Predicted signatures of p-wave superfluid phases and Majorana zero modes of fermionic atoms in RF absorption

0.16 MB·English

by  Eytan Grosfeld

Predicted signatures of p-wave superfluid phases and Majorana zero modes of
fermionic atoms in RF absorption
Eytan Grosfeld,1 Nigel R. Cooper,2 Ady Stern,1 and Roni Ilan1
1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
2T.C.M. Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom
(Dated: February 6, 2008)
7
0 Westudythesuperfluidphasesofquasi-2DatomicFermigasesinteractingviaap-waveFeshbach
0 resonance. Wecalculatetheabsorptionspectraofthesephasesunderahyperfinetransition,forboth
2 non-rotating and rotating superfluids. We show that one can identify the different phases of the

n p-wavesuperfluidfromtheabsorptionspectrum. Theabsorptionspectrumshowsclearsignaturesof
a theexistenceof Majorana zero modes at thecores of vorticesof theweakly-pairingpx+ipy phase.
J

5
1 Theobservationofp-waveFeshbachresonancesincold phase.
fermion gases[1] opens up the possibility of the exper- We consider a gas of identical fermionic atoms in an

] imental realisation of unconventional superfluid states internal state that we denote , interacting via a p-
r |↓i
e with non-zero pairing angular momentum[2, 3, 4]. It wave Feshbach resonance. The phase diagram of the
h
has recently been predicted[3, 4] that a one-component system has the same qualitative form in wide [4] and
t
o fermion gas interacting via a p-waveFeshbach resonance narrow [3] resonance limits. Depending on temperature,
t. showsaseriesofp-wavesuperfluidphases,includingapx andonthe detuning andanisotropicsplitting of the Fes-
a
phase, and both strong- and weak- pairing phases[6] of hbachresonance,there appear superfluid phases with p
m x
p ip symmetry. Thestrong-andweak-pairingphases and p +ip symmetries. For p +ip , there is a transi-
x y x y x y
- ±
differ in the nature of their excitation spectra, and are tion between the weak-pairingphase (chemical potential
d
n separated by a topological phase transition[6, 7]. µ > 0) and the strong-pairing phase (µ < 0) as the res-
o onance is swept from positive to negative detuning[3].
Among these, the weak-pairing p ip phase has re-
c x± y We propose to probe the properties of these phases by
cently received considerable theoretical attention. The
[
vortex excitations of this phase are predicted to hold measuring the absorption spectrum, under excitation of
1 an atom in state to a new internal state via the
gapless Majorana fermions on their cores, which should
v give rise to non-abelian exchange statistics in a quasi2D perturbation |↓i |↑i
6
2 geometry[5, 6]. Such excitations may have their uses in
3 topologically-protected quantum computation [3, 8, 9]. Hpert =η d2reiq·r−iωtψ†(r)ψ (r)+h.c., (1)
1 However, to our knowledge, there as yet are no exper- Z ↑ ↓
0
imental observations of these very interesting physical
where η sets the Rabi frequency of the transition. We
7
propertiesinanyrealisationofap ip superfluid. De-
0 x± y consider to be a hyperfine transition, driven
/ velopingmethodstoexperimentallyprobetheproperties by RF ab|↓soirp→tio|n↑i[10, 11] or a two-photon transition[9];
t
a of p-wavesuperfluid phases in cold atom systems is then measurement of the population of excited (or remain-
m of high and timely interest.
ing)atomsisachievedbyseparateabsorptiononanelec-
- Inthis Letter weshowthatfeatures inthe RFabsorp- tronictransition. Wedenotetheenergysplittingbetween
d
tion spectrum of a fermionic gas provide unambiguous the states and by E ; for a hyperfine transition,
n g
|↓i |↑i
o signatures of the different p-wave superfluid phases. We Eg 109Hz,andwesetthenetphotonmomentumtrans-
∼
c showthattheweak-pairingphasescanbeidentifiedfrom fertozero,q 0. The -atomswillnotparticipateinthe
: → ↑
v the strong-pairing phases in the absorption spectrum of resonant p-wave interaction with the -atoms, so for the
↓
i thesuperfluidatrest. Mostsignificantly,wefocusonthe mostpartweshallconsiderthe -atomstobe free. How-
X ↑
p +ip phaseandshowthatforarotatingfluidatweak- ever, we shall discuss the effects of s-wave interspecies
x y
r
a pairing, the absorption spectrum has clear and striking interactions. We shall focus on a uniform gas in a quasi-
signatures of the zero energy Majorana fermion modes two-dimensional geometry, that is with confinement ap-
on the cores of the vortices. This spectrum is therefore plied in the z-direction such that the associated confine-
a direct probe of the physics underlying the proposed mentenergyislargecomparedtotheFermienergy. This
non-abelianexchange statistics of this phase. (These ex- simplifiestheanalysis,allowinganalyticalcalculationsof
perimentalsignaturesdonotrequiremanipulationofpo- the absorption features. The three dimensional case is
sitionsofvorticesorthe useofotherlocalisedprobes[9].) left for a future work.
Furthermore, we show that the absorption spectrum for Whentheinteractionbetweenatomsmaybeneglected
the rotated p +ip weak-pairing phase depends on the (far from the Feshbach resonance) and they form a sim-
x y
sense of the rotation. This dependence is a direct in- ple Fermi gas, a transition of an atom from to re-
|↓i |↑i
dication of the time-reversal symmetry breaking in this quires an energy of E , leading to a delta-function peak
g
2
of the absorptionatω =E . (We chooseunits for which
g
~=1.) Incontrast,wefindthatinthesuperfluidphases
(close to the resonance) the absorption differs strongly
for weak- and strong-paired superfluids. In the weak-
pairing phase the energy E becomes a threshold energy
g
for absorption, with non-zero weight at ω = E and a
g
continuous spectrum above this (see Eq. (4)). In the
strongpairingphasethethresholdisshiftedtoE +2µ,
g
| |
with weight that is zero at E +2µ and grows linearly
g
| |
with energy. The discontinuity in the absorption spec-
truminthe weak-pairingphaseunambiguouslyidentifies
it from strong pairing.
FIG. 1: Peak distribution associated with the Majorana
In the weak-pairing p ip phase, the introduction
x y
of vortices by rotation lea±ds to the creation of a set of modesonthevorticesoftheweak-pairingpx+ipy phase. The
first peak will be at E ≃Eg−µ, and other peaks will follow
gapless modes on the vortices. In the presence of a vor-
at spacings ωc apart. Continuous absorption will be mea-
tex lattice we find that a series of equally spaced sub- sured startingat E ≃Eg. Fortheoppositesense of rotation,
gap peaks will be present in the absorption spectrum, see the weight of the peaks starts linearly and passes through a
Fig. [1]. These peaks are a result of transitions from maximum. The inset shows the shape of a single peak for a
the zero energy core states of -atoms to various states square (dashed line) and triangular (solid line) lattice, show-
↓ ing a van-Hovesingularity at E =2t (Γ is measured in units
of -atoms, and have a weight that is linearly propor-
tion↑al to the number of vortices N . The peaks start of 2π|η|2NVΘ0/t).
V
at E µ, and their weight decays monotonically in
g
∼ −
energy, becoming small well before the onset of the con- Hpair, = d2kψ†(k)v∆(kx + iky)ψ†( k) + h.c.. The
tinuum at Eg µ+∆ (∆ is the superfluid gap). For values↓of µ and v ↓may be obtained f↓ro−m the mean-field
a dense vortex−lattice, in which there is tunneling be- calculationRs of R∆efs.3, 4. The Hamiltonian for -atoms
tween the Majorana modes, the peaks broaden to reveal is diagonal in the Fock space of αk = ukψ ,k+v↓kψ†, k
a lineshape showing a van-Hove singularity, see inset of ↓ ↓−
Ftiicge.s o[1f]t.heFoorpptohseitesasmeneseweoafkr-optaaitriionng,pwheasfien,dwtihthatvtohre- aonfdthαe†kB=dGu∗keψq↓†u,akt+ionv,k∗ψ|u↓k,−|2k =wh21ere1u+k,v2kkm2a−reµsol/uEti↓o,kns
absorption in the Majorana modes again gives rise to a and v 2 = 1 1 k2 µ /hE (cid:16), and (cid:17)E =i
set of narrow absorption peaks. The peaks still start at | k| 2 − 2m − ↓,k ↓,k
Eg µ, but now their weight rises linearly in energy, k2 µ 2+v2kh2 [3,(cid:16)6]. Trea(cid:17)tingthie excited -atoms
∼pAassswine−gshtharlloudgishcuassmbaxeliomwu,mthweedlliffbeerfoenrecethine csopnetcitnruaufmor. aqs(cid:0)fr2eme,−E↑,(cid:1)k =k∆2/2m−µ, a summation over k↑leads to
the absorption spectrum [19]
the two senses of rotation is a signature of the fact that
px+ipy breaks time-reversal symmetry. S δω+2µ θ(δω), µ>0
We now provide the details of the calculations. The Γ[ω]= η 2 (4)
rate of excitations between the two hyperfine states is | | v∆2 1+ mδvω∆2 2 ×( θ(δω−2|µ|), µ<0,
governedby the Golden Rule h i
whereS istheareaofthesampleandδω ω E . Non-
g
≡ −
Γ[ω]=2π η 2 M 2δ(E +E ω), (2) zero interactions of the excited -atom with the -atoms
| | | ab| ↓,a ↑,b− lead to a (mean-field) shift of bo↑th of these spect↓ra [20].
ab
X
We now turn to discuss the system under rotation, in
where E (E ) is the energy required to produce an
,a ,b a regimewhere a largevortexlattice is formed,N 1.
-excitat↓ion( -↑excitation),and M is the dimensionless V ≫
↓ ↑ ab (The rotation frequency Ω is then close to the trapping
matrix element (measured in units of η) for producing
frequencyω .) Wefirstexamineasuperfluidintheweak
excitations with quantum numbers a and b respectively. T
pairing p +ip phase, and concentrate on the regime
InthefollowingweshallcalculateΓ[ω],firstforasystem x y
Ω µ mv2.
at rest, then for a rotating system. ≪ ≪ ∆
Projected to the low-energy sector of the Majorana
We describe the superfluid phases within the
modes,theHamiltonianforthe -atomsbecomesatight-
Bogoliubov-de-Gennes (BdG) approach. For a uniform ↓
binding Hamiltonian of the form[12]
system, the Hamiltonian is given by
H =H +H =it s γ γ , (5)
N N 0, pair, ij i j
H =H0,↓+Hpair,↓+H0,↑+Eg ↑−2 ↓ , (3) ↓ ↓ ↓ Xhiji ↓ ↓
(cid:18) (cid:19)
where s = , in a way that corresponds to half a flux
where H0,s = d2kψs†(k) 2km2 −µ ψs(k), s =↑,↓, quantumijper±plaquette for a square lattice, and a quar-
R (cid:16) (cid:17)
3
ter of a flux quantum per plaquette for a triangular lat- The spectrum of the -atoms in the rotating frame
↑
tice. Here γ represents a Majorana fermion localized at depends on the strength of the s-wave interspecies in-
i
R . The parameter t describes the tunneling between teractions. Since this depends sensitively on the atomic
i
the Majoranamodes ofnearby vortices;this tunneling is species used, we shall consider two limiting cases: For
assumed small, t µ, which is valid if the separation vanishing interspecies interactions, the -atoms arrange
≪ ↑
between vortices, ℓ 1/(2mΩ), is large compared to into Landau levels due to rotation, with the cyclotron
≡
the spatial extent of the Majoranamode, r . The Majo- frequency ω = 2Ω 2π 102Hz. (The confinement
rana wave function g(rp) has an oscillating0contribution frequency focr - and∼-atom×s is assumed the same.) For
↑ ↓
within the vortex core, and a decaying part outside the stronginterspecies repulsion,thelowenergystatesofthe
core. When µ mv2, the size of the core can be es- -atoms are localized at the cores of vortices where the
≪ ∆ ↑
timated as rcore ∼ mvvF∆2 log−1/2 vv∆F, with vF being the dtiegnhsti-tbyinodfin↓g-aHtoammsiltiosnlioawn., Tanhde rcoatnatiboendthesecnritbreadnsblayteas
Fermi velocity of the underlying fermionic gas,while the
to a magnetic flux threading the lattice plaquettes (the
decay length outside the core is r = v /µ. In the limit
0 ∆
Azbel-Hofstadter problem [15]).
ofsmallµ we haver r , suchthat we mayapprox-
0 core
≫
imate g(r) e−r/r0 , with r being the distance from the
≈ √2πr0r
centerofthevortex[13]. TheanalysisoftheHamiltonian Starting with the case of zero interspecies interactions
(5) is presented in [12, 14]. For a square and triangular (V = 0), the Hamiltonian H can be diagonalized by
0,
lattice the spectra are E(cid:3),k = 2t sin2(akx)+sin2(aky) ex↑panding ψ↑(r) = npφnp(r↑)cnp, where φnp(r) are
wavefunctions for the n’th Landau level, and p is some
andE△,k =√2t 3+cos(2akx)−q2cos(akx)cos(√3aky) quantum number relaPted the degeneracy of the Landau
respectively[12].q level. The rate of energy absorption is determined by
v
|Mαk,np|2 = N λαz(∗j),kλαz(i),keik·(Ri−Rj) d2rd2r′e2i[Pm6=iarg(r−Rm)−Pm6=jarg(r′−Rm)]gi(r)gj(r′)φ∗np(r′)φnp(r),(6)
V ij Z
X
where g (r)=g(r R ). The functions λα are listed Theintegratedabsorptionoverallfrequenciesforexcita-
i − i z(i),k
in Ref.12, z(i)=1,...,v numbers the vorticescontained tions from the zero modes can be found using the com-
in a unit cell, and α = ±1,...,±v/2 is a band index. pleteness of the Landaulevel functions np|Mαk,np|2 =
Since p enumerates degenerate states, we can now sum 1, hence dωΓ[ω]=π η 2N /2.
2 | | V P
overitusingtheadditiontheoremforLandaulevels[16]. When interspecies interaction is strong, the -atoms
We keep only the leading term R = R . Finally we feelaperiRodicpotentialV duetothevortexlattic↑estruc-
i j
expand the phases in Eq. (6) to first order in r r , ture. Due to depletion o↑f -atoms at a vortex core, the
′
and obtain M 2 = r02 1+2n r0 2 −2. −Thus potential V possesses ther↓e a minimum, and may hold
p| αk,np| ℓ2 ℓ several bou↑nd states φ (r) for -atoms. Several tight-
n
theintegratePdabsorptionintothhen’thL(cid:0)and(cid:1)aui levelfalls bindingbandswillbeformed,lab↑elledbyn. The -atoms
monotonically with increasing n. Plugging the matrix ↑
feelthesamefluxperplaquetteastheMajoranafermions
elements into Eq. (2), we arrive at the result
do. Therefore,theHamiltonianforasinglebandoftight-
η 2 ω ω binding atoms in the rotating frame may be written as
n
Γ[ω]=2π| | N Θ F | |− , (7)
V n
t t
Xn (cid:20) (cid:21) H0, =En′ c†n,i cn,i +it′n sijc†n,i cn,j , (9)
whereΘ = r0 2 1+2n r0 2 −2,ω =E µ+ω (n+ ↑ Xi ↑ ↑ Xhiji ↑ ↑
n ℓ ℓ n g− c
1/2), and F(cid:0)is a(cid:1)dhimensio(cid:0)nles(cid:1)sifunction proportional to wheret′n isthetunnelingmatrixelementforthe↑-atoms,
thedensityofstatesofthebandformedbytheMajorana and En′ is an on-site energy. Diagonalizing the Hamilto-
operators, which is nonzero over a range of order unity. nian, one finds that the spectrum of the -atomshas the
It is explicitly calculated below. same k-dependence as that of the Maj↑orana fermions.
Using the results above, assuming t ωc, we can find The resulting absorption is given by Eq. (7) with Θn =
the integrated absorption over a single≪Landau level | d2rg(r)φn(r)|2, ωn =Eg−µ+En′, t→t+t′n.
We now proceed to calculate the function F appear-
ωc R
2 d(ω ωn)Γ=π η 2NVΘn. (8) ing in Eq. (7). This function is F[E/t] ≡ tρ[E]/NV,
Z−ω2c − | | with ρ(E) being the density of states of the Majorana
4
fermions. Starting with a square lattice, F[x] is gapcloses. Whenvorticesarefarawayfromoneanother
and from the edge, the zero energy state exists as long
8x dy
as the gap does not close, i.e., as long as β =0 [18][23].
F [x]= .(10)
(cid:3) (2π)2 ZC x2−y2 4−x2+y2 4−y2 Toconclude,ourresultsshowthattheabs6orptionspec-
trum can be used to distinguish the various p-wave su-
p p p
For 0 < x < 2, C = [0,x], while for 2 < x < 2√2
perfluidphasesofanatomic Fermigas,andto probe the
C = √x2 4,2 . We note the diverging density of
breaking of time reversal symmetry. Furthermore, for a
−
states at E = 2t, and the discontinuity at the top of
rotating superfluid in the weak pairing p +ip phase,
(cid:2) (cid:3) x y
the band E = 2√2t. For a triangular lattice, we again
theabsorptionshowsaseriesofsub-gappeaks,ofunique
find a logarithmic divergence near E = 2t and disconti-
shapeanddistribution,whicharethedirectconsequence
nuities in the response at the top and the bottom of the
of the zero-energyMajoranamodes on the vortices. The
band, E =2√3t and E =√3t respectively.
most favourable parameters for the observation of these
For the opposite sense of rotation, the vortex lat- sub-gappeaksareµ mv2 (toavoidfiniteenergybound
tice consists of an array of vortices of opposite circula- ≪ ∆
states in the vortex core), and temperature T µ (to
tion. These vortices also have zero modes[17]. However, ≪
avoid thermally excited quasi-particles which may wash
the Majorana modes now have a non-trivial dependence
out the absorption peaks). A practical way to observe
around the vortex center, gi(r) e−iarg(r−Ri). (We the peaks would be by comparing measurements of the
∝
again assumed r r .) As a consequence, the inte-
0 core absorption for increasing rotation frequency. This will
≫
gratedweightoftheabsorptionfromtheMajoranamode
vary both the strength of the peaks (due to the increase
intothen’thLandaulevelisproportional,forsmalln,to
of N ) and their spacing (due to the increase of ω ).
V c
9π r 4 1 We acknowledge financial support from the US-Israel
M 2 = 0 n+ . (11) BSF (2002-238) and the ISF, and from EPSRC Grant
| | 64 ℓ 2
Xp (cid:16) (cid:17) (cid:18) (cid:19) No. GR/S61263/01 and the ICAM Senior Fellowship
Programme (NRC).
The weightpassesthrougha maximumatafrequencyof
about µ2 above the onset at E µ.
mv∆2 g−
Thus we find that in the weak pairing p +ip phase,
x y
the spectrum depends strongly on the sense of rotation.
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J. Zhang et al., Phys. Rev. A 70, 030702 (2004); C. H.
the relative sense ofrotationas comparedto the internal
Schuncket al., Phys.Rev.A 71, 045601 (2005).
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[10] C.A. Regal and D.S. Jin, Phys. Rev. Lett. 90, 230404
jorana modes would also distinguish the time-reversal
(2003).
symmetric p phase from the phase of p +iβp , with
x x y [11] C. Chin et al.,Science 305, 1128 (2004).
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6
ken. The latter may appear for an anisotropic Feshbach (2006).
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5
[18] D.A. Ivanov,Phys.Rev.Lett. 86, 268271 (2001). geneous state with domains of thetwo phases.
[19] The unphysical logarithmic divergence of the integrated [22] Inarotatingsystemtime-reversalsymmetryisexplicitly
intensity in (4) is a reflection of the need to introduce a broken,so,ina“rotation cooled” experiment,onephase
UVcut-off[4]toregularise thetheory for∆k ∝kx+iky. may form in preference over theother.
[20] Theremayalsobearenormalisationoftheeffectivemass [23] Wenotethattheanisotropicorderparameterofthepx+
of theexcited particle, and a damping for k6=0 modes. iβpy phasewillchooseapreferreddirectionfortunneling
[21] There is also the possibility of formation of an inhomo- between vortices.