ,l (0) = − Rλ,l [Ψ~k] . (37) k 1 − λwl For example by setting λ = 0 in a first order Born approx- In the partial wave channels of angular quantum number imation gives a correct evaluation of a scattering process different from l, components of the wave function Ψ (~r ) at a vanishly small collisional energy (Wigner threshold). ~k Generalization to the case where the interaction oc- coincide as expected with the partial waves of Ψ(0)(~r ). ~k curs in several partial wave channels is straightforward: All the information on the interaction are gathered in it is obtained after simple addition of the corresponding [αβ... ] the term Sl[αβ... ]Rλ,l [Ψ~k] in Eq.(35). From Eq.(21), pseudopotentials given by Eq.(28). 8
B. Links with other formalisms vector kF . However, in this mean field approach the ap- proximation made on the interaction term is only valid for binary processes of vanishly small colliding energies It is instructive to compute the matrix element of Vλ,l between two states |Φi and |Ψi. A simplifying step is to (equivalent to a Born approximation with the pseudopo- realize that the projection operator in the partial wave tential and the particular choice λ = 0). Consequently > (l) is implicitly present in the expression of the pseudopo- in the resonant regime where kF |w0| ∼ 1, the character- tential Eq.(28). Indeed, assuming that Φ(~r ) is regular at istic collisional energy of pairs of particles on the Fermi 2 2 r = 0, one obtains: surface is ¯h /mw0 and the approximation made is not justified. By the way, it is possible to build a qualitative [αβ... ] l h~r |Πl|Φi and simple mean field model which supports the stability S (∂ Φ)r=0 = l! lim . (46) l [αβ... ] r→0 rl of the gas at the BCS-BEC crossover [46]. Taking two states with angular parts characterized by Moreover from Eqs.(38,46) and Eq.(73) in the Appendix, the same spherical harmonic in Eq.(47), one obtains the one finds: expression of the pseudopotential in radial coordinates: 2 d Ω hΦ|Πl|~r i hΦ|Vλ,l|Ψi = gλ,l lim l gλ,l δ(r) l+1 Z 4π r→0 r vλ,l = lim(Dl + λ)(r . ) . (52) 4π rl+2 r→0 lim(D + λ)rl+1h~r |Π |Ψi . (47) → l l r 0 In the particular case λ = 0, αl = 0, r.h.s. of Eq.(52) coincides with the form of the pseudopotential introduced First, we consider plane waves h~r |~k i = exp(i~k.~r). The in Ref. [37]: matrix element between two such states is: ¯h2w (2l+1)!! δ(r) ′ ′ l ′ l 2l+1 l+1 h~k |V |~k i = g (kk ) P (~n .~n ) , (48) v0,l = lim ∂r (r . ) . (53) λ,l λ,l l k k 2µ(2l)!! rl+2 r→0 ′ ′ ′ with ~nk = ~k /k . For wave vectors of same modulus ′ Note that the delta-shell regularization procedure intro- (k = k = 2µE/¯h) and for the particular value of the duced in Ref. [37] while correct appears not to be essential free parameterp λ given in Eq. (45), comparison with the in the zero-range approach. expression of the partial wave amplitude Eq.(5) shows that Eq.(48) coincides with the on-shell T -matrix t(~k ′ ← ~k ) taken at energy E: C. Regularized scalar product 2 ′ ′ 2π¯h ′ t(~k ← ~k )= h~k |V Born |~k i = − f(~k ← ~k ), (49) λ ,l µ Except for the case of the Fermi pseudopotential which is a particular case of Eq.(28), two scattering states Ψ~k ′ ~ ′ ~ ′ ~ ~ ′ where f(k ← k ) = (2l + 1)Pl(~nk .~nk)fl(k) is the full and Ψ~k (with k 6= k ) of expressions given by Eq.(43) scattering amplitude. are not orthogonal each others with respect to the usual Another interesting application of Eq.(47) is given by scalar product. To examine this point, we exclude the considering spherical outgoing waves: singularity at r = 0 in the scalar product between the two different scattering states by introducing a cut-off r0. lm h~r |klmi = Y (Ω)jl(kr) . (50) We obtain from the Schr¨odinger equation:
In this case, the matrix elements are given by: 2 3 ∗ r0 d ~r Ψ ′ (~r )Ψ (~r )= ~k ~k 2 ′2 ′ l Zr>r0 k − k ′ ′ ′ gλ,l(kk ) ′ ′ hklm|Vλ,l|k l m i = δll δmm . (51) 2 ∗ ∗ 4π(2l+1) d ΩΨ ′ (~r )∂ Ψ (~r ) − Ψ (~r )∂ Ψ ′ (~r ) . (54) ~k r ~k ~k r ~k Zr=r0 For λ = 0, this result coincides with the one obtained in Ref. [45]. Here, the λ-freedom is a new ingredient and This relation is general i.e. it is also valid for scattering shows clearly that mean field calculations in zero-range wave functions associated with a finite range two-body approaches have to be developed carefully. For example, potential whatever the value of r0. In standard situations considering a two-component Fermi gas close to the uni- where the Hamiltonian is hermitian, the r.h.s. of (54) is tary limit with α0 = 0 and a large and negative value of zero for r0 = 0. However in the zero-range approach, the scattering length w0, the mean field model in Ref. [45] for wave functions satisfying the boundary conditions in predicts an instability which is not observed in experi- Eq.(13), as r0 → 0 this term converges toward a constant ments [28, 29, 30, 31, 32]. This failure can be explained value for l = 0 and α0 6= 0 and even worse, it diverges for as follows. The interaction concerns only particles close l> 0. Interestingly, the regular part of Eq.(54) for such to the Fermi surface (Pauli blocking) with a Fermi wave wave functions can be obtained through the following 9 surface integration: • Particles interacting in the s-wave channel only:
3 ∗ ∗ (Ψ|Φ) = lim d ~r Ψ (~r )Φ(~r ) 3 0 → Reg d ~r Ψ ′ (~r )Ψ (~r ) = r0 0 ~k ~k Zr>r0 r0→0 Zr>r0 ∞ 2l+2 2 2 ∗ α r +α0r0 d ΩΨ (~r )Φ(~r ) . (57) l 0 2 ∗ Z − lim d ΩΨ~ ′ (~r )h~r |Πl|Ψ~ki, (55) r=r0 r0→0 ((2l−1)!!)2Z k Xl=0 r=r0 This result can be written more formally by intro- ducing a scalar product of the form: where the operator Reg { . } extracts the regular part at r0→0 3 ∗ r0 = 0. The meaning of this result is that the Lapla- (Ψ|Φ)0 = d ~rg(r)Ψ (~r )Φ(~r ) , (58) cian operator is not Hermitian with respect to the usual Z scalar product when one considers singular functions like with the weight g(r)=1+ α δ(r). the ones in Eq.(43). In another hand, the true scatter- 0 ing states associated with the finite range potential ex- • Particles interacting in the p-wave channel only (for perienced by the particles, are orthogonal to each other example in the case of polarized fermions): so that direct identification of the true scattering states with the states |Ψ~ki is not possible. This feature fol- 3 ∗ (Ψ|Φ)0 = lim d ~r Ψ (~r )Φ(~r ) lows from the fact that the mapping between the two r0→0 Z < r>r0 eigen basis is not valid for r ∼ R while it is justified only 4 3 2 ∗ outside the potential range. Indeed, the singular bound- + (α1r0 − r0) d ΩΨ (~r )Φ(~r ) . (59) ary behavior imposed on wave functions in Eq.(16) is a Zr=r0 way to reproduce the effect of the true finite range poten- This scalar product has been first introduced in tial for r > R but has a formal character for interparticle Ref. [35] in the same form as in Eq.(58) with the distances r < R. In order to understand this issue more ∼ weight g(r)=1+ δ(r) (α r2 − r) . . closely, we consider the sphere of radius R which delim- 1 its the outer parts (region r > R) from the inner parts true (region r < R) of the wave functions. Contribution D. Bound states and their normalizations to the scalar product coming from the outer-part of the true wave functions is given by Eq.(54) with r0 = R and due to orthogonality between the true scattering states, In this section, we consider the normalization of the scalar product of the inner-parts cancels this term. How- two-body bound states of vanishing energy supported by ever in the zero-range potential approach the inner parts the zero-range pseudopotential. We show that using the of the wave functions (r < R) are not properly described, regularized scalar product is equivalent to the method so that compensation of the two contributions does not based on the analyticy of the scattering amplitude [38]. occur. To solve this inconsistency, the idea is to modify This last method was used in Ref. [15] in the context the usual scalar product itself by including implicitly the of s-wave narrow resonances. In this case or also for p- contribution of the inner parts. We define then a regu- wave or higher partial waves channel resonance where bound states are even not square integrable with respect larized scalar product ( . | . )0 by subtracting the surface 2 term (r.h.s. of Eq.(54)) from the usual scalar product as to the usual scalar product (they don’t belong to the L Hilbert space), the regularized scalar product appears r0 → 0: as the natural way to perform the normalization in the configuration space. Moreover, this scalar product allows 3 ∗ for a generalization of the method of Ref. [38] in the case (Ψ |Φ)0 = Reg d ~r Ψ (~r )Φ(~r ) r0→0 Zr>r0 of inhomogeneous situations. ∞ 2l+2 If one considers solutions of the Schr¨odinger equation αl r0 2 ∗ + lim d ΩΨ (~r )h~r |Πl|Φi, (56) with a finite range potential as functions of ~r and of r0→0 ((2l−1)!!)2Z Xl=0 r=r0 the energy E: Ψ(E, ~r ), then a bound state of energy EB and wave function ΨB = Ψ(EB, ~r ) corresponds to so that by construction, two eigenstates of the zero-range a case where Ψ(E, ~r ) is square integrable. Using the Hamiltonian are orthogonal to each other with respect to Schr¨odinger equation, one can show that: this metrics. 2 2 3 2 ¯h r0 2 ∗ In general there are several possible representations of d ~r |ΨB| = − d Ω Ψ (E, ~r )∂r∂EΨ(E, ~r ) Eq.(56) due to the various and equivalent ways of extract- Zr
ACKNOWLEDGMENTS 2 ′ d Ω ′ ′ h~r |Πl|Ψi = (2l+1) Ψ(r, Ω )Pl(~n.~n ) . (72) Z 4π Y. Castin, F. Chevy, M. Holzmann and F. Werner are acknowledged for thorough discussions on the subject. together with Eq.(69) and the definition of the multipoles Laboratoire de Physique Th´eorique de la Mati`ere Con- of the wave function Eq.(20), we obtain Eq.(21). dens´ee is Unit´eMixte de Recherche 7600 of Centre Na- In this paper, we also use an interesting property of tional de la Recherche Scientifique. the symmetric traceless tensors found in Ref. [49]. Let us consider two symmetric and traceless tensors A and APPENDIX: USEFUL PROPERTIES OF THE B of order l, then SYMMETRIC TRACE FREE TENSORS S l ′ 2 ′ ′ [αβ... ] d Ω ′ ′ ′ ′ [α β ... ] A n αn β ...n α′ n β′ B This appendix collects some useful properties of the Z 4π symmetric trace free tensors Sl defined in Eq.(18) [44, l! [αβ... ] α ′ = A[αβ... ] B . (73) 49]. Let us consider the two vectors ~r = x ~eα and ~r = (2l+1)!! ′α x ~eα. By definition: ′′ ∞ As an example, taking A = S and B = S allows one to 1 (2l−1)!! x ′αx ′β ... l l = S . (65) recover from Eqs.(69,73) the known relation: |~r − ~r ′| l! l[αβ... ] rl+1 Xl=0 2 ′ α ′ ′′ d Ω ′ ′ ′′ Introducing the unit vectors ~n = ~r/r = n ~eα, ~n = Pl(~n.~n )=(2l+1) Pl(~n.~n )Pl(~n .~n ). (74) ′ ′ ′α ′ Z 4π ~r /r = n ~eα and the symmetric trace free tensor Sl associated with ~r ′, we have: We close this appendix with the link between the sin- ′α ′β ′ [αβ... ] gularity of the lth partial wave of a wave function with Sl[αβ... ]n n ··· = S l[αβ... ]Sl , (66) the derivatives of the delta distribution. To this end, we so that Eq.(65) can be rewritten use the so-called relation:
∞ ′ l 1 1 ′[αβ... ] (2l−1)!! r 1 ′ = S S . (67) ∆~r ′ = −4πδ(~r − ~r ) (75) |~r − ~r ′| r l[αβ... ] l l! r |~r − ~r | Xl=0 This expression can be identified with the standard ex- together with Eq.(67) and the formal expansion: pansion (for r ′ < r) [50]: ∞ ′ l ∞ ′ l ′ (−r ) l ′α ′β 1 1 r ′ δ(~r − ~r )= (∂ δ)(~r )n n ..., (76) = P (~n.~n ) , (68) l! [αβ... ] |~r − ~r ′| r r l Xl=0 Xl=0 l where Pl(x) is the Legendre polynomial of degree l and where (∂ [αβ... ]δ)(~r ) is the lth partial derivative of the we obtain: delta distribution (see Eq.(19)). Finally, we obtain:
′ (2l−1)!! ′[αβ... ] P (~n.~n )= S S . (69) Sl[αβ... ] ′ ′ l l! l[αβ... ] l ∆ n αn β ··· = ~r rl+1 ′ For example, taking ~n = ~ez shows that: l+1 (−1) l ′α ′β 4π (∂ [αβ... ]δ)(~r )n n .... (77) l! (2l−1)!! S = P (cos θ) , (70) l(zzz...z) (2l−1)!! l This expression justifies the introduction of the potential (i) with θ = (~r, ~ez). Eq.(69) can be used to have an expres- Vl in Eq.(25). Using Eq.(69), one can also write Eq.(77) sion of the projection operator Π over the partial wave in terms of Legendre Polynomials: d l l in terms of the tensor Sl. For that purpose, we recall l+1 the addition theorem [50]: Pl(cos θ) (−1) l ∆~r l+1 =4π (∂z δ)(~r ) . (78) m=l r l! ′ 4π ∗ ′ P (~n.~n )= Y lm(Ω)Y lm (Ω ) , (71) l 2l+1 mX=−l 12
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