Scattering and bound states in two-dimensional anisotropic potentials

0.42 MB·English

by  Matthias Rosenkranz

Scatteringandboundstatesintwo-dimensionalanisotropicpotentials
Matthias Rosenkranz and Weizhu Bao
Department of Mathematics, National University of Singapore, 119076, Singapore
(Dated:January31,2012)
Weproposeaframeworkforcalculatingscatteringandboundstatepropertiesinanisotropictwo-dimensional
potentials. Usingourmethod, wederivesystematicapproximationsofpartialwavephaseshiftsandbinding
energies.Moreover,themethodissuitableforefficientnumericalcomputations.Wecalculatethes-wavephase
shift and binding energy of polar molecules in two layers polarized by an external field along an arbitrary
direction. Wefindthatscatteringdependsstronglyontheirpolarizationdirectionandthatabsoluteinterlayer
bindingenergiesarelargerthanthermalenergiesattypicalultracoldtemperatures.
2
1
PACSnumbers:03.65.Nk,34.50.-s,67.85.-d
0
2

n I. INTRODUCTION is deeper than thermal energies in typical ultracold experi-
a mentsoverarangeofpolarizationangles.
J
Anisotropic interactions are at the heart of many physical We consider two particles interacting via a 2D poten-
0 systems,suchastheatomicnucleus,heteronuclearmolecules, tial V(r). In the center-of-mass frame the scattered state
3
or ultracold dipolar atoms. The anisotropy has a marked in- depends on both the relative vector r = r(cosϕ,sinϕ) be-

] fluences on their scattering and bound state properties. Ad- tween the two particles and the incoming wave vector k =
s ditionally, confining a system to layers often leads to exotic k(cosξ,sinξ). Consequently, we expand the dimensionless
a
effectsandisthoughttobeacrucialingredientfortheelusive wave function in partial waves for both vectors: Ψ(k,r) =
g
nt- theHoreyteorofnhuigclhe-aTrcmsuopleerccuolnesduacrteivaitpyr[i1m]e. candidate for study- r21aπd∑ia∞ml,Sn=c−hr∞o¨dψimn√ng(rke,rr)eeqimuϕaeti−oinnξo.fInthseerttwinogpthairstiecxlepsanressiounltsinitnothe
a inglow-energyanisotropicscatteringinacontrolledmanner.
u Notably,theypossessalargepermanentdipolemoment. Fur- (cid:20) m2 1/4 (cid:21)
.q ther, several species have now been cooled to ultracold tem- ∂2r− −r2 +k2 ψmn(k,r)=∑Vmm(cid:48)(r)ψm(cid:48)n(k,r). (1)
t peratures [2]. These conditions give rise to an anisotropic m
a (cid:48)
dipole-dipole interaction (DDI), which can be controlled by
m (cid:113)
external fields [3]. For example, an electric field polarizes Here, k = E2µ∆2/h¯2 is the collision momentum at en-
-
d the molecules. However, if the DDI dominates the three- ergy E, µ is the reduced mass, and ∆ is the unit of
n dimensional (3D) dynamics, the gas becomes unstable [4].
length. The matrix elements of the potential are V (r)=
o Confiningthemoleculestoatwo-dimensional(2D)layersta- mm(cid:48)
c bilizesthegasforperpendicularpolarization[5]. Perpendicu- µπ∆h¯22(cid:82)02πdϕeiϕ(m−m(cid:48))V(r).
[
larlypolarizedmoleculesintwoparallellayersgiverisetoin- Different physical solutions of Eq. (1), such as scattering
1 terlayerboundstatesandscatteringacrosslayers[6–8]. Vary- orboundstates,fulfilldifferentboundaryconditions. Forour
v ing the polarization direction with an external field controls proposedformalism,weexpandallphysicalstatesψmninaset
7 the anisotropy of their interaction. However, only a very re- ofregularsolutionsofEq.(1),φmn,withwell-definedbound-
6 centworkhasstudiedtheinfluenceofthisDDIanisotropyon aryconditions. Specifically,werequirethattheseregularso-
1
.6 bouInndthsitsatpeasp[e9r],.we propose a general framework for calcu- Hluetiroen,swbeehhaavveeadtethfieneodrigthineasscalilmedr→B0eφsmsenl(kf,urn)c/tijon(nksr)jm=(xδ)m=n.
1 latingscatteringphaseshiftsandbindingenergiesforawide (cid:112)πx/2J (x)andy (x)=(cid:112)πx/2Y (x),whereJ andY are
m m m m m
0
classof2Dpotentialsincludinganisotropicpotentialsandthe Besselfunctionsofthefirstandsecondkind,respectively.For
2 specialcaseofvanishingpotentialvolume(cid:82)d2rV(r)=0.Our 2D scattering, these scaled Bessel functions are the equiva-
1
: methodandcalculationextendtheresultsinacentral2Dpo- lentoftheRiccati-Besselfunctionsfamiliarfrom3Dscatter-
v
tential by Klawunn et al. [6] to the anisotropic case. Bound ing[12]. Ourchoiceoftheboundaryconditionfixesthefree-
i
X states and scattering in other classes of 2D potentials have domintheexpansioncoefficientsoftheregularsolutionand
been studied in the past [10, 11]. On the one hand, our for- itsderivative[13].
r
a malism allows for efficient numerical computations in 2D as We assume lim r2V (r)=0 and lim rV (r)=0.
r 0 mn r ∞ mn
itonlyrequiressolvingafirst-orderdifferentialequation. On Becauseofthelatte→r,atlargedistancessolutio→nsofEq.(1)are
theotherhand,wepresentsystematicapproximationsforscat- proportionaltothescaledHankelfunctionsh (x)= j (x)
±m m
tering phase shifts and binding energy in anisotropic 2D po- iy (x). Thismotivatesustoexpandtheregularsolutionsas±
m
tentials. Specifically, werecovertheBornapproximationfor
thephaseshiftsandanapproximationconnectingthebinding 1
φ (k,r)= [h+(kr)f+(k,r)+h (kr)f (k,r)]. (2)
energy and the total phase shift at low energies. As an ex- mn 2 m mn −m m−n
ample,wecalculatetheinterlayerbindingenergyands-wave
phase shift of polar molecules in a bilayer at arbitrary polar- We insert this expansion into Eq. (1) and require that
ization.Theinterlayerbinding,e.g.,oftwo6Li40Kmolecules, h+(kr)∂ f+(k,r)+h (kr)∂ f (k,r)=0. Thisconditionre-
m r mn −m r m−n
2
ducestheSchro¨dingerequationtothefirstorderequation wefindthebindingenergy
∂rfm±n(k,r)=±h∓mi(kkr)∑Vmm(cid:48)(r)φm(cid:48)n(k,r) (3) Eb=−4e−2γ−21−AB. (6)
m
(cid:48) We obtain the power series A=∑∞ A(j) and B=∑∞ B(j)
j=1 j=1
forthecoefficients fm±n. TheformalsolutionofEq.(3)is by expanding φmn(r) into a power series of the potential
strength V , where j indicates the power of V . For a gen-
0 0
1 (cid:90) r eral (possibly anisotropic) 2D potential the leading terms of
f (k,r)=δ drh+(kr)∑V (r)φ (k,r).
m±n mn±ik 0 (cid:48) m (cid:48) m mm(cid:48) (cid:48) m(cid:48)n (cid:48) theseseries,includingallpartialwaves,aregivenby
(cid:48)
(4)
(cid:90) ∞
Here,wehavefixedtheintegrationconstantssothat f (k,∞) A(1)= drrV (r), (7)
m±n 00
exhibitsthecorrecthigh-energybehavior[13]. − 0
(cid:90) ∞ (cid:90) r (cid:18)r (cid:19)
A(2)= drrV (r) drr ln (cid:48) V (r) (8)
00 (cid:48) (cid:48) 00 (cid:48)
r
0 0
II. BOUNDSTATES 1 (cid:90) ∞ (cid:90) ∞
∑ drrV (r) drrV (r)
00 mm
−m=02m 0 0
A weak 2D potential supports bound states if (cid:82)d2rV(r) (cid:54) (cid:90) ∞
0 [11]. Using the coefficients fm±n we locate bound state≤s + ∑ dr√rV0m(r)um(10)(r),
in the following way. For Im(k)>0 [Im(k)<0] f (k,r) m=0 0
m−n (cid:54)
[f+(k,r)] converges as r ∞ because the right-hand side (cid:90) ∞ 1 (cid:90) ∞
ofmEnq.(3)vanishessufficien→tlyquicklyfortheassumedlong- B(1)= 0 drrln(r)V00(r)−m∑=02m 0 drrVmm(r), (9)
rangebehaviorofthepotential[13]. Thefunctions f (k)= (cid:54)
m±n (cid:90) ∞ (cid:90) r (cid:18)r (cid:19)
lgiemnre→ra∞lfsm±cna(ttke,rri)n|gImt(hk)e<>o0ryar[e12t,he13J]o.stInfutrnocdtuiocninsgfathmeilmiaratfrriocems B(2)=− 0 drrln(r)V00(r) 0 dr(cid:48)r(cid:48)ln r(cid:48) V00(r(cid:48)) (10)
F (k)=[f (k)]welocateboundstatesbyfindingmomenta 1 (cid:90) ∞ (cid:90) ∞
± m±n + ∑ drrln(r)V (r) drrV (r)
00 mm
kb=i|kb|onthepositiveimaginaryaxiswithvanishingdeter- m=02m 0 0
minant,i.e., (cid:54) (cid:90) ∞
∑ dr√rln(r)V (r)u(1)(r)
det[F−(kb)]=0⇔∑n fm−n(kb)cn(kb)=0. (5) −m(cid:54)=0 10 (cid:90) ∞ 0m (cid:90)m0r (cid:18)r (cid:19)
+ ∑ drr1 mV (r) drr1+mln (cid:48) V (r)
− m0 (cid:48) (cid:48) 0m (cid:48)
2m r
IffthedeterminantoftheJostmatrixF− vanishes,thenthere m=0 0 0
(cid:54)
ehxainsdtssaidneonovfanEiqs.hi(n5g).seMtoofrceooevfefir,ciaetntlsarcgnefudlifisltlainncgetshethreigshot-- + ∑ 1 (cid:90) ∞drr1/2−m[Vmm(r)wm(r)+ ∑Vmn(r)u(n1m)(r)].
2m
lutions um(kb,r) = ∑nφmn(kb,r)cn(kb) vanish exponentially m(cid:54)=0 0 nn=(cid:54)=m0
because um(kb,r)=O[h+m(kbr)]=O[e−|kb|r] 0. There- (cid:54)
−r−−→∞
fore, um(kb,r) describes a bound state. Its bin→ding energy is Here, un(1m)(r) = (cid:82)0rdr(cid:48)r(cid:48)1/2+mgn(r,r(cid:48))Vnm(r(cid:48)) +
Eb=−|kb|2inunitsofh¯2/2µ∆2. (1/2n)r1/2+n(cid:82)0∞dr(cid:48)r(cid:48)1+m−nVnm(r(cid:48)) and
Now we are going to derive an approximate expres- w (r) = (cid:82)rdrr1/2+mg (r,r)V (r)
spiootnentfioarl tVhe(r)bi=ndinVg0Ve(rn)e,rgywhinerea Vw0eakchaarnaicstoetrriozpeisc 2thDe s∑ymns(cid:54)=te0m,ma(1ti/c2alnl)yrh1/i2g+hmer(cid:82)o0∞rdder0r(cid:48)rt(cid:48)eVrn(cid:48)mn(cid:48)(srb(cid:48))y.exmpaWndei(cid:48)ngcmdamnet[F(cid:48)cal(cku)l]at−toe
strength of the potential. First, we introduce the ex- −
higher orders in V . Very recently, Volosniev et al. [9] also
plicit expression for the regular solutions φ (k,r) = 0
jn(kr)δmn − (cid:82)0rdr(cid:48)gm(k,r,r(cid:48))∑m(cid:48)Vmm(cid:48)(r)φm(cid:48)n(k,r(cid:48))m,n where f2oDunSdchsuro¨cdhinagneerxepqaunastiioonnodfirtehcetlbyinfodrinbgoeunnedrgstyatbeys.solvingthe
g (k,r,r) = k 1[j (kr)y (kr) j (kr)y (kr)] is the
m (cid:48) − m m (cid:48) − m (cid:48) m Forconcretenessletusnowconsidertwopolarizeddipoles
free Green’s function of Eq. (1) [14]. Furthermore, the
trapped in two parallel layers separated by a distance ∆. We
regular solutions of the Schro¨dinger Eq. (1) for k = 0 are
define the z axis to be perpendicular and the x-y plane to be
φmn(r) = rn+1/2δmn − (cid:82)0rdr(cid:48)gm(r,r(cid:48))∑m(cid:48)Vmm(cid:48)(r)φm(cid:48)n(r(cid:48)), parallel to the layers. Without loss of generality, we assume
where g0(r,r(cid:48)) = √rr(cid:48)ln(r(cid:48)/r) and gm=0(r,r(cid:48)) = thatthedipolesarepolarizedwithinthex-zplaneatanangle
(√rr(cid:48)/2m)[(r(cid:48)/r)m−(r/r(cid:48))m]arecorrespondingf(cid:54)reeGreen’s ϑfromthezaxis. Theyinteractviatheinterlayerpotential
functions. Now √π(k/2)n+1/2φ (r)/n! is the expansion of
mn
φ (k,r)toleadingorderink. InEq.(4)wereplaceφ (k,r) r2+1 3(rcosϕsinϑ+cosϑ)2
anmdn the scaled Hankel function by their respective mlenading V(r)=V0 − (r2+1)5/2 . (11)
order terms. Then we insert the asymptotic form of this ap-
proximationfor f (k)intotheleft-handsideofEq.(5). The Here, r=r(cosϕ,sinϕ) is the dimensionless projected vec-
m−n
resultisdet[F (k)]=A[ln(k)+γ ln(2) iπ/2]+1 B=0, tor between the dipoles in polar coordinates (in units of ∆)
−
where γ is the Euler-Mascheroni−constan−t. Solving t−his for k andV =µd2/2πh¯2ε ∆ is the interaction strength in units of
0 0
3
(a) (b) axisasthepolarizationdirectionchangesfromperpendicular
)
2 1 toparallel[cf. insetsinFig.1(a)]. Thisisbecausetheattrac-
Dµ
2 tive dipole term in the DDI dominates over the quadrupole
/
2of¯h 10−2 tneartmes. aHndowtwevoerp,efaokrsϑde>ve0lo.4p7.πF,othrepaqruaaldlerluppoolleartiezramtiodno,mthie-
s
E(unitb||1100−−046.0 0.1 0.2 VVV0000.===313100.4 0.5 0 2 4 6 JJJ 8===0Jp2c10 dIifcnuiaplfilFollyillgee.odtne1,r(tmtahh)iesvthacoenordiudselhidrsetossaifnmacµnpedmlibft.ehyteIwtfhtewetehoncereptshaeteataisbkoeisnliptboyeefacrkioensmqteuiesrilras(cid:39)eyymem5er∆mnbt,eosttuyraniprcde-.
J /p (rad) V0(unitsofh¯2/2µD 2) statesatparallelpolarizationbecauseoftheirgreateroverlap
than states at perpendicular polarization. The parallel polar-
FIG. 1. (Color online) Binding energy of lowest interlayer bound izationboundstatecanbedistinguishedfromthemainlyper-
states in the bilayer dipolar potential Eq. (11) as a function of (a) pendicularoneinatimeofflightmeasurement. Thetime-of-
the polarization angle from the symmetry axis and (b) interaction flight image reflects the double peak in an asymmetric mo-
strengthV0. The symbols mark numerical results, the dashed line mentumdistribution.
anapproximationinRef.[11]forV0=1,andsolidlinesthecorre-
spondingapproximationEqs.(6)-(10). Thelatterapproximationre-
mainsvaliduptomoderateV aslongas E 1. Theinsetsshow
0 b
| |(cid:28)
densitiesoftheboundstatesattheindicatedpolarizationanglesand III. SCATTERING
V0=10.Thedarkcirclesindicatetheradiusofonelayerdistance∆.
Next we focus on the 2D scattering problem, for which
h¯2/2µ∆2,withε0theelectricconstantanddtheelectricdipole k is real. Then the functions fm±n(k,r) attain the finite
moment(formagneticdipolesV0=µµ0µ2d/2πh¯2∆withµ0the loifmEitq.fm±(3n()kv)a=nislhimesr→a∞t rfm±n(k∞,r.)Ibneocarduesrettohecarpigtuhrte-htahnedmsiidxe-
magneticconstantandµ themagneticdipolemoment). The
d →
potentialfulfillsA(1)=0and(cid:82)d2rV(r)=0sothatatleastone ing of different partial waves, it is necessary to calculate the
full S-matrix. We express the partial wave components of
boundstateexistsforallpolarizationangles. Forϑ=0, this
the scattering solution as a linear combination ψ (k,r) =
potentialreducestothecentralcasediscussedinRefs.[6,7]. mn
Using Eq. (6) and expanding A and B to fourth and second ∑m φmm(k,r)cmn(k)oftheregularsolutions,Eq.(2).Asymp-
(cid:48) (cid:48) (cid:48)
totically,wereplace f (k,r)inφ (k,r)withtheJostfunc-
order,respectively,werecoverthebindingenergyforperpen- m±m(cid:48) mm(cid:48)
dicularpolarizationgiveninRef.[7]. tions fm±m(k). Ontheotherhand,thegeneralasymptoticscat-
In Fig. 1 we plot the binding energy of the lowest lying teringwav(cid:48) efunctionin2DisΨ(k,r) [eikr+a(k,r)eikr]/2π,
interlayer bound states in the potential (11) as a function of → · √r
where a(k,r) is the scattering amplitude. We match the two
the polarization angle and interaction strength. Their bind-
asymptotic expressions for the scattering wave functions by
ing is strongest for perpendicular polarization and weakest
expanding the first exponential in Ψ(k,r) in terms of scaled
for parallel polarization. We observe that the weak poten-
Hankelfunctions. Thisway,weextracttheS-matrix
tial approximation for the binding energy, Eqs. (6)-(10), re-
mainsvaliduptomoderatepotentialstrengthsV (cid:46)3aslong
0 S(k)=F+(k)[F (k)] 1 (12)
asthebindingenergyissufficientlysmall, E 1. Incon- − −
b
| |(cid:28)
trast, the approximation in Ref. [11], E exp(1/c) with
b
c=(1/8π2)(cid:82)d2rd2r(cid:48)V(r)ln r r(cid:48)V(r(cid:48))|, d|e∼scribes the angle andthecoefficientscmn(k)=[F−(k)]−mn1. IfthepotentialV(r)
dependence of the binding e|ne−rgy|only poorly. Our numer- is central, S(k) is diagonal, with elements e2iδm(k) and δm(k)
ical computations are based on Netlib’s ZVODE solver, and them-thpartialwavephaseshift.
we provide an explicit Jacobian for improved stability. We Let us now derive approximate expressions for
includepartialwavesuptosixthorder. the phase shifts of anisotropic 2D scattering at low
Asanexample,weconsiderbosonic6Li40Kmoleculessep- energies. First we introduce an iterative solution
ha¯r2a/t2eµd∆b2y(cid:39)∆1=.2µ2K00(numni.tsTofheknB)thaendeVne0r(cid:39)gy9s.5ca(lcef.inV0F=ig.110iins faorpowtheer sceorieefsficiienntsthe fmp±no(tken,rti)al=str∑en∞jg=t0hfm±Vn(0j).(k,r)Froams
Fig.1),withkB theBoltzmannconstant. Therefore,theinter- Eq. (4) and the expression for φmn(k,r) we obtain
lraaynegreboofupnodlastraizteatoiofnLiaKngmleosleucnudleerstsyhpoiucladlpeexrpseisritmovenertaalwteimde- fm±n(j+1)(k,r) = ±i1k(cid:82)0rdr(cid:48)h∓m(kr(cid:48))∑m(cid:48)Vmm(cid:48)(r(cid:48))φ(mj(cid:48))n(k,r(cid:48))
peraturesinthenano-Kelvinregime. Forfermionic40K87Rb and fm±n(0)(k,r) = δmn, with φm(jn+1)(k,r) =
mofokleBc).ulTeshewreefhoarev,eiVt0ca(cid:39)n1bewsitehenenferorgmyFsciga.le1(cid:39)(a)2t2h0antKthe(uynriets- −jn((cid:82)k0rrd)δr(cid:48)mgnm.(kIn,rt,hre(cid:48))r∑emm(cid:48)aVimnmd(cid:48)e(rr(cid:48)o)fφ(mthj(cid:48))nis(ks,erc(cid:48)t)iona,nwde cφo(mn0n)s(idk,err)lo=w
quiremuchlowertemperatures. Incontrast,fordipolaratoms scattering energies such that only up to two partial waves (cid:96)
with a magnetic dipole moment, V0 1 (e.g., 52Cr). Inter- and (cid:96)(cid:48) dominate the properties of the S-matrix. The phase
(cid:28)
layerdipoleswithV0 1bindtooweaklyforallpolarization shifts are given by tan2δ(cid:96)(k)=ImS(cid:96)(cid:96)/ReS(cid:96)(cid:96), where S(cid:96)(cid:96) is a
(cid:28)
anglestobestableatreasonableexternalparameters. diagonalmatrixelementofSandδ isobtainedbyreplacing
(cid:96)
(cid:48)
Thepeakoftheboundstatewavefunctionshiftsalongthex (cid:96) (cid:96). ByinsertingEq.(12)weobtainthephaseshiftsfrom
(cid:48)
→
4
(a) 00..45 ϑϑ==0π6 (b) VV00==kk==20.5 HeIrne,FEigb.is2(tah)ewbiendpilnogt tehneersg-yw,aEvqe.p(6h)a.se shift of two dipoles
ad) 0.3 ϑ=π2 V0=3,k=0.1 cwoiltlhisVio0n=m1oimnteenrtaucmtinkg,aec.gro.,ss40tKw8o7Rlabyewrsitahsdaifffuenrecntitosnpoinfsthaet
π(r 0.2 ∆=200nm. The sharp increase of the s-wave phase shift at
/
δ0 0.1 k→0(ϑ=0,π/6)isaconsequenceoftheveryweaklybound
states.Ifδ dominatesinEq.(15),averysmallbindingenergy
0.0 0
leadstoaphasejumpinδ closetok=0. Sincethebinding
0
0.1 energydecreasesstronglywithincreasingpolarizationangle,
− 10 2 10 1 100 0.0 0.1 0.2 0.3 0.4 0.5
− −
k(unitsof∆−1) +− z ϑ/π(rad) −x+ elaxrpizreastisoionno(n1l5y)adteuscnrriebaelsistthiceasllcyatstemrianllgcooflmlisaiionnlyeinne-rpgliaense. pOon-
the other hand, the weak potential approximation (13) de-
FIG.2. (Coloronline)S-wavephaseshiftforanisotropicinterlayer
scribes the numerics excellently at all considered momenta.
scattering of dipoles as a function of (a) the collision momentum
ItfailsatlargepotentialstrengthsV 1andsmallmomenta.
forV0=1 (e.g., KRb molecules at ∆=200nm) and (b) the polar- 0(cid:29)
Forlargemomentak 1thisapproximationbecomesidenti-
ization angle. The symbols mark numerical results, solid lines the
(cid:29)
second-order Born approximation Eq. (14), dashed lines Eq. (15), calwiththeBornapproximation(14). InFig.2(b)weobserve
and dotted lines Eq. (13) to second order in fm−n(j). The weak po- thatthes-wavephaseshiftcanvarystronglywiththepolariza-
tentialapproximation(13)describesscatteringwellfrommoderately tionangleatsmallmomenta. Thephasejumpiscausedbya
smallcollisionmomentaanduptomoderatepotentialstrengthsatall weaklyboundstateatϑ 0.3π[seeFig.1(a)].Thisvariability
(cid:39)
polarizations. Thebound-stateapproximation(15)isonlyaccurate shouldbeobservableinthescatteringofpolarmolecules.The
atverylowmomentaandpredominantlyperpendicularpolarization. bound-stateapproximation(15)describesthisbehaviorqual-
itativelyevenforsuchmoderatelylargepotentialstrengthsas
long as k 1. The difference is mainly due to neglecting
theJostmatrixF−sinceF+=F−forrealk. Then higher ord(cid:28)er partial waves in Eq. (15). For a small potential
strength V 1 our approximation Eq. (13) describes scat-
tan2δ (k) 2Ims(cid:96)(cid:96) (13) tering exce0ll(cid:28)ently at all polarizations. We find that this ap-
(cid:96) (cid:39) Res
(cid:96)(cid:96) proximationremainsvalidevenatmoderatelylargepotential
strengthsV (cid:46)3athigherenergiesk2(cid:38)1.
with s(cid:96)(cid:96) = S(cid:96)(cid:96) det(F−)2. Using terms fm−n(j) up to 0
| |
second order and expanding s to second order in
(cid:96)(cid:96)
V we find Im(s ) = I(1) + I(1)I(1) I(1) I(1) +
2I0(1)I(1) + I(2), (cid:96)(cid:96)Re(s−) j(cid:96)=j(cid:96) 1 +j(cid:96)j(cid:96)∑y(cid:96)j(cid:96) −[j(cid:96)2j(cid:96)I(cid:48)(y1(cid:96))(cid:48)j(cid:96) + IV. CONCLUSIONS
2I(p(1I1my(/j)(cid:96)m2qkj)n(cid:96))(+(cid:82)ky(cid:96)0)∞(cid:48)jI(cid:96)d(cid:48)(jm=1r)jp2mn((+k1jr/(cid:96))Iky(∑m)1)m(cid:82)j2m0∞V]nd−mr(pr2(cid:96)m)(cid:96)I((cid:82)(j(cid:96)k10rj)r(cid:96)2d)Vr−(cid:48)mgnm2((rIky()(cid:96)1,q)jr(cid:96)n,(cid:48)(Irky((cid:48)(cid:96)1)mr(cid:48)V))=j(cid:96),m(cid:96)n,+(cid:96)((cid:48)r−(cid:48)4I)pI(j2yn(n(cid:96)1)y(()jm(cid:96)kkIrj)y(m(cid:48)(cid:96)1)(cid:48))y(cid:96)=(cid:48), amtenredWitnhbgeoo.dhuangWvdeeensetpharraatoevlpiezpoersdsoeepdtrheivreateiJdfeorsassmftyosferotwearmmonriaasktloiictfsromoarppkcipcanrloo2cwxDuilnmaptfoairnttoiegomnnstisc3aaDlftsot.ersrOcitanhutger-
and p , q stand in for scaled Bessel functions of order m scatteringphaseshiftsandbindingenergyatlowtomoderate
m n
andn,respectively. Theanisotropyofthepotentialentersthe potentialstrengths. Forweakpotentialswehaverecovereda
phase shift through mixing terms, such as I(1) , and the sum second-orderBornapproximation. Thecentralequation(3)is
in I(2). These terms are absent for a centrajl(cid:96)jp(cid:96)(cid:48)otential so we alsowell-suitedfornumericalcomputations.
pn Wehaveappliedourmethodtopolarmoleculestrappedin
recover a result in Ref. [6]. Furthermore, for small V we
0 a bilayer and polarized along an arbitrary direction. We find
expandEq.(13)tosecondorderinV0: that absolute energies of 6Li40K interlayer bound states are
largerthantheirthermalenergyinultracoldexperimentseven
tan2δ(cid:96)(k)(cid:39)−2I(j(cid:96)1j)(cid:96)(k)+2[I(cid:96)(2)−I(j(cid:96)1j)(cid:96)Iy((cid:96)1)j(cid:96)−I(j(cid:96)1j)(cid:96)(cid:48)Iy((cid:96)1(cid:48))j(cid:96)]. (14) fornonperpendicularpolarization. Thes-wavephaseshiftof
molecules with moderate or large DDI exhibits a strong de-
Thus, we recover a second-order Born approximation of the pendence on the polarization angle. These results are im-
phase shift from our general formalism. We find a fur- portant, e.g., for the BEC-BCS crossover in fermionic polar
ther approximation by considering the sum of all partial molecules in bilayers [15, 16]. Varying the direction of the
waves at small energies. For any S-matrix, cot∑(cid:96)δ(cid:96)(k) = external polarizing field should influence the crossover from
Re(detS)/Im(detS) = Re(detF−)/Im(detF−). As in the interlayerpaircondensationtoBCSpairing.
−
preceding section, we expand detF around k =0 and use
− WethankDieterJakschforhelpfuldiscussions. Thiswork
thefactthatkisrealtoobtain
was supported by the Academic Research Fund of the Min-
1 (cid:18) k2 (cid:19) istry of Education of Singapore, Grant No. R-146-000-120-
cot∑δ (k) ln . (15) 112.
(cid:96) (cid:96) (cid:39) π |Eb|
5
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