Non-Hermitian multichannel theory of ultracold collisions modified by intense light fields tuned to the red of the trapping transition

Non-Hermitian multichannel theory of ultracold collisions modified by intense light fields tuned

to the red of the trapping transition

R. Napolitano

Instituto de Quı´mica de Araraquara, Universidade Estadual Paulista, Araraquara, Sa˜o Paulo 14800-900, Brazil and Instituto de Fı´sica de Sa˜o Carlos, Universidade de Sa˜o Paulo, Sa˜o Carlos, Sa˜o Paulo 13560-970, Brazil

~Received 15 August 1997!

We develop a systematic scheme to treat binary collisions between ultracold atoms in the presence of a strong laser field, tuned to the red of the trapping transition. We assume that the Rabi frequency is much less than the spacing between adjacent bound-state resonances. In this approach we neglect fine and hyperfine structures, but consider fully the three-dimensional aspects of the scattering process, up to the partial d wave. We apply the scheme to calculate the S matrix elements up to the second order in the ratio between the Rabi frequency and the laser detuning. We also obtain, for this simplified multichannel model, the asymmetric line shapes of photoassociation spectroscopy, and the modification of the scattering length due to the light field at low, but finite, entrance kinetic energy. We emphasize that the present calculations can be generalized to treat more realistic models, and suggest how to carry out a thorough numerical comparison to this semianalytic theory.@S1050-2947~98!04902-6#

PACS number~s!: 32.80.Pj, 34.50.2s, 34.50.Rk

I. INTRODUCTION

We can modify and control certain aspects of ultracold collisions using laser fields suitably tuned and polarized. In-deed, recent experimental@1–8# and theoretical @9–11# stud-ies have assured our ability to suppress inelastic ultracold-collision rates by light fields tuned to the blue of the trapping transition. Moreover, in the context of quantum degeneracy, a zero-temperature calculation@12# has shown the possibility of modifying the scattering length associated with the inter-action potential of two ground-state atoms, in the presence of red-detuned photons.

Recent experiments show that Bose-Einstein condensa-tion of trapped alkali-metal atoms occurs at finite ultracold temperatures@13–15#. In this energy regime, we have devel-oped a two-state model for controlling scattering lengths and photoassociation spectral line shapes of alkali-metal atoms by red-detuned laser fields @16#. However, at high enough intensities of the light field, a two-state model cannot de-scribe the collision completely. Due to multiphoton pro-cesses, partial waves higher than the s wave contribute ap-preciably to the scattering @17#. Furthermore, atom-field dressing effects manifest themselves when the collision oc-curs in the presence of a strong enough laser field. Below 1

mK, light shifts become comparable to the entrance collision energies, even for detunings of several tens cm21 from the trapping transition. We expect dressing effects to be espe-cially relevant to photoassociation spectral line shapes of alkali-metal atoms, since this topic has been extensively studied in the past few years mostly in the low-intensity regime@18–26#.

In this paper we present a systematic approach to treat binary ultracold collisions in the presence of an intense laser field, tuned to the red of the trapping transition. The only assumption we make concerning the light intensity is that the corresponding Rabi frequency is much less than the spacing between adjacent bound-state resonances. We predict, in a

unified way, the modification of scattering lengths and pho-toassociation spectral line shapes, regarding only isolated resonances. With this simple model we are able to take into account multichannel and dressing effects in the regime of finite ultracold temperatures. As in Ref.@10#, we take a step beyond the two-state approaches by neglecting fine and hy-perfine structures, but considering fully the three-dimensional aspects of the problem. For simplicity and con-creteness, we introduce partial waves up to the d wave and treat the particular case of a pair of sodium atoms in the presence of linearly polarized, red-detuned photons. In a forthcoming paper we will present numerical calculations based on the present theory and a comparison between the effects of linear and circular polarizations. We emphasize that the present approach is easily extensible to higher partial waves and other alkali-metal species. This treatment can also accommodate more than one ground-state potential curve and retardation effects. Although it may be considered cum-bersome, fine and hyperfine structures can, in principle, also be incorporated into the formalism.

The solution of the coupled equations describing the col-lision process is fully detailed in this paper, in a self-contained manner, for completeness and clarity. Although we present a systematic algorithm to calculate dressing ef-fects to all orders, we illustrate the procedure through ex-plicit calculations for a special situation relevant to sodium, up to the second order in the ratio between the Rabi fre-quency and the detuning. In the present treatment we intend to emphasize multichannel, finite-temperature, and dressing effects, without unnecessary sophistication.

To account for spontaneous emission from the excited bound state, we adopt a non-Hermitian effective Hamiltonian that we describe in detail in Sec. II. Following this specifi-cation of the problem in the undressed basis, including all the coupled states up to the partial d wave, in Sec. III we refor-mulate the collision process in terms of the asymptotic dressed states. In the dressed basis, the Hamiltonian is asymptotically diagonal and this basis represents the actual 57

incoming and outgoing physical states. In Sec. IV we con-sider the closed-channel manifold assuming that it contains an isolated resonance. In this way we simplify the closed-channel problem by regarding only the resonant contribution to the corresponding Green function. We obtain the S matrix elements in Sec. V, up to the second order in the ratio be-tween the Rabi frequency and the detuning. We also illus-trate the application of the procedure by calculating the asymmetric line shapes of photoassociation spectroscopy, and the modification of the scattering length by the light field. In Sec. VI we discuss the region of parameter space where we expect this theory to be valid. Finally, in Sec. VII we conclude by summarizing the present approach and pre-senting some directions we will follow in a forthcoming pa-per.

II. THE UNDRESSED POTENTIAL MATRIX

In this article we consider two sodium atoms in the hy-perfine state uF51,mF521

&

, as in Ref. @15#, so that the lower molecular state involved is the a 3S_u1. From this lower molecular state, the colliding system can absorb a red-detuned photon and be excited to a bound state of symmetry

3_S g

1_{. Asymptotically, that is, when the interatomic distance}

R approaches a large but finite value R_` at which we can

neglect the interaction between the atoms, the a 3S_u1 mo-lecular state corresponds to two atoms in their 2S ground states. The 3Sg1 bound molecule asymptotically dissociates into one atom in its 2S ground state and another in its 2P first excited state. Because in this paper we ignore fine and hyperfine interactions, let us denote by j the electronic or-bital angular-momentum quantum number of the asymptotic two-atom system. Thus, if the two atoms are in their 2S ground states, then j50; and if one of the atoms is in its 2S ground state and the other in its 2P first excited state, then j51. The j50 state has angular-momentum projection mj

50 along a space-fixed z axis, while the j51 state may have

mj521, 0, or 11. Therefore, the asymptotic electronic ba-sis states relevant to this problem are denoted u j,mj

&

, with

mj50 for j50 and mj50,61 for j51.

Two separated atoms can come together at different im-pact parameters. Equivalently, we can describe the rotational state of the approaching atoms by introducing the spherical harmonic functions ul,m_l

&

[Y_lml₍u_,w_{), where l}50,2,4, . . . for bosons@27# and ml52l,2l11, . . . ,l21,l. The angles

u andwdescribe the orientation of the interatomic axis with respect to the space-fixed reference frame. Up to the l52 partial wave (d wave! and for linear light-field polarization, the only coupled undressed states, including the ground state, are given by @10#

u1

&

[uj50,mj50

&

ul50,ml50

&

uN

&

, ~1a!

u2

&

[uj50,mj50

&

ul52,ml50

&

uN

&

, ~1b!

u3

&

[uj51,mj50

&

ul50,ml50

&

uN21

&

, ~1c!

u4

&

[uj51,mj521

&

ul52,ml51

&

uN21

&

, ~1d!

u5

&

[uj51,mj51

&

ul52,ml521

&

uN21

&

, ~1e!

u6

&

[uj51,mj50

&

ul52,ml50

&

uN21

&

, ~1f! and we have also included the radiation-field Fock statesuN

&

anduN21

&

containing N and (N21) photons, respectively. These states are normalized to unity, that is,

^

aub

&

5d_a,b, for a,b51, . . . ,6. The radiation can only couple u1

&

tou3

&

and u2

&

to u6

&

, but there are other couplings that force the introduction of the statesu4

&

andu5

&

, as we show below. The radiative couplings we consider are

^

1uHIu3

&

5

^

2uHIu6

&

5\V, ~2! where V5

A

2VA andVA is the atomic Rabi frequency for the 32_S→32_{P transition of sodium given by V}

A/(2pc)

'1.4796631023

_A

_{I cm}21_{, and I is the laser intensity in} W/cm2@10#. Let us notice that the molecular Rabi frequency

V is a real quantity, independent of R.

The total Hamiltonian for this problem has the form

H52 \ 2 2mR ]2 ]R2R1 L2 2mR21V~R!, ~3!

with the potential operator

V~R!5He~R!1Hf1HI11j, ~4!

where L[2i\R3¹ is the rotational angular-momentum operator satisfying L2ul,ml

&

5\2l(l11)ul,ml

&

and zˆ_•Lul,ml

&

5\mlul,ml

&

, R is the interatomic vector of polar coordinates (R,u,w),m is the reduced mass of the colliding partners, He(R) is the electronic Hamiltonian for a fixed in-teratomic separation R, H_fis the radiation-field Hamiltonian, HI is the operator representing the interaction between the colliding atoms and the electromagnetic field, 1 is the iden-tity operator, andj is an arbitrary shift in energy we specify below. The nonzero matrix elements of HI are given by Eq.

~2!. The action of Hf on a Fock state containing N photons of frequency v is

HfuN

&

5\v~N1

1

2!uN

&

. ~5!

Let E0 be the total electronic energy ~not including atomic kinetic energy! of two separated 2S atoms and E1 the total electronic energy of a 2S atom and a distant 2P atom. Hence, H_e(R) is diagonal in the basis set$u j,L,J,M_J

&

%:

He~R!u j,L,J,MJ

&

5@W_uLu

j _~R!1E

j2i\gd1,j#u j,L,J,MJ

&

,

~6!

where the Born-Oppenheimer potentials W_uLuj (R) satisfy limR_{→`</sub>W<sub>uLu</sub>j (R)50 and R<sub>`} is such that W_uLuj (R'R_`)'0, and the Hund’s-case ~a! eigenfunctions uj,L,J,MJ

&

are de-fined as u j,L,J,MJ

&

[

A

2J11 4p DM_J,L J_* ~w,u,0!u j,L

&

and u j,L

&

[

(

m_j521 1 D_L,m j j_{* ~w,u,0!u j,m} j

&

. ~7! Here,\g is the atomic linewidth~the molecular linewidth is equal to 2\g @28#! of the 32P state of sodium that we

in-troduce at this point in an ad hoc fashion to account for the spontaneous emission from the bound-state resonance

@16,29#. The unnormalized symmetric top eigenfunctions

D_M

J,L

J_{* (w,u,0) and D} L,mj

j_{* (w,u,0) are the Wigner rotation} matrix elements @30#. The functions uj,L

&

refer to a molecule-fixed reference frame that has its z axis parallel to R. Let us notice that L50 for j50 and L50,61 for j

51, given the isotropy of space. As explained in Ref. @10#,

the implicit R dependence of the molecular eigenfunctions

u j,L,J,MJ

&

is of no appreciable effect due to the smallness of the ratio between the electronic and nuclear mass ~Born-Oppenheimer approximation!. The meaning of the quantum numbers J and M_J is clear from the equations J[j1L, J2u j,L,J,MJ

&

5\2J(J11)uj,L,J,MJ

&

, and zˆ•Ju j,L,J,MJ

&

5\MJu j,L,J,MJ

&

, with J5uj2lu,uj2lu11, . . . , j1l21,

j1l and MJ52J,2J11, . . . ,J21,J.

The Born-Oppenheimer molecular basis$u j,L,J,MJ

&

% is related to the asymptotic space-fixed basis $u j,mj

&

ul,ml

&

% through the relation@31#

u j,mj

&

ul,ml

&

5

(

J5uj2lu j1l

(

M_J52J J

(

L52 j j

A

2l11 2J11 3

^

j,l,mj,mluJ,MJ

&

3

^

j,l,L,0uJ,L

&

u j,L,J,MJ

&

, ~8!

adopting the notation of Ref. @30# to write Clebsch-Gordan coefficients as, for example,

^

j,l,mj,mluJ,MJ

&

. In the un-dressed basis, Eqs.~1!, the potential operator of Eq. ~4! can be represented by a 636 complex matrix through the use of Eqs. ~2!, ~5!–~8!: V~R!5

3

W0 0_~R! 0 \V 0 0 0 0 W₀0~R! 0 0 0 \V \V 0 V_a~R!2\D_c D~R!/

A

15 D~R!/

A

15 22

A

5D~R!/15 0 0 D~R!/

A

15 Vb~R!2\Dc 22D~R!/7

A

3D~R!/21 0 0 D~R!/

A

15 22D~R!/7 Vb~R!2\Dc

A

3D~R!/21 0 \V 22

A

5D~R!/15

A

3D~R!/21

A

3D~R!/21 Vc~R!2\Dc

4

, ~9!

where Dc[D1ig is a complex quantity, D[v2(E1

2E0)/\,0 is the laser detuning that we assume to be nega-tive ~red detuning!, and we have defined the functions

D~R![W1 1_~R!2W 0 1_{~R!, ~10a!} Va~R![ 2W₁1~R!1W₀1~R! 3 , ~10b! Vb~R![ 5W₁1~R!12W₀1~R! 7 , ~10c! Vc~R![ 10W₁1~R!111W₀1~R! 21 . ~10d!

To derive this potential matrix we have chosen the energy shift appearing in Eq.~4! to bej[2E02\v(N1

1

2), for the

sake of simplicity.

III. THE CLOSE-COUPLING FORMULATION IN THE DRESSED-STATE REPRESENTATION

In the close-coupling formalism, described in detail in Ref. @10# and references cited therein, we look for wave functions, written as uCb~R!

&

5

(

s51 6 F_s,b~R! R us

&

, ~11!

that satisfy the time-independent Schro¨dinger equation, HuC_b(R)

&

5EuC_b(R)

&

, where H is the total Hamiltonian given by Eq. ~3!, E is an eigenvalue corresponding to the continuous spectrum of H, and the index b enumerates the eigenstates belonging to a given E. The eigenvalue E of H is a complex quantity, since H is not Hermitian, and in the following we show that the suitable choice of asymptotic boundary conditions implies a finite imaginary part of E. It then follows that the solution of the time-dependent Schro¨-dinger equation, HuF_b(R,t)

&

5i\(]/]t)uF_b(R,t)

&

, is given byuF_b(R,t)

&

5exp@2i Re(E)t/\1Im(E)t/\#uC_b(R)

&

. Be-cause Im(E),0, the amplitude of the state uF_b(R,t)

&

de-creases exponentially, due to spontaneous emission. In this section we also show that the magnitude of Im(E) can be made very small by choosing a sufficiently large red detun-ing.

Substituting Eqs. ~3! and ~11! into the time-independent Schro¨dinger equation gives a matrix equation for the radial amplitudes F_s,b(R): 2 \ 2 2m d2 dR2Fb~R!1 L2 2mR2Fb~R!1V~R!Fb~R!5EFb~R!, ~12! where F_b~R![@F1,b~R!F2,b~R!F3,b~R!F4,b~R!F5,b~R!F6,b~R!#t is a 631 column matrix ~the superscript t indicates matrix

transposing!, V(R) is the 636 complex potential matrix of Eq. ~9!, and L2 is a 636 diagonal matrix containing, along its diagonal, the sequence of eigenvalues of the square of the rotational angular-momentum operator, ordered according to

the basis set of Eqs.~1!: (0,6\2,0,6\2,6\2,6\2). Asymptoti-cally, Eq.~12! becomes \2d2F_b(R)/dR222mV_{`</sub>F<sub>b</sub>(R)50, where V<sub>`} is a 636 R-independent complex matrix defined as V_`[

3

2E 0 \V 0 0 0 0 2E 0 0 0 \V \V 0 2\Dc2E 0 0 0 0 0 0 2\Dc2E 0 0 0 0 0 0 2\Dc2E 0 0 \V 0 0 0 2\Dc2E

4

. ~13!

The matrix V_{`</sub>is not diagonal at very large interatomic sepa-ration, meaning that the coupling with the radiation is always present. Therefore, the undressed states, Eqs. ~1!, are not suitable for describing the asymptotic states necessary to de-fine the S matrix. In other words, in the actual physical situ-ation of a collision in the presence of photons, the incoming and outgoing scattering states are the stationary states of the asymptotic Hamiltonian. Accordingly, the normalized eigen-states of V<sub>`} ~the dressed states! are obtained by diagonaliz-ing Eq. ~13! and are given by

uD1

&

[a2u1

&

1b2u3

&

, ~14a!

uD2

&

[a2u2

&

1b2u6

&

, ~14b!

uD3

&

[a1u1

&

1b1u3

&

, ~14c!

uD4

&

[u4

&

, ~14d!

uD5

&

[u5

&

, ~14e!

uD6

&

[a1u2

&

1b1u6

&

, ~14f! with a₆and b₆complex quantities satisfyingua₆u21ub₆u2

51 and defined as a₂[ V

A

uz2u21V2 , b₂[ z2

A

uz2u21V2 , ~15a! a₁[2 V D1ig

A

D2_1g2 uz1u21V2 , b₁[2 z1 D1ig

A

D2_1g2 uz1u21V2 , ~15b! z6[2~D1ig₂ !6 1 2

F

A

uZu1Re~Z! 2 2i

A

uZu2Re~Z! 2

G

, ~15c! Z[~D1ig!214V25~D214V22g2!12igD. ~15d!

Although uD1

&

and uD3

&

are normalized and linearly inde-pendent, they are not orthogonal. The same occurs withuD2

&

and uD6

&

. It is also worth noticing that D,0 and uDu.g

.0, as we assume in the following.

It is easy to show that the quantities a₆5a₆(D,V,g) and b₆5b₆(D,V,g) satisfy the property a₆(lD,lV,lg)

5a6(D,V,g) and b6(lD,lV,lg)5b6(D,V,g) for all

real l.0. Hence, by choosing l51/uDu, we obtain a₆

5a6(21,V/uDu,g/uDu) and b65b6(21,V/uDu,g/uDu), that is, a₆ and b₆ can be viewed as functions of the two independent variables V/uDu and g/uDu. The sign of D is fixed, D,0, thus a₆ and b₆ are fully specified by giving

V/uDu andg/uDu. In this paper we are interested in the case of large red detunings, namely, uDu@g anduDu@V. In par-ticular, we can expand a₆and b₆up to the second order in

V/uDu to obtain a₂'12 V 2 2~D21g2!, b2' V D1ig, ~16a! a₁'2 V D1ig, b1'12 V2 2~D21g2!

F

11 4igD D2_1g2

G

. ~16b!

The scattering boundary conditions are more naturally stated in the dressed-state representation, Eqs.~14! and ~15!, than in the undressed, Eqs. ~1!, as we show below. There-fore, if we define the 631 column matrix G_b(R)

[M21_F

b(R), where M is the 636 complex matrix defined as M[

3

a₂ 0 a₁ 0 0 0 0 a₂ 0 0 0 a₁ b₂ 0 b₁ 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 b₂ 0 0 0 b₁

4

, ~17!

then Eq. ~11! is equivalently written in terms of the dressed states and the radial amplitude 631 column matrix G_b(R) as uCb~R!

&

5

(

s51 6 F_s,b~R! R us

&

5_s51

(

6 G_s,b~R! R uDs

&

, where G_s,b(R)5(_n516 ( M21)_s,nF_n,b(R). By left-multiplying Eq. ~12! by M21and noticing that V(R)F_b(R)

5V(R)M M21_F

b(R)5V(R)MGb(R), we obtain the matrix Schro¨dinger equation in the dressed-state representation:

d2 dR2Gb~R!2 2m \2 M 21_V `M Gb~R!2 L2 \2_R2Gb~R! 2U~R!Gb~R!50, ~18!

where we have added and subtracted M21V_{`</sub>M and intro-duced the 636 complex matrix U(R), satisfying lim<sub>R→`}U(R)50 and U(R'R_`)'0, and defined in terms of the potential matrix V(R) as

U~R![2m

\2 M

21_@V~R!21

6E2V`#M, ~19! with 16 representing the 636 identity matrix. Let us also notice that L2satisfies the property M21L2M5L2, since the radiation does not couple states of different rotational quan-tum numbers@32#.

The states uD₁

&

and uD₂

&

have degenerate eigenvalues

2E1\z2 of V`, the statesuD3

&

anduD6

&

have degenerate eigenvalues 2E1\z₁, and the statesuD₄

&

anduD₅

&

have degenerate eigenvalues2E2\Dc. In this article we assume that the kinetic energy E_k[\2k2/(2m).0 is much less than

2\D.0, since we are considering the regime of ultracold

temperatures and large red detunings. Asymptotically, at t

50, let us suppose we prepare the system in the state uDa

&

, fora51,2; the total Hamiltonian eigenvalue of the system, if the atoms are fixed at rest at position R_`, is equal to\z₂, because then (d2/dR2)G_a,b(R)50. Therefore, if we prepare the system in the state uD_a

&

, for a51,2, but now with a finite kinetic energy Ek, then (d2/dR2)Ga,b(R)5

22mE_k/\2_G

a,b(R) and E5Ek1\z25\2k2/(2m)1\z2. It follows from Eqs.~15c! and ~15d! that the imaginary part of the eigenvalue E is Im(\z₂),0 and that Im(\z₂)

van-ishes as uDu becomes much greater than V. Moreover, choosing a positive real kinetic energy is equivalent to im-posing oscillatory asymptotic solutions for G_a,b(R), that is, G_a,b(R)'c1exp@ikR#1c2exp@2ikR#, with c1 and c2 being independent of R for R'R_`. Now, for example, for r

53,6 and using E5Ek1\z₂, we have the asymptotic radial equation d2G_r,b(R)/dR22YG_r,b(R)50, with the defini-tion Y[2m(2Ek2\z21\z1)/\2. The function Gr,b(R) assumes the asymptotic behavior G_r,b(R)'d1exp@yrR

1iyiR#1d2exp@2yrR2iyiR#, where d1 and d2 are indepen-dent of R for R'R_`, y_r[

A

@uYu1Re(Y)#/2, and y_i[

2

A

@uYu2Re(Y)#/2 @

A

Y56(yr1iyi), since it follows from Eqs. ~15c! and ~15d! that Im(Y),0#. Because yr.0 we must impose d150 for Gr,b(R) to be finite asymptoti-cally; indeed, with d150, Gr,b(R) vanishes for R'R`. A similar reasoning implies that G<sub>r,b</sub>(R), forr54,5, also van-ishes for R'R<sub>`</sub>. Thus only the states uD1

&

and uD2

&

are accessible asymptotically, corresponding to open channels for the collision; the other states are therefore the closed channels in this problem, manifesting themselves only through resonances.

In the dressed-state representation the off-diagonal matrix elements of the Hamiltonian vanish asymptotically and we can apply the analysis of Ref.@16# generalized to the multi-channel case. The first step is then to distinguish between closed and open channels and break Eq.~18! into two matrix equations, one for the open-channel manifold and the other for the closed-channel manifold. As we have discussed in the previous paragraph, the open-channel manifold corresponds to the 232 block of the Hamiltonian comprising the sub-space spanned by the first two dressed statesuD₁

&

anduD₂

&

. The closed-channel manifold is spanned by the last four dressed states, corresponding to a 434 block of the Hamil-tonian. The open-channel 232 block is coupled to the closed-channel 434 block by the off-diagonal blocks: one 234 and the other 432. Thus, we can write the 636 ma-trix U(R) as a 232 matrix whose elements are also matri-ces:

U~R!5

F

Ugg~R! Uge~R!

Ueg~R! Uee~R!

G

, ~20!

where Ugg(R) is a 232 complex matrix, Uee(R) is 434,

Uge(R) is 234, and Ueg(R) is 432. Now we are able to give explicit expressions of the different blocks of the matrix

U(R), Eq. ~20!. From Eqs. ~9!, ~13!, ~17!, and ~19!, it

fol-lows that U_gg~R!5 2m G\2

F

a₂b₁W₀0~R!2a₁b₂Va~R! 2

A

5a₁b₂D~R!/15 2

A

5a₁b₂D~R!/15 a₂b₁W₀0~R!2a₁b₂V_c~R!

G

, ~21! Uge~R!5 2m G\2

F

a₁b₁@W₀0~R!2Va~R!# 2a₁D~R!/

A

15 2a₁D~R!/

A

15 2

A

5a₁b₁D~R!/15 2

A

5a₁b₁D~R!/15 2a₁

A

3D~R!/21 2a₁

A

3D~R!/21 a₁b₁@W₀0~R!2V_c~R!#

G

, ~22!

Ueg~R!5 2m G\2

F

2a₂b₂@W₀0~R!2Va~R!# 22

A

5a₂b₂D~R!/15 b₂D~R!/

A

15

A

3b₂D~R!/21 b₂D~R!/

A

15

A

3b₂D~R!/21 22

A

5a₂b₂D~R!/15 2a₂b₂@W₀0~R!2V_c~R!#

G

, ~23! and, finally, Uee~R!5 2m G\2

F

a₂b₁Va~R!2a1b2W0 0_~R! a₂D~R!/

A

15 a₂D~R!/

A

15 22

A

5a₂b₁D~R!/15 b₁D~R!/

A

15 Vb~R! 22D~R!/7

A

3b₁D~R!/21 b₁D~R!/

A

15 22D~R!/7 Vb~R!

A

3b₁D~R!/21 22

A

5a₂b₁D~R!/15

A

3a₂D~R!/21

A

3a₂D~R!/21 a₂b₁Vc~R!2a1b2W0 0 ~R!

G

, ~24!

whereG[a₂b₁2a₁b₂is the determinant of M . The 636 complex matrix U(R) is not symmetric, reflecting the fact that the Hamiltonian is not Hermitian to account for spontaneous emission.

Let us define the 231 column matrix Ggb(R)[@G1,b(R) G2,b(R)]t and the 431 column matrix Geb(R)[@G3,b(R)

G4,b(R) G5,b(R) G6,b(R)]t. Thus, it is straightforward to derive from Eqs.~18! and ~20! the following matrix equations:

d2 dR2Ggb~R!1

F

k 2₁ 22 L_gg2 \2_R22Ugg~R!

G

Ggb~R!5Uge~R!Geb~R!, ~25a! d2 dR2Geb~R!2

F

k 2₁ 41s1 L_ee2 \2_R21Uee~R!

G

Geb~R!5Ueg~R!Ggb~R!, ~25b!

where k2[2mEk/\2[2m(E2\z2)/\2,k2[22m@Ek1Re(\z22\z1)#/\2, 12 is the 232 identity matrix, Lgg 2

is a 232 diagonal matrix containing the sequence (0,6\2) along its diagonal, 14is the 434 identity matrix, s is a 434 diagonal matrix defined as s[22m \

F

iIm~z₂2z₁! 0 0 0 0 Re~z₁!1D1ig1iIm~z₂! 0 0 0 0 Re~z₁!1D1ig1iIm~z₂! 0 0 0 0 iIm~z₂2z₁!

G

, ~26! and Lee 2

is a 434 diagonal matrix containing the sequence (0,6\2,6\2,6\2) along its diagonal.

Equations~25! are in a form suitable for applying scattering boundary conditions, because Ugg(R), Uee(R), Uge(R), and

Ueg(R) vanish asymptotically and so does the physically acceptable solution for the matrix Geb(R), as we have discussed above. As R tends to R_`, it follows from Eq.~25a! that the asymptotic form of G_gb(R) can be written as@33#

G_a,b~R!'i

A

m 2p\2k

H

da,bexp

F

2i

S

kR2 pl_b 2

DG

2Sa,bexp

F

i

S

kR2 pl_a 2

DGJ

, ~27!

where a,b51,2, l1[0 (s wave! and l2[2 (d wave!, and

S_a,bare the elements of the scattering matrix. From elemen-tary collision theory the meaning of the indexbis now clear:

b51 corresponds to an s-wave in the entrance channel,

whileb52 corresponds to a d-wave. Analogously, the index

a specifies the exit-channel s- or d-wave. In the following sections we establish an algorithm to obtain the 232 S ma-trix for this problem.

IV. THE CLOSED-CHANNEL MANIFOLD

The 434 matrix Uee(R) appearing in Eq. ~25b! can be viewed as a function not only of R, but also of the param-eters a₆ and b₆ defined by Eqs. ~15!. Thus, we can write,

explicitly, Uee(R)[Uee(a2,b2,a1,b1,R). It is apparent from their definition and Eqs.~16! that a₂and b₁tend to 1, and b₂ and a₁ to 0 in the limit of vanishing V or infinite

uDu. Taking the case of zero intensity or infinite red detuning

as reference, we define the 434 matrices U0(R)[Uee(a2

51,b₂50,a₁50,b₁51,R) and U_p(R)[U_p(a₂,b₂, a₁,b₁,R)[Uee(a2,b2,a1,b1,R)2U0(R). Explicit ex-pressions for U0(R) and Up(R) are readily obtained from Eq.~24! and we notice that U0(R) is a real 434 matrix. The eigenvalues of U₀(R) are 2mW₀1(R)/\2, 2mW₁1(R)/\2 ~dou-bly degenerate!, and 2m@3W₀1(R)/714W₁1(R)/7#/\2. Let d0(R) be the 434 diagonal matrix obtained by diagonaliz-ing U0(R). It is easy to verify that there is a 434 orthogonal matrix A (A215At), independent of R, such that d0(R)

5At_U

0(R)A. An explicit form of A can be chosen as

A5

F

A

3/3 0

A

6/3 0 2

A

5/5

A

2/2

A

10/10

A

5/5 2

A

5/5 2

A

2/2

A

10/10

A

5/5 2

A

15/15 0 2

A

30/15

A

15/5

G

, ~28!

whose columns give orthonormal eigenvectors of U0(R). Hence, writing Uee(R)5U0(R)1Up(R) in Eq. ~25b! and left-multiplying the resulting equation by At, we obtain, after rearranging terms and defining G˜_eb(R)[AtG_eb(R),

d2 dR2G ˜_eb~R!2@k2₁ 41d0~R!#G˜eb~R! 5At_U eg~R!Ggb~R!1Atuee~R!Geb~R!, ~29! where, for convenience, we have defined the 434 complex matrix uee(R) as

uee~R![

L_ee2

\2_R21Up~R!1s. ~30!

Following the procedure of Ref. @16#, now we solve Eq.

~29! in terms of its Green function, using its right-hand side

as the source. Accordingly, let fn(R) be the 431 column matrix satisfying the homogeneous differential equation d2fn(R)/dR22d0(R)fn(R)5kn

2_f

n(R), wherekn 2

is a 434 diagonal matrix containing the sequence of positive numbers (k_n,32 ,k_n,42 ,k_n,52 ,k_n,62 ) along its diagonal ~in the closed-channel manifold the indices run from 3 to 6!. The index n

51,2,3, . . . enumerates the discrete spectrum of this

homo-geneous differential equation. Because d0(R) is real, each element of fn(R) can be chosen as a real function. The homogeneous problem has, besides the discrete, also a con-tinuous spectrum. However, as we have discussed in the pre-vious section, we are looking for solutions such that Geb(R) vanishes asymptotically and, therefore, we construct the cor-responding Green function in terms of the discrete eigen-functions only. Since the eigen-functions fn(R) for all natural numbers n form a complete set in the discrete spectrum of the homogeneous problem, the solution of Eq. ~29! can be expressed as G˜_n,b~R!5

(

n gn,n,b kn,n 2 _2k2fn,n~R!, ~31! where the indexn assumes the values from 3 to 6 to specify the 4 matrix elements of G˜eb(R) andfn(R), and the quan-tities gn,n,b involve the source terms on the right-hand side of Eq.~29!: gn,n,b5

E

0 R_` fn,n~R!@AtUeg~R!Ggb~R! 1At_u ee~R!Geb~R!#ndR. ~32! Equations ~31! and ~32! are easily verified by substitution into Eq. ~29! and assuming that fn,n(R) are properly

nor-malized. In the notation of Eq. ~32!, a symbol like @Q#_n stands for the element with index n of the 431 column matrix Q. Let us notice that k_n,42 5k_n,52 and fn,4(R)

5fn,5(R) for all n, since both pairs of quantities

@kn,4 2

,fn,4(R)# and @kn,5 2

,fn,5(R)# correspond to the same eigenvalue of U0(R), namely, 2mW1

1

(R)/\. Moreover, we also assume that there are values of n and n53,4,5,6 for whichk_n,2_n5fn,n(R)50. It is particularly true that a purely repulsive potential W₁1(R) does not support bound states and, in this case, surelyk_n,42 5k_n,52 5fn,4(R)5fn,5(R)50 for all

n;k_n,62 andfn,6(R) are different from zero for some n only if the eigenvalue 2m@3W0

1

(R)/714W1 1

(R)/7#/\2 of U0(R) supports bound states.

We assume that the light field is nearly resonant with an isolated bound state, say n5r, of the attractive W₀1(R) po-tential~of 3_S

g

1_{symmetry!. Thus,k}2_'k r,3

2 _{, sincen53 is the} index corresponding to the eigenstate of U₀(R) with eigen-value 2mW0

1

(R)/\2, as explained above. Now, G_r,b(R)5(_n536 A_r,nG˜_n,b(R) for r53,4,5,6 and, from Eq.

~31!, we can approximate Gr,b(R) by taking into account only the resonant term and neglecting the others:

G_r,b~R!'A_r,3 gr,3,b

kr,3

2 _2k2fr,3~R!. ~33!

Let us notice that in Eq.~28! all the elements A_r,3are differ-ent from zero, showing that the dominant contribution to G_r,b(R) is indeed given by Eq.~33! for allr53,4,5,6. This resonant approximation was checked numerically for the two-state problem of Ref.@20# and turned out to be excellent. After some algebraic manipulation of Eqs.~32! and ~33!, we obtain G_r,b~R!'A_r,3 Qr,3,b kr,3 2 _2k2_2w r,3 fr,3~R!, ~34! where Qr,3,b[

E

0 R_` fr,3~R!@AtUeg~R!Ggb~R!#3dR, ~35a! w_r,3[

E

0 R_` fr,3~R!@A t_u ee~R!A#3,3fr,3~R!dR. ~35b!

The notation@Atuee(R)A#3,3in Eq.~35b! stands for the entry of the 434 matrix Atuee(R)A with row and column indices both equal to 3.

V. THE CONTINUUM GROUND STATES AND THE S MATRIX

Having found a valid approximation to the solution of the closed-channel problem, Eqs. ~34! and ~35!, in this section we consider the open-channel manifold and the obtaining of the corresponding 232 scattering matrix. Similarly to U_ee(R), the 232 potential matrix U_gg(R), Eq. ~21!, is a function also of the parameters a₆ and b₆ of Eqs. ~15!, U_gg(R)[U_gg(a₂,b₂,a₁,b₁,R). Since we are interested

in finding the modification in the collision process caused by the light field, in the previous section we have assumed the case of a₂5b₁51 and b₂5a₁50 as reference. Accord-ingly, let us define the 232 matrices: Uref(R)[Ugg(a2

51,b₂50,a₁50,b₁51,R), that is diagonal, and

Um(R)[Um(a2,b2,a1,b1,R)[Ugg(a2,b2,a1,b1,R)

2Uref(R). It is easy to obtain explicit expressions of the matrices Uref(R) and Um(R) from Eq.~21!. With these defi-nitions, we rewrite Eq.~25a! as

d2 dR2Ggb~R!1

F

k 2₁ 22 Lgg 2 \2_R22Uref~R!

G

Ggb~R! 5Uge~R!Geb~R!1Um~R!Ggb~R!. ~36!

The solution of the homogeneous differential equation

speci-fied by making Uge(R)50 and Um(R)50 on the right-hand side of Eq.~36! gives the phase shifts for the entrance s and

d waves in the absence of radiation. Thus, let G_a0(R), a

51,2, be the real regular solution of the homogeneous

prob-lem satisfying the scattering boundary condition: G_a0(R)

'sin(kR1ha2pla/2) as R tends to R`. Here, l1[0 and l2

[2, andha[ha(k) is the phase shift of the la wave in the absence of the light field. Now let us take G_a1(R), the real irregular solution of the homogeneous problem that has the asymptotic behavior G_a1(R)'cos(kR1h_a2pl_a/2), and de-fine the complex function G_ac(R)[G_a1(R)/k1iG_a0(R)/k that behaves asymptotically as G_ac(R)'exp@i(kR1h_a

2pl_a/2)#/k. Therefore, imposing the scattering boundary conditions of Eq.~27!, it is straightforward to verify that the solution of Eq.~36! is @16# G_a,b~R!52

A

m 2p\2kda,bGa 0_~R!exp@ih a#1

E

0 R_` Ga~R,R

8

!@Uge~R

8

!Geb~R

8

!1Um~R

8

!Ggb~R

8

!#adR

8

, ~37!

whereG_a(R,R

8

) is the Green function given byG_a(R,R

8

)[2G_a0(R)G_ac(R

8

)u(R

8

2R)2G_ac(R)G0_a(R

8

)u(R2R

8

), with the step functionu(x2x

8

)51 if x.x

8

andu(x2x

8

)50 otherwise, for all real x and x

8

. From imposing the scattering boundary conditions and using Eqs.~34! and ~35a!, it also follows that

S_a,bexp@2ih_a#5d_a,bexp@ih_a#2i

A

2p\

2

mk Ia,b, ~38!

where the quantity I_a,b is given by

I_a,b5 Lr,3,a kr,3 2 ₂ k2_2w r,3

E

0 R_` fr,3~R!@AtUeg~R!Ggb~R!#3dR1

E

0 R_` G_a0~R!@Um~R!Ggb~R!#adR, ~39!

and we have defined Lr,3,a as

Lr,3,a[

E

0

R_`

G_a0~R!@Uge~R!A#a,3fr,3~R!dR. ~40!

The parameters b₂and a₁, according to Eqs.~16!, are approximately b₂'V/(D1ig) and a₁'2V/(D1ig) for large red detunings (V/uDu!1). Using Eqs. ~16!, ~21!, ~22!, and ~23!, it follows that Ueg(R) and Uge(R) are proportional toV/uDu and Um(R) is proportional toV2/D2, in the limit of large red detunings. From Eqs.~39! and ~40! it then follows that Ia,b is proportional to V2/D2. Hence, we can iterate Eq.~37! to obtain a perturbation series for Ggb(R) in the parameterV2/D2, using also Eqs.~34! and ~35a!. Substituting this estimate of Ggb(R) into Eqs.~38! and ~39! gives the approximated scattering matrix. In this paper we illustrate this procedure up to the second order inV/uDu. In this case, Eqs. ~37! and ~39! give

I_a,b5

A

2m p\2_k

F

Lr,3,aL˜r,3,b kr,3 2 _2k2_2w r,3 1

E

0 R_` G_a0~R!@Um~R!#a,bG_b 0 ~R!dR

G

exp@ih_b# , ~41!

where we have defined

L ˜ r,3,b[

E

0 R_` G_b0~R!@AtUeg~R!#3,bfr,3~R!dR. ~42!

Using Eqs.~38! and ~41! gives the S matrix elements:

S_a,b5

F

d_a,b2 2iLr,3,aL ˜ r,3,b k~k_r,32 2k22wr,3! 22i_k

E

0 R_` G_a0~R!@Um~R!#a,bG_b 0 ~R!dR

G

exp@i~h_a1h_b!#. ~43!

Using Eqs. ~10!, ~16!, ~21!–~24!, ~26!, ~28!, ~30!, ~35b!, ~40!, and ~42!, we can rewrite the matrix elements of Eq. ~43! explicitly as S_1,15

H

12 2piV 2 ~D1ig!2

F

@Qr,3,1#2 3@~Ek2\Dr!1i\g#

2Pa

G

J

exp@2ih1#, ~44a!

S_1,25S_2,15 4piV 2 3

A

5~D1ig!2

F

PD2 Qr,3,1Qr,3,2 ~Ek2\Dr!1i\g

G

exp@i~h₁1h₂!#, ~44b! S_2,25

H

12 2piV 2 ~D1ig!2

F

4@Qr,3,2#2 15@~Ek2\Dr!1i\g# 2Pc

G

J

exp@2ih2#, ~44c!

where we have defined the quantities

Qr,3,a[

A

2m pk\2

E

0 R_` G_a0~R!@W₀1~R!2W₀0~R!#fr,3~R!dR, ~45a! Pa[ 2m pk\2

E

0 R_` G₁0~R!@W₀0~R!2Va~R!#G1 0 ~R!dR, ~45b! PD[ 2m pk\2

E

0 R_` G₁0~R!@W₁1~R!2W₀1~R!#G₂0~R!dR, ~45c! Pc[ 2m pk\2

E

0 R_` G₂0~R!@W₀0~R!2Vc~R!#G2 0_{~R!dR, ~45d!} e[ \ 2 2m

E

0 R_` fr,3~R! 1 R2fr,3~R!dR, ~45e! Er[ \2_k r,3 2

2m , and \Dr[2\D2Er14e. ~45f!

Equations ~44! are valid up to the second order in V/uDu. These S matrix elements contain information relevant in the context of photoassociation line shapes and modification of scattering lengths. If g50, this theory is Hermitian and the S matrix is unitary, S†5S21, meaning that the total probability is conserved. Thus, in the case ofgÞ0 the total probability is not conserved and this loss is proportional to the amount of probability flux going into spontaneous emission from the bound state, that is, the total probability loss is proportional to the photoassociation line shape. Then we can obtain the line shape for an s wave in the entrance channel by taking the difference Js[12uS1,1u22uS1,2u2, and, for a d wave, we define Jd[1

2uS2,2u22uS1,2u2. From Eqs. ~44! it follows that, up to the second order in V/uDu,

Js' 8pgV2 ~D2_1g2_!2

F

@Qr,3,1#2@2D~Ek2\Dr!1\~D22g2!# 6@~E_k2\D_r!2_1~\g!2_# 2PaD

G

, ~46a! Jd' 8pgV2 ~D2_1g2_!2

F

2@Qr,3,2#2@2D~Ek2\Dr!1\~D22g2!# 15@~Ek2\Dr!21~\g!2# 2PcD

G

. ~46b!

Let us notice that these line shapes are asymmetric and require the knowledge of the k-dependent integrals of Eqs.~45!. We do not attempt to calculate these integrals in the present article, because here we focus only on the theoretical procedure to obtain the line shapes of Eqs.~46!. In a forthcoming paper we will calculate the quantities @Qr,3,1#2, Pa, @Qr,3,2#2, and Pc numerically, and present a comparison of the present theory with a numerical close-coupling calculation, to check the accuracy of this semianalytic approach. In a forthcoming paper we will also investigate the dependence of the collision process on the polarization of the light field. The line shapes of Eqs.~46!, due to their physical relevance, will be granted full attention in this separate paper. Therefore, it is not our purpose here to analyze thoroughly the line shapes above, but it is important to mention that they show the Wigner-threshold-law behavior for low entrance kinetic energies, since in this case the quantities@Qr,3,1#2 and@Qr,3,2#2 are proportional to k2l11 @20#.

For low enough entrance kinetic energies, the dominant contribution to the collision process comes from the s wave alone and we can show that in this caseh1'2kA0, where A0is an energy-independent parameter called the scattering length. In the presence of light, the scattering length is modified according to Eq.~44a!. Thus, by writing S1,1[exp@22ikA#'122ikA, we obtain A'A01 pV2 k~D21g2!2

F

@Qr,3,1#2@~D22g2!~Ek2\Dr!22\Dg2# 3@~Ek2\Dr!21~\g!2# 2~D2_2g2_!P a

G

2 2ipgV 2 k~D21g2!2

F

@Qr,3,1#2@2D~Ek2\Dr!1\~D22g2!# 6@~Ek2\Dr!21~\g!2# 2PaD

G

, ~47!

where A is the modified scattering length that now acquires an imaginary part, reflecting the absorption into an unstable bound-state resonance. We notice also that if g50 the imaginary part vanishes, as we would expect. The quantities

@Qr,3,1# 2 _{and P}

a are proportional to k at low enough kinetic energies, and this cancels the k in the denominator of the second and third terms on the right-hand side of Eq.~47!. As we have mentioned above, in a forthcoming paper we will concentrate on the line shapes of photoassociation. We will also focus on a numerical investigation of the modification of the scattering length by light, given the importance of this subject for experimental purposes.

The present article is to be viewed as a discussion of multichannel predictions expressed by Eqs. ~46! and ~47!. It is worth remarking that these results are consequences of the solution of the Schro¨dinger equation using approaches to dif-ferent stages of the theory already considered usual in the literature. However, not all of these usual approaches are incorporated into a single unified treatment, as we have pre-sented here. The importance of the present procedure resides in this unification of techniques and the possibility of its extension to treat more realistic situations, as the introduc-tion of fine structure, for example. Also, if the magnitude of the detuning is greater than the Rabi frequency, the pertur-bative iteration of Eqs. ~37!–~39! can be easily extended to higher orders in the perturbation parameter, V/uDu, as long as the power broadening is much less than the separation of the adjacent bound-state resonances, that is, as long as each resonance remains isolated.

VI. VALIDITY OF THE THEORY

Our first approximation involves the neglect of fine and hyperfine structures; also the magnitude of the red detuning is assumed to be very large as compared with the Rabi fre-quency. If we choose D/(2pc)'21.0 cm21, we satisfy both requirements: fine and hyperfine structures are small compared with the magnitude of the attractive potential and a Rabi frequency corresponding to 1 cm21implies an intensity of I'23105 W/cm2, meaning that V/uDu!1 for any rea-sonable value of laser intensity. Adjacent 3_S

g

1 _resonances for D/(2pc)'21.0 cm21 are separated by about 0.14 cm21. Thus, imposing V/(2pc)!0.14 cm21 implies that the intensity must satisfy I!4.53103 W/cm2. Such an in-tensity is achievable experimentally, and in this circumstance power broadening is important, requiring changing the present theory to include effects of more than one resonance

being excited. This generalization to overlapping resonances is not impossible to accomplish, but we can still apply this isolated-resonance theory for red detunings of magnitudes satisfyinguDu/(2pc)*10.0 cm21, in which case the separa-tion between adjacent resonances is about 1.0 cm21, imply-ing the condition I!23105 W/cm2.

A typical situation in a trap corresponds to temperatures from a few tens to a few hundreds ofmK, and densities from about 1010to 1014cm23. Let us fix the density to about 1011 cm23. Then we estimate a mean interatomic distance of about 43104a0, where a0 is the Bohr radius. The longest-range Born-Oppenheimer potential curves appearing in Eq.

~9!, for example, are the ones corresponding asymptotically

to one ground-state 2S atom and one first-excited-state 2P atom. These curves behave asymptotically as C₃/R3_{, where}

C3 is on the order of a few atomic units @34#. Thus, the

R-dependent part of the potential matrix is about 331029

cm21. Even at the low laser intensity of 1 W/cm2, the Rabi frequency is about 231023cm21, or about 105_{times greater} than the magnitude of the R-dependent part of the potential matrix, and uDu/(2pc)'1.0 cm21 is 108 _{times greater. The} centrifugal terms appearing in Eq.~12!, for instance, are ei-ther zero or 3\2/(mR2). At R_{`</sub>'43104a0, 3\2/(hcmR<sub>`}

2 )

'231028 _cm21_{. Thus, we can safely choose R}

`

'43104_a

0 as the asymptotic region of this theory. It now remains to determine the minimum asymptotic ki-netic energy for which this theory is valid. We mentioned at the beginning of Sec. III that the stationary states of the total Hamiltonian have an exponentially decreasing amplitude due to loss through spontaneous emission. Accordingly, for a colliding system to survive during the whole process, this decreasing amplitude must remain significant from the start to the end of the scattering encounter. Letv be the initial and final relative speeds of the colliding partners~in this problem the open channels correspond to the same asymptotic kinetic energy!. Thus, the time taken for the atoms to move from a distance about R_{`</sub>to their encounter region is approximately R<sub>`}/v, and from their breaking apart to their reaching the asymptotic region again is on the same order, R_{`</sub>/v. The total duration of the collision process is about 2R<sub>`}/v. It follows from Sec. III that the amplitude of the asymptotic stationary states decays as exp@Im(\z₂)t#, with Im(\z₂)

,0. At the end of the collision, this factor gives

exp@2 Im(\z₂)R_{`</sub>/v# and we impose 2uIm(\z<sub>2</sub>)uR<sub>`}/v!1, orv@2uIm(\z₂)uR_`. Up to the second order inV/uDu, this condition becomes

v@2gR_{`</sub>V 2 D2. ~48! Using g/(2p)'10 MHz, uDu/(2pc)'1.0 cm21, R<sub>`}'4 3104_a 0, and V/(2pc)'1.4796631023

A

2I cm21, we ob-tainv/I@0.12 cm/s, where the intensity I is given in W/cm2_. Using the reduced mass of sodium, m'1.91310223g, and defining the collision temperature as T[mv2/(3kB), where

kB is Boltzmann’s constant, according to Eq. ~48! T must satisfy the inequality

T

I @6.6310

210 _K,

a condition well satisfied in current experiments.

VII. CONCLUSION

In this article we have developed a non-Hermitian multi-channel procedure to treat binary collisions of ultracold at-oms in the presence of a red-detuned laser field. We have obtained the scattering matrix elements up to the second or-der in the ratio between the Rabi frequency and the detuning, and showed that this approach is valid for a significant range of experimental conditions. The perturbation series can be easily extended to higher orders. As practical results of these calculations, we have presented the multichannel line shapes of photoassociation spectroscopy for the s and d wave in the entrance channel, and the light-modified scattering length for low but finite entrance kinetic energies. Although we have

truncated the partial waves at the l52 manifold, the inclu-sion of higher partial waves is straightforward, at least for this case with no fine or hyperfine structure.

In future work, we will be considering the line shapes of Eqs.~46! and the modification of scattering lengths, Eq. ~47!, in greater detail. We will investigate numerically these quan-tities for conditions relevant to actual experiments. The dif-ferences between linear and circular polarization effects will be examined and we expect them to be appreciable, as in the case of blue detuning @10#, due to the inherently different topologies of the respective sets of adiabatic potential curves. Guided by the present model of ultracold collisions, we will develop a fully numerical close-coupling calculation to treat the scattering exactly. The exact numerical results will deter-mine the boundaries of the parameter space within which this semianalytic theory is valid. We expect that these investiga-tions will be relevant to the interpretation of many experi-mental results yet to appear.

ACKNOWLEDGMENTS

This work was supported by FAPESP ~Fundac¸a˜o de Am-paro a` Pesquisa do Estado de Sa˜o Paulo!. The author is very

grateful to Vanderlei S. Bagnato, Paul S. Julienne, and John Weiner for their kind hospitality and fruitful discussions. The author is also indebted to Jayme De Luca for a careful read-ing of the manuscript and helpful suggestions. This work has also been partially supported by Conveˆnio FINEP/PRONEX Grant No. 41/96/0935/00.

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