Quantum coherence of Hard-Core-Bosons and Fermions : Extended, Glassy and Mott Phases

0.66 MB·English

by  Ana Maria Rey

Quantum coherence ofHard-Core-Bosons andFermions : Extended, GlassyandMott Phases
Ana Maria Rey1,2, Indubala I. Satija2,3 and Charles W. Clark2
1 InstituteforTheoretical Atomic, Molecular andOptical Physics, Cambridge, MA,02138,USA
2 National Institute of Standards and Technology, Gaithersburg MD, 20899, USA and
3 Dept. of Phys., George Mason U., Fairfax, VA, 22030,USA
(Dated:February6,2008)
6
WeuseHanbury-Brown-Twissinterferometry(HBTI)tostudyvariousquantumphasesofhardcorebosons
0
(HCBs) and ideal fermions confined in a one-dimensional quasi-periodic (QP) potential. For HCBs, the QP
0
potentialinducesacascadeofMott-likeband-insulatorphasesintheextendedregime,inadditiontotheMott
2
insulator,Boseglass,andsuperfluidphases.Atcriticalfillingfactors,theappearanceoftheseinsulatingphases

n isheraldedbyapeaktodiptransitionintheinterferogram, whichreflectsthefermionicaspectofHCBs. On
a theotherhand,idealfermionsintheextendedphasedisplayvariouscomplexitiesofincommensuratestructures
J
suchasdevil’sstaircasesandArnoldtongues. Inthelocalizedphase,theHCBandthefermioncorrelationsare

6 identicalexceptforthesignofthepeaks. Finally,wedemonstratethatHBTIprovidesaneffectivemethodto
1 distinguishMottandglassyphases.

]
r
e Recent developments in the manipulations of ultra-cold experimentalobservation. Furthermore,we findthatQPdis-
h atoms in optical lattices have opened new possibilities for ordercanbeexploitedtodistinguishMottfromglassyphases.
t
o exploringthe richness and complexityof strongly-correlated Four-point correlations can be experimentally probed by
t. systems. A particulartopicofcontinuoustheoreticalinterest Hanbury-Brown-Twiss interferometry (HBTI) [10] which
a incondensedmatterphysicshasbeenthedifferentphasesin- measures the intrinsic quantum noise in intensity measure-
m
ducedbytheinterplaybetweendisorderandinteractions[1,2]. ments, commonly refereed to as shot noise or noise corre-
d- Incoldatomicgasesdisordercanbeintroducedinacontrolled lations. HBTI is emerging as one of the most important
n mannereitherbyplacingaspecklepatternonthemainlattice toolstoprovideinformationbeyondthatofferedbystandard
o orbyimposingaquasiperiodic(QP)sinusoidalmodulationon momentum-distribution based characterization of phase co-
c theoriginallatticebyusinganadditionalweakerlatticewith herence. The latter is obtained from images of the density
[
the desired wavelength ratio [3]. Furthermore, by changing distributionoftheatomicsampleafteritsreleasefromthecon-

1 the depth of the optical potential it is experimentally possi- finingpotential.HBTIisperformedbymeasuringthedensity-
v bletocontroltheeffectivedimensionalityaswellastheratio density correlations in such images [11, 12]. Sharp peaks
7
between atomic kinetic and interaction energy in these sys- (dips) in these correlations reflect bunching (anti-bunching)
0
3 tems. All these attributes have made cold atomic systems a of bosons (fermions), characterize the underlying statistics,
1 unique new laboratory to test many established results, ex- and reveal information about the spatial order in the lattice
0 plore new phenomenaunderlying disordered systems and to [11,13].
6 confrontopenquestions[4,5].
In our study we find that the QP potential induces in the
0
/ In this paper, we use two and four point correlation func- HBTIpatternanhierarchicalsetofpeaksappearingatthere-
t
a tionstocompareandcontrastthedifferentquantumphasesin ciprocallatticevectorsofbothlatticeswithcompetingperiod-
m a gas of one dimensional(1D)hard core bosons (HCBs) and icities. WeusethesepeakstogetherwiththefirstorderBragg
- fermions subject to a QP potential. In an HCB gas [6], the peaksinthemomentumdistributiontoprovideadetailedchar-
d
repulsiveinteractionsbetweenbosonsmimicthePauliexclu- acterizationofthedifferentmany-bodyphases. Threecharac-
n
o sionprinciple,andasaresultHCBsresembleinmanyrespects teristic phases are known in disordered HCB systems [14]:
c a system of non-interacting fermions. A 1D HCB gas in a i)theincompressibleMottInsulator(MI)phase,ii)thesuper-
v: periodic optical lattice has just recently been experimentally fluidphase,andiii)theinsulatingbutcompressibleBose-glass
i realized [7]. A system of non-interactingfermionsmightbe (BG) phase. Here we also find a number of Mott-like band
X
realized experimentally by reducing interatomic interactions insulator phases. These phases correspond to the filling of
ar viaFeshbachresonances[8]. Herewetreatthecaseofatoms thesub-bandsofafragmentedenergyspectrumthatemerges
confined by a periodic lattice, with an additional QP poten- fromtheQPsuperfluidphase. Transitionstothesephasesare
tialintroducedtoaddpseudo-randomdisorder. AQPsystem signaledbyadecreaseintheintensitiesofthefirstandsecond
is in between periodic and random systems and is known to ordercoherentpeaks.Theyarefoundtobefollowedbyapeak
exhibit localization transitions at a finite depth of the addi- to dip transition in the HBTI pattern when an emptyband is
tionallattice[9]. Hereweshowthattheinterplaybetweenthe populatedwithfewatoms.
effects of disorder, interactions and quantum statistics leads IdealfermionsinthelocalizedMIandglassyphasesexhibit
to new quantumphases, fractalstructuressuch as the devil’s first- and second-order interference patterns similar to the
staircase, and fragmented Fermi distribution functions. Our onesofHCBs,exceptforadifferentsignofthepeaks. How-
studiesrevealtheseeffectsaremanifestedinfirst-andsecond- ever,intheextendedphase,fermionsandHCBsbehavequite
order interferometry, which makes them accessible to direct differently,withfermioninterferencepatternsdisplayingvari-
2
ouscomplexitiesofincommensuratestructures.Whenplotted Inthisletterwetreatthecaseofσ =(√5 1)/2.Innumer-
−
asafunctionofthefillingfactor,theirquasi-momentumdistri- icalstudies, σ isapproximatedbytheratio oftwo Fibonacci
butiondisplaysanArnoldtongue-likestructure;theintensity numbers F /F , (F = F = 1,F = F + F ),
M−1 M 1 0 i+1 i i−1
ofHBTIpeaksexhibitsastep-likepatternwhichresemblesa which describe the best rational approximant to σ. For this
devil’s staircase [15]. Thus, our study points out the poten- rational approximation the unit cell has length F and the
M
tial of ultra-cold gases to provide laboratory demonstrations single-particle spectrum consists of F bands and F 1
M M
−
ofnon-linearandmulti-fractalphenomena. gaps. Thegapsoccurathalf-valuesofreciprocallatticevec-
In a typical experiment, atoms are released by turning off tors, which we label by the Bloch index Q = nσ(mod
n
±
theexternalpotentials.Theatomiccloudexpands,andispho- 1)2π/a, with n an integer and Q a ( π,π). The size of
n
∈ −
tographedafter it enters the ballistic regime. Assuming that the gaps decreases as n increases. In this paper, we present
the atoms are noninteracting from the time of release, prop- ourresultsforσ =55/89.
erties of the initial state can be inferred from the spatial im- Ingeneralthephasediagramofinteractingbosonswithdis-
ages [11, 13, 16]: the column density distribution image re- orderis rather complicated. Howeverthe analysis simplifies
flectstheinitialquasi-momentumdistribution,n(Q),andthe in the HCB limit, because multiple site occupanciesare for-
densityfluctuations,namelythenoisecorrelations,reflectthe bidden and the system is thus ”fermionized”. The ground
quasi-momentumfluctuations,∆(Q,Q′), state energy of an N-particle HCB system is the sum of the
first N single-particle eigenstates, as is the case for ideal
fermions. This is why the localization transition occurs at
1
nˆ(Q) = LXeiQa(j−k)Ψˆ†jΨˆk, (1) λc = J in HCB systems. Two-point and four-point corre-
j,k lationfunctionsfor HCBs canbe expressedas a To¨plitz-like
∆(Q,Q′) nˆ(Q)nˆ(Q′) nˆ(Q) nˆ(Q′) . (2) determinant involving two-point free fermionic propagators:
≡ h i−h ih i g = N−1ψ∗(n)ψ(n). However, in contrast to the ideal
lm Pn=0 l m
In Eq. (2) we have assumed that both, Q,Q′ lie inside the fermion case where a direct application of Wick’s theorem
first Brillouin zone. In this paper we focus on the quantity can be used to express any correlation function in terms of
∆(Q,0) ∆(Q). L isthe numberoflattice sites anda the g , for HCBs the calculations are more elaborate because
lm
latticecon≡stant. Ψˆ isabosonicorfermionicannihilatorop- theyrequiretheevaluationofmultipledeterminants[13,17].
j
erator at the site j. The low energy physics of a 1D gas of
stronglycorrelatedbosonsinan opticallattice modulatedby 1.2
aQPpotentialiswelldescribedbytheHamiltonian: Ν=0.3 Ν=1 Ν=0.7
1
H(HCB) =−JX(ˆb†jˆbj+1+ˆb†j+1ˆbj)+XVjnˆj, (3) HLnQ00..68
j j
0.4
where ˆbj is the annihilation operator at the lattice site j; it -Π -Π(cid:144)2 0 Π(cid:144)2 Π
Qa
satisfiesbosoniccommutationrelations,plustheon-sitecon- 0.1
ditionˆb2 = ˆb†2 = 0whichsuppressesmultipleoccupancies. 0.08 Ν=0.3
j j 0.06 Ν=1
HereJ isthehoppingenergybetweenadjacentsites,andwe 0.04 analytic
haveintroducedVj =2λcos(2πσj)todescribetheadditional DHLQ 0.02
QPpotential. Theparameterλisproportionaltotheintensity 0
of the lasers used to create the QP lattice [3]. Note that the -0.02
primaryopticallatticedefinesthetight-bindingcondition,and -Π -Π(cid:144)2 0 Π(cid:144)2 Π
Qa
theQPpotentialiscreatedbyasecondaryopticallattice.Ideal
fermionsaredescribedbythesameHamiltonian,inwhichthe FIG.1:(coloronline)CorrelationsofHCBsintheglassyphase(λ=
HCBoperatorsarereplacedbyfermionoperators. 2J)Top: Momentumdistribution. Bottom: noisecorrelations. The
For a single atom, the eigenfunctions at site j, ψ(n) and centralpeakatQ=0istruncatedtohighlighttheQPpeaks.
j
eigenenergiesE(n) oftheHamiltonianinEq.(3)satisfy:
We nowdiscussthe phasediagramsforHCBandfermion
J(ψ(n) +ψ(n))+2λcos(2πσj)ψ(n) =E(n)ψ(n). (4) systemes,startingwiththelocalizedphase. Inthelimitλ
− j+1 j−1 j j λ ,thesingleparticlewavefunctionsarelocalizedatindivi≫d-
c
ThisistheHarperequation,aparadigmin1DQPsystems[9]. uallatticesites,ψ(n) δ(j l )andE(n) 2λcos(2πσl ),
j → − n → n
Forirrationalσ,themodelexhibitsatransitionfromextended wherethelocalizationcenterl issuchthatcos(2πσl ) <
n n−1
tolocalizedstatesatλ = J. Belowcriticality,allstatesare cos(2πσl ). In this limit the determinants involved in the
c n
extendedBloch-likestatescharacteristicof a periodicpoten- evaluationofHCBcorrelationsbecometrivialandananalytic
tial. Above criticality, all states are exponentially localized descriptionis possible. The momentumdistributioninterfer-
andthespectrumispoint-like. Atcriticalitythespectrumisa encepatternflattensoutforbothHCBsandfermions,nˆ(Q)=
Cantorset,andthegapsformadevil’sstaircase. ν with ν = N/L the filling factor. The noise correlations
3
wsimheprleifsy=to0∆f(oQr)H=CBνsδQan,0d−s2=ν(11L−fso)r+fer(m−L1i2o)sns(cid:12)(cid:12).PAlpepiaroQxligmll(cid:12)(cid:12)a2t-, 100..8800..66ΝΝ
00..44
00..22
ing the sum by an integral, it is possible to show that ∆(Q)
is nonzero only at the reciprocal lattice vectors, Q , of the
n 4
combinedsuperlattice.Atthesepoints, nHQL
2
∆(Qn)≡∆n ≈νδn,0−2ν(1L−s)+(−1)s(πsinn)22[πνn]. (5) -Π --ΠΠ(cid:144)(cid:144)22 Q00a ΠΠ(cid:144)(cid:144)22 ΠΠ0
1
00..88 ΝΝ
00..66
Eq. (5)illuminatesvariousimportantaspectsofHBTIinthe 00..44
00..22
glassyphase: i)theexplicitdependenceofnoisecorrelations
onquantumstatistics, sincethebosonpeaksarepositiveand 1
0 DHQL
thecorrespondingfermionpeaksarenegative(exceptforthe -1
-2
autocorrelationpeak∆ ); ii)thepotentialofHBTIasanex-
perimentalspectroscopy0tool,sincetheintensityofthen=1 -Π --ΠΠ(cid:144)(cid:144)22 Q00a ΠΠ(cid:144)(cid:144)22
peakisdirectlyrelatedtothegroundstateenergyofthemany-
body system, ∆1 = PNk=−01cos(2πnσlk) = E/(L2λ); iii) FIG.2: (coloronline)HCBsinthesuperfluidphase,λ = 0.5J,top:
the capability of noise-correlationsto clearly distinguish be- Momentumdistributionandbottom:noisecorrelationswherethein-
tween BG and MI phases, since in the Mott phase, ν = 1, tensitiesofthecentralpeaksarescaledbyafactorof1/10toshow
thesatellitedips.
thewholehierarchicalsetofQPpeaksdisappears;iv)theuni-
versal properties of the peaks whose intensities approach an
asymptotic value independentlyof the underlyingcommuta- 0.2
tionrelations(exceptforthe∆ peakwhichisalwayslarger
0
for HCBs). This is a unique feature of the BG phase, as in 0
theextendedphasefermionicdipsareweakerthanthecorre-
-0.2 0.15
spondingbosonicpeaks(seebelow). LQ 0.1
Fig.1 shows the momentum distribution, n(Q) and noise DH-0.4 J1 0.05
0
correlations, ∆(Q), in the localized phase. Both n(Q) and -0.05
-0.6 0 0.2 0.4 0.6 0.8 1
∆(Q) were computednumericallyfor finite λ usingthe the- Ν
Ν=0.3 Ν=0.6 Ν=Σ
oretical framework discussed in Refs. [13, 17]. It should
0 Π(cid:144)2 Π 3Π(cid:144)2 2Π
be noted that even though the analytic results were derived Qa
for λ limit, they provide a fair description of the nu-
→ ∞
mericalresultsforfiniteλ. Themaindifferencebetweenthe FIG. 3: (color online) HCBs noise correlation for λ = 0.5J. In
λ=2J HBTIpatternandtheanalyticsolutionisthenegative theinsetweplotthevisibilityforthedominantpeak(seetext)asa
backgroundwhichislargeratlowquasi-momenta.Theback- functionofν.
groundreflectsthefactthatthesingleparticlelocalizedwave
functionshavefinitelocalizationlengthξ = 1/log(λ). This
by satellite dips, immersed in a negative background. The
also gives rise to the finite width in the momentumdistribu-
later reflects the longrangecoherencein the extendedphase
tion,whichisproportionaltoξ. Asshowninthefigure,QP-
asdiscussedinRef.[13]. Itshouldbenotedthat,incontrastto
induced localization may lead to Friedel oscillations [18] in
glassyphase,fewQPrelatedpeaksarevisible.
themomentumdistribution. Theseoscillationsareduetothe
Astrikingaspectofthisphaseisthecascadeoflobes(seen
two-folddegeneracyoftheeigenstatesandcanbeunderstood
in the Fig. 2) describing a series of transitions to insulat-
in the strongly disordered limit by taking the eigenfunctions
to be a superposition of the two degenerate wave functions: ing phases. These transitions occur at filling factors νc(n) =
onelocalizedatlkandtheotheratL−lk. Foranoddnumber nσ(mod1)andν¯c(n) = 1−nσ(mod1),whichcorrespondto
ofatoms,thisleadston(Q)=1+ 1 cos(2Qal ). thefillingsatwhichthevarioussub-bandsunderlyingtheQP
L N−1
In the extended phase two-point fermion correlations g spectrumarecompletelyfilled. Atthesecriticalfillings,both
lm
arelong-range,theTo¨plitz-likedeterminantsarecomplicated, first and second order correlations depict a reduction in the
and we have found an analytical analysis to be difficult. In centralpeakanddipintensitiesandalsothedisappearanceof
this phase our numerically obtained results are summarized theQPinducedpeaksas shownin Fig. 2. Explicitcomputa-
in Figs. 2 and 3. In contrast to the glassy phase, the mo- tionofthecompressibilityshowsthatthesystemisinfactin-
mentumdistributionexhibitsinterferencepeakswhichsignal compressibleatthesecriticalfillings. Thesefeaturesarerem-
thequasi-long-rangecoherencecharacteristicofthisphase: a iniscentoftheMottinsulatingphaseatν = 1,sowereferto
large peak at Q = 0 and quasiperiodicity induced peaks at thesephasesasMott-likeband-insulatorphases. However,in
reciprocal lattice vectors ~Q whose intensity depends on contrasttotheMottphase,numberfluctuationsinthesephases
n
±
the filling factor. These peaks also exist in the noise inter- donotcompletelyvanish,butareonlysomewhatreduced.
ferencepatternwheretheyarenarrowerandareaccompanied A rather interesting consequence of the Mott-like band-
4
insulating transitions is the fact that as ν is increased be- small λ << J, first order perturbation theory explains the
yondthecriticalvalue,thepeaksatthereciprocallatticevec- single step observed at ν(1) = Q a/π, as only z(k1) with
tors associated with the corresponding filled band become k = ν(1)L is nonzero. jAus λ i|ncr1e|ases further, m0ore steps
dips. This change in the sign of the peaks beyond the crit- 1 ju
areobservedasothereigenvectors,acquireanonzeroQ = 0
icalfilling can be seen by plottingthe fringevisibility ϑ =
n component. ThestepscanalsobeseenintheQ =0planeof
∆(Q ) [∆(Q +δQ) ∆(Q δQ)]/2,withδQ=4π/aL.
n − n − n− n(Q). Atcriticality,thewholehierarchyofstepsresemblesa
Fig. 3 illustrates this for the dominant peaks that occur at
devil’sstaircase, mimickingthe multi-fractalstructureof the
Q . Herethepeaktodiptransitionatν(1),issignaledbya
1 c gapsinanenergyspectrumthatisaCantorset[9].
±
jumpinthevisibilityfrompositivetonegative.Thisdipinthe
secondorderinterferencepatterncanbeinterpretedasaman-
ifestationoffermionization. Physicalinsightintothisbehav-
ior can be gained by looking at the noise correlation pattern
of a single particle which can be shown to exhibit negative
fermion-type dips: ∆(Q) = z0 2δQ0 zQz0 2 with zQ is 1
| | −| | nHQL 1
theFourier-transformofthegroundstate wavefunction. For 0.5 0.8
HCBsthesenegativefringessurviveevenfortwoatoms. This 0 0.6
Ν
impliesthatthepeaktodiptransitioncanbeunderstoodasthe --ΠΠ(cid:144)(cid:144)22 0.4
00 0.2
consequenceofanemptybandpopulatedwithfewatoms. QQaa ΠΠ(cid:144)(cid:144)22
Π
IncontrasttothesuperfluidbehaviorofHCBs,fermionsin
the extendedphase havemetallic propertiesand differentin-
terferometric pattern. For λ = 0, the sharply peaked HCB
momentumdistributionisreplacedbyastep-functionprofile
for fermions: n(Q) = 1 for Q Q and n(Q) = 0 for 0.02
F
| | ≤ 1
|Q| > QF, with QF = πν/a being the Fermi momentum. DHQL0.01 0.8
For 0 < λ < λ this distribution retains part of the step- 0 0.6
c ΠΠ(cid:144)(cid:144)22 0.4 Ν
like profile of the free fermion gas. However, the Fermi sea ΠΠ
gets fragmented and additional step-function structures cen- QQaa 33ΠΠ(cid:144)(cid:144)22 0.2
teredatdifferentreciprocallatticevectorsofthesecondlattice
develop(SeeFig.4). Wecallthefilledstatescenteredaround
Q = 0 the main Fermi sea and those around the QP related FIG.4: (color online) Fermionsinthemetallicphase, λ = 0.5J.
reciprocallatticevectorsthequasi-Fermiseas. Top:MomentumdistributionandBottom:noisecorrelations.
The quasi-Fermi seas emanating from the QP reciprocal
lattice vectorsformtongue-likestructureswhichresemblein In summary, quasiperiodicity can be exploited to distin-
manyrespectstheArnoldtonguesthatdescribemode-locked guishglassyandMottphasesandinduceaseriesofMott-like
periodic windows in the circle map [15]. As the number band insulating phases. Furthermore, it provides an explicit
of atoms increases, the width of the tongues increase. Pre- probetowitnessfermionizationofbosons,andfacilitatesthe
cisely at the critical fillings νc(n) and ν¯c(n), at which a quasi- experimentalobservationofcomplexfractalstructures.
bandis filled upand the system becomesa band-insulator,a
quasi-Fermi sea merges with the main Fermi sea. As λ in-
creases, more tonguesbecomevisible, and with the increase
ofν theyoverlapinacomplexpattern. AsseeninFig.4,we
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5
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