Table Of ContentProbing non-Abelian statistics of Majorana fermions in ultracold atomic superfluid
Shi-Liang Zhu,1 L.-B. Shao,2 Z. D. Wang,2 and L.-M. Duan3
1Laboratory of Quantum Information Technology and SPTE,
South China Normal University, Guangzhou, China
2Department of Physics and Center of Theoretical and Computational Physics,
The University of Hong Kong, Pokfulam Road, Hong Kong, China
3Department of Physics and MCTP, Michigan University, Ann Arbor, USA
1
We propose an experiment to directly probe the non-Abelian statistics of Majorana fermions by
1
braiding them in an s-wave superfluid of ultracold atoms. We show different orders of braiding
0
operations give orthogonal output states that can be distinguished through Raman spectroscopy.
2
Realization of Majorana states in an s-wave superfluid requires strong spin-orbital coupling and a
n controllableZeemanfieldintheperpendiculardirection. Wepresentasimplelaserconfigurationto
a generatetheartificialspin-orbitalcouplingandtherequiredZeemanfieldinthedarkstatesubspace.
J
0 PACSnumbers: 03.75.Mn,03.75.Lm,05.30.Pr,03.67.Lx
1
] Quantumstatisticsisafundamentalconceptinphysics the s-wave superfluid state is stable and has been ob-
s
a that distinguishes fermions from bosons. The discov- served in many experiments [14]. It has been noted re-
g ery of the fractional quantum Hall (FQH) effects opens cently that an s-wave superfluid with spin-orbital (SO)
- up the possibility to study exotic quasi-particles beyond coupling emulates properties of the p-wave superfluid
t
n bosons and fermions, which are called anyons [1, 2]. [10, 11, 15, 16]. Realization of Majorana fermions in
a
There are two types of anyons, with Abelian or non- an s-wave superfluid state requires SO coupling and a
u
q Abelian statistics. For Abelian anyons, the winding of controllable Zeeman field on the effective spins [11]. For
. onequasi-particlearoundanothergivesaphasefactorto ultracold atoms, an effective SO coupling can be gener-
t
a the wave function which is a fraction of 2π. For Non- atedthroughgradientofspatiallyvaryingBerryphasein-
m
Abelian anyons, a winding of one quasi-particle around ducedfromlaserbeams[17–19]. Weproposeaschemeto
- anotherisassociatedwithaunitarytransformationinthe realize both SO coupling and the requiredperpendicular
d
subspace of degenerate ground states [3]. Such unitary Zeeman field on the effective spin, which a laser configu-
n
o transformationsformarepresentationofthebraidgroup rationcompatiblewiththecurrentexperimentandsignif-
c wherethe elementsin generalarenon-commutative. Re- icantlysimplerthanthepreviousproposal[19]. TheMa-
[ cently, the systems with non-Abelian anyons have at- jorana fermions can be adiabatically movedaround each
2 tractedconsiderableinterestascandidatesforrealization other through control of the pinning potentials from fo-
v of fault-tolerant quantum computation [4, 5]. cusedlaserbeams. Weconstructexplicitlytwocomposite
1 Probing quantum statistics is a challenge for both braiding operations A and B, for which the different or-
6
Abelian and non-Abelian anyons. Recent theoretical ders of braiding operations AB and BA, acting on the
7
5 workssuggestthatinterferenceexperiments inawellde- initial vacuum, give orthogonal output states that can
. signed point contact geometry may serve as a probe of be distinguished through the Raman spectroscopy. The
2
1 Abelian[6]ornon-Abeliananyons[7]intheFQHsystem. proposedexperimentthusprovidesapromisingapproach
0 Besidesofthequasi-particlesintheFQHsystem,avortex to directly probe the fundamental non-Abelian statistics
1 excitationinp +ip superconductortrapsazero-energy of the anyons. Realization of Majorana fermions has
x y
v: boundstatewhichrepresentsakindofself-conjugatepar- been proposed in other systems [10, 11], however, for
i ticles called Majorana fermions [8]. Majorana fermions typical condensed matter materials, it is very challeng-
X
havethefeatureofnon-Abelianstatisticsandhaveraised ing to directly demonstrate the non-Abelian statistics of
r significantinterestincondensedmatterphysics[5,8–12]. the Majorana fermions by braiding them in space. The
a
Several inspiring proposals have been suggested very re- proposed realization of Majorana fermions in an s-wave
centlytorealizeMajoranafermionsindifferentmaterials atomic superfluid therefore offers a unique opportunity
[10,11]. AschemetoobserveMajoranafermionshasalso to adiabatically rotate them in space with focused laser
beenproposedinthep-wavesuperfluidstateofultracold beams to observe the non-Abelian fractional statistics,
atoms [12]. which is the most interesting property associated with
these exotic quasi-particles.
Inthispaper,weproposeanexperimenttoprobenon-
Abelian statistics of the Majorana fermions with an ul- We consider a cloud of ultracoldfermionic atoms with
tracoldatomicgasinthes-wavesuperfluidstate. Incon- threerelevantspincomponentsintheground-stateman-
tract to the p-wave superfluid state, which still remains ifold, label as 1 , 2 , 3 (e.g., they can be taken as
| i | i | i
tobe realizedexperimentallydueto challengeassociated the lowest three Zeeman levels of 6Li atoms near the
with instability of the p-wave Feshbach resonance [13], broad s-wave Feshbach resonance). To generate the
2
effective SO coupling, we apply three traveling wave field has been demonstrated in recent experiments [21].
laser beams in the x and y directions which couple the To get the standard form of the SO coupling, we choose
atomic levels 1 , 2 , 3 resonantly to the excited state the value ofβ to satisfysin2β =2cosβ =2(√2 1)and
| i | i | i −
e through the standard tripod configuration as shown observe the system in the moving frame with the veloc-
| i
in Fig. 1 [17–19]. The single-particle Hamiltonian for ity v = (h¯k/2m)(1+cos2β), so that we can drop the
y
each atom takes the form H = p2/2m+H , where p constant term (h¯k/2)(1+cos2β) in A .
s ej y
denotes the momentum operator and m is the atomic The effective Hamiltonian H has the required SO
so
mass. The light-atom interaction Hamiltonian H is coupling, but we still need to generate a perpendicular
ej
given by H = ¯h j=3(Ω e j +h.c.) = ¯hΩ e B + Zeeman field on the effective spin , with the cou-
ej j=1 j| ih | | ih | |↑i |↓i
h.c., where the corPresponding Rabi frequencies take the pling term hzσz. For this purpose, we apply two addi-
form Ω = Ω sinβ cosφe−ikx, Ω = Ω cosβe−iky, and tionallaserbeamsthatcouplethestates 1 and 3 tothe
1 2 | i | i
Ω3 = Ω sinβ sinφeikx with k the wave number of the excited state |ei off-resonantly with a large detuning ∆d
light, Ω = Ω 2+ Ω 2+ Ω 2, and the bright state (see Fig.1(a)), with the corresponding Rabi frequencies
1 2 3
|Bi ≡ (Ω∗1|p1i|+Ω|∗2|2i|+Ω| ∗3|3i|)/Ω| . We assume uniform denoted by Ω1z =i|Ωz|e−ikx and Ω3z =|Ωz|eikx. With
plane wave laser beams so that Ω,β,φ are all constants the driving Ω1z , Ω3z ∆d, it leads to a perturbation
| | | |≪
and the values of β,φ will be specified below. The di- HamiltoniangivenbyHz = i¯hΩpe2ikx 1 3 +h.c,where
− | ih |
agonalization of Hej yields two orthogonal dark states Ωp = |Ωz|2/∆d [22]. We assume Ωp ≪Ω, so the Hamil-
D1 = sinφe−ik(y−x) 1 cosφe−ik(y+x) 3 , D2 = tonian Hp can not pump the atoms outside of the dark
| i | i − | i | i
cosβcosφe−ik(y−x) 1 sinβ 2 +cosβsinφe−ik(y+x) 3 , state space spanned by the and levels, however,
| i− | i | i |↑i | ↓i
which are not coupled to the excited state e and thus it splits the degeneracy of these two levels and leads to
| i
immune to spontaneous emission. We assume the atoms a coupling Hamiltonian of the form Hz = (h¯Ωpcosβ)σz
are initially pumped to these dark states and they re- in the dark state space, which gives exactly the effective
main in the dark states as long as the typical velocity v Zeeman field term hzσz with hz = ¯hΩpcosβ. The Zee-
of the atoms satisfy the condition ηvk Ω [20], where man field hz/¯h can be continuously tuned from zero to
≪
the prefactor η 1.3 for the isotropic SO coupling dis- about 2π 50 kHz (enough for our later applications),
≈ ×
cussed below. In typical experiments, the wave number and under this condition, the requirement Ωp Ω is
≪
k 107 m−1 and the Rabi frequency Ω of the order of well satisfied with the off-resonant excitation probabil-
ma≈gnitude 2π 20 MHz can be readily obtained, so the ity (Ωp/Ω)2 < 10−5. Note that in this scheme we need
×
condition ηvk Ω is fully satisfied for ultracold atoms only two laser beams shined from the same direction as
≪
with the typical atomic velocity about several centime- Ω1,Ω2 to create the required Zeeman field. This signifi-
tersper second. The probabilityofnon-adiabaticexcita- cantlysimplifiestheexperimentalrequirementcompared
tion, estimated by (ηvk/Ω)2, is negligible (about 10−6). with the scheme in Ref. [19] which needs five additional
In a resonant coupling configuration as shown in Fig. laser beams and more complicated optical configuration
1a, the effective atomic spontaneous emission rate is re- to create the Zeeman field.
duced from the original rate γ by a factor of the non-
s (cid:3)(cid:4)(cid:5)
(cid:3)(cid:6)(cid:5)
adiabatic excitation probability and thus estimated by D
(cid:2) (cid:3)
γs(ηvk/Ω)2, where is at a few Hz level. So the atomic W (cid:4) (cid:1)
spontaneous emissions induced by these laser beams are s + s + s - s -
negligibleiftheexperimentaltimeiswithin100ms. The p
(cid:2)
ereffdeuctcievdeiaftoonmeicussepsoonffta-rneesoounsaenmtcisosuiopnlinragtceocnafingubreaftuiortnhoerr W (cid:1)(cid:1) W (cid:1) W (cid:4)W (cid:3) W (cid:3)(cid:1) W W(cid:3)(cid:2) (cid:3)(cid:1) W W(cid:1)(cid:2) (cid:1)(cid:1)
increases the Rabi frequency Ω with more laser power.
Under the above condition, we can project the Hamil- (cid:1) (cid:4) (cid:3)
tonian H to the subspace spanned by these two dark
s
states, and the projected Hamiltonian takes the form FIG. 1: Schematic representation of the light-atom interac-
tion configuration for generation of theeffectiveHamiltonian
(px Ax)2 (py Ay)2 with SOcoupling anda perpendicularZeeman field. (a) The
H = − + − , (1)
so 2m 2m relevant atomic levels and the coupling Rabi frequencies to
corresponding laser beams with specified polarizations σ+,
where Ax = ¯hkcosβσx, Ay = (h¯k/2)sin2βσy + σ−, or π. (b) The spatial configuration and propagating di-
−
(h¯k/2)(1+cos2β). In the derivation, we have assumed rection of the laser beams.
φ = π/4 and defined an effective spin with the basis
vector = (D + iD )/√2 and = (iD + Now we have the effective single-particle Hamiltonian
1 2 1
| ↑i | i | i | ↓i | i
D )/√2. The Pauli matrices σ ,σ are defined in as- H + H of the right form. The atomic interaction
2 x y so z
| i
sociation with these spin basis vectors. Equation (1) de- is described by the s-wave scattering. In the second
scribes the atom in a spin-dependent synesthetic gauge quantization formalism, it is described by the Hamil-
field[17–19],andaspin-independentversionofthisgauge tonian H = g Ψ†(r)Ψ†(r)Ψ (r)Ψ (r)d3r,
int i<j ij i j j i
P R
3
where Ψ (r) denote the fermionic field operators, i,j h >h canbe easily satisfied since we assume h /¯h can
i z c z
are summarized over the three spin components 1,2,3, betunedupto2π 50kHz,whileE /¯hisaround2π 10
f
× ×
and g are proportionalto the s-wavescattering lengths kHz [14]. A vortex is needed for emergency of the Ma-
ij
in the corresponding channel. In the s-wave super- jorana fermion in a central potential because the Hamil-
fluid state, the Hamiltonian H can be replaced by tonian has the particle-hole symmetry [11]. Vortices in
int
the effective mean-field BCS type Hamiltonian H = atomic superfluid can be created with several methods
BCS
∆ (r)Ψ (r)Ψ (r)d3r+h.c., where ∆ (r) and have been observed in many experiments [23, 24].
i<j ij j i ij ≡
Pg ΨR†(r)Ψ†(r) correspondtothesuperfluidorderpa- Forinstance,therequiredangularmomentumassociated
ijD i j E with the vortices can come from initial rotation of the
rameters. The typical s-wave pairing potential ∆ in
ij atomic cloud [23] or be imprinted from nonzero orbital
experiments is about 2π 3.5 kHz [14], which is much
× angular momentum of laser beams [24].
smaller than Ω, so the interaction Hamiltonian H
BCS Wenowturntotheimportantquestionofprobingnon-
can not pump the atoms outside of the dark-state sub-
Abelian statistics associated with Majorana fermions.
space. Under this condition, we can project H to
BCS For an unambiguous demonstration of the non-Abelian
the dark-state subspace, and the projected Hamiltonian
statistics, we need to construct two braiding operations
H takestheformH = ∆(r)Ψ (r)Ψ (r)d3r+h.c.,
P P ↑ ↓ Aand B. When appliedto the initial state,the compos-
where ∆ is a linear sRuperposition of ∆12,∆13,∆23 ite braidings AB and BA give orthogonal output states
with the superposition coefficients determined from
whichcanbedistinguishedthroughtheexperimentalde-
the transition matrix between 1 , 2 , 3 and ,
| i | i | i | ↑i tection. In the following, we explicitly construct these
, B . If we write the wave-function Φ(r) as
| ↓i | i braiding operations for Majorana fermions and discuss
Φ(r) = [u (r),u (r),v (r),v (r)]T in the basis speci-
↑ ↓ ↓ ↑ the experimental detection of the output states.
fiedbytheoperators(Ψ (r),Ψ (r),Ψ†(r), Ψ†(r))T,the
↑ ↓ ↓ − ↑ Let us consider 2N Majorana fermions confined in
Schrodinger equation for the total Hamiltonian H =
2N vortex cores, denote by γ (i = 1,2, ,2N). The
i
H +H +H with eigen-energy E takes the form of ···
so z P Majorana fermions can be combined pairwise to form
the Bogoliubov-de Gennes (BdG) equation
N complex fermionic states c = γ + iγ . Each
j 2j 2j−1
fermionicstatecanbeoccupiedorunoccupiedandthere-
H +H ∆(r)
so z Φ(r)=EΦ(r). (2) fore the ground state is 2N-fold degenerate. If we ex-
(cid:18) ∆∗(r) −σy(Hs∗o+Hz∗)σy (cid:19) change one Majorana fermion j with its nearest neigh-
bor j + 1 by adiabatically moving the pinning poten-
If we introduce τ to denote Pauli matrices in the
x,y,z
tial,wegetanunitaryoperationτ =exp(πγ γ /4)=
Nambu spinor space, the Hamiltonian H can be written j j+1 j
(1+γ γ )/√2actingonthe2N-dimensionaldegenerate
in the form j+1 j
Hilbertspace[9]. Theelementaryinterchangeoperations
τ (j = 1,2, ,2N 1) form a complete set of genera-
j
p2 torsfortheb·r·a·idgro−up. Todemonstratethenon-Abelian
H = [ µ]τ +α(σ p +σ p )τ +h σ
z x x y y z z z
2m − statistics, we need at least four Majorana fermions due
+(τxRe∆ τyIm∆), (3) to existence of the superselection rule. For four Majo-
−
rana fermions, they combine into two complex fermions
wherethe laserinducedSOcouplingrateα= ¯hk(√2
c ,c . The ground state has degeneracy four, but the
− − 1 2
1). All the constant potentials have been included into
Hamiltonian H conserves parity of the fermion number,
the definition of the chemical potential µ which is self- and the even-number subspace spanned by 0 ,c†c† 0
consistently determined from the particle density. For {| i 1 2| i}
is decoupled from the odd-number subspace spanned by
ultracold atoms in a global spin-independent confining c† 0 ,c† 0 . At low temperature (temperature signifi-
trap V(r) , under the local density approximation, the { 1| i 2| i}
cantly below the superfluid gap), the initial state is typ-
globaltrapcanalsobeincludedintothe definitionofthe
ically a vacuum state 0 [12], and in the even-number
cheTmhiecaHlapmoitletnotniiaalnwHithinthEeqs.(u3b)shtiatsutbioeennµin→vesµti−gaVte(dr)i.n subspace {|0i,c†1c†2|0i},|oinly two of the three generators
τ ,τ ,τ are independent with
1 2 3
the context of a recent proposal for realization of non-
Abelian anyons in a solid state heterostructure consist- 1 1 i 0 1 1 i
ing of a superconductor, a 2D electron gas with Rashba τ1 =τ3 = √2(cid:18) −0 1+i (cid:19), τ2 = √2(cid:18) i −1 (cid:19).
−
SO coupling, and a magnetic insulator [11]. It is known (4)
thatforthisHamiltonian,aMajoranafermionzeromode From these generators, we construct two composite
emerges from a vortex core if the parameters in the braiding operations
Hamiltoniansatisfytheconditionhz >hc =qµ2+|∆|2 A=τ1τ2τ1−1, B =τ1−1τ2τ1−1 (5)
[11, 16]. For ultracold atoms, h is comparable with the
c with the property
Fermi energy E since the chemical potential µ E
f f
| | ∼
and the pairing potential ∆ 0.35E . The condition AB =iσ , BA= iσ , (6)
f z x
| | ∼ −
4
where τ−1 represents a braiding along the reverse direc- [2] F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982); F.
1
tion. With the initial state 0 , the braiding operations Wilczek, Fractional Statictics and Anyon Superconduc-
AB and BA yield two ortho|gional output states, either tivity (World Scientific, Singapore, 1990).
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oftheanyons. Duringbraiding,weneedtofulfilltheadi-
[5] C. Nayak et al., Rev.Mod. Phys.80, 1083 (2008).
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core is estimated by ω0 ∆2/Ef 2π 1.2 kHz. If the mon et al., Phys.Rev.B 55, 2331 (1997).
∼ ∼ ×
braiding time tb is of the order of tens of ms, the non- [7] A.SternandB.I.Harperin,Phys.Rev.Lett.96,016802
adiabatic excitation probability estimated by 1/(ω0tb)2 (2006);S.DasSarma,M.Freedman,andC.Nayak,ibid.
is negligible. 94, 166802 (2005); D. E. Feldman and A. Kitaev, ibid.
Thetwooutputstatesi0 and ic†c† 0 canbedistin- 97, 186803 (2006).
| i − 1 2| i [8] N. Read and D.Green, Phys. Rev.B 61, 10267 (2000).
guished through the Raman (or r.f.) spectroscopy [25].
[9] D. A. Ivanov,Phys. Rev.Lett. 86, 268 (2001).
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zero energy trapped at the vortex cores. We can apply (2008).
a Raman π-pulse to resonantly bring these zero-energy [11] J. D. Sau et al., Phys. Rev. Lett. 104, 040502 (2010);
atoms to a different hyperfine state D [12], which is Phys. Rev.B 82, 214509 (2010).
outside of Feshbach resonance and thu|sionly weakly in- [12] S. Tewari et al., Phys.Rev.Lett. 98,010506 (2007).
[13] J. P. Gaebler et al., Phys.Rev. Lett.98, 200403 (2007).
teracting with other atoms. If we choose the Rabi fre-
[14] W. Ketterle and M. W. Zwierlein, arXiv:0801.2500.
quency of the Raman pulse to have a Gaussian shape
[15] C.Zhangetal.,Phys.Rev.Lett.101,160401 (2008); A.
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and the center of the vortex core is at r =0), it will not [16] M.Sato et al., Phys.Rev.Lett. 103, 020401 (2009).
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× 0 B. Anderson, and V. Galitski, Phys. Rev. A 78, 023616
Ω = 1.7ω , t = 5/ω and w = 1.5µm. If the two laser
0 0 f 0 (2008).
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rections, it also applies a momentum kick to the target [20] This condition is obtained by requiring the off-diagonal
atoms to push them outside of the atomic cloud. These coupling matrix element oftheHamiltonian Hs between
target atoms in a different hyperfine state D can then the dark states and the bright state much smaller than
| i the energy difference h¯Ω.
be detected with the standardquantumjump technique.
[21] Y. J. Lin et al., Nature(London) 462, 628 (2009).
Insummary,wehaveproposedanexperimentalscheme
[22] There is also a constant a.c.-Stark shift term
to create and observe Majorana fermions in an s-wave |Ωz|2/∆d (|1ih1|+|3ih3|) associated with the laser
superfluidofultracoldfermionicatomswithlaserinduced
(cid:0)beams Ω1z(cid:1),Ω3z. However, this constant frequency shift
SOcoupling. Thenon-AbelianstatisticsoftheMajorana can be easily canceled with a corresponding frequency
fermions can be directly probed through laser-controlled offset of the laser beam Ω2 applied to the level |2i, so it
braidinganddetectionbasedontheRamanspectroscopy. has no physical effect on the ground levels.
This work was supported by the NSF of China (No [23] M.W.Zwierleinetal.,Nature(London)435,1047(2005).
[24] M.F.Andersenetal.,Phys.Rev.Lett.97,170406(2006).
10974059), the SKPBR of China (Nos.2011CB922104
[25] T.-L. Dao et al., Phys. Rev.Lett. 98, 240402 (2007); A.
and 2007CB925204), the RGC of Hong Kong, Schirotzek et al., Phys. Rev.Lett. 101, 140403 (2008).
the DARPA OLE Program under ARO Award
W911NF0710576,and the AFOSR MURI program.
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