Analytical coupled-channels treatment of two-body scattering in the presence of three-dimensional isotropic spin-orbit coupling

0.19 MB·

by  Q. Guan

Analytical coupled-channels treatment of two-body scattering in the presence of
three-dimensional isotropic spin-orbit coupling
Q. Guan1 and D. Blume1
1Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA
(Dated: January 17, 2017)
It is shown that thesingle-particle spin-orbit coupling terms,which—in thecold atom context—
7 are associated with synthetic gauge fields, can significantly and non-trivially modify the phase
1 accumulation at small interparticle distances even if the length scale (kso)−1 associated with the
0 spin-orbitcouplingtermissignificantlylargerthanthevanderWaalslengthrvdWthatcharacterizes
2 thetwo-bodyinteractionpotential. Atheoreticalframework,whichutilizesageneralizedlocalframe
n transformationandaccountsforthephaseaccumulationanalytically,isdeveloped. Comparisonwith
a numerical coupled-channels calculations demonstrates that the phase accumulation can, to a very
J good approximation, be described over a wide range of energies by the free-space scattering phase
5 shifts—evaluated at a scattering energy that dependson kso—and the spin-orbit coupling strength
1 kso.

PACSnumbers:
]
s
a
g The tunability of low-energy scattering parameters space scattering length a and scattering volume V re-
s p
- such as the s-wave scattering length a and p-wavescat- main good quantities provided(k )−1 is largerthan the
t s so
n tering volume V by means of application of an external two-body van der Waals length r . Indeed, model
p vdW
a
magnetic field in the vicinity of a Feshbachresonance[1] calculations for a square-well potential in the presence
u
has transformedthe field of ultracoldatom physics, pro- ofthree-dimensionalisotropicspin-orbitcouplingsuggest
q
. viding experimentalists with a knob to “dial in” the de- thattheabovereasoningholds,provided1/as andVp are
t
a sired Hamiltonian. This tunability has afforded the in- small [17].
m vestigation of a host of new phenomena including the This work revisits the question of how to obtain and
- BEC-BCS crossover [2, 3]. Most theoretical treatments parameterizetwo-bodyscatteringobservablesinthepres-
d of these phenomena are formulated in terms of a few ence of three-dimensional isotropic spin-orbit coupling.
n
o scattering quantities such as as and Vp, which properly Contrary to what has been reported in the literature,
c describethelow-energybehaviorofthetwo-bodysystem. our calculations for Lennard-Jones and square-well po-
[ The recent realization of spin-orbit coupled cold atom tentials show that the three-dimensional isotropic spin-

systems [4] is considered another milestone, opening the orbit coupling terms can impact the phase accumulation
1
door for the observation of topological properties and in the small interparticle distance region where the two-
v
9 providing a new platform with which to study scenar- body interaction potential cannot be neglected even if
7 ios typically encountered in condensed matter systems (kso)−1 is notably larger than rvdW. We observe non-
9 with unprecedented control [5–7]. An assumption that perturbative changes of the scattering observables when
3 underlies most theoretical treatments of cold atom sys- k changes by a small amount. An analytical treat-
so
0
tems with synthetic gauge fields is that the spin-orbit ment, which reproduces the full coupled-channels results
.
1 couplingterm,i.e.,theRamanlaserthatcouplesthe dif- suchastheenergy-dependenttwo-bodycrosssectionsfor
0 ferent internal states or the shaking of the lattice that the finite-range potentials with high accuracy, is devel-
7 couples different bands, leaves the atom-atom interac- oped. Our analytical treatment relies, as do previous
1
tions “untouched”. More specifically, mean-field treat- treatments [9–13, 15–17], on separating the short- and
:
v ments “simply” add the single-particle spin-orbit cou- large-distance regions. The short-distance Hamiltonian
i pling term to the mean-field Hamiltonian and parame- is treated by applying a gauge transformation, followed
X
terize the atom-atom interactions via contact potentials byarotation,that“replaces”thep-dependentspin-orbit
r
a with coupling strengths that are calculated for the two- coupling term by an r- and p-independent diagonal ma-
bodyvanderWaalspotentialwithoutthespin-orbitcou- trix (r and p denote the relative position and momen-
pling terms [7, 8]. tum vectors, respectively). The diagonal terms, which
Consistentwithsuchmean-fieldapproaches,mosttwo- can be interpreted as shifting the scattering energy in
body scattering studies derive observables based on the eachchannel, can introduce non-perturbative changes in
assumption that the two-body Bethe-Peierls boundary the scattering observables for small changes in kso, es-
condition, derived in the absence of single-particle spin- pecially when Vp is large. We note that our derivation
orbit coupling terms, remains unaffected by the spin- of the short-distance Hamiltonian, although similar in
orbit coupling terms, provided an appropriate “basis spirit, differs in subtle but important ways from what is
transformation” is accounted for [9–16]. The underlying presented in Ref. [10, 12].
premise of these two-body and mean-field treatments is Our analytical framework also paves the way for de-
rooted in scale separation, which suggests that the free- signing energy-dependent zero-range or δ-shell pseudo-
2
potentials applicable to systems with spin-orbit cou- h(J)u(J) = Eu(J), where h(J) [20] denotes the scaled ra-
pling. While energy-dependent pseudo-potentials have dial Hamiltonian for a given J (note that the Hamilto-
proven useful in describing systems without spin-orbit nian is independent of the M quantum number). For
J
coupling [18, 19], generalizations to systems with spin- r >r , the interaction potential V can be neglected
max int
orbit coupling are non-trivial due to the more intricate and u(J) is matched to the analytic asymptotic V = 0
int
nature of the dispersion curves. Our results suggest a solution [13, 15, 16]
paradigm shift in thinking about spin-orbit coupled sys-
tems with non-vanishing two-body interactions. While u(J) −−−−−→r J(J)−N(J)K(J) , (3)
the usual approach is to assume that the short-distance r>rmax (cid:16) (cid:17)
behavior or the effective coupling strengths are not im- where J(J) and N(J) are matrices that contain the reg-
pacted by the spin-orbit coupling terms, our results sug-
ular and irregular solutions for finite k (for J = 0 and
so
gest that they can be for specific parameter combina-
1, explicit expressions are given in Ref. [16]). Defin-
tions. Eventhoughouranalysisiscarriedoutforthecase
ing the logarithmic derivative matrix L(J)(r) through
ofthree-dimensionalisotropicspin-orbitcoupling,ourre- (u(J))′(u(J))−1, where the prime denotes the partial
sults point toward a more general conclusion, namely
derivative with respect to r, the K-matrix is given by
that spin-orbit coupling terms may, in general, notably
modify the phase accumulation in the short-distance re- ′ ′
K(J) = rN(J) −L(J)(r) rN(J) ×
gion.
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
We consider twoparticleswith positionvectorsr and
j ′
masses mj (j = 1 and 2) interacting through a spheri- rJ(J) −L(J)(r) rJ(J) , (4)
callysymmetrictwo-bodypotentialV (r)(r =|r −r |). (cid:20)(cid:16) (cid:17) (cid:16) (cid:17)(cid:21)(cid:12)r=rmax
int 1 2 (cid:12)
Both particles feel the isotropic spin-orbit coupling term (cid:12)
the S-matrix by S(J) =(I +ıK(J))(I −ıK(J))−1, where
with strength k , V(j) =~k p ·σσσ(j)/m , where p de-
so so so j j j I denotes the identity matrix, and the cross sections by
notesthecanonicalmomentumoperatorofthejthparti-
σ = 2π|S(J)−δ |2/k2, where α and β each take the
cle andσσσ(j) the vectorthat contains the three Paulima- αβ βα αβ α
trices σσσ(j), σσσ(j) and σσσ(j) for the jth particle. Through- values 1,2,···.
x y z In general,the K-matrix has to be determined numer-
out, we assume that the expectation value of the total
icallyviacoupled-channelscalculations. Inwhatfollows,
momentumoperatorPofthe two-bodysystemvanishes.
we addressthe questionwhether K can,atleastapprox-
In this case, the total angular momentum operator J,
imately, be described in terms of the logarithmic deriva-
J=l+S, of the two-particlesystem commutes with the
tive matrix of the free-space Hamiltonian H . If the
system Hamiltonian and the scattering solutions can be fs
spin-orbitcouplingtermV vanishedinthesmallrlimit,
labeledbythequantumnumbersJ andM ; M denotes so
J J one could straightforwardly apply a projection or frame
the projection quantum number, l is the relative orbital
angular momentum operator, and S=~(σσσ(1)+σσσ(2))/2. transformation approach [21–24] that would project the
inner small r solution,calculatedassumingthatV van-
Separating off the center of mass degrees of freedom, so
ishes in the inner region, onto the outer large r solution,
the relative Hamiltonian H for the reduced mass µ par-
calculated assuming that V vanishes in the outer re-
ticle with relative momentum operator p can be written int
gion [25]. The fact that V does not vanish in the small
asasumofthe free-spaceHamiltonianH andthe spin- so
fs rlimitrequires,asweshowbelow,ageneralizationofthe
orbit coupling term V , H =H +V , where
so fs so frame transformation approach.
We start with the Hamiltonian H and define a new
p2
H = +V (r) I ⊗I (1) HamiltonianH˜ throughT−1HT,whereT isanoperator
fs int 1 2
(cid:20)2µ (cid:21) tobedetermined. ThesolutionΨ˜ ofthenewHamiltonian
isrelatedtothesolutionΨofH throughΨ˜ =T−1Ψ;here
and V =~k ΣΣΣ·p/µ with Σ=(m σσσ(1)⊗I −m I ⊗
so so 2 2 1 1 and in what follows we drop the superscripts “(J, M )”
σσσ(2))/M. Here, I denotes the 2×2 identity matrix that J
j and “(J)” for notational convenience. The operator T
spans the spindegreesof freedomofthe jth particle and
reads RU, where R = exp(−ık Σ·r); the form of U is
so
M the total mass, M = m1 + m2. For each (J,MJ) introduced later. To calculate H =R−1HR, we use
channel,the r-dependent eigenfunctions Ψ(J,MJ) areex- R
panded as [13, 15, 16] R−1H R=H −V −E [Σ·r,Σ·∇]+O(r) (5)
fs fs so so
Ψ(J,MJ)(r)= r−1u(J)(k,r)|J,M ;l,Si, (2) and
l,S J
Xl,S R−1V R=V +2[Σ·r,Σ·∇]+O(r), (6)
so so
where the sum goes over (l,S) = (0,0) and (1,1) for where −ı~∇ = p and E = ~2k2 /(2µ) and where the
so so
(J,M )=(0,0)andover(l,S)=(J,0),(J,1),(J−1,1), notationO(r) indicates that terms of orderr and higher
J
and (J +1,1) for J > 0. In the |J,M ;l,Si basis (us- are neglected (r “counts” as being of order r and p as
J
ing the order of the states just given), the scaled radial beingoforderr−1). AddingEqs.(5)and(6)andneglect-
set of differential equations for fixed J and M reads ing the O(r) terms, we find that the spin-orbit coupling
J
3
term V is replaced by a commutator that arises from differs from the “normal” free-space Schr¨odinger equa-
so
the fact that the operator Σ·p does not commute with tion by channel-specific energy shifts. These shifts in-
the exponent of R, troduce a non-trivial modification of the phase accumu-
lation in the short-distance region and—if a zero-range
HRsr =Hfs+Eso[Σ·r,Σ·∇]. (7) or δ-shell pseudo-potential description was used—of the
boundary condition. While the energy shifts do, in
Here,thesuperscript“sr”indicatesthatthisHamiltonian many cases, have a negligible effect, our analysis below
is only valid for small r. shows that they can introduce non-perturbative correc-
Our goal is now to evaluate the second term on the tions in experimentally relevantparameter regimes. The
right hand side of Eq. (7). Defining the scaled short- channel-specific energy shifts are not taken into account
distance Hamiltonian hsRr through rHRsrr−1 and express- in Ref. [12].
ing hsRr in the |J,MJ;l,Si basis, we find To relate the logarithmic derivative matrix L˜sr(r) =
hsRr =(cid:18)−2~µ2∂∂r22 +Vint(r)(cid:19)I1⊗I2+V +ǫ, (8) vlo′gva−r1itohfmthicesdcearlievdatsihveortm-daitsrtiaxncLe(rH)a,mtihlteon“iTan-oph˜sTerratotiotnh”e
needs tobe “undone”. Assumingthatthe short-distance
where V is a diagonal matrix with diagonal elements Hamiltonian provides a faithful description, i.e., assum-
~2l(l+1)/(2µr2). For J = 0, the matrix ǫ is diagonal ing that the higher-order correction terms can, indeed,
with diagonal elements −3Eso and Eso. For J > 0, in be neglected for r <rmax, we obtain
contrast,the11and22elementsare,ingeneral,coupled:
L(r )≈ TL˜sr(r)T−1−T T−1 ′ . (10)
ǫ=E c/−M32 −(∆c/MM/2M)2 00 00 , (9) max n (cid:0) (cid:1)o(cid:12)(cid:12)(cid:12)r=rmax
so 0 0 d /M2 0 To illustrate the results, we focus on the J = 0 sub-
1
 0 0 0 d /M2  space. Denoting the usual free-space phase shifts at
 2 
scattering energy ~2k2/(2µ) for the interaction poten-
dw1he=re−∆JMM2=−m(J1 −+m1)2∆,Mc 2=, a2npdJd(2J=+14)m(m1m22 2−−md211),. δtipa(lkV),inrtefsoprecthtievesl-yw,atvheeasnhdorpt--rwaanvgeecKha-mnnaetlrsixbyK˜δssr(fko)ratnhde
Since the r-dependent 11 and 22 elements of V are iden- Hamiltonian h˜sr has the diagonal elements tan(δ (k ))
T s s
tical (recall l = J for these two elements), the ma- and tan(δ (k )), where ~2k2/(2µ) = E + 3E and
trix U, which is defined such that U−1ǫU is diagonal, ~2k2/(2µ)p= Ep −E . If we nsow, motivated by thsoe con-
also diagonalizes hsRr, i.e., the short-range Hamiltonian ceptp of scale separastoion, make the assumption that the
h˜sr = U−1hsrU is diagonal. This implies that the scaled phaseshiftstan(δ (k ))andtan(δ (k ))areaccumulated
T R s s p p
radialshort-distanceSchr¨odingerequationh˜sTrv =Evcan at r =0 and correspondingly take the rmax →0 limit of
be solved using standard propagation schemes such as Eq.(4)withL(J) givenbytherighthandsideofEq.(10),
the Johnson algorithm [26]. This Schr¨odinger equation we obtain the following zero-range K-matrix,
Kzr =− as(ks) k+2 k+k− − Vp(kp) k+2(k−−kso)2 k+k−(k+−kso)(k−−kso) , (11)
k+−k− (cid:20) k+k− k−2 (cid:21) k+−k− (cid:20) k+k−(k+−kso)(k−−kso) k−2(k+−kso)2 (cid:21)
where ~k± =± 2µ(E+Eso)−~kso. large Vp(0), respectively. The dashed lines show the re-
p sults predicted by our zero-range model that accounts
To validate our analytical results, we perform numer- for the spin-orbit coupling induced energy shifts. This
ical coupled-channels calculations. Since the wave func- model provides an excellent description of the numeri-
tion in the J = 0 subspace is anti-symmetric under the cal results for the Lennard-Jones potential, provided the
simultaneous exchange of the spatial and spin degrees length (k )−1 associated with the spin-orbit coupling
so
of freedom of the two particles, the solutions apply to term is not too small compared to the van der Waals
twoidenticalfermions. TheSchr¨odingerequationforthe length r , where r is given by (2µC /~2)1/4 (in
vdW vdW 6
Lennard-Jonespotential V (r)=C /r12−C /r6, with Figs.1and2,thelargestk r consideredcorresponds
LJ 12 6 so vdW
C and C denoting positive coefficients, is solved nu- to 0.4913 and 0.4171,respectively).
6 12
merically [27]. The solid lines in Figs. 1 and 2 show
the partial cross section σ and the K-matrix element The dash-dotted lines in Figs. 1 and 2 show σ and
22 22
K as a function of k for vanishing scattering en- K for the zero-rangemodel when we set the spin-orbit
22 so 22
ergy E for a two-body potential with large a (0) and couplinginducedenergyshiftsartificiallytozero. Inthis
s
4
π) 1 (a) π) 1 (a)
2 2
2k)/(so0.5 2k)/(so0.5
σ(22 σ(22
0 0
50 2
(b) (b)
K22 0 K22 0
-50 -2
4 15
k)/a(0)ss 02 (c) k)/V(0)pp 0 (c)
a(s-2 V(p
-4 -15
0 4 8 12 0 0.1 0.2 0.3 0.4
ksoas(0) ksoas(0)
FlinIGe.sh1o:ws((Ca)oltohreosnclainleed) pLaarrtgiaelacsr(o0s)s cseacstei.onTσh2e2(bklsaoc)k2/s(o2lπid) FlinIGe.sh2o:ws((Cao)ltohreosnclainleed) pLaarrtgiealVcpr(o0s)s cseacstei.onTσh2e2(bklsaoc)k2/s(o2lπid)
and (b)theK-matrixelement K22 for E =0as afunction of and (b) the K-matrix element K22 for E = 0 as a
k24so.4a2s(a0n)dfoVrpt(h0e)/L(revndnWar)d3-=Jon−e0s.2p3o8t0en(ttihailswpiothtenatsi(a0l)/survpdpWort=s afusn(0c)t/iornvdWof=ks0o.a9s5(901)afnodr Vtph(e0)L/e(rnvndaWrd)3-J=on2e6s.6p1o,tceonrtrieaslpownitdh-
two s-wave bound states in free space). The red dashed line ing to as(0)/(Vp(0))1/3 = 0.3213 (this potential supports 4
shows the result for the zero-range model developed in this four s-wave bound states in free space). The red dashed
work [see Eq. (11)]; the numerical results for the Lennard- line shows the result for the zero-range model developed in
Jones potential and the model are indistinguishable on the this work [see Eq. (11)]; the model reproduces the numer-
scaleshown. Toillustratetheimportanceoftheenergyshifts, ical results excellently for ksoas(0) . 0.3. The blue dash-
thebluedash-dottedlineshowstheresultsforthezero-range dotted line shows the results for the zero-range model that
model that artificially neglects the energy shifts. The solid artificially neglects the energy shifts. The solid line in (c)
linein(c) showsthescaled energy-dependents-wavescatter- shows the scaled energy-dependentp-wavescattering volume
ing length as(ks)/as(0), where ~2ks2 =6µEso. Vp(kp)/Vp(0), where ~2kp2 =−2µEso. The green circles mark
threeof thefour ksoas(0) valuesconsidered in Fig. 4.
case, the divergence in the K matrix element at finite
22
iments.
k is not reproduced. For large a (0) [see Fig. 1(a)],
so s
the model without energy shifts introduces deviations at To further explore the two-particle scattering proper-
the few percent level in the cross section σ . For large tiesinthepresenceofspin-orbitcouplingforshort-range
22
V (0) [see Fig. 2(a)], in contrast, the model without the potentials with large free-space scattering volume V (0),
p p
energy shifts provides a quantitatively and qualitatively Figs. 3(a) and 3(b) show the partial cross section σ
22
poor description of the cross section σ even for rela- as a function of the scattering energy −E ≤ E ≤ 0
22 so
tively small k (k a (0)& 0.05). Figures 1(c) and 2(c) and 0≤0≤400E , respectively, for a (0)/(V (0))1/3 =
so so s so s p
demonstrate that the divergence of the K matrix ele- 0.3213 and a (0)k = 0.07673. The results for the
22 s so
mentoccurswhenthefree-spacescatteringlengtha (k ), Lennard-Jones potential (dashed line) and square-well
s s
calculated at energy 3E , or the free-space scattering potential (solid line) are essentially indistinguishable on
so
volume V (k ), calculated at energy −E , diverge. We the scale shown. To assess the accuracy of our zero-
p p so
find that this occurs roughly when a (0)k ≈ 10 and range model, we focus on the Lennard-Jones potential
s so
(V (0))1/3k ≈ 0.21; we checked that this holds quite and compare the numerically determined partial cross
p so
generally, i.e., not only for the parameters considered in section (σ )exact with the partial cross section (σ )zr
22 22
the figures. In Figs. 1(c) and 2(c), the “critical” k predicted using Eq. (11). Solid lines in Figs. 3(c) and
so
values correspond to k r = 0.1423 and k r = 3(d) show the normalized difference ∆, defined through
so vdW so vdW
0.1462,respectively. For comparison,using the k value ∆ = |(σ )zr−(σ )exact|/(σ )exact. The deviations are
so 22 22 22
for the one-dimensional realization of Ref. [4] and as- smaller than 1.3%for the scattering energies considered.
suming r = 100a , one finds k r ≈ 0.03. This Neglecting the spin-orbit coupling induced energy shifts
vdW 0 so vdW
suggests that the phenomena discussed in the context of in our zero-range model and calculating the normalized
Figs. 1 and 2 should be relevant to future realizations difference, we obtain the dashed lines in Figs. 3(c) and
of three-dimensional isotropic spin-orbit coupling exper- 3(d). Clearly, the zero-range model provides a faith-
5
100 0.04
π) (a) (b)
2
2)/(o10-1 0.02 π2) 1 (a) (b) 1
σ(k22s 2k)/(so0.5 0.5
10-2 0 (2
100 (c) (d) 100 σ2
0 0
-1 -0.5 0 -3 0 3
10 10 10
∆ 10-2 10-2 E/Eso E/E
so
FIG. 4: (Color online) Scaled partial cross sec-
10-4-1 -0.5 0 -2 0 210-4 tion σ22(kso)2/(2π) for the Lennard-Jones potential with
E/Eso 10 E/E10 10 as(0)/(Vp(0))1/3 = 0.3213 (Vp(0) > 0) and as(0)/rvdW =
so 0.9591 for four different kso [the green dotted, blue dash-
dotted, black solid, and red dashed lines correspond to
FIG. 3: (Color online) Large V (0) case. (a) and (b): The
p
reddashed andblacksolid linesshowthescaled partialcross ksorvdW = 0.1, ksorvdW = 0.12, ksorvdW = 0.14, and
section σ22(kso)2/(2π)fortheLennard-Jonesandsquare-well ksorvdW = 0.146, respectively] as a function of the scatter-
ingenergyE [panel(a)coversnegativeE (linearscale) while
potential, respectively, as a function of the scattering energy
E. For both potentials, we have a (0)/(V (0))1/3 = 0.3213 panel (b) coverspositive E (logarithmic scale)].
s p
[Vp(0) >0] and ksoas(0) =0.07673. The length scale associ-
ated with the spin-orbit coupling is notably larger than the
rangeofthepotential(ksorvdW =0.08 fortheLennard-Jones dispersion relations. Restricting ourselves to three-
potentialandksorsw =0.07676 forthesquare-wellpotential).
dimensionalisotropicspin-orbitcouplingtermsandspin-
(c) and (d): The solid and dashed lines show the normal-
independentcentraltwo-bodyinteractions,wedeveloped
ized difference ∆ (see text) between the cross section for the
an analytical coupled-channels theory that connects the
Lennard-Jones potential and the zero-range model, obtained
using Eq. (11), and between that for the Lennard-Jones po- short- and large-distance eigenfunctions using a gener-
tential and the zero-range model that neglects thespin-orbit alized frame transformation. A key, previously over-
coupling induced energy shifts, respectively. The zero-range looked result of our treatment is that the gauge trans-
model derived in this work (solid line) provides an excellent formation that converts the short-distance Hamiltonian
description (the deviations are smaller than 1.3 % for the to the “usual form” (i.e., a form without linear momen-
data shown) over the entire energy regime. Panels (a) and tum dependence) introduces channel-dependent energy
(c) cover negative E (linear scale) while panels (b) and (d)
shifts. These energy shifts were then shown to appre-
cover positive E (logarithmic scale)].
ciably alter the low-energy scattering observables, espe-
cially in the regime where the free-space scattering vol-
umeislarge. Toillustratethis,the(J,M )=(0,0)chan-
ful description of the full coupled-channels data for the J
nel wasconsideredexemplarily. Our frameworkprovides
Lennard-Jones potential only if the spin-orbit coupling
thefirstcompleteanalyticaldescriptionthatconsistently
induced energy shifts are included.
accounts for all partial wave channels. Moreover, the
Figure 4 demonstrates that the non-quadratic single-
first numerical coupled-channels results for two-particle
particle dispersion relations have a profound impact on
Hamiltonian with realistic Lennard-Jones potentials in
the low-energy scattering observables for a large free-
thepresenceofspin-orbitcouplingtermswerepresented.
space scattering volume. Specifically, the lines in Fig. 4
The influence of the revised zero-range formulation put
showthenumericallyobtainedpartialcrosssectionσ as
22 forwardinthispaperontwo-andfew-bodyboundstates
afunctionofthescatteringenergyforthesameLennard-
and on mean-field and beyond mean-field studies will be
Jonespotentialas thatused inFigs.2 and3 forfour dif-
the topic of future publications.
ferent spin-orbit coupling strengths, namely k r =
so vdW
0.1, 0.12, 0.14 and 0.146 [Fig. 3 used k r = 0.08;
so vdW
three of the four k values considered in Fig. 4 are
so
marked by circles in Fig. 2(c)]. Figure 4 shows that the
I. ACKNOWLEDGEMENT
partialcrosssectiondependssensitivelyonthespin-orbit
coupling strength k . This can be understood by real-
so
izing that a change in the spin-orbit coupling strength Support by the National Science Foundation through
leads to a significant change of the k -dependent scat- grant number PHY-1509892 is gratefully acknowledged.
so
tering volume V (k ). Theauthorsacknowledgehospitalityofandsupport(Na-
p p
This paper revisited two-body scattering in the pres- tional Science Foundation under Grant No. NSF PHY-
ence of single-particle interaction terms that lead, in 1125915)by the KITP. We thank J. Jacob for providing
the absence of two-body interactions, to non-quadratic us with a copy of his coupled-channels code.
6
[1] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fesh- Raman coupling, Sci. Rep.4, 4992 (2014).
bachresonances in ultracold gases, Rev.Mod. Phys.82, [15] S.-J. Wang and C. H. Greene, General formalism for
1225 (2010). ultracold scattering with isotropic spin-orbit coupling,
[2] I. Bloch, J. Dalibard, and W. Zwerger, Many-body Phys. Rev.A 91, 022706 (2015).
physics with ultracold gases, Rev. Mod. Phys. 80, 885 [16] Q. Guan and D. Blume, Scattering framework for two
(2008). particles with isotropic spin-orbit coupling applicable to
[3] Theory of ultracold atomic Fermi gases, S. Giorgini, L. all energies, Phys. Rev.A 94, 022706 (2016).
P.Pitaevskii,andS.Stringari,Rev.Mod.Phys.80,1215 [17] Y.WuandZ.Yu,Short-rangeasymptoticbehaviorofthe
(2008). wave functions of interacting spin-1/2 fermionic atoms
[4] Y.-J. Lin, K. Jim´enez-Garc´ıa, and I. B. Spielman, Spin- with spin-orbit coupling: A model study, Phys. Rev. A
orbit-coupled Bose-Einstein condensates, Nature (Lon- 87, 032703 (2013).
don) 471, 83 (2011). [18] D. Blume and C. H. Greene, Fermi pseudopotential ap-
[5] J. Dalibard, F. Gerbier, G. Juzeliu¯nas, and P. O¨hberg, proximation: Two particles under external confinement,
Colloquium: Artificial gauge potentials for neutral Phys. Rev.A 65, 043613 (2002).
atoms, Rev.Mod. Phys. 83, 1523 (2011). [19] E. L. Bolda, E. Tiesinga, and P. S. Julienne, Effective-
[6] V. Galitski and I. B. Spielman, Spin-orbit coupling in scattering-lengthmodelofultracoldatomiccollisionsand
quantumgases, Nature 494, 49 (2013). Feshbachresonancesintightharmonictraps,Phys.Rev.
[7] H.Zhai, Degenerate quantum gases with spin-orbit cou- A 66, 013403 (2002).
pling: a review, Rep.Prog. Phys. 78, 026001 (2015). [20] Here and in what follows, underlined quantities are ma-
[8] H. Zhai, Spin-Orbit Coupled Quantum Gases, Int. J. trices.
Mod. Phys. B 26, 1230001 (2012). [21] U.Fano,StarkeffectofnonhydrogenicRydbergspectra,
[9] X. Cui, Mixed-partial-wave scattering with spin-orbit Phys. Rev.A 24, 619(R), (1981).
coupling and validity of pseudopotentials, Phys. Rev. A [22] D.A.Harmin,TheoryoftheNonhydrogenicStarkEffect,
85, 022705 (2012). Phys. Rev.Lett. 49, 128 (1982).
[10] P.Zhang,L.Zhang,andY.Deng,ModifiedBethe-Peierls [23] C. H. Greene, Negative-ion photodetachment in a weak
boundary condition for ultracold atoms with spin-orbit magnetic field, Phys.Rev.A 36, 4236 (1987).
coupling, Phys. Rev.A 86, 053608 (2012). [24] B. E.Granger andD.Blume,TuningtheInteractionsof
[11] Z. Yu, Short-range correlations in dilute atomic Fermi Spin-Polarized Fermions Using Quasi-One-Dimensional
gases with spin-orbit coupling, Phys. Rev.A 85, 042711 Confinement, Phys.Rev. Lett.92, 133202 (2004).
(2012). [25] S.-J. Wang, Ph.D thesis, PurdueUniversity (2016).
[12] L. Zhang, Y. Deng, and P. Zhang, Scattering and effec- [26] B.R.Johnson,TheMultichannelLog-DerivativeMethod
tive interactions of ultracold atoms with spin-orbit cou- for Scattering Calculations, J. Comput. Phys. 13, 445
pling, Phys. Rev.A 87, 053626 (2013). (1973).
[13] H.Duan,L.You,andB.Gao, Ultracold collisions inthe [27] F.Mruga laandD.Secrest,Thegeneralizedlog-derivative
presence of synthetic spin-orbit coupling, Phys. Rev. A method for inelastic and reactive collisions, J. Chem.
87, 052708 (2013). Phys. 78, 5954 (1983).
[14] Y.-C.Zhang,S.-W.Song,W.-M.Liu,Theconfinement
inducedresonancein spin-orbitcoupled cold atoms with