Bound states in a quasi-two-dimensional Fermi gas

0.41 MB·English

by  Jesper Levinsen

Quasi-two-dimensional bound states
Jesper Levinsen1 and Meera M. Parish1,2
1T.C.M. Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom
2London Centre for Nanotechnology, Gordon Street, London, WC1H 0AH, United Kingdom
(Dated: July 3, 2012)
We consider the problem of N identical fermions of mass m and one distinguishable particle
↑
of mass m interacting via short-range interactions in a confined quasi-two-dimensional (quasi-2D)
↓
geometry. For N = 2 and mass ratios m /m < 13.6, we find non-Efimov trimers that smoothly
↑ ↓
evolvefrom2Dto3D.Inthelimitofstrong2Dconfinement,weshowthattheenergyoftheN+1
systemcanbeapproximatedbyaneffectivetwo-channelmodel. Weusethisapproximationtosolve
2 the3+1problemandwefindthataboundtetramercanexistformassratiosm /m aslowas5for
↑ ↓
1 strong confinement, thus providing the first example of a universal, non-Efimov tetramer involving
0 three identical fermions.
2

l
u Anunderstandingofthefew-bodyproblemcanbeim-
J portant for gaining insight into the many-body system. 6

2 In dimensions higher than one, few-body bound states
can, for instance, impact the statistics of the many-body
]
s quasiparticle excitations. Indeed, for fermionic systems, 5
a the two-body bound state is fundamental to the under-
g
standingoftheBCS-BECcrossover[1–4],whiletheexis-
-
t tence of three-body bound states of fermions [5, 6] with
n 4
a unequal masses can lead to dressed trimer quasiparti-
u cles in the highly polarized Fermi gas [7]. Even in one
q dimension (1D), few-body bound states can impact the
. 3
t many-body phase: It has already been shown that one
a 0 0.1 0.2
can have a Luttinger liquid of trimers [8].
m
Ingeneral,attractivelyinteractingbosonsreadilyform
-
d bound clusters, with the celebrated example being the
FIG.1: (Coloronline)Criticalmassratiofortheappearance
n Efimov effect in 3D [9], where there is a universal hierar-
oftrimersandtetramersinquasi-2D,wherethe2Dlimitcor-
o
chyoftrimerstatesforresonantshort-rangeinteractions. responds to (cid:15) /ω →0. The solid line follows from the solu-
c b z
[ However, for identical fermions it is less clear whether tionofthefullthree-bodyquasi-2Dproblem,Eq.(8). Dashed
bound states exist, since these are constrained to have linesfollowfromaneffectivetwo-channelmodel. Thevertical
1 odd angular momentum owing to Pauli exclusion. For dotted line marks unitarity, where the 3D scattering length
v diverges.
short-range interactions in 3D, trimers consisting of two
9
5 identical fermions with mass m↑ and one distinguishable
4 particle with mass m can only exist above the critical
↓
0 mass ratio m /m (cid:39) 8.2 [5], while Efimov trimers only Here we show that tetramers involving N = 3 identical
↑ ↓
7. appear once m↑/m↓ (cid:38) 13.6 [10]. However, the existence fermions can appear for m↑/m↓ as low as 5 in quasi-2D
0 oflarger(N+1)-bodyboundstatesinvolvingN >2iden- (see Fig. 1), thus putting it within reach of current cold-
2 ticalfermionsremainslargelyunknown—ithasonlyre- atom experiments.
1
centlybeenshownthatEfimovtetramersexistin3D[11]. Weconstructthegeneralequationsfortheboundstate
:
v In this Letter, we investigate the problem of N iden- of the N +1 system in quasi-2D and we reveal how to
Xi tical fermions interacting with one distinguishable par- simplify the problem in the case of the trimer (N = 2).
r ticle in a confined quasi-2D geometry, where the cen- In the limit of strong 2D confinement, we show that the
a trifugal barrier is reduced and the binding of fermions N + 1 problem can be described by an effective two-
should be favored. Such 2D geometries have recently channel model, analogous to that used for Feshbach res-
been realised in ultracold atomic Fermi gases [12–16], onances. Thisimportantsimplificationallowsustosolve
where the fermions are confined to 2D with an effec- the aforementioned N =3 problem in quasi-2D.
tive harmonic potential. In addition to allowing one to In the following, we assume the two atomic species
explore the 2D-3D crossover, the harmonic confinement {↑,↓} to be confined to a quasi-2D geometry by an ap-
can strongly modify the scattering properties of atoms proximately harmonic potential along the z direction,
viaconfinement-inducedresonances[6,17,18]. Ithasal- V (z)= 1m ω2z2. Here,werestrictourselvestoequal
↑,↓ 2 ↑,↓ z
readybeendemonstratedthatstablenon-Efimovtrimers confinementfrequenciesforthetwospeciessinceitallows
canexistforlowermassratiosm /m inquasi-2D[6,19]. a separation of the relative and center of mass motion
↑ ↓
2
along the z-direction, as we discuss below. Such a sce-
nario can, in principle, be engineered experimentally us-
ingspin-dependentopticallattices. However,eveninthe
=
case where the confinement frequency is species depen-
dent, regimes exist in which the few-body properties are
only weakly affected by this dependence. For instance,
forlargemassratiosandonthemolecularsideoftheFes-
hbach resonance, once the ↑↓ dimer is smaller than the
light atom oscillator length, l↓ = (cid:112)(cid:126)/m ω , the light
z ↓ z FIG. 2: The diagrams which give the binding energy of the
atom is essentially confined by its interaction with the
N+1boundstateinquasi-2D.Blackdotsindicatetheinitial
heavy atoms [6]. interaction inside f.
The starting point of our analysis is the T-matrix de-
scribing the repeated two-body interspecies interaction.
In the ultracold gases, the interaction is described by a even, and 0 otherwise. The Clebsch-Gordan coefficients
zero-range model as the van der Waals range of the in- Cnnn0nr1(m↓,m↑) ≡ (cid:104)n0n1|nnr(cid:105) were obtained in Ref. [22]
teratomicpotentialismuchsmallerthanallotherlength and vanish unless n0+n1 =n+nr.
scalesintheproblem,includingtheconfinementlengths. Wenowturntothequestionoftheexistenceofbound
TheT-matrixmaybeconsideredinthebasisoftheindi- states consisting of N spin-↑ atoms and a single spin-↓
vidual motion of a spin-↓ and ↑ atom. However, due to atom. Tothisend,weconstructthesumofconnecteddi-
the restriction to equal confinement frequencies for the agramswithN+1incomingatoms(Fig.2). The↑atoms
two species, the center of mass and relative motion sep- are considered on-shell with 2D momenta ki, harmonic
arates and it is advantageous to work in this basis. In oscillatorquantumnumbersni,andcorrespondingsingle-
the center of mass frame of the harmonic oscillator po- particleenergies(cid:15)kini↑ =ki2/2m↑+niωz fori=1,...,N.
tential, atenergy(cid:15)belowthetwo-bodythresholdω (we Weconsiderscatteringinthecentreofmassframeofthe
z
set (cid:126) = 1) and at total 2D momentum q, the T-matrix 2D motion and at a total energy E below the N + 1
takes the form [20] atom threshold (N +1)ωz/2. Thus, the ↓ atom has 2D
T(q,(cid:15))= √2π (cid:26)lzr −F(cid:18)−(cid:15)+q2/2(m↑+m↓)(cid:19)(cid:27)−1, mtuommnenutmubmerkn00,≡an−d(cid:80)enNi=er1gkyi,Eh0a≡rmEon−ic(cid:80)oNis=c1ill(cid:15)aktionri↑q.uTahne-
mr as ωz sum of diagrams with N +1 incoming particles in which
(1) the ↓ atom interacts first with the ↑ atom numbered 1
where the zero-range interaction is renormalized by the is denoted fn0...nN. Note that there is no dependence
use of the 3D scattering length, as. Here, mr = on k1 as thekin2.i.t.kiaNl interaction depends only on the total
m↑m↓/(m↑ + m↓) is the reduced mass and lzr = momentum of the two atoms.
(cid:112)
1/2mrωz is the confinement length corresponding to The occurrence of a bound state corresponds to a sin-
the relative motion. We use the definition of F [21] gularity of f at its binding energy. This singularity re-
(cid:32) (cid:33) sults from the summation of an infinite number of dia-
(cid:90) ∞ du e−xu gramsand,atthepole,f satisfiesthehomogeneousinte-
F(x)= √ 1− . (2)
(cid:112)
0 4πu3 [1−exp(−2u)]/2u gralequationillustratedinFig.2: Theinitialinteraction
isdescribedbyaT-matrix,andthenthespin-↓atomsub-
The two-dimensional scattering always admits a two- sequently interacts with another of the ↑ atoms. Thus,
body bound state of mass M = m↑ +m↓ and binding therighthandsidecontainsN−1termsandtheintegral
energy (cid:15)b >0 satisfying lzr/as =F((cid:15)b/ωz). equation satisfied by the bound state energy is (setting
The T-matrix in the basis of individual motion is re- the volume to 1):
lated to T by the change of basis
n(cid:48)n(cid:48)
Tnn0(cid:48)0nn1(cid:48)1(q,(cid:15)) = n(cid:88)nrn(cid:48)rCnnn0nr1(m↓,m↑)Cnnn(cid:48)0n(cid:48)r(cid:48)1(m↓,m↑) fkn20......knNN = −k(cid:48)1(cid:88),n(cid:48)0n(cid:48)1 ET0n0+0n11(cid:15)k(k1n01+↑−k1(cid:15),kE(cid:48)0n0(cid:48)0↓+−(cid:15)k(cid:15)1kn(cid:48)11n↑(cid:48)1)↑
×ψnr(0)ψn(cid:48)r(0)T(q,(cid:15)−nωz). (3) ×(cid:110)fkn(cid:48)1(cid:48)0kn32.n..(cid:48)1knN3...nN +...+fkn2(cid:48)0.n..NkNn2−.1..kn(cid:48)1N−1n(cid:48)1(cid:111), (4)
Here, n and n are the quantum numbers labelling the
0 1 wherek(cid:48) =k +k −k(cid:48),andtheminussignonther.h.s.
eigenstates of the single-particle Hamiltonians H = 0 0 1 1
↓,↑ appears because f is antisymmetric under the exchange
−2∇m20↓,,1↑ + 12m↓,↑ωz2z02,1 while nr and n are the quan- of incoming fermions. Equation (4) embodies a simple
tum numbers in the basis of relative, z = z − z , and generic formulation for the (N +1)-body problem
01 0 1
and center of mass, Z = (m z + m z )/M, coordi- in quasi-2D, which in principle allows us to capture the
01 ↓ 0 ↑ 1
nates. The wavefunction of the relative motion takes crossover from 2D to 3D. Indeed, for the case of N = 2,
(cid:112)
the value ψnr(0) = (−1)nr/2 (nr−1)!!/nr!! if nr is it is a generalization of the Skorniakov-Ter-Martirosian
3
equationforatom-dimerscattering[23],whileforN =1, -1
Eq. (4) simply reduces to the condition for the two-body
binding energy. Finally, we note that Ref. [24] derived
-1.05
an expression similar to our Eq. (4) for the 3D N +1
problem.
An important simplification to Eq. (4) becomes possi- -1.1
bleinthelimitofstrongquasi-2Dconfinement, ω (cid:29)(cid:15) .
z b
Here, the function F can be expanded as
-1.15
1 ln2
F(x)≈ √ ln(πx/B)+ √ x+O(x2) (5)
2π 2π
0 0.05 0.1
withB ≈0.905[20,21]. Ontheotherhand,considerthe
denominatoronther.h.s. ofEq.(4)whichweshallwrite
FIG. 3: (Color online) Energy of the trimer in quasi-2D for
for simplicity as (cid:15)−nω . Here, the typical energy scale
z mass ratios m /m = 3.5, 4, 5, 6.64 (from top to bottom).
(cid:15)∼(cid:15) since, for bound states, the function f is strongly ↑ ↓
b √ The solid lines correspond to the full calculation, while the
peaked at momenta ∼ 2mr(cid:15)b, while it quickly decays dashedlinesarederivedfromtheeffectivetwo-channelmodel,
for large momenta. Now, if we expand the denominator Eq. (6).
in powers of (cid:15) /ω (assuming n (cid:54)= 0), then the lowest
b z
ordertermvanisheswhenintegratedovermomentumdue
to the antisymmetry of fk(cid:48).... Consequently, the lowest and N012, we adopt the new basis:
1
non-vanishing contribution from the denominator is of
onrodne-rze(r(cid:15)ob./ωWze)2cownhcelnudtehethhaatrtmoolniniceaorsocirldlaetroirnin(cid:15)bd/eωxznthies χnk2021 = ψn011(0)n0(cid:88)n1n2(cid:104)N012n021n01|n0n1n2(cid:105)fkn20n1n2. (7)
integral equation for the N +1 bound state reduces to
Here,wehavedroppedn fromthel.h.s. astheresulting
01
f = T˜(k +k ,E˜ +(cid:15) ) equation will be independent of this index. Also, the
k2...kN 0 1 0 k1↑ centerofmassquantumquantumnumber,N ,hasbeen
×(cid:88)fk(cid:48)1k3...kN +...+fk2...kN−1k(cid:48)1, (6) neglected as it only causes a shift in the ener0g12y; since we
k(cid:48)1 E˜0+(cid:15)k1↑−(cid:15)k(cid:48)0↓−(cid:15)k(cid:48)1↑ consider the lowest lying trimer, N012 will be set to 0 in
the following. In the new basis, Eq. (4) for the trimer
with the single particle energies (cid:15) ≡ (cid:15) , E˜ = E − becomes
k k0 0
(cid:80)Ni=1(cid:15)ki↑, and T˜ obtained from Eq. (1) using the linear χn021 =T (cid:0)k ,E−(cid:15) −n01ω (cid:1)
expansion of F, Eq. (5). As the effects of confinement k2 2 k2↑ 2 z
iTn-mthaitsrilxim, Eitqa.r(e6)comntaayinbede osobltealiynewditthhinrouthghe lainsetarriiczteldy × (cid:88) ψn02(0)ψn(cid:48)01(0)(cid:104)n021n(cid:48)01|n012n02(cid:105)χnk(cid:48)1012 ,(8)
2D2-channelmodel[25],wheretheclosedchannelcorre- k(cid:48)1,n012n02nE(cid:48)01−(cid:15)k(cid:48)1↑−(cid:15)k2↑−(cid:15)k(cid:48)1+k2↓−(n012+n02)ωz
sponds to excited harmonic oscillator modes. Thus, the
confinement length lr plays the role of an effective range where we can see that the r.h.s. indeed does not depend
z
inthismodel,withthe2Dlimitlr/a →0corresponding on n01. The matrix element in Eq. (8) may be evaluated
z s
by a series of coordinate transformations:
to a single-channel model. This simplification crucially
dependsontheantisymmetryresultingfromFermistatis- (cid:88)
(cid:104)n01n |n02n (cid:105)= CN01n2(M,m )
ticsanditthusdoesnotapplytoboundclustersinvolving 2 01 1 02 0n01 ↑
2
bosonsconfinedto2D,asconsideredinRefs.[26,27]. Fi- n0n1n2N01N02
nally, we note that a similar simplification was recently × CNn00n1n101(m↓,m↑) CNn00n2n202(m↓,m↑)C0Nn00122n1(M,m↑) ,
obtained for quasi-1D atom-dimer scattering [28].
Wenowproceedtosolvethethree-bodyproblemusing whereseveralsumscanbedroppedduetotheconstraints
theabovemethods. First,notethatusingthezero-range on the Clebsch-Gordan coefficients.
condition and removing the center of mass generally al- Since the trimer consists of identical fermions, it must
lows one to reduce the number of harmonic oscillator necessarily have odd angular momentum L in the x-y
quantum numbers by two in Eq. (4) [6]. For the three- plane of the 2D layer. Thus, the lowest-energy trimer
body problem, this is achieved by changing coordinates has L = 1, and it can be regarded as a p-wave pairing
totherelativemotionofthetwoatomsinitiallyinteract- of ↑ fermions mediated by their s-wave interactions with
tinhgir,dz0a1to=mz,0z−01z=1,(tmhezrel+atmivezm)o/t(imon+ofmthe)−pazir,aannddtthhee twhheelrieghφt↓isptahreticalneg.lIenotfhkiscwaisteh,wreespheacvtetχonkt202h1e=x-χ˜ankx202i1seaiφn2d,
2 ↓ 0 ↑ 1 ↓ ↑ 2 2 2
centerofmassZ =(m z +m z +m z )/(m +2m ). χ˜isafunctionofk ≡|k |. Integratingoverφ inEq.(8)
012 ↓ 0 ↑ 1 ↑ 2 ↓ ↑ 2 2 2
Defining the corresponding quantum numbers n , n01, then leaves an integral equation that only depends on k
01 2 2
4
barrierbetweenheavyparticles(whichgoesas1/m )di-
↑
-1
minishescomparedwiththeeffectiveattractivepotential
induced by the light particle (∼1/m ).
↓
Perturbing away from the 2D limit, we find that the
0
trimer-tetramer transition shifts to larger m /m with
↑ ↓
increasing (cid:15) /ω , as shown in Fig. 1. Eventually, we ex-
b z
pect to encounter the four-body Efimov effect in 3D for
-0.005 m↑/m↓ > 13.4 [11]. However, it remains an open ques-
tion whether our quasi-2D tetramers exist in 3D below
5 10 the critical mass ratio for Efimov physics.
-1.5 To conclude, we have provided the first example of
0 5 10
a universal, non-Efimov tetramer involving three iden-
tical fermions. Since this quasi-2D tetramer exists for
FIG.4: (Coloronline)Energyofthedimer(solidline),trimer mass ratios m /m as low as 5, it could potentially be
↑ ↓
(dotted), and quadrumer (dashed) in 2D as a function of probed with ultracold 6Li-40K mixtures. Its small bind-
mass ratio. The trimer binds when m /m > 3.33, consis-
↑ ↓ ing energy (Fig. 4) suggests that it could appear as a
tent with Ref. [19], while the trimer-tetramer transition oc-
resonance in atom-trimer interactions. For instance, if
curs at m /m = 5.0. Inset: The difference between trimer
↑ ↓ a cloud of trimers were prepared under strong quasi-2D
and quadrumer energies, E −E .
3 4
confinement, signatures of the resonance could be ob-
served by colliding a cloud of heavy atoms with a cloud
and n01. The same applies for the two-channel model of trimers, similar to the proposal of Ref. [29] for detect-
2
Eq.(6)withN =2,wherenowthereisonlyadependence ing an atom-dimer resonance. In addition, the presence
on k . of trimers and tetramers has implications for the many-
2
Wehavecalculatedthetrimerbindingenergyasafunc- body phases in quasi-2D, particularly for the highly po-
tionofconfinementforarangeofmassratios,asdepicted larized Fermi gas [30, 31].
inFig.3. Weseethatthebindingenergydecreasesaswe We emphasize that although we have focussed on the
perturb away from 2D and the centrifugal barrier is in- N +1 problem in quasi-2D, the form of Eq. (4) is com-
creased. Correspondingly, we find that the critical mass pletely general and may be extended to other shapes of
ratio m /m for trimer binding smoothly evolves from the confining potential and/or different dimensionalities.
↑ ↓
the2Dlimitof3.33[19]towardsthe3Dresultof8.2[5],as Forinstance,inquasi-1DonewouldusetheT-matrixde-
showninFig.1. Forthespecialcaseof6Li-40Kmixtures, rivedinRefs.[17,21],alongwithappropriatelyredefined
where m /m =6.64, our results agree with Ref. [6]. In harmonicoscillatorandmomentumindices. Futhermore,
↑ ↓
the limit of strong 2D confinement, we see that the two- theproblemmaybestudiedclosetonarrowFeshbachres-
channel model captures the lowest-order dependence on onances,characterizedbyalargeeffectiverange,byusing
(cid:15) /ω of the trimer energy and critical mass ratio. an energy-dependent scattering length [32]. Finally, our
b z
We can exploit the two-channel model (6) to solve the work suggests that a two-channel model may be used to
more complicated four-body (N = 3) problem in quasi- model strongly confined quasi-2D Fermi systems in gen-
2D. Once again, the presence of identical fermions re- eral.
quires us to consider total angular momentum L = 1. We gratefully acknowledge fruitful discussions with
Thus, we have for the tetramer Stefan Baur, Andrea Fischer, Pietro Massignan, Wave
Ngampruetikorn, and Dmitry Petrov. MMP acknowl-
fk2k3 =f˜(k2,k3,∆φ32)eiφ2 =−f˜(k3,k2,−∆φ32)eiφ3, edges support from the EPSRC under Grant No.
EP/H00369X/1. JL acknowledges support from a Marie
where ∆φ =φ −φ . We note that a similar equation
32 3 2 Curie Intra European grant within the 7th European
forthetetramerenergywasobtainedforthe3Dproblem
Community Framework Programme.
in Ref. [11].
Beginningwiththe2Dlimit((cid:15) /ω =0),wedetermine
b z
the energy of the tetramer compared to the trimer and
dimerenergies(seeFig.4). Followingthetransitionfrom
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