Topological superfluid in one-dimensional spin-orbit coupled atomic Fermi gases

1.1 MB·English

by  Xia-Ji Liu

Topological superfluid in one-dimensional spin-orbit coupled atomic Fermi gases
Xia-Ji Liu1 and Hui Hu1
1ARC Centre of Excellence for Quantum-Atom Optics,
Centre for Atom Optics and Ultrafast Spectroscopy,
Swinburne University of Technology, Melbourne 3122, Australia
(Dated: January 30, 2012)
We investigate theoretically the prospect of realizing a topological superfluid in one-dimensional
2 spin-orbit coupled atomic Fermi gases under Zeeman field in harmonic traps. In the absence of
1 spin-orbit coupling, it is well-known that the system is either a Bardeen-Cooper-Schrieffer (BCS)
0 superfluidoraninhomogeneousFulde-Ferrell-Larkin-Ovchinnikov(FFLO)superfluid. Hereweshow
2 that with spin-orbit coupling it could be driven into a topological superfluid, which supports zero-
energy Majorana modes. However, in the weakly interacting regime the spin-orbit coupling does
n
notfavorthespatiallyoscillating FFLOorderparameter. Asaresult,itseemsdifficulttocreatean
a
J inhomogeneous topological superfluidin current cold-atom experiments.

6
PACSnumbers: 03.75.Ss,71.10.Pm,03.65.Vf,03.67.Lx
2

]
s I. INTRODUCTION of balanced and imbalanced spin-populations. Here we
a show that by adding spin-orbit coupling both superflu-
g
ids can turn into a topological superfluid. We discuss
- Topological superfluids are new states of matter that
t in detail the resulting zero-energyMajoranaedge modes
n attract intense attentions in recent years [1, 2]. They
and their possible experimental signature. We also ex-
a havea full pairinggapin the bulk and exotic gaplessex-
u citations at the edge - the so-called Majorana fermions plore the possibility of creating an inhomogeneous topo-
q logical superfluid with spatially oscillating FFLO order
- which obey non-Abelian statistics [3, 4]. These excita-
. parameter. Unfortunately, the spin-orbit coupling seems
t tions are immune to decoherence caused by local per-
a to suppress the FFLO order parameter. As a result, in
m turbations. By properly braiding excitation quasipar-
the weakly interacting regime we always find the same
ticles, topological quantum information might be pro-
- topological superfluid with a uniform order parameter,
d cessed. As aresult,topologicalsuperfluids couldprovide
whatever the initial state is a BCS or FFLO super-
n an ideal platform for topological quantum computation
fluid. Ourstudyisbasedontheself-consistentsolutionof
o [5, 6]. Because of this potential application, the realiza-
c tion of topological superfluids in a well-controlled envi- fullymicroscopicBogoliubov-deGennes(BdG)equations
[ [25, 28]. It enables ab-initio simulations under realistic
ronment is highly desirable.
experimental conditions.
1 Theoretically, there are a number of proposals on re-
The paper is organized as follows. In the next section
v alizing a topological superfluid in two-dimensional (2D)
3 (Sec. II),wepresentthemodelHamiltonianandtheBdG
settings, including the use of 2D p-wave pairing [7, 8],
6 equations. InSec. III,wediscussthe phasediagramata
proximity coupling to a conventional s-wave supercon-
6 sufficiently large spin-orbit coupling and the phase tran-
ductors for the surface state of three-dimensional (3D)
5 sitionfromBCSsuperfluidtotopologicalsuperfluid. The
. topologicalinsulators[9–11], and 2D atomic Fermi gases
1 wave-functions of Majorana edge modes are shown and
with strong Rashba spin-orbit coupling [12, 13]. It is
0 their possible experimental detection is considered. In
also possible to create a topological superfluid in one-
2 Sec. IV, we present the phase diagram at a given Zee-
1 dimensional (1D) solid-state systems by suitably engi-
manfield andshowthe transitionfromFFLO superfluid
: neering spin-orbit coupling of electrons, such as InAs
v to topological superfluid. Finally, in Sec. V we provide
wires and banded carbon nanotubes [14–17]. The pur-
i conclusions and some final remarks.
X pose of this work is to examine the possibility of observ-
r ing topological superfluids in 1D ultracold atomic Fermi
a gases [18], which may be regardedas highly controllable
II. MODEL HAMILTONIAN AND BDG
quantum simulators of the corresponding 1D solid-state EQUATIONS
systems. We note that 1D atomic Fermi gases can now
be routinely created in cold-atom laboratories [19]. The
We consider a trapped two-component 1D atomic
spin-orbit coupling for neutral atoms may also be gener-
Fermi gas under a non-Abelian gauge field (spin-orbit
atedby using the so-called“non-Abeliansynthetic gauge
coupling) and Zeeman field, described by the model
fields” technique [20, 21].
Hamiltonian,
Even in the absence of spin-orbit coupling the 1D ul-
tracold atomic Fermi gas is of great interest. It hosts = dxψ†(x) S(x) hσ +λkσ ψ(x)
a Bardeen-Cooper-Schrieffer (BCS) superfluid and an H ˆ H0 − z y
(cid:2) (cid:3)
exotic inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov
+g dxψ†(x)ψ†(x)ψ (x)ψ (x), (1)
(FFLO) superfluid [19, 22–28], respectively, in the case 1Dˆ ↑ ↓ ↓ ↑
2
where ψ†(x) [ψ†(x),ψ†(x)] denotes collectively the gases can now be manipulated using 2D optical lattices
≡ ↑ ↓
creationfieldoperatorsforspin-upandspin-downatoms. [19]. Thegeneralizationofthesyntheticgaugefieldλkσ
y
In the single-particle Hamiltonian (i.e., the first line hasalreadybeendemonstratedina 3DBosegasof87Rb
of the above equation), S(x) (~2/2m)∂2/∂x2 + atoms[20]. Inaddition,itsrealizationinfermionicatoms
H0 ≡ −
mω2x2/2 µ describes the single-particle motion in a has been proposed [21]. Therefore, all the techniques re-
harmonict−rappingpotentialmω2x2/2andinreferenceto quiredtosimulateEq. (1)arewithincurrentexperimen-
thechemicalpotentialµ,thestrengthoftheZeemanfield tal reach.
is denoted by h, λkσy iλ(∂/∂x)σy is the spin-orbit To understand the 1D superfluidity in the presence of
≡ −
coupling term with coupling strength λ, σy and σz are spin-orbit coupling, we calculate elementary excitations
the 2 2Paulimatrices. The secondline ofthe equation within the mean-fieldBdG approach[25, 28]. The wave-
×
is the interaction Hamiltonian, with the (attractive) in- function of low-energy fermionic quasiparticles Ψ (x)
η
teractionstrengthgivenby the s-wavescatteringlength: with energy E is solved by,
g = 2~2/(ma ). η
1D 1D
−
The model Hamiltonian Eq. (1) can be realized
Ψ (x)=E Ψ (x), (2)
straightforwardly with cold fermionic atoms. It is a di- HBdG η η η
rect generalization of the standard model Hamiltonian
for a 1D spin-imbalanced Fermi gas, through the inclu- where Ψ (x) [u (x),u (x),v (x),v (x)]T in the
η ↑η ↓η ↑η ↓η
≡
sionofanon-Abeliansynthetic gaugefield λkσ . Exper- Nambu spinor representation and the BdG Hamiltonian
y
imentally, a bundle of 1D spin-imbalanced atomic Fermi reads accordingly,
BdG
H
S(x) h λ∂/∂x 0 ∆(x)
H0 − − −
 λ∂/∂x S(x)+h ∆(x) 0 
HBdG = 0 H0∆∗(x) S(x)+h λ∂/∂x . (3)
 −H0 
 ∆∗(x) 0 λ∂/∂x S(x) h 
 − − −H0 − 
Here ∆(x) = (g /2) [u v∗ f(E ) + Fermi gas without spin-orbit coupling [25, 28]. We in-
− 1D η ↑η ↓η η
uf↓(ηxv)↑∗ηf(−E1η)/][ex/(iksBT)+th]e is ortdheerPFeprmariamdeitsetrributainond ctraoldduecnesiatyhaipghproexniemrgaytiocnut(-LoDffAE)c,isaubsoevdefowrhtichhe haiglho--
≡ lying energies and wave-functions. This leads to an ef-
function at temperature T. The order parame-
fective coupling constant in the gap equation, ∆(x) =
ter is to be solved self-consistently together with
the number equation for the chemical potential, −[g1eDff(x)/2] η[u↑ηv↓∗ηf(Eη) + u↓ηv↑∗ηf(−Eη)], where
dr[n↑(r)+n↓(r)] = N, where N is the total number η is now rePstricted to |Eη| ≤ Ec. We refer to Ref.
´of atoms and the density of spin-σ atoms is given P[25] for further details of geff(x) and the LDA atomic
by, nσ(x) = (1/2) η[|uση|2f(Eη) + |vση|2f(−Eη)]. density. 1D
We note that the usPe of Nambu spinor representation In harmonic traps, it is useful to characterize the in-
leads to an inherent redundancy built into the BdG teraction strength by using a dimensionless interaction
Hhoalme itlrtaonnsiafonrm[2a].tioHn:BduGσi(sx)in→varivaσ∗n(txu)nadnedr tEhηe →par−ticEleη-. npa0raismethteerz[2e5ro],-tγem≡p−ermatgu1rDe/(c~en2nte0r) =de2n/si(tny0ao1fDa),nwihdeerael
Therefore, every eigenstate with energy E has a partner two-component Fermi gas with equal spin populations
at E. These two states describe the same physical N/2. In the Thomas-Fermi approximation (or LDA),
−
degrees of freedom, as the Bogoliubov quasiparticle n = 2N1/2/(πa ) and a = ~/(mω) is the char-
operators associated with them satisfy ΓE = Γ†−E. This ac0teristic oscillatohor lengthhoof theptrap. Therefore, the
redundancy has been removedby multiplying a factor of dimensionless interaction parameter is given by,
1/2 in the expressions for order parameter and atomic
1 a
density. ho
γ = . (4)
πN1/2 (cid:18)a (cid:19)
The BdG equation (2) can be solved by expanding 1D
u (x) and v (x) in the basis of 1D harmonic oscil- We note that, for a 1D atomic Fermi gas created us-
ση ση
lators. On such a basis, Eq. (3) is converted to a secu- ing 2Dopticallattices,the typicaldimensionless interac-
lar matrix. A matrix diagonalization then gives the de- tion strength is about γ = 3 5 [19, 25]. Through-
∼
sired quasiparticle energy spectrum and wave-functions. out the paper, we shall take a slightly smaller value
Numerically, we have to truncate the summation over of γ = π/2 1.6, in order to validate the mean-field
≃
the energy levels η. For this purpose, we adopt a hy- treatment. It is also convenient to use the Thomas-
brid strategy developed earlier by us for an imbalanced Fermi energy E = (N/2)~ω and Thomas-Fermi radius
F
3
x = N1/2a as the units for energy and length, re- system, however, the critical Zeeman field may become
F ho
spectively. For the spin-orbit coupling, we use a dimen- position dependent. As a result, in harmonic traps we
sionless parameter λk /E , where k = √2mE is the would have a mixed phase with both conventional and
F F F F
Thomas-Fermi wavevector. We have performed numeri- topological superfluid components, which separate spa-
cal calculations for a Fermi gas of N = 100 fermions in tially in real space. Without confusion, we shall still
traps at both zero temperature and finite temperature. refer to such a mixed phase as a topological superfluid.
In the following, we present only the zero-temperature
results,astheinclusionofafinite butsmalltemperature
(i.e., T = 0.1TF) essentially does not affect the results. A. Phase diagram at λkF/EF =1
The Fermi energy is E = (N/2)~ω = 50~ω. We have
F
takena cut-offenergyE =4E =200~ω andhaveused
c F
3N =3001Dharmonicoscillatorsastheexpansionfunc-

tions. These parameters are already sufficiently large to 10-1 0.1 E/EF
ensure the accuracy of calculations.
0.0
-3
EF10 -0.1
/ 590 595 600 605 610
III. PHASE DIAGRAM AT A GIVEN }
| -5
SPIN-ORBIT COUPLING E 10 BCS Topological Normal state
{|
n superfluid superfluid
i -7
The most salient feature of a spin-orbitcoupled Fermi m 10 0.1 E/EF
gas is the appearance of topological superfluidity and 0.0
-9
zero-energyMajoranafermionmode, under anappropri- 10 -0.1
ate Zeeman field. The quasiparticle operators of Ma- 590 595 600 605 610
jorana fermions are real and satisfy γ = γ†, which
0.2 0.3 0.4 0.5 0.6 0.7 0.8
means that a quasiparticle is its own antiparticle [3, 4].
h/E
Mathematically, we can always write a complex ordi- F
nary fermion operator c in terms of two real Majorana Figure 1: (color online) Phase diagram at a given spin-orbit
fermions γ and γ , such as c = γ iγ . An ordinary coupling λk /E = 1, determined from the behavior of
1 2 1 2 F F
−
fermion may therefore be viewed as a bound state of thelowest eigenenergy ofBogoliubov quasiparticlespectrum,
two Majorana fermions, which in general can not be de- min{|Eη|}. AstheZeemanfieldincreases,thesystemevolves
confined. However, the deconfinement does happen in a from a conventional BCS superfluid to a topological super-
fluid,andfinallytoanormalstate. Thetwoinsetsinthemid-
topologicalsuperfluid,leadingtotwoMajoranafermions
dleandright showthequasiparticle spectrumath/E =0.3
localized respectively at the two edges of topological su- F
and 0.5, respectively.
perfluid. This can be clearly seen with the help of the
particle-holeredundancyoftheBdGequation[2,13]. Let
In Fig. 1 we report the phase diagramat a fixed spin-
us imagethatwehaveazero-energysolution E =0. Be-
orbit coupling strength λk /E = 1. The emergence of
causeoftheparticle-holeredundancyΓ =Γ† ,wewill F F
E −E a topological superfluid can be clearly revealed by the
immediately have Γ0 =Γ†0 - exactly the defining feature behavior of the lowest eigenenergy of the quasiparticle
of a Majorana fermion. We note that, zero-energy Ma- energy spectrum. As shown in the middle inset, at a
jorana fermions should always come in pairs, since the smallZeemanfieldthe energyspectrumisgapped. How-
original model Hamiltonian describes ordinary fermions ever,byincreasingtheZeemanfieldaboveacriticalvalue
only and each Majorana fermion is just a half of ordi- ofh 0.35E ,thelowesteigenenergybecomesexponen-
F
∼
nary fermion. It is straightforward to check from the tially small. Four quasiparticle modes with nearly zero
BdG Hamiltonian that the wave functions of two paired energy appear, as seen clearly from the right inset. By
Majorana fermions should satisfy uσ(x) = vσ∗(x) and further increasing the Zeeman field (h > 0.65EF), the
uσ(x) = −vσ∗(x), respectively. The former follows the system will be driven into a normal state with negligible
particle-hole symmetry, while the later is required to ex- superfluid order parameter.
pressanordinaryfermionbytwoMajoranafermions[32]. The appearance of the topological superfluid can also
Tosatisfytheprescriptionofazero-energysolutionfor be monitored by the calculation of h h (x), where
c
a topological superfluid, the quasiparticle energy spec- h (x)= µ2(x)+∆2(x)isthelocalcriti−calZeemanfield
c
trum must become gapless at a certain point. In the foralocapluniformcellatpositionxwiththelocalchem-
case of a homogeneous spin-orbit coupled Fermi gas un- ical potential µ(x) µ mω2x2/2 and order parameter
der a Zeeman field, this happens at a critical Zeeman ∆(x). The local un≡ifor−m cell would be in the topologi-
field [14, 15], cal superfluid state if h > h (x). In Fig. 2, we present
c
h h (x) and ∆(x) at different phases. In accord with
hc = µ2+∆2. (5) Fi−g. 1c, at a small field h=0.3E (Fig. 2(a)), h<h (x)
p F c
Thesystemwillbeinaconventionalsuperfluidath<h foranypositionxandthewholeFermicloudisinthecon-
c
and in a topological superfluid at h > h . For a trapped ventional superfluid. At the field h=0.5E (Fig. 2(b)),
c F
4

0.4 (a) h/E = 0.3
F 0.5
0.2
s
e
0.0 d
o
m 0.0
-0.2 a
n
-0.4 ra
F o -5
)/E -1.5 -1.0 -0.5 0. 0 0.5 1.0 1.5 Maj-0.5 (a) EZES=-3.3x10 EF
x
and ( 00..24 (b) h/EF = 0.5 ns of -0.5 0. 0 0.5
x)]/E F 0.0 unctio 0.5 uu((xx)) vv((xx))
( -0.2 f
hc e
h - -0.4 Wav 0.0
[
-1.5 -1.0 -0.5 0. 0 0.5 1.0 1.5
-5
0.4 (c) h/EF = 0.8 -0.5 (b) EZES=+3.3x10 EF
0.2
-0.5 0.0 0.5
0.0 x/ xF
-0.2 Figure 3: (color online) Wave functions of the paired Majo-
rana modes at the inner wing of the trap, x ≃ ±0.5x : one
-0.4 has the energy E ≃ −3.3×10−5E (a) and theFother
ZES F
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 EZES ≃+3.3×10−5EF (b). Bothmodessatisfy thesymme-
x/xF tryrequirementforMajoranawave-functions. Thewavefunc-
tionsareinunitsofa−1/2. Hereh=0.5E andλk /E =1.
Figure 2: (color online) Spatial dependence of the critical ho F F F
The system is in thetopological superfluidphase.
Zeeman field h−h (x) (solid lines) and the superfluid order
c
parameter ∆(x) (dot-dashed lines), at λkF/EF = 1 and at
three Zeeman fields h/E = 0.3, 0.5, and 0.8. The cross-
F
patterns highlight the area in which the atoms are in the -10
topological superfluid state. 0.5 (a) EZES=-7.2x10 EF
s
e
d
o
awnedfitnhderhef>orhect(hxe)raetatrheettwwoobwloinckgss oofftthoepohlaorgmicoanliscutprearp- na m 0.0
a u(x) v(x)
fluid, as highlighted by the cross-pattern. At an even r
large Zeeman-field (Fig. 2(c)), the area of h > hc(x) ajo-0.5 u(x) v(x)
M
extends over the whole system. However, the superfluid
order parameter becomes so small, the system can no of -1.2 -1.0 -0.8 0.8 1.0 1.2
longer be viewed a superfluid. We note that, at large ns
attractive interactions where the order parameter is not ctio 0.5 (b) EZES=+7.2x10-10EF
destroyed by large Zeeman field, it is possible to have a n
u
singletopologicalsuperfluidthroughoutthe wholeFermi f
e
cloud. v
a 0.0
W
B. Majorana fermions
-0.5
Ineach ofthe topologicalsuperfluidphases,we should
-1.2 -1.0 -0.8 0.8 1.0 1.2
find two Majorana fermion modes, well-localized at the
two edges respectively. At the Zeeman field h = 0.5E , x /xF
F
we therefore could have four Majorana fermions, as in- Figure 4: (color online) Wave functions of the paired Majo-
dicated by the energy spectrum in the right inset of Fig. rana modes at the outer wing of the trap, x ≃ ±1.1xF: one
1. The wave functions of these Majorana fermions are has the energy EZES ≃ −7.2×10−10EF (a) and the other
showninFigs. 3and4forstateslocalizedatx 0.5x EZES ≃+7.2×10−10EF (b). Otherparametersarethesame
F
≃± as in Fig. 3.
and 1.1x , respectively. It is interesting that the wave
F
±
5
functions of two paired Majorana fermions, for example,
0.8
these located at x 0.5x and x +0.5x (Fig. 3),
F F
≃ − ≃ (a) h/E = 0.3
tend to interfere with each other [13, 29]. This quasi- F
0.6 n(x)
particleinterferenceortunneling leadstothe splitting of
degenerate zero energy Majorana modes to a finite but
0.4
exponentially small energy: E 3.3 10−5E . n(x )
ZES F
≃ ± ×
The tunneling between the paired Majorana fermions at 0.2 n(x)
the outer wing of the trap, x 1.1x , is more diffi-
F
≃ ±
cult (see Fig. 4),so the energy splitting is muchsmaller, 0.0
i.e., E 7.2 10−10E . It is readily seen that the -1.5 -1.0 -0.5 0. 0 0.5 1.0 1.5
utpraσyi(roxefd)ZM=wEaSa−jvo≃verσ∗af±(nuxan)c,wtai×aosvneasnfutsianctciiptsiafFoytnesde.itbhyerthueσr(exq)ui=redvσs∗y(mx)moer- x), n(x) 00..68 (b) h/EF = 0.5
(
n 0.4
),
x
C. Density distribution and local density of states ( 0.2
n
0.0
We now consider the possible experimental signature -1.5 -1.0 -0.5 0. 0 0.5 1.0 1.5
for observing topological superfluid and the associated 0.8
Majorana fermions. The useful experimental tools in- (c) h/EF = 0.8
0.6
clude in-situ absorption imaging and spatially resolved
radio-frequency(rf)spectroscopy[30],whichgiverespec-
0.4
tively the density distribution and the local density of
states of the Fermi cloud [31].
0.2
In Fig. 5, we plot the spin-up n (x) and spin-down
↑
n↓(x) density distribution and their difference ∆n(x) = 0.0
n (x) n (x) at different phases. While the shape -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
↑ ↓
− x/x
of the spin-up density distribution n (x) is nearly un- F
↑
changedacrossdifferentphases,inthe topologicalsuper- Figure 5: (color online) The spin-up and spin-down density
fluid phase (see Fig. 5(b) at h = 0.5E ) the spin-down distribution,n (x)(dashedlines)andn (x)(solidlines),and
F ↑ ↑
density distribution n↓(x) shows an interesting bi-modal their difference ∆n(x) = n↑(x)−n↓(x) (dot-dashed lines),
structure. It decreases rapidly when the atoms enter the are shown at the conventional superfluid phase (a), topolog-
topological area from the center. Accordingly, a broad ical superfluid state (b), and normal state (c). The den-
sity distributions are in units of the Thomas-Fermi density
dipappearinthe density difference aroundthe trapcen-
ter. Thebi-modaldistributioninn (x) maybe regarded n0 =2N1/2/(πaho). Thespin-orbit coupling is λkF/EF =1.
↓
asausefulandconvenientfeaturetoidentifythetopolog-
icalsuperfluid. However,itisnotacharacteristicfeature
for identifying the Majorana modes, as the contribution
oftheMajoranamodestothedensitydistributionisneg-
ligibly small, i.e., relatively at the order of N−1/2. 1.0 1.0
A practical way to probe the Majorana fermions is to (a) (x,E) (b) (x,E)
measure the local density of states using the spatially 0.5 0.5
resolved rf spectroscopy [30, 31], with which we antici- x/xF0.0 0. 0
pate thatthe contributionsofMajoranafermions willbe
well-isolated in both energy domain and real space. The -0.5 -0.5
local density of states for spin-up and spin-down atoms
is defined by, -1.0 -1.0
-0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2
ρ (x,E)= 1 u 2δ(E E )+ v 2δ(E+E ) . E/EF E/EF
σ ση η ση η
2Xη h| | − | | i Figure 6: (color online) Linear contour plot (in arbitrary
(6) units)forthelocaldensityofstatesofspin-upatomsρ↑(x,E)
InFig. 6,wereportthelocaldensityofstatesinthetopo- (a) and of spin-down atoms ρ↓(x,E) (b). The contributions
from Majorana fermions are highlighted by circles. Here the
logicalsuperfluidstate. Near the zeroenergy,the contri-
Fermi cloud is in the topological superfluid state with pa-
butions from Majorana fermions are clearly visible and
rameters h = 0.5E and λk /E = 1. In the calculation,
arewell-separatedfromotherquasiparticlecontributions F F F
the δ-function in ρ (x,E) has been simulated by a Lorentz
σ
by an energy gap ∆ 0.1E . It is interesting to note
F distribution with a small energy broadening Γ=0.01E .
∼ F
that the Majorana modes at x 1.1x and 0.5x
F F
≃ ± ±
contribute to ρ (x,E) and ρ (x,E), respectively. This
↑ ↓
6

can be understood from the wave function of Majorana
modes, as shown in Figs. 3 and 4. The wave-functions 0.6 (a) kF/EF = 0.2 (b) kF/EF = 0.6 (c) kF/EF = 1.0
at 0.5x are dominated by the spin-down component,
wh±iletheFwave-functionsat 1.1xF havemainlythespin- (x) n(x)
up component. ± x), 0.4
(
x), n n(x)
(
n0.2
IV. PHASE DIAGRAM AT A GIVEN ZEEMAN (x)
FIELD
0.0
We now turn to consider the possibility of observing
-1 0 1 -1 0 1 -1 0 1
a topological superfluid with spatially oscillating order
x/xF x/xF x/xF
parameter[24,25]. Intheabsenceofspin-orbitcoupling,
Figure 8: (color online) Density distributions and order pa-
it is known that the ground state of an imbalance 1D
rameter at h = 0.4E and at three different spin-orbit cou-
Fermi gas under Zeeman field can be an inhomogeneous F
plings: λk /E = 0.2 (a), 0.6 (b), and 1.0 (c). The density
FFLO superfluid with oscillating order parameter. It is F F
distribution n (x) is in units of the Thomas-Fermi density
σ
thereforenaturaltoask: whatisthefateofsuchaFFLO n0 =2N1/2/(πaho). The order parameter ∆(x) is in unitsof
superfluid when we switch on the spin-orbit coupling?
theFermi energy E .
F

0.02
FFLO Topological
small, suggesting a topological superfluid. However, in
superfluid superfluid
this case,the orderparameterno longeroscillatesinreal
EF space, as shown in Figs. 8(b) and 8(c). Therefore, we
/ 0.4
} h-h(x) conclude that it seems impossible to create an inhomo-
| c
{|E 0.01 0.2 geneous topological superfluid with spatially oscillating
n order parameter in 1D spin-orbit coupled Fermi gas, if
mi 0.0 we do not tailor specifically the geometry or other pa-
(x)
rameters of the Fermi cloud.
-0.2
-1 0 1
x/x
F
0.00
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
k /E V. CONCLUSIONS
F F
Figure 7: (color online) Phase diagram at a given Zeeman
field h= 0.4EF, determined from the behavior of the lowest In conclusions, we have investigated theoretically the
eigenenergyofBogoliubovquasiparticlespectrum,min{|Eη|}. properties of a 1D imbalanced Fermi gas under non-
Asthespin-orbitcouplingincreases,thesystemevolvesfroma
Abelian synthetic gauge field. We have predicted that
FFLOsuperfluidtoatopologicalsuperfluid. Theinsetshows
bysuitablytuningthestrengthofspin-orbitcouplingand
the critical Zeeman field h−h (x) and the order parameter
c Zeeman field, it is possible to create a topologicalsuper-
∆(x) at λk /E =0.3, where the Fermi gas is in the FFLO
F F
fluid, which hosts Majorana zero-energy fermions at its
superfluid state.
edge. The order parameter in the topological superfluid
is always of the conventional Bardeen-Cooper-Schrieffer
In Fig. 7, we present the phase diagram at a given
type,asthespin-orbitcouplingtendstodestroyinhomo-
Zeeman field h = 0.4E , determined again by tracing
F
geneous Fulde-Ferrell-Larkin-Ovchinnikov pairing. Ex-
the behavior of the lowest eigenenergy of the quasiparti-
perimentally,thetopologicalsuperfluidmaybeidentified
clespectrumasafunctionofthespin-orbitcoupling. The
from the bimodal distribution of the spin-down atomic
densitydistributionsandorderparameterarereportedin
density by using in-situ absorption imaging. The as-
Fig. 8 for three values of spin-orbit coupling. At small
sociated Majorana fermions may be detected by apply-
spin-orbitcoupling,wefindastableFFLOorderparame-
ing the spatially resolved radio-frequency spectroscopy,
ter which is modified slightly by the spin-orbit coupling.
which would show a well-isolated signal at zero energy.
However, in the area where ∆ (x) is nonzero, the
FFLO
criterion for a topological superfluid h > h (x) is al- At the end of this paper, we would like to emphasize
c
ways not satisfied, as seen from the inset for the case of that the ultracold atomic Fermi gas with non-Abelian
λk /E = 0.3. This excludes the coexistence of FFLO synthetic gauge field is an ideal platform for creat-
F F
superfluid and topological order. As a result, the energy ing topological superfluid and manipulating Majorana
spectrum is gapped and min E > 0. With increas- fermions,becauseofitsunprecedentedcontrollabilityand
η
{| |}
ing the spin-orbit coupling above λk /E 0.6, we ob- flexibility. Thissystemcannowbe readilyrealizedinul-
F F
≃
serve that the lowest eigenenergy becomes exponentially tracold atom laboratories.
7
Acknowledgments NFRP-China (Grant No. 2011CB921502).
This work was supported by the ARC Discovery
Project (Grant No. DP0984637 and DP0984522) and
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