The Anyon Hubbard Model in One-Dimensional Optical Lattices

2.5 MB·English

by  Sebastian Greschner

Anyon Hubbard Model in One-Dimensional Optical Lattices
Sebastian Greschner1 and Luis Santos1
1Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover, Appelstr. 2, DE-30167 Hannover, Germany
Raman-assisted hopping may be used to realize the anyon Hubbard model in one-dimensional
optical lattices. We propose a feasible scenario that significantly improves the proposal of [T.
Keilmann et al., Nature Commun. 2, 361 (2011)], allowing as well for an exact realization of
the two-body hard-core constraint, and for controllable effective interactions without the need of
Feshbach resonances. We show that the combination of anyonic statistics and two-body hard-core
constraintleadstoarichgroundstatephysics,includingMottinsulatorswithattractiveinteractions,
5 pair superfluids, dimer phases, and multicritical points. Moreover, the anyonic statistics results
1 in a novel two-component superfluid of holon and doublon dimers, characterized by a large but
0 finite compressibility and a multipeaked momentum distribution, which may be easily revealed
2 experimentally.

l
u PACSnumbers: 37.10.Jk,67.85.-d,05.30.Pr
J

2 ((aa)) (b)
1 Particlesareclassifiedasbosonsorfermionsdepending
L, L
on whethertheir wavefunction is symmetricor antisym- AA 2 3 AA
s] metric under exchange. Other types of quantum statis- LLL11 LL44 L(i,) L
2 4
ga taibclsy,ar2eD, hsoywsetevmers, paollsoswiblfeorinthloeweexrisdtiemnecnesioofnas.nyRoenms,air.ke-. LLL2222 LLL333 AA AA (ii) AABB
- AAA B L1, L3
t particles with fractional statistics interpolating between AA AA AA
n bosons and fermions [1–3]. Anyons play a fundamental AAA ∆ UUAABB BB (iii)
a L, L
1 4
u role in key areas of modern physics, as fractional quan- AAABB AA (iv) AA AAB
q tum Hall effect [4–7] and topological quantum comput-
t. ing [8]. Fractional statistics is, however, not exclusive of
a FIG.1: (a)Ramanschemeproposedfortherealizationofthe
2Dsystems[9]. Inparticular,1Danyonshaveattracteda
m AHM;(b)Ramanassistedhops(i)–(iv)discussedinthetext.
large deal of interest [10–23], although the experimental
-
realization of a 1D anyon gas is still lacking.
d tics, 2BHCC, and controllable interactions, results in a
n Ultracoldatomsofferextraordinarypossibilitiesforthe far richer physics for the AHM, including pair super-
o
analysis of interesting many-body systems [24]. In par- fluid (PSF), a dimer (D) phase, and an exotic partially
c
[ ticular,severalideashavebeenproposedforthecreation paired (PP) phase. The latter constitutes, to the best of
and manipulation of anyons in cold gases [25–29]. Par- ourknowledge,anoveltwo-componentsuperfluidcharac-
2
ticularlyinterestingistherecentproposalfortherealiza- terized by a large, but finite, compressibility and a mul-
v
2 tion of the anyon Hubbard model (AHM) using Raman- tipeaked momentum distribution, which may be readily
6 assisted hopping in 1D optical lattices [30]. In this pro- revealed in time-of-flight (TOF) experiments.
4 posal the anyonic statistics may be controlled at will, Anyon-Hubbard model.– We briefly describe our pro-
7 opening the possibility for the observation of statisti- posal for the realization of the AHM [32]. We consider
0
cally induced quantum phase transitions [30], asymmet- atoms, bosons or fermions, with states A and B , in
1. ric momentum distributions [21], and intriguing particle a deep tilted spin-independent 1D lattice| wiith no|diirect
0 dynamics in the lattice [20, 22, 23]. hopping. For the specific case of 87Rb, the choice would
5
In this Letter we first discuss a scheme for realizing be A F = 1,m = 1 and B F = 2,m =
1 F F
| i ≡ | − i | i ≡ |
: the AHM that, although following Ref. [30], solves cru- 2 . Both states are coupled far from resonance by D1
v − i
cial drawbacks that would render the original proposal, lasers L (with linear polarization) and L (with cir-
i 1,4 2,3
X in general, unfeasible. The scheme also allows for con- cularσ polarization)(Fig.1(a)). Duetoselectionrules,
−
r trollable effective interactions without the need of Fes- B is just affected by lasers L1,4. In contrast, both L2,3
a | i
hbach resonances, and for an exact two-body hard-core and L couple with A , but the coupling with L can
1,4 1,4
| i
constraint (2BHCC), contrary to the approximate real- be made much smaller than that of L , and does not
2,3
ization of 2BHCC resulting from Zeno projection due affect the main conclusions of this Letter [33]. We hence
to large three-body loss rates [31], where non-negligible assume below that A is just affected by L . These
1,2
| i
losses are typically present. Neither the controllable in- lasers, with frequencies ω , induce four Raman-
j=1,...,4
teractions nor the inherent 2BHCC were considered in assisted hops (Fig. 1(b)): (i) (A,0) (0,A), given by
→
Ref. [30], which focused exclusively on statistically in- L ,suchthatω ω = ∆(with∆thelatticetilting);
2,3 2 3
− −
duced superfluid (SF) to Mott insulator (MI) transi- (ii) (A,A) (0,AB), assisted by L with ω ω =
2,4 2 4
→ −
tions. We show that the interplay of anyonic statis- ∆+U +U; (iii) (AB,0) (A,A), given by L
AB 1,3
− →
2
2 (a) 2 (b) 3 (c) 0 (d) 4/3
MI
0 SF 0 SF 0 MI
U / t-2 U / t-2 PP U / t-3 SF D / tµ-1 PP 1
U
PSF FR PSF 2/3
-4 -4 -6 PSF
1/3
-2
0 1 φ / π 2 0 1 φ / π 2 0 1 φ / π 2 0 1 φ / π 2
FIG. 2: Phase diagram of the AHM: (a) dilute limit (ρ 0), (b) incommensurate intermediate fillings (here ρ=1/2) and (c)
→
unitfilling(ρ=1). ThePSF-SFtransitionlines(boxesorsolidlinein(a)),extractedfromthecrossingoftwoandoneparticle
excitations coincides with the onset of parity order. At incommensurate fillings for small φ the system exhibits a first order
transition from SF to PSF phase through a forbidden region (FR, dotted area). For φ/π & 0.6 we find the PP phase. For
unit filling the Ising-type PSF-SF transition and the Kosterlitz-Thouless SF-MI transition lines cross around φ/π 0.8 in a
≈
multicriticalpoint. ForlargervaluesofφthesystemexhibitsadimerizedgappedphaseD.(d)Grand-canonicalphasediagram
and lines of constant density ρ for U =0 as function of µ and φ. Solid lines denote the MI and PP phase, the rest is SF. For
µ/t< 2 (µ/t>4) the vacuum ρ=0 (fully occupied phase, ρ=2) is realized.
−
withω ω = ∆ U U; (iv)(AB,A) (A,AB), with J the lattice hopping in the absence of tilting, and
1 3 AB
− − − − →
assisted by L , such that ω ω = ∆. The notation δ U ,∆,Ωthedetuningfromthesingle-photontran-
1,4 1 4 AB
− − (cid:29)
(ηL,ηR) denotes the state of neighboring sites L and R: sitions; b†j (bj) are creation (annihilation) operators at
ηaLn,dRa=B0p(avratciucluem);)U,AAB(scihnagrleacAteoriczceusptahteioAnB)oirnAteBra(catinonA. sointee jo,btaanidnsntjh=e abn†jbyjo.nI-nHturobdbuarcdinmg αodje≡l (eAiφHPM1≤)l[≤3j0−]1:nlbj,
The frequency differences compensate the tilting, avoid-
U
ingBlochoscillations;moreover,thedetuningU actsasa H = −t (αj†αj+1+H.c.)+ 2 nj(nj −1) (2)
controllableeffectiveon-siteinteraction,asshownbelow.
j j
X X
We assume the width of the Raman resonances, W,
is small-enough such that each Raman process may be where αj,αj† satisfy anyonic commutations [45]: αjαk† −
addressed independently. If the atoms are fermions, the e−iφsgn(j−k)αk†αj =δjk and αjαk−e−iφsgn(j−k)αkαj =0.
only undesired process (A,0) (0,B) (where B denotes None of the Raman-assisted hoppings leads to triple
a site with a single B atom)→may be avoided if UAB (cid:29) occupation, resulting in an effective 2BHCC, (b†j)3 = 0.
W. For bosons, (A,A) (0,AA) and (AB,A) This inherent constraint turns out to be crucial. It pre-
→ →
(B,AA)(withAAasitewithtwoAatoms)mustbealso vents collapse for a large U/t < 0, leading to a rich
avoided. This demands U U , U , U W, physics of quantum phases (see Fig. 2) [46].
AA AB AA AB
− (cid:29)
where U characterizes AA interactions [32]. Note that although hops (i) and (iv) share the same
AA
The laser L is characterized by a Rabi frequency energy difference ∆, they may be independently ad-
j −
Ω|Ωj1e||iΩφ4j|, =wit|Ωh2|φ|Ω13|== −|Ωφ1|,|Ωa3n|d=φ|jΩ6=21||Ω4=| =0.Ω2W(eforim87pRosbe; dsorlevsisnegd,theevemnaijforδd(cid:29)rawUbAacBk,∆of. RTefh.i[s30p]oiinntwihsicchruocnially,
4 3 2√3 2√3 one ground state component was considered, and hence
for other species Clebsch-Gordan coefficients may dif-
hops (i) and (iv) cannot be independently addressed un-
fer [32]). Ω is chosen such that they mimic bosonic
j
| | less δ U ,∆, which for typical experimental param-
enhancement, absent otherwise due to the distinguisha- (cid:28) AB
eters would lead to large heating. An exception may
bilitybetweenAandB;(cid:15)=1( 1)forbosons(fermions).
− be given by the use of clock transitions in alkaline-earth
Under these conditions the system reduces to an effec-
atoms[47]);however,theadiabaticeliminationoftheex-
tivesingle-component1DBose-Hubbardmodel[32],with
cited state demands Ω δ, and hence t δ U ,∆,
on-site Fock states 0 , 1 A , 2 AB , and an (cid:28) (cid:28) (cid:28) AB
{| i | i ≡ | i | i ≡ | i} would lead to very long experimental time scales [48].
occupation-dependent Peierls phase [30, 32]:
Effective repulsion.– The dilute limit (lattice filling
ρ 1) is best understood from the analysis of the
U (cid:28)
H = −t (b†jeiφnjbj+1+H.c.)+ 2 nj(nj −1)(1) tinwgo-lpenargttihcl(einpruonbiltesmof[4th9]e,lcahttaircaectsepraicziendg)by the scatter-
j j
X X
(1+cosφ)
whereU isaneffectivecouplingconstantwhosesignand a= − (3)
4(1 cosφ)+2U/t
magnitude may be tailored without the need of Fesh- −
bach resonances by properly choosing the laser frequen- Bycomparisontoa1Dgasofbosonswithmassmoneex-
cies; t J(Ω2/δ)/2∆ is the effective hopping rate [34], tractsaneffectiveinteractionstrengthg = 2/(am)[49].
’ −
3
. . .
2
3
(a) 3 (b) 0.15 (c)
ρ
c SvN
1
2 L
1 10 20 x 40 N / a,d
0.5 1
PSF PP SF PSF PP SF
-1.1 -1
0 0 0
. -2 0 µ / t 2 4 . -2 0 U / t 2 . -2 0 U / t 2
FIG.3: Phasetransitionsforφ=π. (a)Equationofstate,densityρasfunctionofchemicalpotentialµ(calculatedforL=48
sites, U = 0). The inset shows a detailed view on the PP region for L = 240 sites. (b) Estimation of the central charge for
L = 24 (dashed line) and L = 48 (solid line) sites and periodic boundary conditions (ρ = 1/2). The inset shows fits to the
block-entanglement entropy S for (from bottom to top) U = 2, 2, 1.0. (c) Sum of local single- (N ) and two-particle
vN a
− −
correlations (N ) for L=60 sites and ρ=1/2, see text.
d
Thus the AHM behaves as a repulsively interacting sys- J /t = 1+eiφ. For φ π correlated hopping dom-
2
− ≈
tem for small ρ and U 0, since even in that limit g inates, J = 2J , and doublons, or more precisely
2 1
→ −
remains finite and positive for any φ > 0, approaching dimers, d =α 11 +β 20 , gainalargebind-
| ij | ij,j+1 | ij,j+1
Tonks limit as φ π. This is consistent with a weak ing energy √2J . However, dimer hopping, J /√2,
couplinganalysis[→50],inwhichtheLuttingerliquid(LL) is reduced c−ompar2ed to single particles. For a c1ertain
parameterK =π/(φ2+ U )1/2,showingthatφhasqual- range of small U and ρ it is energetically favorable to
2ρt
itativelythesameeffectasarepulsiveU >0. InFig.2(a) occupy both doublon dimers d and atomic dimers,
| ij
we depict the phases for ρ 0 [51]: a SF phase, and a a = α˜ 01 +β˜ 10 . Neglecting interactions
PSFofboundpairsatU <→0. Duetotheeffectiverepul- | ij | ij,j+1 | ij,j+1
between these two quasiparticles, reasonable for small
sion the SF extends into the U <0 region for φ>0.
doublonandatomdensities,ρ andρ ,withρ +ρ 1,
d a a d
(cid:28)
Incommensurate fillings.– Bose models (φ = 0) with one arrives at an effective model with two independent
2BHCC have been recently studied in different scenar- hardcore bosons, H =H +H , with (for U =0)
ad a d
ios [31, 52, 53]. A SF to MI transition results for ρ = 1
andlarge-enoughU >0; forU <0atransitiontoaPSF Ha = J1 (a†jaj+1+H.c.)−J1 a†jaj, (4)
occurs, which at incommensurate ρ may become first or- j j
X X
der[54]. Thisforbiddenregion(FR)occupiestheshaded J
area in Fig. 2(b). At fixed ρ the ground state is charac- Hd = −√12eiargJ2 d†jdj+1−√2J2 d†jdj (5)
terizedbytheformationofamacroscopicallyboundstate Xj Xj
with a phase separation of doublons and holes. The FR
shrinks, however, fast with increasing φ. where aj (a†j) and dj (d†j) are annihilation (creation) op-
erators of hardcore atom and doublon dimers on sites
For φ & 0.6π a novel gapless phase, henceforth called j,j + 1, and ρ + 2ρ = ρ. In spite of being a gross
a d
partially paired (PP) phase, appears between the PSF
oversimplification, H captures the qualitative features
ad
and SF phases. As depicted in Fig. 3(a), a sharp kink
of the PP phase. For U 0, at low ρ the ground state
in the ρ(µ) curve (at µ t) accompanies the onset of ≈
’ − only contains a dimers, whereas for higher ρ (& 0.3 for
the PP phase. Whereas SF (PSF) is characterized by
U =0)bothdimersarepresent. Theresultingρ(µ)curve
steps of 1 (2) particles in the ρ(µ) curve, the PP phase
obtainedfromH (notshown)exhibitsanirregularpat-
ad
exhibitsacomplexsequenceofstepsof2and1particles.
tern of steps of 2 or 1 particles. The two-component
Evenmorerelevantisthebehaviorofthecentralchargec,
phase extends for values around U = 0, whereas for
whichwecalculatedfromtheconformalexpressionofthe
U J (U J ) only a (d) dimers are present [57].
von Neumann entropy, S (l) = c ln Lsin πl +γ, (cid:29) 1 (cid:28)− 1
vN,L 3 π L The atom and doublon dimerizations, Na =
fγoraacosnusbtsaynstte[5m5,o5f6l]e.nAgtshshloiwnnainsyFstiegm. (cid:2)3(obf)L, c(cid:0)sointetrs(cid:1)a,(cid:3)rwyittho ihb†ibi+1i, and Nd = ih(b†i)2(bi+1)2i, approximately
measureρ andρ (Fig.3(c)). WhereastheSFandPSF
a d
the SF and PSF phases (with c = 1), the PP phase is a P P
phasesareLLsofalmosthardcoresingleparticlesorpairs
two-component gapless phase characterized by c=2.
(for φ = π), in the PP phase both components coexist.
Nature of the PP phase.– The features of the PP Note the existence of many sharp jumps to states with
phase may be understood from the following simpli- approximately finite doublon density.
fied picture. Due to the 2BHCC the hopping term in The AHM is characterized by a density- and U-
Eq. (1) is J1b†jbj+1 + J2b†jnjbj+1, with J1/t = 1 and dependent drift of the quasimomentum distribution [30,
4
50], evident in Fig. 4 for the SF. Interestingly, whereas π 0.06 (b) 0.1
the SF presents an usual single-peaked distribution, the
PP phase shows a multipeaked distribution. This is be- k)
k (
cause the a and d hop with a different Peierls phase, and n
hencetheoverall(a+d)momentumdistributionpresents
0 0
multiple peaks. For PSF, single-particle correlations de- 0.05
(c)
cay exponentially, and the peak structure blurs.
Unitfilling.–Weconsidernowthespecialcaseofρ=1. )
k
(
Tuning φ may induce a SF to MI transition [30, 50]. n
(a)
This is so even for U < 0 (Fig. 2 (c)) due to the effec-
tive on-site repulsion discussed above. This Kosterlitz- -π 0 0
Thouless transition may be determined as the curve at -2 0 U / t 2 -π 0 k π
which K = 2 [58], which is consistent with the open-
ing of the energy gap. As mentioned above, for U < 0 FIG. 4: (a) Quasimomentum distribution n(k) for ρ = 1/2
an Ising-type transition occurs from SF to PSF. How- as function of U/t for φ=0.9π (L=60 sites). Dashed lines
denote phase transitions between PSF, PP and SF phases.
ever,inthevicinityofφ π,thecombinationofanyonic
≈ (b) and (c) show the momentum distribution for U =2t and
statistics and 2BHCC results in a dimerized phase, D.
U =0fortheAHM(solidline)andthecomponentsA(dashed
The latter is characterized by a finite dimer-order pa-
line)andB (dottedline)separately(forbosonicatoms). The
rameter OD = TL/2−TL/2+1 with Ti =b†ibi+1+H.c.. latterareobtainedbydirectsimulationofthetwo-component
The Ising transition line is determined by the crossing of modelwithRaman-inducedhopping(seeSuppl. Materialfor
(cid:10) (cid:11)
single- and two-particle excitations, and by a sharp peak a discussion of the fermionic case).
in the fidelity susceptibility [59]. Interestingly, the phase
diagram presents a SF-MI-D-PSF multicritical point.
Probingtheanyon-Hubbardmodel–Adiscussiononex- maybeextendedto2Dlattices. Althoughfor2Dlattices
perimentaldetectiondemandsagrand-canonicalanalysis theboson-anyonmappingislost, a2Dsetupwouldopen
since local density arguments translate the dependence the exciting possibility of inducing a density-dependent
on the chemical potential, µ, into a spatial dependence magnetic field, contrary to the static field created in re-
of the phases in the presence of an overall harmonic con- cent experiments [34, 63–65]. This possibility will be in-
finement. Crucially, the PP phase has a finite extension vestigated in a forthcoming work.
inbothφandµ(Fig.2(d)). Hubbardmodelswithcorre- Acknowledgements.–WethankA.Eckardt,C.Klempt,
latedhopping(bothforfermions[60,61]andbosons[62]) D. Poletti, M. Roncaglia, E. Tiemann, T. Vekua for
may present a similar (but insulating) phase with coex- discussions. We acknowledge support by the Cluster
istent doublons and holons, which is, however, infinitely QUEST and the DFG Research Training Group 1729.
compressible,i.e. itisasinglepointasafunctionofµ. As Simulations were carried out on the cluster of the Leib-
a result these phases cannot be observed in trapped ex- niz Universit¨at Hannover.
periments, just leading to abrupt density jumps. On the
contrarythenovelPPphase,havingafiniteµextension,
maybeexperimentallyobservablebyproperlysettingthe
central µ in the PP region (adjusting the particle num-
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[33] Thespuriouscouplingof B withL2,3leadstoadditional sition between XY and XY2 phases (i.e. SF and PSF
| i
small terms in the final Hamiltonian, which as discussed phase) is first order passing a region of macroscopic
in the Supplementary Material do not affect the main jumpsinmagnetizationasfunctionofmagneticfield(i.e.
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Supplementarymaterialfor”Theanyon-Hubbardmodelinone-dimensionalopticallattices”
S. Greschner and L. Santos1
1Institutfu¨rTheoretischePhysik,LeibnizUniversita¨tHannover,Appelstr. 2,DE-30167Hannover,Germany
InthisSupplementaryMaterialwediscussinmoreextentthecreationofdensity-dependentgaugefieldsusing
Raman-assistedhopping,andadditionaldetailsaboutdetectionusingthequasi-momentumdistribution.
I. CREATIONOFDENSITY-DEPENDENTGAUGEFIELDS F'=2
δ F'=1
5
A. Two-componentsystem
1
0
2 Weconsidertwohyperfinestates,AandB,ofabosonicor
χ<<1
l fermionicatom. Bothcomponentsareconfinedtothelowest L
u bandofatilted1Dopticallatticealongx,ofspacingD,and 1,4
L
J depth V0 = sER, with ER = ~2π2/2mD the recoil energy. 2,3
2 WithouttiltingthereisahoppingrateJ tonearestneighbors.
1 Thelatticetiltinginducesanenergyshift∆fromsitetosite. F=2 B
Wedenoteasw(x jD)theWannierfunctionmaximally B≡ F=2,mF=−2
] − ω
s localized at site j. Due to the tilting, it is convenient to use HFS
a Wannier-Stark states. For J ∆, the Wannier-Stark state F=1
g (cid:28) A
centered at site l may be approximated as ψ (x) w(x A≡ F=1,m =−1
t- lD)+ J(w(x (l+1)D) w(x (l 1)Dl)). T’he3Don−- F
n ∆ − − − −
site wavefunction at site j is Φ (r) = ψ (x)ϕ(y,z), where
a j j FIG.1:(Coloronline)Left:Transitionsconsideredfortherealization
u ϕ(y,z) is given by the strong transversal confinement. For
oftheRaman-assistedhoppings,withπ polarizedlasers(blue)and
q simplicityweassumebelowϕ(y,z) w(y)w(z).
’ σ polarizedlasers(red);right:ifχ 1(seetext),onemayassume
. On-site interactions between α and β (α,β =A,B) − (cid:28)
t inaverygoodapproximationthattheπlasersaffectjusttheAstate,
a are characterized by the coupling constant U =
α,β whereastheσlasersaffectjusttheBstatebyconstruction.
m 4π~2aα,β d3rw(r)4, with a the corresponding scatter-
- m | | α,β
ing length. For a sufficiently deep lattice, the evaluation of
d
R
n the on-site interactions is simplified by means of the har- same as that of the σ laser from the F = 1,m = 1
F
[co ml =onDicsa−p1p/r4o/xπi.mUatsiionng:thΦis(arp)pr’oxi(m√aπtiol)n−w3/e2eo−btra2i/nl2:,Uwα,hβe’re |tFun0ed=. N2o,tme0Ftha=t du−e2tio tsrealnescittiioonn.rulB|esot|hFla=ser2s,marFe −r=edi−d→2ei-
2 π(2π)3/2s3/4~m2aDα3,β. Of course, for fermions only inter- ilsindkasrakswwietlhl rFesp=ec1t,tmo the=σ l1asetor. FIn =con2t,rmast, t=he π1laasnedr
v specieson-siteinteractionsarepossible. F =1,m |= 1 ,wiFthde−tuniings|,re0spectivel0Fy,δ+−ωi
2 | 0 0F − i HFS
andδ+ω ω ,whereω andω arethehyper-
6 HFS− H0 FS HFS H0 FS
4 B. Lasertransitions fine splittings of the 2S1/2 and 2P1/2 manifold, respectively.
The fact that the π laser affects F = 1,m = 1 may
7 F
| − i
0 Before discussing Raman-assisted hopping, we comment introducespuriouseffects(seeSubsec.ID).
1. on specific transitions in alkali atoms. We analyze first the Fermionsmaybetreatedinasimilarway. Forthespecific
0 caseofbosons,whichweillustratewiththeexampleof87Rb case of 40K, we choose A F = 9/2,mF = 9/2 and
≡ | − i
5 (belowwecommentonotherspecies). WechooseA F = B F = 7/2,mF = 7/2 . Wemaythenemployasimilar
1 1,m = 1 and B F = 2,m = 2 . Note ≡tha|t one π–≡σ|laserconfiguration−asiniFig.1.SinceF =9/2liesbelow
F F
v: couldinpr−inciipleswa≡pt|hedefinitionofA−aindB.Theoppo- F = 7/2 the lasers must be now blue detuned. The above
i sitechoicewouldbehoweverproblematic,sincethescattering mentionedchoiceofAandBisbetterthantheopposite,since
X
lengthsfulfilla(F = 1,m = 1 , F = 2,m = 2 ) = it minimizesthe spurious effectsintroduced thethe coupling
F F
ar a(|F =2,mF =|−2i,|F =2,m−Fi=|−2i)[1].Henc−eswiap- of|F =7/2,mF =−7/2ibytheπlaser(seeSubsec.ID).
ping A and B would lead to a = a , which would not
AA AB
allow to remove some spurious Raman transitions (see Sub-
sec.IH).Notealsothatsinceweconsidermaximallystretched
C. Raman-assistedhopping
states,hyperfine-changingcollisionsareabsent.
We consider two lasers (see Fig. 1), one with linear po-
larization(π laser), andotherwithσ polarization(σ laser). Directhoppingispreventedforalarge-enoughtilting,J
We define as δ the detuning of the−π laser from the F = ∆.Ramanassistedhoppingmaybeinducedbytheabsorpti(cid:28)on
|
2,m = 2 F = 2,m = 2 transition, which ofaphotonoflaserL andtheemissionofaphotonoflaser
F − i → | 0 0F − i m
is (up to slight detunings of the order of kHz or smaller) the L .TheamplitudeoftheRaman-assistedhopping,J ,from
n nm
2
with the example of 87Rb, with A F = 2,m = 2 ,
Species ωHFS (GHz) ωH0 FS (GHz) Γ(MHz) χ and B F = 1,m = 1 . Let≡us|consider inFdetai−l thie
F
Na 1.722 0.189 9.8 0.21 ≡ | − i
different Raman-assisted hops we would like to induce (the
87Rb 6.834 0.814 5.75 0.059
Clebsch-Gordanfactorsmayvariateforotheralkalispecies):
85Rb 3.036 0.361 5.75 0.13
Cs 9.193 1.167 4.57 0.046 (i)(A,0) (0,A). Thistransitionleadstoanenergy
• →
shift∆E = ∆. Forω ω = ω ω = ∆,the
2 3 1 4
− − − −
transitionisrealizedbyeitherL andL ,orbyL and
1 4 2
TABLEI: ValuesofωHFS,ωH0 FS,thelinewidthΓ,andχ(evaluated L ,withamplitudeV = 1Ω2Ω∗3 + Ω1Ω∗4 1χ,with
forδ=0.5GHz)fordifferentatomicspecies.ForNaand87Rb,A 3 23 2 δ δ 6
≡
|aFsi=nE2q,.m(4F).=Fo−r28i5,Rabn,dAB≡|FF ==13,,mmFF==−13i,,anadndχBisdeFfine=d χ=6 1 δ + 1 δ . (4)
≡ | − i ≡ | 4δ+ω 12δ+ω ω
2,mF =−2i,andχ=6 257δ+ωδHFS + 247δ+ωHFSδ−ωH0 FS . For (cid:20) HFS HFS − H0 FS(cid:21)
Cs, A≡ |F = 4,mF = −h4i, andB≡ |F = 3,mF = −3ii, and (ii) (A,A) (0,AB). This transition has ∆E =
χ=6 7 δ + 3 δ . • →
48δ+ωHFS 16δ+ωHFS−ωH0 FS UAB−∆,andinvolvesabsorbingaσphoton,andemit-
h i tingaπone.Weassumeω ω =U ∆+U,where
2 4 AB
− −
U is a small detuning (within the Raman resonance
sitej tositej+1,maybeevaluatedfollowingRef.[2]: width).Asdiscussedbelowthistunnelingplaystherole
ofeffectiveon-siteinteractionsinthefinalmodel. The
V
Jnm = nm d3rΦj+1(r)∗eiδknm·rΦj(r) (1) transition is quasi-resonantly provided by L2 and L4,
4 Z withamplitudeV24 = √16Ω2δΩ∗4.
where δknm = k k is the difference between the laser
wave vectors of thne−lasemrs, and Vnm is the amplitude of the • (ainiid)(iAs Bpe,r0fo)r→me(dAb,yA)a.bFsoorrbtihnigscaasπe,p∆hEoto=n,UaAnBd−th∆en,
Ramancoupling(seeSubsec.ID).ForJ ∆:
(cid:28) emittingaσone. Forω1 ω3 = UAB ∆ U,the
− − − −
transition is quasi-resonantly provided by L and L ,
dxψj+1(x)∗eiδkxnmxψj(x)’ withRamanamplitudeV13 = √16Ω1δΩ∗3. 1 3
Z
eiδkxnmD(j+1/2)(cid:20)Z dxw(x−D)∗w(x)eiδkxnmx • (iv∆),(AanBd,Aca)n→be(oAn,lAyBre)a.liFzeodr tuhsisintgraπnsiltaiosenrs∆. EFo=r
−
J δknmD ω ω = ∆,thetransitionisrealizedbyL ,with
+2i sin x dxw(x)2eiδkxnmx . (2) 1− 4 − 2,3
∆ 2 | | amplitudeV = 1Ω1Ω∗4.
(cid:18) (cid:19)Z (cid:21) 14 3 δ
Fors(cid:29)1(l(cid:28)D): For |Ω1|4|Ω4| = |Ω2|3|Ω3| = |Ω21√||Ω33| = |Ω22√||Ω34| = Ω2,
V δknmD Ω1 = Ω1 e−iφ, and Ωj=2,3,4 = Ωj , one obtains V23 =
Jnm i nm Jsin x eiδkxnmD(j+1/2). (3) Ω2(1 +| χe| iφ), V = √2Ω2, V| | = √2Ω2e iφ, and
’ 2∆ 2 δ − 24 δ 13 δ −
(cid:18) (cid:19) (cid:18) (cid:19) V = 2Ω2e iφ. The term proportional to χ in (i) is not
Note that δkxnm 6= 0 is necessary to establish a significant di1s4cussed iδn t−he main text, for simplicity of the exposition.
hopping[2–4]. It results in an additional term in the final Hamiltonian, as
showninSubsec.IE.However,χcanbekeptsmall. Specific
examplesfordifferentspeciesforδ = 0.2GHzareshownin
D. Four-laserscheme Table I [5–8]. We show below that for small χ the spurious
termdoesnotmodifytheconclusionsofthemaintext.
We employ the notation of the main text, (η ,η ), to
j j+1
denote the states of neighboring sites, where η may be
j,j+1
0 (empty site), A or B (singly occupied site with an A or B 2. Fermioniccase
component),andAA,ABandBBfordoubly-occupiedsites.
ForfermionsitisbettertochooseAasthestatewithlarger
F, e.g. A F = 9/2, 9/2 and B F = 7/2, 7/2
≡ | − i ≡ | − i
1. Bosoniccase for40K.IntheschemeofFig.1ofthemaintext, wechoose
L asπlasers,andL asσlasers. Wemayproceedasfor
2,3 1,4
Letusfirststudythebosoniccase. WeconsidertheRaman bosons, but choosing Ω1 = Ω1 e−iφ. Due to the inverted
−| |
scheme of Fig. 1 of the main text, where L are σ lasers, choice of A and B, we obtain: V = Ω2, V = √2Ω2,
2,3 23 δ 24 δ
and L1,4 are π lasers. The lasers are characterized by their V = √2Ω2e iφ, and V = 2Ω2(e iφ + χ). For 40K,
Rabi frequency Ω proportional to the square root of 13 δ − 14 δ −
j=1,...,4 χ = 32 δ + 49 δ , with ω =
their intensity, their frequency ωj, and their wave vector kj. 243δ+ωHFS 243δ+ωHFS−ωH0 FS HFS
Asmentionedabove,weillustratethecaseofbosonicspecies 1.286GhHz, ωH0 FS = 0.155GHz(thelineiwidthisΓ = 5.96
3
.
MHz)[9]. Forδ = 0.2GHz,χ = 0.072. Notethatinverting (a) 2
(a) (b)
thechoiceofAandB,wouldleadtoV ofthesameformas
nm
inthebosoniccasebutwithχfourtimeslarger(χ=0.28for
δ =0.2GHz). ThelatterjustifiesourchoiceofAandB.
1
E. EffectiveHamiltonian ρ 1
χ = 0
Weconsiderthatonlyprocesses(i)to(iv)areresonant,and χ=0.05
postponetotheSubsec.IHthediscussionaboutspuriouspro-
χ=0.1
cesses. We denote as c the bosonic operator corresponding
j χ=0.2
totheFock-statemanifold 0 , 1 A , 2 AB . Let
{| i | i≡| i | i≡| i}
usconsiderk1,2 =key,andk3,4 =kex,andk =π/D. 0 0
-2 0 0 4
. µ / t µ / t
1. Bosoniccase
FIG.2:Equationofstateρ=ρ(µ)oftheAHM(U =0,φ=π,L=
48)fordifferentvaluesofχforthespuriouscorrelatedtunnelingin
For the bosonic case, the Raman-assisted hopping of the
(a)Eq.(6)and(b)Eq.(7).Smallarrowsdepictthekinksintheρ(µ)
previoussubsectionresultintheeffectiveHamiltonian:
curveforχ=0.2characterizingthePP-SFtransition.
H = −t (−1)j c†j+1eiφnjcj +H.c. F. Effectofafiniteχ>0
Xj h i
+ χ (−1)j eiφc†j+1(1−nj)(1−nj+1)cj +H.c. For χ = 0, Hamiltonians (6) and (7) reduce to Eq. (1) in
Xj h i themaintext,andhencetotheanyon-Hubbardmodel(AHM).
U χ>0resultsinspuriouscollisionally-assistedhoppingterms.
+ nj(nj 1). (5) However,ascommentedabove,χcanbekeptsmall.
2 −
j Figure2(a)showstheeffectofthespuriouscorrelatedtun-
X
nelinginEq.(6)whichbasicallyeffectsthelow-densitypart
with nj = c†jcj, and t = Ω22∆/δ J. Typical values of the ρ < 1andresultsinasmallshiftoftheSF-PPtransition(in-
dicatedbysmallarrows)tolowerρ. InterestinglyaPPphase
Raman-assisted hopping rat(cid:16)e, t, a(cid:17)re of the order of few tens
of Hz. Note that the factor ( 1)j, which results from the maybeobservedalsoforρ>1forsufficientlylargeχ.
− Figure2(b)showstheeffectofthespuriouscorrelatedtun-
x projection of δk, may be easily eliminated by redefining
nelinginEq.(7). Theeffectisnegligibleforsmalllatticefill-
thebosonicoperatorsintheform: b = c , b = c ,
4l 4l 4l+1 4l+1 ings ρ < 1. Due to its ρ2 -dependence even for much larger
b = c , b = c , with l an integer. In this
4l+2 4l+2 4l+3 4l+3
− − valuesofχasdiscussedintheprevioussubsectionthecurves
wayweobtain:
of Fig. 2 (b) basically overlap with the pure AHM-case. For
densitiesρ 1thephaseboundariesareslightlymodified. In
H = −t b†j+1eiφnjbj +H.c. particular th≥e size of the Mott-insulator region increases and
Xj h i the PP phase may now extend to a small region above unit
+ χ eiφb†j+1(1−nj)(1−nj+1)bj +H.c. filling(seearrowsinFig.2(b)).
Note that, interestingly, the choice of a larger detuning δ
Xj h i
U andthuslargervaluesofχactuallyfavorstheobservationof
+ nj(nj 1). (6) thePPphaseinexperiments.
2 −
j
X
G. ComparisontothemodelofKeilmannetal.
2. Fermioniccase
Note that (i) and (iv) are energetically degenerated, but
Proceedinginexactlythesamewayforthefermioniccase, (apart from the small spurious effect discussed above) they
weobtain maybeaddressedwithdifferentlasersduetoselectionrules.
This point constitutes the major drawback of the proposal of
H = −t b†j+1eiφnjbj +χb†j+1njnj+1bj +H.c. Ref.[10]. Inthatproposal,asinglecomponent(A)wascon-
Xj h i sidered, and process (iv) was of the form (AA,A)→(A,AA),
U which cannot be resolved from process (i). As a result, both
+ 2 nj(nj −1). (7) L2andL3,andL1andL4addressboth(i)and(iv),preventing
j therealizationofthedesireddensity-dependentPeierlsphase.
X
4
I. Initialpreparation
Species aAA/aB aAB/aB δU/h (kHz)
| |
Na 54.53 64.5 0.72
87Rb 100.36 99.08 0.23 If the spurious processes can be neglected, only on-site
85Rb 469 386 1.40 states0, A,andABmaybereachedbyRaman-assistedhop-
− −
Cs 1980 2436 8.89 ping. However, B, AA, BB, and even sites with occupation
−
higherthantwo,mayinitiallyexist. Siteswithanoccupation
larger than two are eventually destroyed by three-body pro-
TABLE II: Values of aAA, aAB, and δU/h for different bosonic cesses, and are hence of no major concern. This is however
species, for s = 20 and D = 532nm (except for 87Rb where we
not the case of B, AA and BB, which should be avoided in
employ s = 25 and D = 266nm) [14]. For Na and 87Rb, A
F = 2,mF = 2 and B F = 1,mF = 1 . For 85R≡b, the initial preparation, since they would induce immobile
| − i ≡ | − i spurious defects. A straightforward way to avoid immobile
A F = 3,mF = 3 , andB F = 2,mF = 2 . ForCs,
A≡ |F = 4,mF = −4i,andB ≡F| = 3,mF = −3 .iNotethat defects is to consider a low-density gas of maximally singly
ne≡gat|ivescatteringlen−gthisareof≡no|concerninourse−t-uip. occupiedsitesbyAparticles(doublyoccupiedAAsitesmay
beeliminatedusingphotoassociationpulses[13]). Onemay
then compress this dilute gas increasing the density until
Thetwoprocessesmaybejustdiscernedbyconsideringavery reaching the desired density. Since A can just move into
smalldetuningδ U ,∆. SinceU and∆areoftheor- AB,doublyoccupiedsiteswillbeguaranteedtohaveanAB
AA AA
(cid:28)
derofkHz,δ . 100Hzisnecessary. HoweverΓistypically occupation,andhencetheon-sitestatesremainalwayswithin
oftheorderoffewMHz(seee.g. TableI),renderingunfeasi- the 0,A,AB manifold.
{ }
bletheproposalofRef.[10].
H. Spuriousprocesses
SpuriousRamanprocessesmustbekeptoutofresonance:
II. QUASI-MOMENTUMDISTRIBUTION
(v)(A,0) (0,B):∆E = ∆
• → −
(vi)(A,A) (0,AA):∆E = ∆+U
• → − AA Asdiscussedabove,theAHMmayberealizedexperimen-
(vii)(AA,0) (A,A):∆E = ∆ U tally with both fermionic and bosonic species. While many
AA
• → − −
observables,mostrelevantlythedensity,donotdependonthe
(viii)(AB,A) (B,AA):∆E = ∆+δU,withδU = particularchoiceofconstituents,others,inparticularthemo-
• → −
(UAA UAB) mentumdistribution,arenotgaugeinvariant.
−
(ix)(AA,B) (A,AB):∆E = ∆ δU As shown in Fig. 4(b) and (c) of the main text, bosonic
• → − −
speciescloselymimicthequasi-momentumdistributionofthe
Note that (vi), (vii), (viii) and (ix) are not possible for
effective model. As mentioned above, the oscillating factors
fermionic atoms. Process (v) is just possible with J24 or ( 1)j resulting from the necessary x projection of δk, may
J . But these laser combinations are (quasi-)resonant with −
13 beeasilyeliminatedbyagaugetransformation. However,this
∆ U . ForasufficientlylargeU process(v)ishence
− ± AB AB transformation must be undone in order to recover the mo-
far from resonance with either J or J . As an illustra-
24 13 mentum distribution of components A and B. As a result, A
tionweconsiderthecaseof40Katoms,witha 170a ,
AB ’ B andBcomponentsexhibitadoubledunit-cell,andthequasi-
with a the Bohr radius, and a lattice with D = 532 nm,
B momentumdistributionn(k)isconsequentlytriviallyshifted
and s = 20, U /h 7 kHz. Note that typical values for
AB ’ and duplicated. For clarity we only depict half the Brillouin
the width of the Raman resonance, W . 50 Hz [2]. Hence zoneforAandBinFig.4ofthemaintext.
U W,andhenceprocess(v)maybesafelyignored.
AB
(cid:29)
For bosons, (vi), (vii), (viii) and (ix) must be also consid- It is interesting to depict as well the momentum distribu-
ered. TheseprocessescanbeonlyrealizedusingJ . Tone- tion for the fermionic case. Figures 3 (a) and (b), which de-
23
glectthe(vi)and(viii)processesoneneedsU W. This pictn(k)neglecting the( 1)j factors, show asingle Fermi-
AA
(cid:29) −
is the case in experiments, where typically U 1 to few surface in the SF, two for the PP, and a blurred n(k) for the
AA
∼
10skHz. Incontrast,toavoid(viii)and(ix)onemustdemand PSF. The oscillating ( 1)j factors obscure this simple inter-
−
δU W. Thelatterconditionismorestrict,butmaybeat- pretation (as shown in Figs. (c) and (d)). However PSF, PP
(cid:29)
tainedaswellinexperiments. Examplesfordifferentbosonic and SF phases may be still clearly discriminated from each
speciesareillustratedinTableII[14,15]. otherbytheircharacteristicmomentumdistributionfunction.
[1] Thes-wavecollisionbetween F = 1,mF = 1 and F = the triplet curve. Hence, the scattering length at zero field is
| − i |
2,mF = 2 for homonuclear collisions can occur only on
− i
5
(a) π 0.02 (b) (c) π 0.02 (d)
k k
u.] u.]
0 k) [a. 0 k) [a.
n( n(
-π 0 -π 0
-2 0 2 -π 0 π -2 0 2 -π 0 π
U / t k U / t k
FIG.3:Quasi-momentumdistributionn(k)oftheAcomponentforthecaseoffermionicatomsasfunctionofU/t(φ=π,ρ=1/2,L=60
sites). (c)and(d)shown(k)includingtheoscillatingfactors( 1)j createdbytheRaman-assistedhopping;(a)and(b)showthesamedata,
−
butexcludingtheoscillatingterms. (b)and(d)shown(k)forU =2(solidline),U =0(dashedline)andU = 2(dottedline). In(a)and
−
(c)thedashedlinesdenotethephasetransitionsbetweenPSF,PPandSFphases.
thesameasthatofthecollisionof F = 2,mF = 2 with http://steck.us/alkalidata(revision2.1.4,23December2010).
| − i
F =2,mF = 2 . [9] T.G.Tiecke.Feshbachresonancesinultracoldmixturesofthe
| − i
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