Dark Solitons with Majorana Fermions in Spin-Orbit-Coupled Fermi Gases

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by  Yong Xu

Dark Solitons with Majorana Fermions in Spin-Orbit-Coupled Fermi Gases
Yong Xu1,∗ Li Mao2,1,∗ Biao Wu3,4, and Chuanwei Zhang1†
1Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
2School of Physics and Technology, Wuhan University, Wuhan 430072, China
3International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China and
4Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
We show that a single dark soliton can exist in a spin-orbit-coupled Fermi gas with a high spin
4
imbalance, where spin-orbit coupling favors uniform superfluids over non-uniform Fulde-Ferrell-
1
Larkin-Ovchinnikovstates, leading todark soliton excitations in highly imbalanced gases. Abovea
0
2 critical spin imbalance, two topological Majorana fermions (MFs) without interactions can coexist
insideadarksoliton,pavingawayformanipulatingMFsthroughcontrollingsolitons. Atthetopo-
p logical transition point, the atom density contrast across the soliton suddenly vanishes, suggesting
e
a signature for identifyingtopological solitons.
S

6 PACSnumbers: 03.75.Ss,03.75.Lm
2

] Solitons, topological defects arising from the inter- tolerant quantum computation [45]. In this context, SO
s
a play between dispersion and nonlinearity of underly- coupled fermionic superfluids have their intrinsic advan-
g ing systems, are significant for many different physical tages for MFs because of their disorder free [46] and
- branches [1–4]. The realization of cold atomic super- highlycontrollablecharacteristics. Thereforeanotherim-
t
n fluids provides a clean and controllable platform for ex- portant question is whether topological Majorana exci-
a
ploring soliton physics. In cold atomic gases, dark soli- tations can exist inside dark solitons if such topological
u
tons represent quantum excitations of a superfluid with defects do exist.
q
t. the superfluid order parameter vanishing at the soliton In this Letter, we address these two important ques-
a center in conjunction with a phase jump across the soli-
tionsbystudyingdarksolitonsindegenerateFermigases
m
ton. Dark solitons have been extensively investigated
(DFGs) trapped in 1D harmonic potentials with the ex-
- in cold atoms [5–13]. In particular, dark solitons have
perimentally already realized SO coupling and spin im-
d
recently been experimentally observed in strongly inter-
n balances. Herethe spinimbalanceisequivalenttoaZee-
acting spin balanced Fermi gases [14], where the Cooper
o man field. In the absence of SO coupling, the FFLO
c pairing wave-function has a phase jump across the soli- state [22] with an oscillating order parameter amplitude
[ ton [15–21]. However, dark solitons in the presence of a
is the ground state [24] with a large Zeeman field, which
2 large spin imbalance have not been well explored. cannotsupportdarksolitons. WithSOcoupling,wefind
v With a large spin imbalance, the ground state of the
(i) SO coupling suppresses the FFLO state, leading
7
superfluid is theoretically predicted to be the spatially
7 to uniform BCS superfluids that support dark solitons.
non-uniform Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)
7 The parameter region for dark solitons and their spatial
3 phase[22]withfinitemomentumpairingin1Dandquasi- properties are obtained.
. 1D [23–28], which is partially verified by the experi-
1 (ii)Forsubstantiallylargespinimbalances,wefindre-
ment [29]. This spatially non-uniform phase does not
0
markably that two MFs cancoexist inside a dark soliton
4 support dark soliton excitations that usually occur in
without any interaction, beyond the general expectation
1 BCS-type uniform superfluids. On the other hand, uni-
v: formsuperfluidscanexistinlargespinimbalancedFermi that two MFs with overlapping wave-functions interact,
leading to energy splitting that destroysMFs. Such soli-
i gases [30] in the presence of spin-orbit (SO) coupling.
X tonsaretopologicalsolitonstobedistinguishedfromsoli-
Since SO coupling for cold atoms has been experimen-
r tons without MFs.
tally generated recently for both bosons and fermions
a
[31–36], a natural question is whether such SO coupled (iii)TheexperimentalsignatureofMFsinsidethedark
superfluids with large spin imbalances can also support solitoninthelocaldensityofstates(LDOS)ischaracter-
dark solitons. ized,whichshowanisolatedzeroenergypeakatthecen-
ter of the soliton. Moreover, the density contrast across
More interestingly, it is well known that defects (vor-
the soliton suddenly decreases to zero at the topological
tices, edges, etc.) in SO coupled fermionic superflu-
transition point, which may be used to experimentally
ids with large spin imbalances can accommodate Ma-
detect topological solitons.
jorana fermions (MFs) [37–42], topological excitations
that satisfy exotic non-Abelian exchange statistics [43]. SystemandHamiltonian: ConsideraSOcoupledDFG
Recently MFs have attracted tremendous attention in confinedina1Dharmonictrapwiththe transversalcon-
various physicalsystems [44] because of their fundamen- finement provided by a tightly focused optical dipole
tal importance as well as potential applications in fault- trap. The many-body Hamiltonian of the system can
2
be written as
(cid:3)=0.65π 1 ξ
=π
H = dxΨˆ†(x)H Ψˆ(x) (1) (cid:3)=1.5π
Z s (cid:3)=2.5π 0.5
−gZ dxΨˆ†↑(x)Ψˆ†↓(x)Ψˆ↓(x)Ψˆ↑(x), /EF 0 (cid:3) /a 0 KF(cid:1)1
−0.5
−
where the single particle grand-canonical Hamiltonian −1
(a) (b)
tHrasp=pi−ng~2p∂ox2t/e2nmtia−l µV+(xV) (=x)m+ωH2SxO2/C2+, µHzis, tthheehcahremmoincaicl (c)
potential, g is the attractive s-wave scattering interac- 10 1
tion strength between atoms that can be tuned through
0.8
Feshbach resonances, m is the atom mass, and ω is the s a
trapping frequency. Ψˆ(x) = [Ψˆ↑(x),Ψˆ↓(x)]T with the n x 6 n/n 0.6 5 nn
atom creation (annihilation) operator Ψˆ†(x) (Ψˆ (x)) at ↓
spinν andpositionx. WeconsidertheeqνualRasνhbaand 2 0.4 (d) 0 n↑
Dresselhaus SO coupling H = i~α∂ σ , where σ (cid:0)1 0 1
SOC − x y i 10 0 10 1 0 1
are Pauli matrices. The Zeeman field Hz = Vzσz gener- x/x x/x
s s
ates the spin imbalance. This type of SO coupling and
Zeeman field have been realized experimentally [31–36]
FIG.1: (Color online)Profiles oftheorderparameter∆((a)
forcoldatomFermigasesusingtwocounter-propagating and(b))andthetotaldensityn((c)and(d))withincreasing
Raman lasers that couple two atomic hyperfine ground atominteractions. In(b)and(d),thedetailsofdarksolitons
states (i.e., the spin). inasmall region areplotted,where∆ andnarerespectively
Within the standard mean-field approximation, the scaled by ∆a and na, which are the asymptotic value to the
origin point without the soliton. In (c), lines corresponding
fermionicsuperfluidscanbedescribedbytheBogoliubov-
to Γ = 1.5π,π,0.65π are moved up 1.5xs,3.0xs,4.5xs with
de Gennes (BdG) equation
respect to the line for Γ = 2.5π. In the inset of (d), the
H ∆(x) u u densityprofiles(nσ forspinσandn=n↑+n↓)foragaswith
s n =E n , (2) Γ=π without SOcoupling areplotted. HereαkF =EF and
(cid:18)∆(x)∗ −σyHs∗σy (cid:19)(cid:18) vn (cid:19) n(cid:18) vn (cid:19) Vz =0.16EF.
where u = [u (x),u (x)]T, v = [v (x), v (x)]
n n↑ n↓ n n↓ n↑
−
are the Nambu spinor wave-functions for the
energy) in the absence of the SO coupling and Zeeman
quasi-particle excitation energy E , the or-
n field and the harmonic oscillator length x = ~/mω
der parameter ∆(x) = g Ψˆ (x)Ψˆ (x) = s
− h ↓ ↑ i as the units of energy and length. For 1D Fermpi gases,
−(g/2) |En|<Echun↑vn∗↓f(En)+un↓vn∗↑f(−En)i, the interaction parameter g = −2~2/ma1D with an ef-
and Pthe atom density ρσ(x) = fective 1D scattering length a1D [29]. A dimensionless
(1/2) |En|<Ec |unσ|2f(En)+|vnσ|2f(−En) with parameter Γ = −mg/n(0)~2 = πxs/√Na1D, which is
the enPergy cut-(cid:2)off E . f(E) = 1/(1 + eE/k(cid:3)BT) is the proportionalto the ratio betweenthe interactionand ki-
c
quasi-particle Fermi-Dirac distribution at the temper- netic energy at the center, can be used to characterize
ature T. With the constraints of a fixed total number the interaction strength [24]. Here n(0) is the density
of atoms N = dx[ρ (x)+ρ (x)] and the definition at the trap center in Thomas-Fermi approximation. In
↑ ↓
of the order paRrameter, Eq. (2) can be solved self- experiments [29], this value can be as large as 5.2. We
consistently. To obtain a stationary soliton excitation choose Γ=π in most of our calculations.
in the superfluid, we choose ∆tanh(x/ξ) with coherent Dark solitons in spin imbalanced DFGs: In Fig. 1,
length ξ = ~v /∆ and Fermi velocity v = ~K /m we plot the order parameter and density profiles for an
F F F
as the initial order parameter, and then solve the BdG imbalanced SO coupled Fermi gas with different inter-
equationsself-consistentlyuntil the orderparameterand action strengths. With increasing interactions, both ∆
density converge. and the depth of the soliton increasewhile µ (not shown
To solve the BdG equation, we expand u and v on here) decreases, signalling the crossover from BCS su-
n n
the basis states of the harmonic oscillator to convertthe perfluids to BEC molecule bound states. Without SO
equationtoadiagonalizationproblemofasecularmatrix. coupling, the existence of a dark soliton leads to the
We consider N = 100 atoms with 300 harmonic oscilla- depletion (enhancement) of spin ( ) component even
↑ ↓
tor states for the wave-function expansion. The energy with small Zeeman fields, thus the total atom density
cut-off E = 240~ω, which is large enough to ensure the only has a small depletionat the solitoncenter as shown
c
accuracyofthecalculation[24]. Wechoosethesinglepar- in the inset of Fig. 1(d). This is in sharp contrast to
ticle Fermi energy E = N~ω/2 (neglecting zero point Fig. 1(c,d) with SO coupling, where a strong deple-
F
3
0.8 FFLO PTS TS 0.18 ∆1 ∆0 0.3 (a) u↓ u↑ 10 4500
0.6 0.6 0 x/xs0 30
V/EzF00..24 NS 00..24 NS PTS TS e−functions−00..33 −4 −2001.5 0 −0.052 4 −10 (c) 1200
v −1 20
(a) 0 (b) a −15 −10 10 15 10
w
00 0.5α K /1E 1.5 0 V0.5/E 1 0 x/xs0 1105
F F z F −10 5
−0.3 (b) (d)
FIG.2: (Coloronline)(a)Phasediagraminthe(α,Vz)plane. −0.1 0 0.1
PTS: partial topological superfluid; TS: topological super- −4 −2 0 2 4 E/E
x/x F
fluid; NS: normal superfluid. (b) Quasi-particle excitation s
energies (allotherlines exceptlight green dashed-dottedand
magentadashedlines)and∆0(maximumoftheorderparam- FIG. 3: (Color online) The left panel shows the wave-
eter in whole region), ∆1 (minimum of the order parameter functions (solid red line for u↑ and dashed green one for u↓)
in region (2xs < x < 9xs), where the topological superfluids of four MFs: two inside the dark soliton plotted in (a) and
exist). αKF =EF and Γ=π. (b) and two at the edges in the insets. (c) and (d) show the
localdensityofstatesforspin↓andspin↑respectively. Here
Vz =0.768EF, αKF =EF, and Γ=π.
tion of the total density (also for each spin component)
in the dark soliton is observed. In Fig. 1(b), we ob-
serve two length scales for the dark soliton. One is by a TS, and TS phase, where the whole region is topo-
K−1 defined using the local density approximation, cor- logical. ThephaseseparationinthePTSphaseoriginates
F
responding to the steep slope and the oscillation wave- from the harmonic trapping geometry, where the chemi-
length. The otheris the coherencelength,corresponding cal potential is replaced by a local one µ¯(x) = µ V(x)
−
to the smoother oscillationslope [16, 47], which is about using the local density approximation. For a homoge-
5K−1,2.2K−1,1.2K−1,K−1 for Γ=0.65π,π,1.5π,2.5π, neous gas, superfluids become topological when Vz >
F F F F µ2+∆2 [38, 39]with Majaronazero modes locatedat
respectively. These two scales are equivalent in the BEC
limit, where the oscillation structure of the soliton van- pthe edges. For a harmonically trapped gas, as the chem-
icalpotential decreasesfromthe trapcenter to edge, the
ishes.
condition V > µ¯2(x)+∆2 [41] (See supplementary
The physicalmechanismfor the dark solitonin anim- z
materials)is firstpsatisfiedatthe wingsofthe superfluids
balanced Fermi gas roots in the SO coupling. With-
out SO coupling, the ground state of a 1D imbalanced (Vz < µ2+∆2 at the trap center) for the PTS phase.
Fermigaseinaharmonictrapispredictedtopossessthe Thistrpansitionischaracterizedbytheappearanceoftwo
two-shell structure: partially polarized core (i.e. FFLO zero energy modes (cyan line and red one hidden be-
state) surrounded by either a paired or a fully polarized hindthecyaninFig.2(b))locatedaroundtheplacewith
phase [24, 28]. This two-shell structure has been par- Vz = µ¯2+∆2, andthe sharpdecreaseof the order pa-
tially verified in the experiment [29]. With SO coupling, rametperinthetopologicalregion(∆1inFig.2(b)). AsVz
the FFLO state is dramatically suppressed as shown in is furthur increased with Vz > µ2+∆2, the whole su-
Fig. 2(a) because the BCS type of zero total momen- perfluidsbecometopological(TpSphase)asthemaximum
tum Cooper pairing can be formed in the same helicity of the order parameter (∆0) drops suddenly. Without a
band, which is energetically preferred than the FFLO soliton, there is only one zero energy mode in TS phase.
state that is formed through the pairing between atoms However, a soliton induces another zero energy mode.
intwodifferenthelicitybands[48,49]withnon-zerototal This additional zero energy mode is different from con-
momentum. A dark soliton can be created in the BCS ventional local gapped excitations (blue and green lines
type of phases, but not in the FFLO phase, where the inFig. 2(b)) thataresimilar to Andreev bound states in
stable state generated by phase imprinting is also an os- a vortex [16, 47]. It appears only when the local super-
cillating FFLO state with a sinusoidal-like form and has fluid, where the soliton is located, becomes topological.
lower energy than a soliton state. In this sense, SO cou- To confirm that the additional zero energy state is
pling makes it possible to generate a single dark soliton MFs inside the soliton, we consider a linear combina-
excitation with a high spin imbalance. tion of the Bogoliubov quasi-particle operators γ0n for
Two MFs inside a dark soliton: There are two topo- states with En 0 (here E2 > E1 correspond to
∼
logical superfluid (TS) phases in Fig. 2(a): partial TS cyan and red lines respectively) to obtain spatially lo-
(PTS) phase, where the superfluid has a phase separa- calized states: γL = (γ0+2 + γ0+1 + γ0−2 + γ0−1)/2,
tion structure with a normal superfluid core surrounded γR =(γ0+2 γ0+1+γ0−2 γ0−1)/2,γS1 =(γ0+2 γ0−2)/2i,
− − −
4
1 sudden decrease of the density contrast at the topologi-
NS PTS TS
caltransitionpointismainlyduetothesharpdecreaseof
5
0.8 thepairingorderparameter,asshownby∆0 inFig.2(b),
0 leading to the sudden decrease of the number of atoms
0.6 participatinginthe pairingaroundthe soliton. Thesud-
P dendisappearanceofthedensitycontrastacrossthedark
0.4 soliton might provide an experimental signature for the
5 appearance of MFs inside the dark soliton. From the in-
0.2 density5 nnn↓↑ 0 tsuetrse,fwoersVezet>ha0t.5t3hEesFo,liwtohnerdeenthsietydnen↓shitaysaincsoidnevetxhestsroulci--
0 ton is largerthan its surrounding. The transitionto this
0
0 0.2 0.4 0.6 0.8 1
convex structure leads to the kinks around V =0.53E
V/E z F
z F observed in Fig. 2(b) and Fig. 4. This convex structure
ofsolitondensityiscausedbythelocalquasi-particleex-
FIG. 4: (Color online) Density contrast Pσ as a function of
citations. In fact, without taking into account of such
Vz forthespin ↓(circle green line), spin ↑(squareblueline),
and total (diamond red line) densities in the soliton region quasi-particle contributions to the density, P↓ is almost
−1.5xs <x<1.5xs. Theinsetsshowthedensityprofiles(nσ zero.
forspinσ andn=n↑+n↓)ofthesolitonineachregionwith Experimental signature of topological solitons and
the unit 1/xs, corresponding to Vz =0.3EF,0.7EF,0.768EF MFs: In the right panel of Fig. 3, we present the lo-
respectively. Here αKF =EF and Γ=π. cal density of states [24] of the Fermi gas, ρ (x,E) =
σ
u 2δ(E E )+ v 2δ(E+E ) /2 with
|En|<Ec | nσ| − n | nσ| n
σP= , , w(cid:2)hich reflect the zero energy excitatio(cid:3)ns at the
γS2 = (γ0+1 γ0−1)/2i. Due to the particle-hole sym- plac↑es↓where MFs are locally accommodated. Clearly,in
−
metry γ0n = γ0†−n, we obtain γL† = γL, γR† = γR, and theTSregion,thezeroenergyMFstatesappearatx=0,
γ† =γ with σ =1,2, indicating that γ , γ and γ where the dark soliton locates. In experiments, the
Sσ Sσ L R Sσ
areselfHermitianMajoranaoperators. InFig.3(a,b),we LDOS could be measured using spatially resolved radio-
plotthewave-functionsofMFs,showingthattwoMFsat frequency(rf)spectroscopy[52]withthespaceresolution
the left and right edges (two insets) and two MFs inside about 1.4µm and spectral resolution about 0.5kHz that
the soliton ((a) and (b)). The wave-functionsof MFs in- are smaller than the space resolution 5.4µm and energy
side the solitonbehave like cos(πKFx/2)exp( x2/ξ02) resolution 3kHz shown in Fig. 3(c, d) [29]. The LDOS
and sin(πKFx/2)exp( ∼x2/ξ02), similar to A−ndreev in a dark soliton in the trap center provides a stronger
∼ −
bound states [16] but with ξ0 larger than ξ. They are andstablersignalthanthatattheedgeswithsmallatom
very different from widely known MFs in vortices or densities around MFs [41].
nanowire ends that interact due to the wave-function The1DSOcouplingandZeemanfieldsconsideredhere
overlap,leadingtoenergysplittingthatdestroysthezero have already been achieved for 40K and 6Li fermionic
energystates. ThevanishinginteractionbetweenMFsin- atomsbycouplingtwohyperfinegroundstatesusingtwo
side the dark soliton is due to the intrinsic property of a Raman laser beams in experiments [32, 33, 36]. In ex-
dark soliton: a sharp phase change. We can understand periment,theSOcouplingandZeemanfieldstrengthcan
this throughKitaev’stoymodel[50], andthe interaction betunedthroughvaryingthelaserintensityorthe setup
is proportional to i2t cos(δφ/2)γS1γS2 if written in the of the laser beams. The SO coupling can be as large as
−
Majorana fermion representation with the phase differ- αK E and a Zeeman field can be readily tuned to
F F
∼
ence δφ between two sides of the soliton. For a dark V E . Therealizationof1DFermigasesandthedark
z F
∼
soliton, δφ = π and there is no interaction. Such coex- soliton can be similar as that in recent experiments [14],
istence of two MFs inside a soliton makes it possible to where an elongated 1D Fermi gas is confined in a har-
dragMFstoanewpositiongiventhatadarksolitoncan monic trap with cylindrical symmetry (radial trapping
be manipulated by means of optical lattices [51]. frequency much larger than axial one) using a combina-
Evolution of soliton structure: The soliton structure tion of weak magnetic trap (axial) and tightly focused
can be characterized by the density contrast P = optical trap (radial). Dark solitons can be experimen-
σ
(nσ nσ )/nσ with the maximum nσ and min- tally created via phase imprinting [11, 12, 14], where a
max min max max
−
imum nσ of the density for spin σ (P for the total half of the cloud is shortly interacted with a laser beam
min t
density) in the soliton region. In Fig. 4, we plot P as a to acquire the phase difference.
σ
function of V . We see both P and P decrease, while Discussion: Themean-fieldBdGtheoryusedheremay
z ↓ t
P is almost a constant with increasing V before the giveaqualitativedescriptionofthe1Dphysics. Inpartic-
↑ z
appearance of MFs inside the soliton in the TS region. ular, for 1D Fermi gases with weak and moderate inter-
However,the solitonstructurealmostvanishesinthe TS actions,theenergyandchemicalpotentialobtainedfrom
region with two MFs accommodating the soliton. The mean-fieldtheoryandexactBetheansatzwerecompared
5
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12-1-0334),AFOSR (FA9550-13-1-0045),and NSF-PHY (2012).
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SUPPLEMENTARY MATERIALS
In the maintext, we show that there are four phases: NS, FFLO, PTS, and TS, instead ofthree in a homogeneous
space. This extral PTS phase arises from the existence of a harmonic trap, where the chemical potential is replaced
by a local one µ¯ = µ V(x) with the local density approximation. It is well known that a homogeneous spin-orbit
−
coupledsuperfluidbecomestopologicalwhenV > µ2+∆2. Inaharmonictrap,thisrelationcanbeapproximately
z
written as V > µ¯2+∆(x)2. Based on this relatipon, a superfluid can have a phase separationstructure (i.e. PTS):
z
a topological supperfluid wing (V > µ¯2+∆(x)2 because of the smaller effective chemical potential µ¯) surrounding
z
a normal superfluid core (V < µp2+∆(x)2). This phase exists when ∆ < V < µ2+∆(0)2. In Fig. S1, we
z z
| |
present the order parameter profiple ∆(x) and ∆V =V µ¯2+∆2 in a large regionpfor three phases: (a) NS; (b)
z
| | −
PTS; (c) TS. In the NS phase, V is so small that V < µ¯2p+∆(x)2 in the whole space and there are no topological
z z
superfluids. As V is increased that V > µ¯2+∆(x)2pin the two shells as shown in Fig. S1 (b) so that superfluids
z z
become partially topological associated wipth two zero energy states. This transition can also be characterized by a
sharpdecreaseofthe minimum ofthe orderparameter(i.e. ∆1)atthe topologicalsuperfluids region(2xs <x<9xs)
as illustrated in Fig. 2(b) in the main text. As V is furthur increased, V > µ2+∆(x)2 and the whole superfluids
z z
become topological. In this phase, a soliton in the trap center hosts two Mapjorana fermions. Across the transition,
the maximum of the order parameter (∆0) drops suddenly as shown in Fig. 2(b) in the main text.
7
0.8 (a)
∆
0
∆
−0.8
.8 (b)−10∆ =m−in5(∆) 0 5 10
1 x/x
s
TS NS TS
− .8
0.5 (c)− −5 5
x/x
s
0
TS
−0.5
−10 −5 0 5 10
x/x
s
.
FIG. S1: (Color online) Plot of the order parameter profile |∆(x)| and ∆V =Vz−pµ¯2+∆(x)2 in (a) for theNS phase with
Vz =0.3EF, in(b) forthePTS phasewith Vz =0.7EF, andin (c) fortheTS phasewith Vz =0.768EF. HereαKF =EF and
Γ = π. ∆1 is the minimum of the order parameter in region 2xs < x < 9xs. The depletion at the center originates from the
existence of a soliton.