Simulation and measurement of the fractional particle number in one-dimensional optical lattices

0.36 MB·English

by  Dan-Wei Zhang

Simulation and measurement of the fractional particle number in one-dimensional
optical lattices
Dan-Wei Zhang,1,∗ Feng Mei,2 Zheng-Yuan Xue,3,1 Shi-Liang Zhu,2,4,† and Z. D. Wang1,‡
1Department of Physics and Center of Theoretical and Computational Physics,
The University of Hong Kong, Pokfulam Road, Hong Kong, China
2National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China
3Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,
5
SPTE, South China Normal University, Guangzhou 510006, China
1
4Synergetic Innovation Center of Quantum Information and Quantum Physics,
0
University of Science and Technology of China, Hefei 230026, China
2
(Dated: June29, 2015)

n
u Weproposeaschemetomimicanddirectlymeasurethefractionalparticlenumberinageneralized
J Su-Schrieffer-Heegermodelwithultracoldfermionsinone-dimensionalopticallattices. Weshowthat
the fractional particle number in this model can be simulated in the momentum-time parameter
6
2 space in terms of Berry curvature without a spatial domain wall. In this simulation, a hopping
modulation is adiabatically tuned to form a kink-typeconfiguration and the induced current plays

] therole of an analogous soliton distributingin thetime domain, such that themimicked fractional
s particle numberis expressed by the particle transport. Two feasible experimental setups of optical
a
lattices for realizing the required Su-Schrieffer-Heeger Hamiltonian with tunable parameters and
g
time-varyinghoppingmodulationarepresented. Wealsoshowpracticalmethodsformeasuringthe
-
t particle transport in the proposed cold atom systems by numerically calculating the shift of the
n
Wanniercenterand thecenter of mass of an atomic cloud.
a
u
PACSnumbers: 03.75.Ss,03.67.Ac,03.65.Vf
q
.
t
a
I. INTRODUCTION sic FPN has never been experimentally detected due to
m
the spin-doubling problem in these materials: two spin
- orientations are present for each electron and thus a do-
d
n Particle fractionalization has been recognized as a re- main wall in the polyacetylene carries an integer charge
o markableandfundamentalphenomenoninbothrelativis- [4]. Inspired by the newly discovered quantum spin Hall
c tic quantum field theory and condensed matter systems insulators [14], it was theoretically proposed to realize
[ [1–10]. The first physicaldemonstrationof fractionaliza- SSH model in an edge of this two-dimensional insulator

2 tion is the celebrated Su-Schrieffer-Heeger (SSH) model by bringing a magnetic domain wall there, and the edge
v of one-dimensional (1D) dimerized polymers [3, 4], such electronswiththeinherentchiralsymmetrymayprovide
9 as polyacetylene. In this model, a kink domain wall in a direct signature of FPN [15, 16]. However, creation
5 the electronhopping configurationinduces a zero-energy of such a magnetic domain wall acting only on the edge
0 soliton state carrying a half-charge [3, 4]. The basic elections is experimentally challenging and the proposed
0 physics of fractionalization in SSH model is governed by schemes are yet to be demonstrated.
0
a low-energy effective Dirac Hamiltonian with topologi-
. Inthepastyears,alotoftheoreticalandexperimental
2 cally nontrivial background fields, which was firstly pro-
0 posed by Jackiw and Rebbi [1, 2]. Subsequent achieve- workhasbeencarriedouttosimulatetheDiracequation
5 ments were made to generalize the original SSH model and the involved exotic effects by using ultracold atoms
1 [17–25]. Especially, it has been proposed to realize the
to exhibit an irrational (arbitrary) fermion number by
: (generalized) SSH model associated with effective Dirac
v breaking the conjugation symmetry [11–13]. The frac-
i tional particle number (FPN) in these systems can be Hamiltonian using ultracold atomic gases in the contin-
X
uum [26] and in optical lattices [27–30]. The detection
understoodintermsofglobaldeformationsoftheholesea
r of FPN in these cold atom systems was also suggested
a (or the valence band) due to the nontrivial background
by optical image of the density distribution of soliton
fields.
modes [26, 30]. Since single-component fermionic gases
SSH model has achieved great success in describing or component-dependent optical lattices are used in the
transport properties of polymers, and some novel phe- realization of FPN in atomic systems, the spin-doubling
nomenaassociatedwiththetopologicalsolitonshavealso problem encountered in condensed matter systems can
been explored in experiments [4]. However, the intrin- be avoided. In a recent experiment with a 1D optical
superlattice,SSHmodelintheabsenceofspatialdomain
walls was realized and its topological features were also
probed[31], making direct measurementofFPN in opti-
∗Electronicaddress: [email protected] cal lattices to be feasible and timely.
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] In this work, we propose a new scheme to mimic and
2
(a)
directly measure FPN in the generalized SSH model us-
ing ultracold fermions in 1D optical lattices. Firstly, we
show that FPN in this model can be simulated in the V(x)
momentum-time parameter space in terms of Berry cur-

vature without creating a spatial domain wall. In this
x
simulation, a hopping modulation is adiabatically tuned
to form a kink-type configuration and the induced cur- (b) (c)
rentplaystheroleofananalogoussolitondistributingin d 1
tehteertsimpaecedoismeaxinp,ressosetdhabtythtehemaimdiiacbkaetdicFpPaNrtiicnleptarraanms-- ! d11 2 g g 2 ! 2
port. Furthermore, we explore how to implement this 1
new scheme with ultracold fermions in 1D optical lat-
R (x)
1
tices. We propose two experimentally setups to realize
the required SSH Hamiltonian with tunable parameters
FIG. 1: (Color online) Two experimental setup for simulat-
andhoppingmodulations,andthenshowpracticalmeth-
ingSSHHamiltonianin1Dopticallattices. (a)Adouble-well
ods for measuring the particle transport in the proposed
opticalsuperlatticetrappingnoninteractingsingle-component
coldatomsystemsbynumericallycalculatingtheshiftof fermionic atoms. A unit cell contains two nearest lattice
the Wannier center and the center of mass of an atomic sites with energy offset ∆ (for atoms denoted by dark and
cloud. Some possible concerns in realistic experiments, light blue small balls) and the atomic hopping exhibits stag-
such as the energy scales, the adiabatic condition and gered modulation configuration. (b) A state-dependent opti-
the effects of an external harmonic trap, are also consid- cal lattice trapping noninteracting two-component fermionic
ered. In comparison with the previous proposals of re- atoms, where atomic states |↑i and |↓i are denoted by blue
and red small balls. The block dotted line represents a Ra-
alizing and detecting FPN [15, 26–30], one advantage of
the presented scheme is that it does not require the spa- manfield R1(x). (c)Raman-assisted tunneling. Theuniform
atomic hopping between the nearest-neighbour in the state-
tialdomaininthehoppingconfiguration,whichisusually
dependentopticallatticeisrealizedbyalarge-detunedRaman
hard to create. Another advantage is the adiabatic par-
transitionwithdetuning∆d1 andRabifrequenciesoftheRa-
ticletransportcorrespondingtothe valueofPFNcanbe manbeamsΩ1,2,whilethehoppingmodulationisrealizedby
directly measured in our proposed cold atom systems. another pair of Raman beams g1,2 with large detuning ∆d2.
The rest of this paper is organized as follows: Section TheZeemansplittinggivesrisetothe∆terminthissystem.
II presents a brief review on fractionlization in a gener-
alized SSH model. In Sec. III, we elaborate our scheme
of simulation and measurement of FPN in this model, odd number sites form two sublattices with modulating
based on the Berry curvature and adiabatic transport hoppingamplitudes(on-siteenergies)andthusaunitcell
approaches. In section IV, we propose two feasible ex- contains two nearest lattice sites, which are used to con-
perimental setups of 1D optical lattices to realize the re- stitute a pseudo-spin. When the inversion (conjugation)
quired Hamiltonian, and then discuss how to measure symmetry preserveswith ∆=0, the system corresponds
the atomic particle transport in the proposed systems. to the original SSH model of polyacetylene.
Finally, a short conclusion is given in Sec. V. ByemployingFouriertransformationinthespinbasis,
we can obtain the Bloch Hamiltonian of the model as
H=d~(k)·~σ, (2)
II. FRACTIONALIZATION IN SSH MODEL
where ~σ = (σ ,σ ,σ ) are the Pauli matrices acting on
x y z
Beforedescribingourscheme,webrieflyreviewthear- the pseudo-spin, d~(k) = (Jcoska,−δsinka,∆) is the
2 2
bitraryFPNinthegeneralizedSSHmodelinthissection. band vector with a being the lattice spacing as shownin
We start with this model described by a tight-binding Fig. 1(a). By linearizing the Blochbands nearthe Dirac
Hamiltonian [7, 12] pointk =π/a,Hamiltonian(2)canbetransformedinto
D
an effective low-energyrelativistic Hamiltonian [1, 2, 11]
H = [J +(−1)nδ] cˆ†cˆ +H.c.
n n+1 ∆
Xn (cid:0) (cid:1) HD =vFpˆxσx−2δσy + 2σz (3)
∆
+ 2 (−1)ncˆ†ncˆn, (1) inthecontinuum,wherevF =Ja/~istheFermivelocity,
Xn pˆx is the momentum operator measured from the Dirac
where cˆ (cˆ†) is the fermion annihilation (creation) op- point, δ and ∆ act as two backgroundfields [11].
n n It has been widely studied that for a kink-type back-
eratorinsite n, J is the uniformhopping amplitude, δ is
ground potential with δ(x → ±∞) = ±δ , an unpaired
the dimerized hopping modulation, and ∆ is a staggered 0
soliton state appears at the kink carrying FPN [2, 11]
potential breaking the inversion symmetry [the conjuga-
tionsymmetryinthe low-energyDiracHamiltonian(3)]. 1 4δ
0
Inthislatticesystem,asshowninFig. 1(a),theevenand Ns =− arctan( ), (4)
π ∆
3
which may exhibit arbitrary fractional eigenvalues. The (a) 0 0.2 0.4 (b) 0 0.5 1
minus sign in FPN is due to the fact that the physical 1.2
1.4
fermion number in the soliton sector is defined as be-
ing measuredrelative to the free sector without the kink 1.1 1.2
background,andthisfractionalpartofthefermionnum-
t0
ber actually comes from the global contribution (polar- t/ 1 1
ization)ofthevalenceband[2,11,26]. Inaddition,ithas
a topological character in the sense that it is dependent 0.9 0.8
onlyontheasymptoticbehaviorofthebackgroundfields
0.6
insteadoftheirlocalprofiles. When∆→0,itrecoversto 0.8
the half fermion number ±1 for the zero-energy soliton 0.9 k (1π/a) 1.1 0.8 k (1π/a) 1.2
2
mode in the original SSH model [1]. (c) 2 (d)
0.54
It is interesting to note that fractionalization also ex-
hibits in many low-dimensional correlated electron sys- 1 0.52
tems. Forinstance,awell-knownexampleisthatoffrac- y odd lattice site
tional excitations in the fractional quantum Hall regime E0 nsit 0.5
e
[5, 9], which is a consequence of strong Coulomb inter- d even lattice site
actionamong2DelectronsinpartiallyfilledLandaulev- −1 0.48
els. In addition, the collective excitations in some 1D
interacting fermion systems amay be characterized by −2 0.46
effective fractional charges via the spin-charge separa- −1 δ(0t)/δ 1 −1 δ(0t)/δ 1
0 0
tion mechanism [10]. Fractionalization in these systems
isbasicallyduetoelectioncorrelationsandhenceiscom- (e) 1 0.04
pletely different from that in SSH model of noninteract-
0.5 0.02
ing fermions. The fractional charges in some correlated
electron systems have been directly observed in experi- 0
ments [9, 10]. However, despite the fractionalization in δδ/ 0 0 −jc
SSH model being investigated for decades, FPN there
−0.5 −0.02
is yet to be directly measured in solid state materials
(mostly due to the spin-doubling problem) or in artifi- −1 −0.04
cial systems, even for the simplest half-fermion-number 00 00..55 11 11..55 22
t/t
case with ∆ = 0. Therefore, experimentally feasible 0
schemes for direct measurement of the intrinsic FPN in
SSH model would be of great value. FIG. 2: (Color online) (a) Berry curvature distribution of
the ground band F in the center region of the k-t parame-
kt
ter space. The Berry curvature outside is almost vanishing,
and integration over the whole parameter space (0 6 k 6
III. SCHEME TO DIRECTLY MEASURE PFN
2π/a,0 6 t 6 2t0) gives the particle transport. (b) Spin
textureofSSHmodel, interpretedas amappingfrom thek-t
Several schemes have been proposed to realize SSH parameterspaceontotheBlochspherebyEq. (5). Thecolors
model and to probe the soliton modes with FPN in cold show the Sz-component cos[θ(k,t)] and the arrows show the
atom systems [27–30]; however,the realizationof the re- azimuthalcomponent(Sx,Sy)=(sinθcosφ,sinθsinφ). Here
quired kink background and the local detection of the the lengths of all the arrows has been divided by a factor π
fermion number of a soliton state therein are still chal- for visibility. (c) The energy spectrum for a lattice system
withsizeL=100. (d)Thevariationofdensityineachlattice
lenging in practical experiments. In the following, we
site with respect to the time-varying hopping modulation at
propose a simpler scheme to mimic and then directly
hall filling. (e) The analog of kink potential and soliton-like
measure PFN via the adiabatic particle transport.
current in the time domain under adiabatic conditions. The
The energy bands of SSH model described by Hamil-
parameters in (a-e) are J =1, δ0 =∆=0.1 and t0 =5/ξ.
tonian (2) can be mapped onto a two-level system in
the Bloch sphere with the parameterized Bloch vector
S~ = h (sinθcosφ,sinθsinφ,cosθ), where θ and φ are
0
respectivelythepolarandazimuthalangles. Inthismap- 0. We consider the fractionalization in this band insula-
ping, we have tor at half filling [26], corresponding to the low-energy
level with the eigenstate |u i = (sinθe−iφ,−cosθ)T
− 2 2
h = 4J2cos2(ka/2)+4δ2sin2(ka/2)+∆2/4, with T being the transposition of matrix. In this
0
θ =arqccos(∆/2h0), (5) framework, one can define the Berry connection Aθ =
φ=−arctan δsin(ka/2) . hu−|i∂θu−i=0 and Aφ =hu−|i∂φu−i=sin2 2θ.
Jcos(ka/2)
Insteadofconsideringthekinkbackgroundfieldinthe
h i
Thedegeneracypointlocatesatk =π/a, δ =0and∆= spatialdomain,hereweintroduceatime-varyinghopping
4
modulation with a kink-type ramping configuration The particle transport for the ground band in this 1D
bandinsulatorovertherampingprogressionofparameter
δ(t)=δ0tanh[ξ(t−t0)], (6) δ(t) from t = 0 to t = tf is given by the integration of
the Berry curvature
wheret denotesthecenterofthetimedomainwallandξ
0
representstherampfrequency. Weassumet0 ≫1/ξ (ac- 1 tf 2π
tually t0 =5/ξ is large enough) such that δ(0)≃−δ0 at Q=−2π dt dkFkt
thebeginningt=0,andthenadiabaticallyrampthesys- Z0 Z0
1 2π
temtot=t =2t withδ(t )≃δ attheend. Thatisto
f 0 f 0 = dk[A (k,t )−A (k,0)] (8)
k f k
say, we simulate a kink potential in the time domain in- 2π
Z0
steadof creatingit in realspace. In this dynamicalcase,
1 4πJδ
the bulk gap is E = 2 ∆2+16δ(t)2, which will close =− arctan 0
att=t0 for origingalSSH modelwith ∆=0. Toguaran- π ∆ π2J2+4δ02+∆2/4!
p
tee the adiabatic condition for original SSH case, which 1 4δp0
≃− arctan =N ,
requires a gapped bulk band, we can use a time-varying π ∆ s
staggered potential with the form ∆(t)=δ sin(πt/2t ). (cid:18) (cid:19)
0 0
Weconsidertheadiabaticevolutionofthesystemwith where the approximation satisfies well for J ≫ δ ,∆,
0
the ramping parameter δ(t) (and ∆(t) for original SSH and becomes exact when ∆ = 0. In the calculation, the
case)andprovidethattheFermilevelliesinsidetheband integration of ∂ A over k vanishes due to the periodic
k t
gap in the whole progression. The Berry phase effect of condition in the Berry vector potential [7]. Here the un-
this 1D band insulator can be measured from the par- quantized adiabatic particle can be regarded as the po-
ticle transport [7]. This is an analog of the adiabatic larizationchangeinthis 1Dbandinsulator[7,41],which
chargepumping proposedby Thouless[32]; however,the has also been discussed in the context of nanotubes and
parametric driving in our case does not form a closed ferroelectricsmaterials[42]. Ifoneconsideranadditional
cycle but only a half one. The topological pumping in anti-kink-type modulation to form a full cycle, then the
cold atom systems and photonic quasi-crystals has been particle transport will be quantized (±1 or 0 depending
discussed in the contexts of SSH model [33–35] and 1D on the loop of the cycle) after one period as the inte-
quasi-periodicHarpermodel[36–40],wherethepumping gration of the Berry curvature in the extensional region
particleisshowntobequantizedoneoveroneperiodand contribute the other fractional portion [7, 15].
can be fractional over a fraction of one period [40]. There are actually close connections between FPN of
In the momentum-time (k-t) parameter space, we can a soliton state and the adiabatic transport via the Berry
rewrite the Berry connection as A = ∂ φA +∂ θA phaseapproachinEq. (8). Wecanconsidertheresponse
k k φ k θ
and A = ∂ φA + ∂ θA . Thus the Berry curvature equation in the progression of dynamical generation of
t t φ t θ
F =∂ A −∂ A in the k-t space can be obtained as the backgroundfield [6, 11]:
kt k t t k
∆Jsin2 ka∂ δ(t) 1 1
F = 2 t , (7) ρs = ∂xΘ(x,t), jc = ∂tΘ(x,t), (9)
kt 2[4J2cos2 ka +4δ(t)2sin2 ka +∆2/4]23 2π 2π
2 2
where Θ(x,t) represents the generalized angular angle
where J and ∆ are assumed to be constants here. We
of the background field, ρ denotes the soliton density
s
notethatBerrycurvaturedistributiongivenbyEq. (7)is
distribution near the spatial domain wall, and j is the
c
modifiedfortime-varying∆(t)=δ sin(πt/2t )discussed
0 0 induced current. In the presentmodel with J ≫δ,∆ (in
previously,however,thecorrespondingPFNgivenbyfol-
whichcasethesamepolarizationvariationisobtainedby
lowing equation (8) remains since it just depends on the
the band Hamiltonian and the low-energy Dirac Hamil-
boundaries of the background fields [4].
tonian), the angular angle Θ = −arctan(4δ/∆) is just
Figure 2(a) shows an example of the Berry curvature
time-dependent. Therefore, the induced current mimics
distribution in the center regionof the k-t space for typ-
an analog of kink-soliton in the time domain, as shown
ical parameters, while the Berry curvature outside is
in Fig. 2(e). The induced current over the whole time
almost vanishing. In small ∆ limit, a sharp peak ex-
domain gives the transferred particle
hibits in the Berry curvature distribution at the posi-
tion of (k,t) = (π/a,t0), which is the dominant contri- tf
Q= j (t)dt, (10)
bution to its integration over the parameter space. Fig- c
ure 2(b) shows the corresponding spin texture, which is Z0
interpreted as a mapping from the k-t parameter space whichtakesthe valuegivenbyEq. (8)anddepends only
onto the Bloch sphere by Eq. (5). Figures 2(c) and on the boundaries of the angular angle under the adia-
2(d) show the static energy spectrum and the adiabatic batic condition.
density variation in each lattice site with respect to the Sofar,we havedescribedourscheme to simulateFPN
hopping modulation for the lattice size L=100, respec- in SSH model in a parameter space and to directly mea-
tively. It looks like a density kink-soliton (anti-soliton) sure it via the adiabatic transport. In contrast to the
configuration appears in the time-domain. previous schemes [15, 26–30], the presented scheme does
5
not involve the spatial kink domain in the hopping con- beams with a relative polarized angle [46]. The separa-
figuration, which is usually hard to realize and (or) con- tionandpotentialdepthfordifferentatomiccomponents
trol in experiments. In addition, the particle transport can be well controlled by the angle and the laser inten-
corresponding to the value of PFN can be directly mea- sity,withasimpleexampleofsucha1Dlatticepotential
sured in cold atom systems, such as from the measure- as shown in Fig. 1(b)
ment of atomic density distribution and atomic current
V (x)=V sin2(k x±π/4). (12)
[43], which will be discussed in 1D optical lattices in the σ 0 s
next section.
HereV isthelatticepotentialdepth,k isthewavelength
0 s
of the laser beams (the lattice spacing a = 2π/k ), and
s
±π/4 are the polarization angles for atomic states | ↑i
IV. EXPERIMENTAL IMPLEMENTATION IN
and |↓i, respectively.
OPTICAL LATTICES
For sufficiently deep lattices, the atoms in the system
mustaltertheirinternalstatesinordertotunnelbetween
In this section, we turn to discuss the implementation
two nearest-neighbor lattice sites. This can be achieved
of our scheme of mimicking and measuring FPN in 1D
by the Raman-assisted tunneling method [43, 47–50], as
opticallattices. Wefirstproposetwoexperimentalsetups
showninFig. 1(c). Theenergyoffsetbetweentwoatomic
torealizetherequiredSSHHamiltonianwithtunablepa-
statesarisefromtheexternalZeemanfieldandthenplay
rameters, and then discuss how to measure the particle
the role of tunable parameter ∆ in this system. Two
transport in the proposed cold atom systems by numeri-
pairsofRamanbeamswithRabifrequenciesΩ andg
1,2 1,2
cally calculating the shift of the Wannier center and the
are used to induce two large-detuned Raman transitions
center of mass of an atomic cloud.
with detuning ∆ and ∆ , respectively. One can use
d1 d2
the former pair of Raman beams to realize the uniform
nearest-neighbor hopping
A. Two experimental setups
J =A w∗(x−x )eikxxw (x−x )dx, (13)
Thefirstexperimentalsetupweproposedisa1Dopti- 0 ↑ n ↓ n+1
Z
calsupperlatticetrappinganoninteractingatomicgasof
where A =|Ω Ω∗|/∆ is the effective Raman strength
single-component fermions, as shown in Fig. 1(a). Such 0 1 2 d1
constant,k isthemomentumdifferencealongthexaxis
anopticallatticehasbeenwidelyrealizedinexperiments x
betweenthetwobeams,andw (x)aretheWannierfunc-
[31, 43–45]. It is generatedby superimposing two lattice σ
tions of the lowest Bloch band for atomic state |σi.
potentialswithshortandlongwavelengthsdifferingbya
To realize the time-varying hopping modulation term,
factor of two, with the optical potential given by
one can use another pair of laser beams with a resulting
V(x)=V sin2(k x+ϕ)+V sin2(2k x). (11) Raman field, as shown in Fig. 1(b),
1 1 2 1
Here k1 is the wave vector of the short wavelength trap- R1 =g1g2∗/∆d2 =A1(t)sin(2ksx+π/2), (14)
pinglasers(thelatticespacinga=2π/k ),ϕandV are
1 1,2
where A is a time-dependent constantcontrolledby the
respectively the relative phase and the strengths of the 1
laser intensities or detuning ∆ . In this way, the hop-
two standing waves. By varying the laser intensity and d2
ping modulation (−1)nδ(t) is given by
the phase, one can fully control the lattice system with
ease[31,43–45],andthenmakethesystemwelldescribed
by Hamiltonian (1) of SSH model in the tight-binging A1(t) w↑∗(x−xn)sin(2ksx+π/2)w↓(x−xn+1)dx.
regime [31]. In the experiments [31, 43–45], the hopping Z
configurationJ+(−1)jδ can be adjusted by varying po- Herethe staggeredhoppingmodulationis aconsequence
tential strengths V or swapping the relative phase ϕ, ofthe relativespatialconfigurationofthelattice andthe
1,2
andthestaggeredpotential∆canbetunedbythephase. Raman field: the period of R (x) is double of the lattice
1
Thereforeinthissystem,astraightforwardwaytorealize period and R (x) is antisymmetric corresponding to the
1
the required hopping modulation with kink-type config- center of each lattice site, as shown in Fig. 1(b). Thus,
uration in the time domain is by changing these tun- to realize the proposed particle transport scheme in this
able parameters of the optical superlattice with a well- system, we can adjust the Zeeman field and the Raman
designed sequence [31]. field R to tune the parameter ∆ and J in SSH Hamil-
0
Anotherexperimentalsetup,whichwouldbemorecon- tonian, and then independently vary the intensity of an-
venient as we will see in the following, is loading an ul- other Raman field A (t) in time with a kink-type form.
1
tracold Fermi gas of two-component (internal states |σi The case of time-varying ∆(t) can also be achieved in
with σ =↑,↓) atoms in a state-dependent optical lattice a similar way. The time modulation of the Raman cou-
[46]. It has been proposed to realize SSH Hamiltonian plinginultracoldatomshasbeendemonstratedinrecent
with a spatial domain wall in this system [27], and such experiments [51].
a state-dependent optical lattice have been experimen- Considering40Katomsandtypicallattice spacinga=
tally created by superposing two linearly polarized laser 532 nm, one has the recoil energy E /~ ≈ 30 kHz. For
R
6
the optical superlattice system with intermediately deep (a) (b)
4
lattice,atypicaluniformhoppingstrengthisJ ∼0.1E , 4
R
and the other parameters δ and ∆ can be tunable in a
3
3
wide regime [31]. For the state-dependent optical lattice
c
system, the uniform hopping strength given by Eq. (13) x 2 2
isproportionaltotheeffectiveRamanintensityA ,which
0
istypicallyinorderofmegahertz,andtheoverlapintegral 1 1
of Wannier functions between neighbor lattice sites can
be about 10−2 [52]. Thus the Raman-induced uniform −01 −0.5 0 0.5 1 −01 −0.5 0 0.5 1
hopping strength in this system is J ∼ 0.4E , and the δ(t)/δ δ(t)/δ
R 0 0
nature (next nearest-neighbor) hopping t within sub-
N
lattices can be effectively suppressedby sufficiently deep FIG.3: (Coloronline)TheshiftoftheWanniercenterineach
lattice V . For example, the numericalcalculation shows latticeasaresponsetotheadiabaticallytuninghoppingmod-
0
that t .10−3E for V ≈22E [52]. We can consider ulation. (a)Thesymmetriccasewith∆=0+(solidblueline)
N R 0 R
typical parameters δ and ∆ (the minimum bulk gap in and ∆(t) = δ0sin(πt/2t0) (dashed red line) with δ0 = 0.1J.
0
Aftertuningthehoppingmodulation overthekinkform,the
thedynamicalprogressionis2∆)inthe orderof0.1J. In
Wanniercenterfor each lattice site shift downwards onesite,
this case,the adiabaticapproximationworkswellforthe
ramptimet ≫~/0.1J ∼1ms. Thusonecanchoosethe i.e. one-half of the unit cell, corresponding to Q = −1/2 as
f expected for the half-charge in the original SSH model. (b)
ramp frequency ξ = 0.01J/~ and t = 50 ms, which is
0 The symmetry-breaking case with ∆ =4δ0 = 0.1J, the shift
wellshorterthanthetypicalcoherencetimeincoldatom
oftheWanniercenterisnearlyhalfalatticesite,correspond-
experiments. Thenon-adiabaticLandau-Zenertransition ing to Q=−1/4 expected in this case. Other parameters in
fromthe groundbandto the excitedbandforthe chosen (a) and (b) are J =1 and t0 =5/ξ.
parameters is then given by P ≈ e−π(0.1J)2/4~δ0ξ ≈ 0.
LZ
Inaddition,thefinitetemperatureeffectsdonotinterfere
with the particle transport progression for temperatures
and (16), the adiabatic particle transport is Q = −1/2
smaller than the energy gap [33]. This requires the tem-
(the sign depends on the shift direction), as expected
perature of the order of 0.08E /k ∼ 20 nk (k is the
R B B for the half-charge in the original SSH model. For the
Boltzmannconstant),whichhasbeenachievedincurrent
symmetry-breaking case with ∆ = 4δ = 0.1J as shown
0
experiments with, e.g., 40K atoms. So we can conclude
in Fig. 3(b), the shift of the Wannier center is nearly
that the required Hamiltonian with tunable parameters
half a lattice site, which is consistent with the expected
andthe adiabaticconditionareableto be realizedunder
Q=−1/4 in this case.
realistic circumstances.
TheshiftofWanniercentershowninFig. 3impliesthe
appearanceofatomiccurrent(thetransportofparticles)
ineachunitcell,whichflowthroughthewholelatticesys-
B. Experimental measurement methods
tem. Under the adiabatic condition, the atomic current
will take the solitonic form in the ramping progression,
It has been shown that the particle transport can be
which is similar to the example shown in Fig. 2(e). In
connectedwiththe Wannier centerbasedonthe modern
principle,the transportdynamics canbe detected by us-
theoryofchargepolarization[7,41]. Especially,theshift ing the single-atomin situ imaging technology in optical
of the Wannier center in each unit cell is
lattices [53]. Thus, the variation of atomic density dis-
tribution in a unit cell associated with induced current
X =x (t=t )−x (t=0), (15)
d c f c can be experimentally extracted out in this way. In our
proposedsystems,it wouldbe moreconvenientto detect
where x ≡ hw |xˆ|w i is the Wannier center, with
c n n
|w (x)i = 1 π e−ik(n−x)|u (k)i being the Wannier the global current through the whole lattice by measur-
n 2π −π − ing the time evolution of the atom fractions in the even
function of the ground band in the n-th unit cell. The
R and odd sublattices, instead of using in situ detection in
shiftoftheWanniercenterencodestheadiabaticparticle
a single unit cell. For the double-well superlattice sys-
transport (the variation of polarization) as [33, 39, 40]
tem in Fig. 1(a), the atomic current associated with
X /a=Q. (16) the atom fractions of the even/odd sublattices has been
d
measured in the experiment by transferring the atoms
In Fig. 3, we have numerically calculated the variation to higher-lying Bloch bands and applying a subsequent
oftheWannier centerineachlattice site intheproposed band mapping technique [43]. For the state-dependent
system. For the symmetric case with ∆ = 0+ (the solid optical lattice system in Fig. 1(b), the even and odd
blueline)and∆(t)=δ sin(πt/2t )(thedashedredline) sublattices trap |↑i and |↓i atoms, respectively. There-
0 0
shown in Fig. 3(a), after adiabatically tuning the hop- foreinthissystem,onecansimplymeasuretheevolution
ping modulation over the kink form, the Wannier center of atom fractions via optical imaging the up-component
in each lattice site shift downwards one site, that is, ex- (down-component) atoms, such as using state-resolved
actly one-half of the unit cell. According to Eqs. (8) time-of-flight measurements [54]. Therefore, one can ob-
7
(a)−0.2 (b) 0 effects andbecomesmallerandsmallerwiththe increase
(N/L)δx (N/L)δx
a com a com of the lattice size in our simulations, which will be invis-
Q by Eq.(8) −0.1 Q by Eq.(8)
xcom−0.3 Ns by Eq.(4) −0.2 Ns by Eq.(4) ipblloetitnhepmraicmtiiccakledexFpPerNimNenstgs.iveFnorbycoEmqp.a(r4i)soinn,twhies asylsso-
δL) tem in Fig. 4 (the dashed black line). From Fig. 4,
N/a−0.4 −0.3 one can see that the corresponding FPN is nearly equal
( δ = 0.1 ∆ = 0.1 to the particle transport within the parameter regimes.
0 −0.4
Incurrentexperiments,thecenter-of-masspositionofan
−0.5 −0.5 atomic cloud can be directly and precisely measured, ei-
0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4
∆ δ ther by using in situ measurement of the atomic density
0
distribution in the optical lattices [53] or deduced from
FIG. 4: (Color online) The shift of the center-of-mass of an the time-of-flight imaging [54].
atomic cloud asa function of(a) theparameter∆ with fixed Finally, we note that a shallow-enough harmonic trap
δ0 = 0.1; and (b) the parameter δ0 with fixed ∆ = 0.1, re- inpracticalexperimentswillnotaffecttheparticletrans-
spectively,forafinitelatticesystemwithL=400sitesandat port[33,39]andthemainresultsofthispaperremainin-
half fillingNa/L=1/2. Thesolid (blue)lineand thedashed tact. Inordertotaketheeffectoftheharmonictrapinto
(black) line denotes the corresponding particle transport Q account, we can add a term H = V (n−L/2)2cˆ†cˆ
obtained by Eq. (8) and fractional particle number Ns ob- intoHamiltonian(1),whereV tisthettranpstrengthanndLn
tainedbyEq. (4) inthetext,respectively. Otherparameters t P
is the lattice size. Within a localdensity approximation,
in (a) and (b) are J =1 and t0 =5/ξ.
thelowerbandwillstillbefilledatthecenterofthe trap
and thus the shift of the Wannier center in these lattice
sites is expected to be nearly the same as those shown
tain the particle transport Q by its integration over the
in Fig. 3; while the band is only partially filled near the
time domain, as given by Eq. (10), which corresponding
edge with the local trap energy V (n−L/2)2 &E , such
to the mimicked FPN in this system. t g
thatthepumpingargumentdoesnotapplytothisregion
Inaddition,theparticletransportcanbedirectlymea-
[33]. Thereforeinpracticalexperiments,onemayempha-
sured from the shift of the center of mass of an atomic
size on the shift of Wannier center in the central region
cloudinafinitelatticesystem[33,35,40]. Inoursystem,
or turn the trap strength to a small value V ∼4E /L2.
the center of mass of an atomic cloud in the lattice with t g
For the shift of the center of mass of an atomic cloud,
L sites x (t) is given by
com
our numerical simulations demonstrate that the results
L shown in Fig. 4 preserve with a deviation less than 2%
1
xcom(t)= |ψoc(n,t)|2n, (17) for Vt ≈ 0.6×10−5J, while Vt ≈ 10−4J is enough if the
Na latticesizereducestoL=100,whichareconsistentwith
nX=1Xεoc
the estimates in the local-density analysis.
where N =L/2 is the atomic number at half filling, ε
a oc
denotes the occupied state of the fermionic atoms, and
ψ is the corresponding wave function. Under the adia-
oc V. CONCLUSIONS
batic evolution with δ(0) → δ(t ) described by Eq. (6),
f
the center of mass of the system shift from the position
Insummary,wehaveproposedaschemetomimic and
x (0) to x (t ). It can be proved that the shift of
com com f measuretheFPNinthegeneralizedSSHmodelwithcold
the center of mass δx = x (t )−x (0) is pro-
com com f com fermions in 1D optical lattices. It has been shown that
portional to the particle transport in the infinite L limit
FPN in this model can be simulated in the momentum-
[33, 40]
time parameter space in terms of Berry curvature with-
N out a spatial domain wall. In this simulation, a hopping
a
δxcom =Q. (18) modulation is adiabatically tuned to form a kink-type
L
configuration and the induced current plays the role of
If L is largeenough, suchthat the bulk properties of the soliton in the time domain, so FPN is expressed by the
system are almost not affected by the edges, N δx /L particle transport. We have also proposed two experi-
a com
in the above equation will be approximated to be the mental setups of optical lattices to realize the required
ideal particle transport Q in infinite system. In Fig. 4, Hamiltonian with tunable and time-varying parameters,
we have calculated the shift of the center of mass of an and considered the energy scales and the adiabatic con-
atomic cloud for a lattice system with L = 400 sites, dition under practical circumstances. Finally we have
N δx /L (the red circles) as a function of the param- discussed how to directly measure the particle transport
a com
eters ∆ [Fig.4 (a)] and δ [Fig.4 (b)], respectively. As in the proposed systems by numerically calculating the
0
shown in Fig. 4(a) and 4(b), the calculated shift of the shift of the Wannier center and the center of mass of an
center ofmass is welldescribedby the particletransport atomiccloud. Consideringthatallthe ingredientstoim-
Q (the solid blue line) obtained by Eq. (8) with small plementourschemeinopticallatticeshavebeenachieved
deviations. These deviations are due to the finite size in the recent experiments, it is anticipated that the pre-
8
sented proposal will be tested in an experiment in the parameter space using cold atoms.
nearfuture. Thedirectmeasurementofsuchamimicked
FPNincoldatomexperimentswillbe animportantstep
towardexploring fractionalizationand topologicalstates
VI. ACKNOWLEDGEMENTS
in cold atom systems. Extensions of this work can en-
able to simulate andmeasureFPN emergingintwo- and
three-dimensional Dirac Hamiltonian with topologically This work was supported by the RGC of Hong Kong
nontrivial(vortexandmonopole)backgroundfields[1,8], (Grants No. HKU7045/13P and HKU173051/14P), the
which has been theoretically studied but remains elusive SKPBR of China (Grants No. 2011CB922104 and
innature. Itwillbe alsointerestingtosimulateavariety 2013CB921804), the NSFC (Grants No. 11125417 and
of topologicalstates [6]and study their properties in the 11474153),and the PCSIRT (Grant No. IRT1243).
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