A general T-matrix approach applied to two-body and three-body problems in cold atomic gases

0.37 MB·English

by  Xiaoling Cui

A general T-matrix approach applied to two-body and three-body problems
in cold atomic gases
Xiaoling Cui
Institute for Advanced Study, Tsinghua University, Beijing, 100084, China
(Dated: January 26, 2012)
We propose a systematic T-matrix approach to solve few-body problems with s-wave contact
interactions in ultracold atomic gases. The problem is generally reduced to a matrix equation
2 expanded by a set of orthogonal molecular states, describing external center-of-mass motions of
1 pairs of interacting particles; while each matrix element is guaranteed to be finite by a proper
0
renormalizationforinternalrelativemotions. Thisapproachisabletoincorporatevariousscattering
2
problemsandthecalculationsofrelatedphysicalquantitiesinasingleframework,andalsoprovides

n aphysicallytransparentwaytounderstandthemechanismofresonancescattering. Forapplications,
a we study two-body effective scattering in 2D-3D mixed dimensions, where the resonance position
J andwidthare determinedwith highprecision from only afewnumberof matrixelements. Wealso

5 study three fermions in a (rotating) harmonic trap, where exotic scattering properties in terms of
2 mass ratios and angular momenta are uniquely identified in theframework of T-matrix.

]
s
I. INTRODUCTION pingpotentials. The keypointofthis methodis tomake
a
g use of the interaction property and introduce a set of
- Interacting ultracold atoms have gained a lot of re- orthogonalmolecular states, which describe the external
t
n searchinterestsfortheir interactionstrengthanddimen- center-of-mass(CM) motions of pairs of interacting par-
a sionality are highly controllable by making use of Fesh- ticles; then the problem is generally reduced to a matrix
u bach resonance and external confinements[1, 2]. In such equation,andaproperrenormalizationschemeforthein-
q
diluteatomicgases,theinteractionbetweenatomscanbe ternalrelativemotionsensuresfinitevalueofeachmatrix
.
at well approximated as contact potential which is charac- element. Using this method, we obtain the bound state
m terizedby the s-wavescatteringlength[1,2]. In this con- solution,scatteringmatrixelementandreducedcoupling
text, the few-body problems play very important roles constantinthelow-dimensionalsubspace. Moreover,the
-
d in studying many-body properties. For instance, solu- treatment has a lot more physical meanings and allows
n tions of these problems determine effective interactions us to make analyticalpredictions to various induced res-
o between atom-atom, atom-dimer and dimer-dimer[3, 4], onances under confinement potentials. In all, T-matrix
c which are fundamental elements to formulate the many- approachis able to unify many studies ofdifferent issues
[
body effective Hamiltonian; moreover, the consideration in the single framework.

4 oftwo-bodyshort-rangephysicsleadstoaseriesofexact To show the efficiency of this approach, we apply it
v universal relations for a many-body system, as first pro- to study two-body effective scattering in 2D-3D mixed
4
posedbyTan[5–7]andrecentlyverifiedinexperiment[8]. dimensions, where T-matrix not only provides a trans-
4
0 Previous studies of few-body systems have revealed parent way to understand the mechanism of multiple
0 many nontrivial effects. One famous example is the Efi- resonances, but also gives explicit expressions for the
. mov effect for three atoms[9, 10], depending closely on resonance position and the width. Particularly, its ef-
0
the short-range interacting parameter apart from sin- ficiency lies in that each resonance can be determined
1
0 gle s-wave scattering length. Another typical effect is accurately by only calculating a few number of matrix
1 the two-body induced resonance scattering and induced elements. Moreover,weapplythismethodtostudythree
: boundstateduetoexternalconfinements[11–24]. Among two-component fermions in a (rotating) harmonic trap,
v
most of previous studies, the problems were solved in and show its unique advantage in identifying scatter-
i
X the framework of two-channel models[25] or by using ing properties in different angular momentum channels.
r pseudopotentials[26, 27]. In this article, we present us- The ground state is obtained for a rotating and trapped
a
ing T-matrix approach to solve few-body problems with system, which gives important hints for quantum Hall
s-wavecontactinteractioninthefieldofultracoldatoms. physics in the fermionic atom-dimer system.
Compared with other methods, T-matrix is able to sys- The rest of this paper is organized as follows. In sec-
tematically provide exact solutions for few-body prob- tion II, we introduce the renormalization concept and
lems, and more importantly, is able to work in a much present T-matrix formulism to solve a general few-body
efficient and physically transparent way. problem. Its relations to other approaches, advantages
In this article, we shall first formulate T-matrix and limitations are also discussed. Section III is the ap-
method and introduce its essential concept, i.e., the plication of T-matrix approach to two-body problems,
renormalization idea to integrate out all high-energy(or where specifically we study the scattering resonances in
short-range) contributions for relative motions. Then a 2D-3D mixed dimensions. Section IV is the application
systematic treatment is presented to a general N-body to three fermions in a (rotating) harmonic trap, where
systemwith contactinteractionsandwith possibly trap- the energyspectrumand scatteringpropertyare studied
2
for different mass ratios and angular momenta of three here V is the volume, and ǫk = k2/(2µ) is the energy
fermions. We summarize the paper in the last section. for relativemotionoftwoparticles with reducedmassµ.
The resultant RG flow equation reads[28]
1 δU 1 δ 1
II. T-MATRIX APPROACH
= ( ) (6)
U2 δΛ V δΛ ǫk
In this section we give a systematic prescription of T- |kX|<Λ
matrix approach to solve few-body problems. The re- relatesthebareinteractionU0tothezero-energyeffective
sulted matrix equation is given by Eq.10 in Section IIA, one(T0 =2πas/µ) via,
from which we extract three important physical quanti-
µ 1 1 1
ties as given by Eqs.(12,14,18) in Section IIB. We also = + , (7)
discuss its relation to other widely used methods in Sec- 2πas U0 V k ǫk
X
tionIIC,anddemonstrateitsuniqueadvantagesandlim-
with a the s-wave scattering length. For a general N(>
itations in Section IID. s
2)-bodyproblem,theideaofrenormalization,thoughnot
as explicitly shownas above,alwaysservesas the under-
lying principle through the whole scheme(see below).
A. Basic concept and general formulism
Now we proceed with the general T-matrix approach.
Suppose a Q species system, and the i-th (i = 1...Q)
WestartfromtheLippmann-Schwingerequationbased species has N− identical particles residing at xi,...xi ;
on standard scattering theory, i 1 Ni
U and U (i <j <Q) are respectively the bare interac-
i ij
ψ = ψ +Gˆ (E)Tˆ ψ . (1) tion strengths between particles within the i th species
0 0 0 −
| i | i | i and between different species (i and j); U (U ) is re-
i ij
Here ψ0 is the eigen-state for non-interacting system, lated to the corresponding scattering length ai (aij) via
with H| amiiltonian Hˆ = Hˆ +Vˆ composed by kinetic Eq.7. Takingadvantageofthezero-rangepropertyofthe
0 kin T
interaction
termandexternaltrappingpotential; ψ isthescattered
| i
state in the presence of interaction potential Uˆ; Q Ni Q Ni Nj
Uˆ = U δ3(xi xi)+ U δ3(xi xj),
1 i m− n ij m− n
Gˆ0(E)= E Hˆ0+iδ (2) Xi=1mX<n Xi<jmX=1nX=1 (8)
−
andthusthesamepropertyofT-matrixgivenbyEq.3,we
is the Green function; the scattering matrix can be ex- expandU andT byasetofmolecularstates xi xj =
pandedasseriesT =U+UG0U+UG0UG0U+...which 0,λ . This state is defined such that one{|pamir−ofnin-
leads to teraic}ting particles(xi , xj) locate at the same site, and
m n
T = (1 UG0)−1U. (3) λ is the energy-level index for the remanent degrees
− of freedom. More detailed description of the molecu-
To this end, G , T and U are all matrixes expanded by lar state is given in Appendix A. Further, for identi-
0
certain complete set of states. If we use ψ to ex- cal bosons/fermions the molecular state should further
0
pand T-matrix, then each T-matrix elemen{t| ini}Eq.3 di- be symmetrized/antisymmetrized by superpositions of
rectlyrelatestothescatteringamplitudeinthescattered above individual ones. Explicitly we have
wavefunctionandthereforerepresentsthe effectiveinter-
Tˆ ψ = fI r =0,λ , (9)
action in the low-energy space. To obtain the effective | 0i λ| I i
interaction, a physically insightful way is to employ the XIλ
concept of renormalization. For the fundamental two- here each state, r = 0,λ (r = xi xj ) with I
body s-wave scattering with contact interaction, | I i I | m − n| ≤
max(I) = Q(Q+1), corresponds to one interaction term
2
Uˆ(r)=U0δ3(r), (4) in Eq.8. The coefficients {fλI} satisfy
µ
one can resortto a simple momentum-shell renormaliza- fλI′′ 2πaI δII′δλλ′ −CλIλI′′ =hrI =0,λ|ψ0i(10)
tionscheme[20]. Thespirithereistotakeallintermediate I′λ′ I
scattering from low-k(momentum) space to the shell in X (cid:2) (cid:3)
high-k space as virtual processes, which in turn modify with
bthyeaeffsmecatilvleδUin;teprearcttuiornbasttirveenlygtδhU(Ui)nintertmhesloofwt-hkessphaeclel CλIλI′′ = V1 ǫ1kδII′δλλ′ +hrI =0,λ|Gˆ0|rI′ =0,λ′i.
k
momentum(δΛ) follows X
(11)
Detailed derivation of Eq.10 is given in Appendix B.
U2 1
δU = , (5) For the two-particle scattering in free space, the en-
Λ−δΛX<|k|<Λ− V ǫk ergy level {λ} is characterized by the CM momentum,
3
which is conserved by the interaction and thus irrele- where U (t,s = P,Q ) is the bare coupling strength
ts
{ }
vant to the scattering problem for relative motions. The between two particular channels;
molecular state is then as simple as r = 0 , with r
| i 1
the distance between two particles. Eq.9 is then re- GQ = (16)
duced to Tˆ ψ = f r = 0 , and f is given by Eq.10 0 E HQ+iδ
as f 1 a|10+i ik, re|produciing the well-known relation − 0
betw−een∝th−se scattering amplitude and the s-wave scat- is the Greenfunction forHamiltonianHQ thatonly acts
0
tering length (a ). The applications of this approach to on closed-channel states. Again due to the property of
s
other few-body systems will be introduced in Section III Uˆ(Eq.8), we insert into Eq.15 a set of molecular states
and IV. r = 0,λ r = 0,λ[30]. Assuming two column
Iλ| I ih I |
vectors
P
B. Calculation of physical observables ξ = rI =0,λψ0 , ζ = rI =0,λUˆψ0 , (17)
{h | i} {h | i}
we then obtain
With the information of molecular states, all physical
quantities can be deduced straightforwardly. We shall g = ξTζ+ξT(UGQ)ζ +ξT(UGQ)2ζ+...
enumerate below three quantities that are detectable or eff 0 0
observable in experiments. = ξT(1−UGQ0)−1ζ
Eq(.I1)0Biosugnidvesntabtyetshoeluptioolne.ofITn-tmhaistrcixa(sseee|ψE0qi.3is),aib.es.e,nt, = hψ0|(1−UGQ0)−1U|ψ0i, (18)
here UGQ is a matrix expanded by molecular states.
0
Det(1−UG0(Eb))=0, (12) Note that geff is different from ψ0 Tˆ ψ0 in (II) only by
h | | i
the Green function therein. Obviously g is the renor-
whereE isthebindingenergy,andtheeigen-vector fI eff
b { λ} malizedcouplingstrengthinthe openchannelby allvir-
gives the bound state as
tual scattering to states in closed channels.
ψ = fIGˆ (E )r =0,λ . (13)
| bi λ 0 b | I i
XIλ C. Relations with other methods
For two-particle scattering(with scattering length a ) in
s
free space, Eqs.(12, 13) give E = 1/(2µa2) and ψ In this section, we analyze the intrinsic relation be-
1 k . b − s | bi∝ tween T-matrix and other widely used methods, such as
(kIIE)bT−-ǫmk|atirix element. Generally, T-matrix element thoseintheframeworkoftwo-channelmodels[17,19]and
Pbetween ψ anditselfcharacterizesthescatteringprop- pseudopotentials[11–13, 18, 31, 32].
0
erty of lo|w-einergy particles. With Eqs.(9,10), we obtain In two-channel models, the closed-channel molecules
are explicitly included in the Hamiltonian; these
ψ Tˆ ψ = fI ψ r =0,λ , (14) molecules couple to atoms in open-channelandthus me-
h 0| | 0i λh 0| I i diate interactions between the atoms. In the present T-
Iλ
X matrixmethod,themolecularstates(definedinEq.9)can
which can also be obtained from Eq.3 as ψ (1 be considered as the analog of closed-channel molecules
0
UG ) 1U ψ . For two-particle scattering in frhee s|pac−e, in two-channel models. The similarities lie in that they
0 − 0
| i
fI f is proportional to the scattering amplitude. arebothconstructedinawaythatfollowsthezero-range
λ ≡
(III)reduced interaction. If trapping potentials con- property of interaction, and describe the CM motion of
fine atoms in a lower dimension, there are two distinct two interacting particles.
scattering channels for the low-energy state, namely the Inpseudopotentials,theproblemissolvedbyapplying
open(P) or closed(Q) channel, depending on whether its Bethe-Peierlsboundaryconditiontothewavefunctionat
wavefunction propagates or decays at large interparti- short inter-particle distance, i.e.,
cle distance in the lower dimension. The effective in-
1 1
teraction strength for low-energy particles in the open lim ψ(x ,x ,x ...)=( )f(x =x ,x ...),
1 2 3 1 2 3
channel is modified from the original bare one by vir- rI=|x1−x2|→0 rI−aI
tual scatterings to the closed channels. We assume g (19)
eff
as the modified interaction strength in open channel, here the index I denotes the pair x1, x2 and aI is the
{ }
which, for instance, has been defined in the reduced 1D scattering length between them. On the other hand, we
Hamiltonian[12, 13, 21, 29] under tight transverse har- notice that above asymptotic behavior can be automati-
monic traps. Following the same procedure in obtaining callysatisfiedbythepresentschemeofT-matrixmethod.
T-matrix element in (II), we have Toshowthis,weexamineEq.1atshortinter-particledis-
tance by projecting it to certain molecular state,
g =U +U GQU +U GQU GQU +...,
eff PP PQ 0 QP PQ 0 QQ 0 QP lim r ,λψ = r =0,λψ +
(15) rI→0h I | i h I | 0i
4
fλI′′hrI =0,λ|Gˆ0|rI′ =0,λi ordinatemethodinunitarylimit[33,34],orbyemploying
I′λ′ athree-bodyforcetoeliminatethecutoffdependence[35].
X
µ 1 1 The extensionofthe presentT-matrix approachto iden-
= fI( I )
λ 2πaI − V Xk ǫk toiffythsiuscphapneorn.trivial few-body effects is out of the scope
µ 1 1
= lim fI I( ). (20)
rI→0 λ2π aI − rI
III. APPLICATION TO TWO-BODY PROBLEM
To derive Eq.20 we have used Eq.10 and the Fourier
transformation of zero-energy Green function in free
Inthis sectionweconsiderthe two-bodysystem. First
space. Remarkably, Eq.20 shows that the asymptotic
we present the formulism of physical quantities intro-
formisgivenbytheinverseofbarepotential1/U ,which
I
ducedinSectionIIB,forbothcaseswhentheCMmotion
is universal regardless of any trapping potential. In this
and relative motion can or cannot be decoupled. Finally
sense, the Bethe-Peierls boundary condition (Eq.19) in
we apply this method to address the effective scattering
theframeworkofpseudopotentialsisequivalenttorenor-
oftwoparticlesin2D-3Dmixeddimensions. Particularly
malization equation (Eq.7) in T-matrix method.
we emphasize the physical insight given by T-matrix to
understand the resonance mechanism, as well as its effi-
ciency in realistic calculation of scattering parameters.
D. Advantages and Limitations
In principle, the T-matrix fomulism presented in Sec-
A. Formulism
tionIIA is appliableto a generalfew-body problemwith
zero-range interactions. It gives a unified treatment to
1. r and R decoupled system
different scattering issues, as shown in Section IIB. Be-
sides, T-matrix approach has the following advantages.
First, we compare this scheme, using molecular states For trapping potential VT(x1,x2) = VT(R)+VT(r),
to expand T-matrix (Eq.10, with that using original the molecular state is simply r = 0,λ0 , with λ0 char-
| i
N particle states. The latter requires a matrix dimen- acterizing the CM motion and staying unaffected by the
sio−n as large as ΓN (Γ is the cutoff index for single par- interaction. The bound state solution Eb is determined
ticle energy levels), while the former could reduce it to from a single equation as
at most Q(Q+1)ΓN 1 for a general case and further to
2 − µ
Q(Q2+1)ΓN−2 for the special case when CM motion can 2πas =C(Eb), (21)
be separated out.
Second, this scheme is physically insightful in that it with
reveals general couplings between the CM and relative
1 1 φ (0)2
motions. Specifically,CMservesasexternalindicesofthe l
C(E) = + | | , (22)
matrix,whilerelativemotionscontributetoeachelement V k ǫk l E−El+iδ
by the renormalizationof internal degree offreedom. As X X
we shall see in Section III, this picture is essentially im- here φ (r) is a complete set of eigen-states only for
n
{ }
portant to understand the mechanism of resonance scat- therelativemotion. AconvergentsolutionofE requires
b
tering in the two-body system. that the ultraviolet divergence in each term of Eq.22 be
More advantages related to the realistic calculations exactly cancelled with each other. This is actually satis-
willbeshowninSectionIIIBandIV,whenapplyingthis fied by a regular potential V (x) without singularity at
T
method to specific two-body and three-body problems. any position x[36].
However, the present T-matrix method still has lim- TheT-matrixelement, T mTˆ(E)n ,whichrep-
mn
itations under certain circumstances. When examining resents the scattering amplitud≡e hfrom| init|ialistate n to
Eqs.(10,11), one can see that the convergence of the so- final state m , is given by | i
lution from the matrix equation generally requires two | i
conditions. First,eachmatrixelementisfinite; secondly, φ (0)φ (0) µ
∗m n = C(E). (23)
thesolutionisindependentofthematrixsize. Anyviola-
T 2πa −
mn s
tion of above two conditions implies that the zero-range
model is insufficient to characterize the interacting sys- Compared with Eq.21, this shows that a bound state
tem, such as when Efimov physics emerges in the three- emerges when all elements T simultaneously diverge.
mn
body sector[9, 10]. T-matrix is able to identify the vio- Similarly for reduced interaction in the lower dimen-
lation of the first condition (see Section IV). However,it sion, we obtain
wouldbe quite involvedfor itto identify the secondcon-
dition. Alternatively, when Efimov physics appear, one ψ (0)2 µ
| 0 | = CQ(E), (24)
can solve the problem by resorting to hyperspherical co- g 2πa −
eff s
5
where CQ follows the form of Eq.22 with the second AccordingtoT-matrixmethod, assumingEq.11yields
summation over all closed channel states. In this sense
the confinement induced resonance(CIR), referring to Cλλ′ λ′ν =cν λν, (28)
F F
g , occurs at
eff
→∞ with c (ν = 1,2...) the eigenvalue and the corre-
ν ν
F
µ sponding eigenvector, then we have
=CQ(E). (25)
2πa
s
ψ ψ
0 ν ν 0
ψ T ψ = h |F ihF | i. (29)
This in turn determines a closed channel bound state h 0| | 0i µ c
with the same energy as ψ . Compared to existing ex- Xν 2πas − ν
0
| i
ploration of CIR mechanism in quasi-1D system[12, 13], This equation explicitly predicts an infinite number of
T-matrix shows in a general way how the divergence of resonances (a ) when each discretized c individ-
eff ν
reducedinteractionintheopenchannelisassociatedwith ually match with→∞µ by tuning a in realistic experi-
the emergence of a closed-channel bound state at the ments. The resona2πnacse position ands width can be con-
same energy level. veniently extracted from the exact diagonalization of C-
Finally, Eqs.(22,23,24) indicate the way how the trap- matrix.
ping potentials modify the low-energy scattering the- To explore the mechanism of these resonances, first
ory. The modification is in fact through the intermedi- we only focus on the diagonal elements of C-matrix.
atevirtualscatteringprocesses,i.e.,byredistributingthe Within eachmolecularchannel,allrelativemotionlevels
energy levels and changing coupling strengths between are coupled together by the attractive interaction and
these states. Above formula can be applied to ordinary this potentially leads to a bound state. This bound
harmonic confinements studied before[11–13, 15–17]. state(relativemotion)combinedwiththemolecularchan-
nel(CM motion) tend to produce the zero total energy,
andthusgiverisetothedivergentT-matrixora .
eff
2. r and R coupled system By tuning the interaction or a , the zero-energy→sta∞te
s
will emerge in order from each molecular channel and
For a general trapping potential, the two-body non- cause the resonanceofaeff. The width of eachresonance
interacting Hamiltonian can be divided to three pieces, is determined by the coupling between the zero-energy
describing the relative motion H (r), CM motion scatteringstateandeachmolecularstate,whichbecomes
rel
H (R), and couplings in-between H (r,R). The narrowerfor higher levels of molecular states.
cm cp
molecular state is then introduced as r = 0,λ , and However, above understanding of multiple resonances
Φ (R) Rλ is the eigen-state of | i is not rigorous, because different molecular channels
λ
≡h | i
could also couple with each other by the combination
2 of H (r) and H (r,R). This additional coupling, as
H (R)= ∇R +V (R)+V (R). (26) rel cp
cm −2M T,1 T,2 shown by off-diagonal elements of C-matrix, would give
a correction to the ideally predicted resonance position.
Combining with Eq.10, one can obtain all solutions cor- In the following section, we shall address these issues by
responding to (I,II,III) in Section IIB. studying a specific system with multiple resonances,i.e.,
Inthiscase,thetrappingpotentialcaninducemultiple two atoms scattering in 2D-3D mixed dimension.
two-body scattering resonancesas revealedpreviouslyin When external trapping potentials are applied such
several settings[21, 23, 24] by numerical calculations in that at low energies, ψ(x ,x ) only propagates as two
1 2
coordinate space. Next we show that these resonances particles are far apart in a lower dimension(open chan-
can be analytically figured out in the framework of T- nel), then the same analysis can be applied to the ef-
matrix method. We classify the situations by whether fective interaction g in this channel. Now C-matrix is
eff
the effective scattering is in 3D space or in the reduced definedbytheclosed-channelGreenfunctionGQ(Eq.16),
lower dimension. 0
which equally results in the matrix equation
Whenexternaltrappingpotentialsareappliedbutthe
low-energy scattering wavefunction, ψ(x1,x2), can still CQ Q =cQ Q. (30)
behave in a propagating way at large 3D inter-particle λλ′Fγλ′ γFγλ
separations x x ,thenits asymptoticformcan Combined with Eq.18, it gives
1 2
| − |→∞
be written as
ψ Q Q ψ
ψ(x1,x2)∼1− d(xa1e,ffx2). (27) geff =Xγ h 0|2Fπµγasih−FcγQγ| 0i. (31)
Here d is the modified interparticle distance according Therefore geff will go through a resonance as long as
to the confinement(see also Ref.[23] and discussions in one cQ is matched with µ by tuning a . This cor-
β 2πas s
Appendix C); a is the effective scattering length, and responds to the energy of dressed bound state in each
eff
can be directly related to T matrix element for zero- closed molecular channel moves downwards and touches
−
energy scattering state[37]. the threshold energy of open channel. The resonance
6
width would be narrower for higher molecular channels to strong coupling(+ ) side. The (N +1)-th resonance
∞ 2
duetothesmalleroverlapwiththelow-energyscattering of a is characterized by the position (a /a ) = e
eff 0 s res N
state in open channel. and the width W .
N
Amazingly, we find good accordance between e
N
and each diagonal matrix elements C˜ , as shown by
NN
B. Results of scattering in 2D-3D mixed Fig.1(b)andFig.2(b). Thatmeansthecorrectioncaused
dimensions by off-diagonal couplings between different molecular
channels are actually negligible. There are mainly two
We consider one atom (41K or 40K, labeled by A) is reasons for this. First, the amplitudes of off-diagonal
axially trapped by a tight harmonic potential with fre- C matrix elements (CNN′) are much smaller than di-
quency ωA, while the other atom (87Rb or 6Li, labeled ag−onal ones, and decrease rapidly as N N′ increases.
| − |
by B) is free in 3D space. The Hamiltonian reads Secondly, it can be atetributed to the destructive inter-
ference among couplings with different molecular chan-
H(r ,r )= ∇2rA +1m ω2z2 ∇2rB +U δ3(r r ). nels. Tosee this, wecarryouta perturbativecalculation
A B −2mA 2 A A A−2mB 0 A− B in terms of off-diagonal couplings between neighboring
(32) molecular states, and get the relative correction to the
As shown in Appendix C, the effective scattering length N th eigen-value as
a forzero-energyscatteringis definedbythe two-body −
weaffvefunction when dAB =qmµBρ2AB +zaB2 →∞, ∆N =N′X=N±2C˜N|NC˜N−NC˜′|N2′N′/C˜NN. (38)
ψ(ρ ,z ,z ) φ (z ,a )(1 eff ), (33) Note that N 2 and N + 2 terms contribute to ∆
AB A B → 0 A 0 − d − N
AB with opposite signs, which results in further suppressed
with ∆N. InsetsofFig.1(b)andFig.2(b)show∆N forthefirst
elevenresonances;we see that the most significant effect
2πa
eff =V ψ T ψ . (34) of off-diagonal couplings occurs for the first resonance
0 0
µ h | | i but is still negligibly small ( 1.5% for K-Rb mixture
∼
and 0.1% for K-Li mixture).
Here φn(z,a0) is the eigen-state of 1D harmonic os- Th∼e vanishingoff-diagonalcouplingsbetweendifferent
cillator with characteristic length a = 1/(m ω );
0 A A molecular channels establish the unique advantage of T-
µ = m u/(1+u) is the reduced mass and u = m /m
A p B A matrix scheme, i.e., the resonance can be accurately de-
is mass ratio.
termined by only a few number of related matrix ele-
NThdeenmootelescaunlaerigsteant-estaintethofisthcaesfeolilsow|rin=g H0,aNmii,ltownhiearne ments. In the limit of zero couplings, we have F˜N′N ≈
| i δN′N and therefore eN = CNN, WN = fN;0,0 2(see
| |
1 ∂2 1 Eq.C4). In this limit and particularly for resonances at
Hcm(Z)=−2(m +m )∂Z2 + 2mAωA2Z2, (35) as > 0 side, the resonance peosition can be determined
A B
by matching bound state energy in each CM molecular
which describes the CM motion along (trapped) z di- channel with the threshold energy of two particles, i.e.,
rection, with oscillation frequency ω = ωA/√1+u and 1 1 ωA
characteristic length a = a /(1+ u)1/4. According to (N + )ω = , (N =2,4,6...) (39)
0 2 − 2µa2 2
Eqs.(29,34) we obtain s
as shown by the dashed lines in Fig.1(b) and Fig.2(b).
aeff WN This equation, as is the direct outcome of T-matrix
= , (36)
a a0 e method, has been usedpreviouslyto determine the reso-
0 N=X0,2,... as − N nance positions[24].
here e is the eigen-value of C˜ matrix (Eq.C6) deter- In addition, Fig.1 and Fig.2 show that the first
N
mined by C˜NMF˜MN′ = eN′F˜NN−′; the resonance width rtweseoennanreclea,tivwehmichotiiosnmleavienlslywidtuheinttohethloewecsotumploinlegcublaer-
is given by
channel(N = 0), always occurs at a < 0 side regardless
s
WN = ˜N′NfN′;0,0 2, (37) of u > 1 or u < 1. The resonance position, however,
| F | sensitively depends on the value of u, as shown by the
N′
X vertical dashed lines in Fig.1(a) and Fig.2(a). This phe-
with f(N;n ,k ) defined by Eq.C4. Due to the nomenon is closely related to the distinct behaviors of
A z
contact interaction and reflection symmetry of trap- C˜ for different u. On one hand, when u 1 we can
NN
≪
ping potential, only molecular states with even parity omit the n -dependence in ln function in Eq.C6, then
A
(N = 0,2,4...) are relevant in this case. Fig.1(a) and usingEq.C5weobtainC˜ −0foranyN. Thisimplies
NN
→
Fig.2(a) shows the first five resonances of a /a for that when trapping the heavy atom, the resonance po-
eff 0
41K-87Rb(m < m ) and 40K-6Li(m > m ) mix- sitions [(a /a ) ] tend to highly aggregate around un-
A B A B 0 s res
tures, when tuning a /a from the weak coupling( ) shifted position(a /a = 0). On the other hand, when
0 s 0 s
−∞
7

4 4
2 2
a0 a0
a/eff0 a/eff0
-2 -2
-4 (a) -4 (a)
-2 -1 0 1 2 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
a0/aS a0/aS

2.5 C~NN 1.5 C~NN
2.0 eN eN
wN 1.0 wN
1.5 1.5
N (%) 0.10 N (%)
1.0
1.0
0.5 0.5 0.05
0.5 0.0
0.00
-0.5
0 4 8 12 16 20
0.0 0 4 8 12 16 20
0.0
(b) (b)
-0.5
0 2 4 6 8 0 2 4 6 8
N
N
FIG. 1: (Color online) (a).Effective scattering length aeff/a0 FIG. 2: (Color online) Same as Fig.1 except for 40K(2D)-
as functions of a0/as for 41K(2D)-87Rb(3D) mixture. The 6Li(3D) mixture. The dashed line in (a) denotes the first
dashed line denotes the first resonance at a0/as = −0.43. resonance at a0/as =−0.02.
(b)Diagonal matrix element C˜NN of Eq.C6(denoted by ×),
thecorrespondingresonanceposition(a0/as)res(+)andreso-
nancewidth(◦). Reddashed lineisthefunctionfitaccording far away to a = 0 side. On the contrary, if the heav-
s −
to Eq.39. Inset shows the relative correction ∆N as defined ier atom is trapped, µ and V would be little affected by
c
in Eq.38. N = 0,2,4... in (b) respectively correspond to the
thetrappingpotential,givingalmostunshiftedresonance
(N/2+1)−th induced resonance in (a) from left to right.
near a = . This analysis leads to similar conclusions
s
∞
for the resonance scattering in other mixed-dimensional
systems, such as 1D-3D mixtures.
u 1, f is vanishingly small for finite N, then
≫ N;nA,kz
the first term dominates in Eq.C6. This predicts res-
onances approaching a = 0 . Only for large enough IV. APPLICATION TO THREE-BODY
s −
N the rest terms in Eq.C6 would dominate and predict PROBLEM
resonances at a >0 side.
s
Besidesthetwo-bodysystem,thegeneralformulismof
Wenotethatthedependenceofthefirstresonancepo-
T-matrix approach allows its straightforward extension
sition(ata <0side)onthemassratiouisinqualitative
s
to other few-body systems. In this section we focus on a
agreement with that for 0D-3D mixtures[22]. Actually
three-bodysystemcomposedbytwo-componentfermions
we cangainthe physicalinsightof such feature from the
in a (rotating) harmonic trap. We shall first present
analysis ofinteractionpotentials affected by the confine-
theformulismandthenexploretheinterestingscattering
ment. Suppose a square-well interaction potential U(r),
property and identify the ground state level crossing in
which is V at r < r and zero otherwise, between A
0 0
− this system.
and B atoms. As V increases, the first scattering res-
0
onance occurs at the critical value V = (π/2)2/(2µr2)
c 0
with µ the reduced mass. When A or B is trapped and
becomes localized, µ will be effectively enhanced, which A. Formulism
reducesthe criticalV andgivesnewresonanceata <0
c s
side. µandV canbesubstantiallymodifiedifthelighter We consider three fermions with one spin- (x ) and
c 1
atom is trapped, and the resonance position will move two identical spin- (x ,x ) in an isotropic↓harmonic
2 3
↑
8
trap. According to Eqs.(A10-A13), we transform the here
vector X = ( 2m x , 2m x , 2m x ) to Y =
1 2 3
(√2MRR,√2µrp,√↓2µρp) by↑Y pT =↑A XT. ±Here α= Mm↓ , β = m↑ , (47)
r(Resp,re−ct,iρve−l)ycaonrdr±e(sRpo,nrd+i,nρ±g+t)otahreeea±ffllecJtaivceobmi±acsosoMrdin,aµt,eµs;, mp↑+m↓ m↑+m↓
the CM coordinate R and its mass M follow ERq.A11; and α2 + β2 = 1. To obtain Eq.46 we have inserted
R
into the Green function a complete set of eigen-states
the other Jacobi coordinates are
r = x2 x1, {inνd,uλc′}e ftohretchoeupmliontgiobnestwofee(nr+d,iρff+er)e.ntFmλλo′le=culFaλr∗′λlevheelrse,
− −
Mm m x +m x and non-zero Fλλ′ require azimuthal quantum number
ρ = ↓ [x3 ↓ 1 ↑ 2], (40) lm beconserved. Notethattheoff-diagonalcouplingof
− mp↑+m↓ − m↑+m↓ {mole}cularstates here is due to the many-bodystatistics,
with the same mass µ = m↑m↓ ; the transfer matrix incontrarytotheprevioustwo-bodycasewhichisdueto
m↑+m↓ the external trapping potentials. More details regarding
reads
to the evaluation of Eq.45 are presented in Appendix D.
m↓ m↑ m↑
M M M
A− = − qm↑m+↓m↓ qm↑m+↑m↓ q0 ; B. Results
 q m↑ q m↑m↓ m↑+m↓ 
 −√M(m↑+m↓) − M(m↑+m↓) M  In the first part of this section we use T-matrix
 q q (41)
methodtoanalyzetheexoticscatteringpropertyofthree
we further obtain ρ ,r by exchanging x x in
r ,ρ , and obtain A+ +by exchanging the 2sec↔ond3and fermions in different limits of mass ratios and in differ-
+ ent angular momenta channels. In the second part, we
th−ird−column of A .
presenttheenergyspectrumandidentifytheenergylevel
Taking advanta−ge of the property of transfer matrix
crossingbetweendifferentangularmomentastatesforthe
(Eq.A14), we can see that with the same trapping fre-
(rotating) trapped system.
quency ω, all three Jacobi coordinates (R,r ,ρ ) can
be well separated from each other. Indepen±dent±ly one
can also prove that the total angular momentum is
1. Scattering property
also separable as Lˆ (x ) = Lˆ (R)+Lˆ (ρ )+
i=1,2,3 α i α α
Lˆα(r ) (α = x,yP,z). Therefore for a trapped sy±stem ByanalyzingEqs.(45,46),wefindnontrivialscattering
with±rotating frequency Ω around z-direction, the rele- properties at two limits of mass ratio u=m /m . Fig.3
vant Hamiltonian in the rotating frame reads showsthe schematicplotsofJacobicoordina↑tes(↓r ,ρ )
H(ρ ,r )=H0(ρ )+H0(r )+U0δ(r+)+U0δ(r ), in both limits of u→0 and u→∞. − −
± ± ± ± −(42)
here
(cid:1)(cid:1)(cid:2)(cid:2)↓↓ (cid:1)(cid:1)(cid:4)(cid:4)(cid:3)(cid:3) (cid:1)(cid:1)(cid:2)(cid:2)↓↓ (cid:1)(cid:1)(cid:2)(cid:2)(cid:3)(cid:3)
2 1
H0(r)= ∇r + µω2r2 ΩLz(r). (43) (cid:1)
−2µ 2 − (cid:1)
ρ(cid:5)α −
The molecular state is defined with respect to Fermi − −
statistics,
(cid:3)(cid:2)↑ (cid:4)(cid:2)↑ (cid:3)(cid:2)↑ ρ(cid:5)α (cid:4)(cid:2)↑
1
−
λ = (r =0,λ r =0,λ ). (44)
+
| i √2 | − i−| i
Herethefirstandsecondλrepresenttheidenticalenergy FIG. 3: (Color online) Jacobi coordinates (r−,ρ−) for three
level nlm for the motions of ρ and ρ under Hamil- fermions(↑↑↓) in the limit of u → 0 (a) and u → ∞ (b).
tonian{ H0.} ( n and lm are−respecti+vely the radial u=m↑/m↓ is themass ratio. α is given by Eq.47.
{ } { }
and azimuthal quantum number). Then we obtain the
C matrix element as
First, when u 0 as shown by Fig.3(a), α 1, β
−
→ → →
1 1 ψ (0)2 0, we have
ν
Cλλ′ =( + | | )δλλ′ Fλλ′,
V Xk ǫk Xν E−Eλ−Eν +iδ − (45) Fλλ′ ∼δl,0δl′,0. (48)
with Therefore the diagonal C-matrix for l = 0 indicates the
6
Fλλ′ = hr− =0,λ|Gˆ0|r+ =0,λ′i Eatdo.mP-dhiymsiecralulyncwoerrcealantesdeesyfrsotmemFwigi.t3h(ae)nethrgayt,Eth=e dEimae+r
ψ (0)ψ ( αρ)
= Z dρψλ∗(ρ)ψλ′(−βρ)Xν E−νEλ′ −ν E−ν +i(δ4,6) fsoinrmgleed↓baytoaml,igshotth↑eaontdhehrea↑vhya↓s si-swaalvmeoisntteerqaucitviaolnenwtittho
9
this dimer only when l=0. Here the heavy dominates
↓
the whole physics.
Second, in the opposite limit when u as shown 2.0 (a) u=2
byFig.3(b),theresultiscompletelydiffer→ent∞. Wefindin u=1
u=0.5
this limit, 1.6 u=0
2)/w L=0
Cλλ′ =(−V1 Xk ǫ1k−Xν E−E|ψλν−(0)E|2ν +iδ[1−(−1)l])δλλ′. (E-E-3w/d01..82 u=2
(49) u=1
0.4 u=0.5
There are two direct consequences as follows. u=0
(i)for odd l, Cλλ′ = , i.e., unphysical divergence in 0.0 L=1
thehigh-energyspaceca∞nnotbeproperlyremoved. This -10 -5 0 5 10
a0/aS
is exactly the evidence of Efimov effect for large u where
anothershort-rangeparameterisrequiredto helpfix the

three-body problem[9, 10]. 1.0
(b)
(ii)for even l, Uˆ takes no effect and the sys-
0.8
tem just behaves like non-interacting. This result is
L=m=1
consistent with that obtained by Born-Oppenheimer
w 0.6
approximation(BOA)[40]. UnderBOA,thewavefunction /
is given by 0.4 u=2
u=1
ψ(x1,x2,x3)=[ϕ(|x2−x1|)+γϕ(|x3−x1|)]f(x2,x3), 0.2 uu==00.5 L=m=0
(50)
where the first part describes the light particle moving 0.0
aroundtwostaticheavyparticles,andf(x ,x )describes -2 0 2 a0/aS 4 6 8
2 3
for two heavy particles afterwards. By imposing Bethe-
Peierls boundary conditions one can find γ = 1, and FIG. 4: (Color online) (a).Energy of three fermions(↑↑↓) vs
the energy of the first part just depends on x± x . interactionstrengthinanon-rotatingisotropicharmonictrap,
2 3
| − |
Therefore the wavefunction is reduced to with a0 = 1/(µω) the confinement length. Different mass
ratios u = m↑/m↓ for total angular momentum l = 0(red)
ψ(x1,x2,x3)=[ϕ(r ) ϕ(r+ )]f1(x2 x3)f2(R), and l = 1(pblack) are plotted. The energy is shifted by the
| −| ± | | − (51) ground-state atom-dimer energy, Ed+3ω/2(Ed is the dimer
and then angular momentum l is determined only by energy). (b)Phase diagram in a rotating harmonic trap with
f (x x ). For γ = 1, the Fermi statistics require l rotating frequency Ω. The ground state is |l=m=1i above
1 2 3
be odd−; for γ = 1, l is even but in this case one can thecurvesand |l=m=0i below.
−
easily check that the resultant wavefunction automati-
cally get rid of the interaction, i.e., Uˆψ =0. Here Fermi
statistics of two spins take the crucial role. The system in weak interacting limit(as 0−) be-
↑ →
Notethattheresultspresentedin(i,ii)uniquelybenefit haves as non-interacting while in molecule limit(a
s
fromtheconceptofrenormalizationandtheprocedurein 0+) as a single dimer plus anatom. This directly resul→ts
momentumspacetoeliminatetheultravioletdivergence. intheinversionofgroundstatefromangularmomentum
Theseanalysesofscatteringpropertiesfordifferentmass l = 1 to l = 0 as 1/a increases. As shown in Fig.4(a),
s
ratios and different angular momenta will be helpful to theinversionisdenotedbythe energylevelcrossing,and
understand the ground state level crossing in the follow- the positionoflevelcrossingcloselydepends onmassra-
ing section. tio u. As expected, when u increases from 0 to , all
∞
energy levels with even-l move upwards and the system
evolves from decoupled atom-dimer(except for l = 0) to
2. Energy level crossing three atoms that are immune from interactions; while
all odd-l move downwards until Efimov physics show up
First, we identify the energy level crossing between and invalidate the present T-matrix method. Therefore
different angular momenta states for a non-rotating sys- byincreasingu,thepositionoflevelcrossingwillmoveto
tem. The energy spectrum of a non-rotating system was strongcouplingsideasshowninFig.4(a). Intuitively,one
previously studied for equal mass[31, 32], and unequal can also attribute this to the enhanced s-wave repulsion
massesusingGaussianexpansiontechnique[38]andadia- between atom and dimer[3].
batichypersphericalmethod[39]. InFig.4(a)weshowthe In unitary limit, our numerical results are in good
spectrum for angular momenta l = 0,1 and for different accordance with those obtained by using hyperspheri-
mass ratiosusing T-matrix method. We also checkedfor cal coordinates. Previously, hyperspherical coordinate
higher l 2 and confirm those states are less modified methodhasbeenappliedtoatrappedsystemwithequal
≥
by the interaction and thus not shown here. mass[33, 34]. In Appendix E we extend this method to
10
arbitrary mass ratios. Note that the maximum mass ra- Jason Ho for valuable suggestions on the manuscript.
tio considered in Fig.3 is much less than critical value, ThisworkissupportedbyTsinghuaUniversityBasicRe-
u = 13.6[3], for the emergence of Efimov state in l = 1 search Young Scholars Program and Initiative Scientific
c
channel(asalsopredictedbyEq.E7whensetting s=0). ResearchProgramandNSFCunderGrantNo. 11104158.
In this regime, as the matrix size increases we get con- The authorwouldlike to thank the hospitality ofthe In-
vergent result for the energy spectrum. stitute for Nuclear Theory at University of Washington,
Finally, we present the ground state for the trapped wherethisworkisfinallycompletedduringtheworkshop
system with rotation(Ω > 0). By comparing energies on”FermionsfromColdAtoms to NeutronStars”in the
of all different angular momenta l we obtain the ground spring of 2011.
state as shown in Fig.4(b). We find l = m = 1 state is
gradually favored by the rotation. The energy gain of
this state is analyzed to be partly from the reduction of
kinetic energywith respectto l =0,and partly fromthe Appendix A: Construction of individual molecular
avoided s-wave repulsion between atom and dimer. As state
Ω increases,the system evolvesto the atom-dimer quan-
tum Hall state; particularly at Ω = ω, all states with In this appendix, we show how to construct an indi-
odd-l degenerate. FinallyweexpectabovequantumHall vidual molecular state in the most efficient way. Let us
physics of fermionic system could be studied in experi- consider a system of N particles with masses m ,...m
1 N
ment,asrecentlyrealizedinarotatingfew-bodybosonic and coordinates x ,...x . For a generalcase, the molec-
1 N
system[41]. ular state is written as
x =x ,λ= N,n ,n ,...n . (A1)
1 2 3 4 N
| { }i
V. SUMMARY
In coordinate space it can be factorized as
Φ (X) N φ (x ), where φ (x ) is the eigen-state of
In conclusion, we present a systematic T-matrix ap- N i=3 ni i ni i
single-particle Hamiltonian
proachto solve few-body problems with contact interac-
Q
tionsinthefieldofultracoldatoms. Takingadvantageof
2
zero-rangeinteractions,the key ingredientofthe present Hˆ (x )= ∇xi +V (x ) (A2)
0 i T,i i
−2m
T-matrix method is to project the problem to a sub- i
space that is expanded by orthogonal molecular states, and Φ (X) the eigen-state of
and meanwhile take careful considerations of the renor- N
malizationforrelativemotions. Thismethodsuccessfully 2
unifies the calculations of various physical quantities in Hˆ0(X)= ∇X +VT,2(X)+VT,2(X). (A3)
−2(m +m )
a single framework, including the bound state solutions, 1 2
effective scattering lengths and reduced interactions in
Here V is the trapping potential for the i th particle.
T,i
the lower dimension. −
The overlapbetween the molecular state(Eq.A1) and N-
We present two applications of this approach, namely, particle state( N l ) is
two-body scattering resonances in 2D-3D mixed dimen- j=1| ji
sions and properties of three fermions( ) in a 3D (ro- Q
↑↑↓ N
tating) trap. For the two-body problem, we show that x =x ,λl ,l ,l ,...l =γ δ (A4)
T-matrixprovidesaphysicallytransparentwaytounder- h 1 2 | 1 2 3 Ni N;l1,l2 nili
i=3
stand the mechanism of induced scattering resonances. Y
Besides, it also gives explicit expressions for the reso- with
nance positions and widths. Due to the separate treat-
ment of relative motions from CM motions, each reso- γ = dXΦ (X)φ (X)φ (X). (A5)
N;l1,l2 ∗N l1 l2
nance can be determined accurately by only consider-
Z
ing a few related matrix elements. For the three-body
TheGreenfunctionterminEq.11canthenbecomputed
problem, T-matrix enables us to identify exotic scatter-
efficiently as
ing properties of three fermions in different angular mo-
mtaetnintgumsycshteamnn,etlhseasnedpwroitpherdtiiffeserpernotvmidaessimraptoiortsa.nItnhainrots- hrI =0,λ|Gˆ0(E)|rI′ =0,λ′i
for the quantumHalltransitionfromzeroto finite angu- = hrI =0,λ|l1...lNihl1...lN|rI′ =0,λ′i.(A6)
lmarasmsormateionst,uamndstbatoes.onOicv/efrearlml,itohneicexstteartnisatliccsonalfilnpelmayenimts-, l1X...lN E−(El1 +...+ElN)+iδ
portant roles and give rise to very rich phenomenon in To this end we have showna generalway to construct
these few-body systems. an individual molecular state. Furthermore, for special
The author thanks Hui Zhai, Shina Tan, Fei Zhou, trapping potentials which enable the decoupling of CM
Hui Hu and Doerte Blume for useful discussions, and motionfromothermotions,itisconvenienttoremovethe