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Hanbury Brown-Twiss Interferometry for Fractional and Int eger Mott Phases

Ana Maria Rey1, Indubala I. Satija2,3 and Charles W. Clark21 Institute for Theoretical Atomic, Molecular and Optical Physics, Cambridge, MA, 02138,USA

2 National Institute of Standards and Technology, Gaithersburg MD, 20899, USA and3 Dept. of Phys., George Mason U., Fairfax, VA, 22030,USA

(Dated: April 7, 2006)

Hanbury-Brown-Twiss interferometry (HBTI) is used to study integer and fractionally filled Mott Insulator(MI) phases in period-2 optical superlattices. In contrast to the quasimomentum distribution, this second orderinterferometry pattern exhibits high contrast fringes in the insulating phases. Our detailed study of HBTI sug-gests that this interference pattern signals the various superfluid-insulator transitions and therefore can be usedas a practical method to determine the phase diagram of the system. We find that in the presence of a confiningpotential the insulating phases become robust as they existfor a finite range of atom numbers. Furthermore, weshow that in the trapped case the HBTI interferogram signalsthe formation of the MI domains and probes theshell structure of the system.

PACS numbers: 71.30.+h, 03.75.Lm, 42.50.Lc

I. INTRODUCTION

Cold atoms in optical lattices are becoming ideal many-body systems to attain laboratory demonstrations of modelquantum Hamiltonians due in part to the dynamical experi-mental control of the various parameters at a level unavail-able in more traditional condensed matter systems. Recentexperiments with bosonic atoms have been able to simulateeffective one dimensional systems [1, 2, 3, 4, 5], to enterthe strongly correlated Tonks Girardeau (TG) regime [2, 3]predicted many years ago [6] and to realize the superfluid toMott insulator(MI) [7] transition by tuning the lattice param-eters [8]. In fact, one of the most active frontier in cold atomsystems is to explore the possibility of creating new quan-tum phases[9]. The capability to physically create them de-mands sophisticated diagnostic tools for their characterizationand actual observation in the laboratory. In this paper, weuse second order interference techniques, namely the HanburyBrown and Twiss interferometry[10] to describe the rich phasediagram of interacting bosonic atoms in the presence of twocompeting lattices.

Multiple well super-lattices have been experimentally real-ized by superimposing two independent optical lattices withdifferent periodicities[11, 12]. Recent theoretical studies ofthe many-body Bose-Hubbard (BH) Hamiltonian in the pres-ence of an additional superlattice [13, 14, 15, 16, 17, 18] havecentered mostly on the detailed phase diagrams of the system.The focus of this paper is to characterize the different phasesof the system by analyzing the four and two point correlationsthat can be extracted from the time of flight images. Our studyis confined to period-2 superlattices, the simplest example of asystem exhibiting almost all the key aspects of a more generalperiod-m lattices.

The HBTI technique measures second order correlationsfrom the shot-noise density fluctuations in the absorption im-ages of expanding atomic gas clouds. This technique has beenshown [19, 20, 21] to be particularly suited for probing many-body states of cold atomic systems, as it provides complemen-tary information to the first-order correlations inferred fromthe average density distribution in the absorption images.For

example, cold bosonic atoms in the MI regime do not exhibitfirst order interference pattern but they do have sharp second-order Bragg peaks which reflect the spatial periodicity of thelattice. The interference peaks are the manifestation of the en-hanced probability for simultaneous detection of two bosons(bunching) due to the Bose-Einstein statistics.

The phase diagram of interacting bosons in the presenceof 1D superlattices is landscaped by various quantum phases.Transitions to the different phases can be driven either bychanging the ratio between the three sets of energy scalesin the system: the interaction energyU , the tunneling rateJ and the the lattice modulationλ or by varying the fillingfactorν. The lattice modulation induces additional fractionalMI phases that occur at filling factors commensurate with theperiodicity of the combined superlattice. These phases canbe understood in the hard core boson (HCB) limit where thestrongly interacting bosons can be mapped to noninteractingfermions. The superlattice fragments the single particle spec-trum, inducing band gaps. The filling of a sub-band at a crit-ical filling factor in the fermionic system results in a band in-sulator state that corresponds to a fractionally filled MI stateof the bosonic atoms. The fractional MI phases survive in thesoft core boson limit beyond a critical value of the interactionenergyU .

Besides the fractional Mott phases, the interplay betweenthe interaction energy and the superlattice potential can alsolead to a new type of integer-filled Mott phases which cannotbe understood in any special limit such as the HCB limit orthe pure periodic case as they appear at finiteU values and forfinite strengths of the superlattice potential. These insulatingphases are characterized by a modulated density profile andhence will be referred asstaggered Mott phases.

Here we demonstrate that HBTI provides definite means tomonitor the various superfluid to MI transitions and we use itas a method to obtain the phase diagram. Firstly, for a fixedU , J andλ we varyν, and find that the onset to the transi-tion is accompanied by a change in the sign of the superlatticeinduced Bragg peaks as the filling factor is increased beyondthe critical ones. Secondly, for a fixed filling factorν and fixedλ/J , we change the ratioU/J across the transitions and show

2

that they are signaled by a sharp maximum in the intensitiesof the Bragg interference peaks. We show that the phase dia-gram obtained from HBTI is in qualitative agreement with therecently reported phase diagram calculated from Monte-Carlosimulations [14, 15, 16, 17, 18] and with a mean field phasediagram that we derive analytically.

Another important aspect of our study is to investigate theeffects of a harmonic trap on the various phases. In contrastto the translationally invariant system where the transition toinsulating phases occurs only at critical filling factors, theparabolic confinement allows for the formation of fractionaland integer filled MI domains in a finite window of fillings.We show that harmonically confined superlattices in the HCBlimit develop a shell structure [22] analogous to the one ob-served in the absence of any lattice modulation at moderatevalues of the interaction. The HBTI interferogram providesclear signature of the formation of the different Mott domains.

The paper is organized as follows. In Sec. II we define noisecorrelations and the model Hamiltonian used in our study. InSec. III, we calculate two and four-point correlations in theHCB case. In Sec.IV, we consider the BH system for finitevalues of interaction. We first analytically calculate the phasediagram by using mean field theory ( whose details are de-scribed in Appendix A) and compare it with the phase diagramobtained from noise correlations. These higher order correla-tions are calculated by numerical diagonalization of the BHHamiltonian. In Sec.V, we discuss the effects of the harmonictrap in the HCB limit. In Sec.VI we state our conclusions.

II. NOISE CORRELATIONS

In a typical experiment, atoms are released by turning offthe external potentials. The atomic cloud expands, and is pho-tographed after it enters the ballistic regime. Assuming thatthe atoms are noninteracting from the time of release, proper-ties of the initial state can be inferred from the spatial images[13, 19, 23]: the column density distribution image reflectsthe initial quasimomentum distribution,n(Q), and the densityfluctuations, namely thenoise correlations, reflect the quasi-momentum fluctuations,∆(Q,Q′),

n(Q) =1

L

∑

j,k

eiQa(j−k)a†j ak, (1)

∆(Q,Q′) ≡ 〈n(Q)n(Q′)〉 − 〈n(Q)〉〈n(Q′)〉. (2)

In Eq. (2) we have assumed that bothQ,Q′ lie inside the firstBrillouin zone. HereL is the number of lattice sites andathe lattice constant. In this paper, for simplicity, we focus onthe quantity∆(Q, 0) ≡ ∆(Q). In our discussion below,Ndenotes the number of particles in the system andν = N/Lis the filling factor. In this paper, we will confine ourselvestothe case withν ≤ 1.

The BH Hamiltonian describes bosons in optical latticeswhen the lattice is loaded in such a way that only the lowestvibrational level of each lattice site is occupied and tunnelingoccurs only between nearest-neighbor sites [22]. The 1D BH

Hamiltonian in the presence of a period 2-superlattice is givenby

H = −J∑

〈i,j〉

a†i aj +U

2

∑

j

nj(nj − 1)

+∑

j

2λ cos(πj)nj +∑

j

Ωj2nj. (3)

Here aj is the bosonic annihilation operator of a particle atsitej, nj = a†j aj , and the sum〈i, j〉 is over nearest neighbors.The hopping parameterJ , and the on-site interaction energyU are functions of the lattice depth. The cosine term describesthe onsite potential generated by the additional lattice withtwice the periodicity of the main lattice. Here the main lat-tice creates the tight-binding system and the second lattice isassumed to be a weak perturbation. In this case atoms in thesuper-lattice have a hopping parameter which is independentof the lattice position. The parameterλ is almost proportionalto the depth (in recoil units of the main lattice) of the addi-tional lattice (see Ref. [11] for details). The last term takesinto account the parabolic potential withΩ proportional to theparabolic trapping frequency.

III. HARD CORE LIMIT

In the strongly correlated regime, whenU → ∞, the BHHamiltonian can be replaced by the HCB Hamiltonian [25],

H(HCB) = −J∑

j

(b†j bj+1 + b†j+1bj) +

∑

j

2λ cos(πj)nj +∑

j

Ωj2nj . (4)

Herebj is the annihilation operator at the lattice sitej whichsatisfies[bi, b

†j ] = δij , and the on-site conditionsb2j = b†j

2 =0, which suppress multiple occupancy of lattice sites.

HCB operators can be linked to spin-1/2 operators bymeans of the Holstein-Primakoff transformation [24], whichmaps bosonic operators into spin operators. The Holstein-Primakoff transformation maps the HCB Hamiltonian into theXY spin- 1/2 Hamiltonian. The latter can in turn be mappedonto a spinless fermion Hamiltonian by means of the Jordan-Wigner transformation [25, 26].

The Bose-Fermi correspondence can be used to calculatevarious many-body observables of the strongly interactingbosonic system in terms of the ideal fermionic two-point func-tions which can be written asglm =

∑N−1k=0 ψ

∗(k)l ψ

(k)m . Here

ψ(k)j are the amplitudes of the single atom eigenfunctions at

sitej andE(k) are the single atom eigenenergies:

−J(ψ(k)j+1 +ψ

(k)j−1)+2λ cos(πj)ψ

(k)j +Ωj2ψ

(k)j = E(k)ψ

(k)j .(5)

The single-particle spectrum of period-2 superlattice can beobtained by decimating every other site of the tight binding

3

Eq. (5). The renormalized system at even or odd sites is de-scribed by

J2(ψ(k)j+2 + ψ

(k)j−2) + (2J2 + 4λ2)ψ

(k)j = E2

(k)ψ(k)j . (6)

Assuming periodic boundary conditions, the eigenenergiesare then given by

E(k) = ±2

√

J2 cos

(

4πk

L

)

+ λ2, (7)

wherek = 0, 1, . . . , L/2. The effect of the additional latticepotential is to split the band into two different sub-bands,eachwith band-width2

√J2 + λ2−2λ, separated by an energy gap

of 4λ.Local observables such as the density distribution and ener-

gies are identical for the HCB and non-interacting fermionicsystems. For example, the HCB ground state energy corre-sponds to the sum of the firstN single-particle eigenstates.On the other hand, fermions and HCBs posses differ non-localcorrelation functions. A general formulation to calculateHCBtwo-point correlations has been developed by Lieb and Mattis[26]. In our earlier studies [23], we have generalized Lieb andMattis formalism and have obtained explicit formulas, involv-ing multiple Toplitz-like determinants, to compute the four-point correlation functions required to calculate noise correla-tions in HCBs. The evaluation of these determinants, whoseorder scales with the size of the system, is in general compli-cated and therefore an analytical treatment is difficult. Belowwe will describe the results obtained by numerical computa-tion of the explicit formulas discussed in our earlier study. Forinformation about the exact formulas and various other rele-vant details we refer the readers to our earlier paper [23].

FIG. 1: Quasimomentum distribution as a function of the filling fac-tor. The calculations were done for a system withλ/J = 1, L = 80and boxed-like boundary conditions

Figs. 1 and 2 show the quasimomentum distribution andthe noise correlations for all filling factors,0 ≤ ν ≤ 1. Thequasimomentum distribution exhibits interference peaks at thereciprocal lattice vectors of the combined superlattice: alargepeak atQ = 0 and somewhat weaker peaks atQa = ±π,induced by the period two modulation. The peaks are a mani-festation of the quasi-long-range coherence of the system.At

FIG. 2: Noise interference pattern as a function of the filling factorfor a system withλ/J = 1 , L = 80 and boxed-like boundaryconditions. In the inset we truncated the central peak to make thenegative background and the peak to dip transition more visible

ν = 1/2 and1 the first order interference pattern flattens out,signaling the insulating character of the system at these criti-cal fillings.

The peaks atQa = 0 and±π also exist in the noise inter-ference pattern (Fig.2), where they are narrower and are ac-companied by adjacent satellite dips, immersed in a negativebackground. These satellite dips are the signatures of the longrange coherence in the second order pattern as they disappearin the insulating phases (see detailed discussion in Ref.[23]).The satellite dips are clearly seen in the inset. In contrasttothe quasimomentum distribution, the peaks at reciprocal lat-tice vectors continue to exist at the critical filling factors inthe HBTI pattern. However, their intensity is strongly reducedcompared to the intensity at other filling factors.

To highlight the signatures of the insulating phases in thenoise correlations, we plot in Fig. 3 the visibility of the cen-tral and superlattice induced peaks. We define the visibil-ity as the intensity of the second order Bragg fringes (noise-correlations) normalized with respect to the quasimomentumdistribution:

V(Q) ≡ ∆(Q, 0)

n(Q)n(0)(8)

The visibility V(Q) is a relevant experimental quantity as thenormalization procedure filters some of the technical noisein-troduced during the measurements.

In the visibility figure one observes the development of verysharp peaks at the critical fillings. In addition, an interestingdefinitive signal of the fractional insulating phase is a changein the sign of the intensity of the superlattice induced peaks:as the filling factor is increased beyond half filling, the peakatQa = ±π becomes a dip. This effect, which we will referto aspeak to dip transition, has its origin in the occupation ofthe second band with few atoms, once the first band is fullyoccupied. In fact, it can be interpreted as a manifestation ofthe fermionization of HCBs. The peak to dip transition is a

4

0 0.2 0.4 0.6 0.8 1Ν

0

0.5

1

1.5

2

Vis

ibili

ty

FIG. 3: Visibility (see text for definition) of theQ = 0 (dotted line)andQa = π (solid lines) peaks for a HCB system withλ/J = 1 andL = 80.

generic feature of strongly correlated bosons confined in a su-perlattice. It is seen for all values ofλ and, as we discusslater, this signature also accompanies the onset to fractionalMott state for finiteU , i.e. in the soft core boson case.

A further insight for this peak to dip transition can be gainedby looking at the noise correlations asλ → ∞ . In thislimit, the correlations can be calculated analytically forν = 1,ν = 1/2 and also for the case of one extra atom beyond halffilling (N = L/2 + 1). At unit filling the ground state in theHCB limit corresponds to a unit filled Fock state. In the half-filled case, to a good approximation, the ground state can beassumed to be a1/2-filled Fock state|Ψg〉 = |1010 . . .10〉.This is due to the reduced number fluctuations exhibited inthis limit, as the tunneling between every two sites scalesas J2/(4λ). At filling factor N = L/2 + 1 the groundstate can be approximated by a1/2-filled Fock state with anextra delocalized particle at the different unoccupied wells,|Ψg〉 = 1/

√L∑N

i=1 |i〉, with |i〉 = |10 . . . 11 . . .10〉. Usingthese ansatzs, it can be shown that

∆(Q)N=L/2 = − 1

L+

1

4(3δQ0 + δQπ), (9)

∆(Q)N=L/2+1 = − 1

L+

1

2(3δQ0 − δQπ), (10)

∆(Q)N=L = − 2

L+ 2δQ0. (11)

As shown by these equations, just beyond half-filling, theQ =0 andQa = π fringes have opposite sign. The sign differencecan be understood by calculating∆(Q) for a single particle(N = 1) or a single hole (N = L − 1). In this simple case,noise correlations can be calculated explicitly for all values ofλ:

∆(Q, 0)N=1=L−1 =1

4

( λ2

λ2 + 1

)

(δQ0 − δQπ). (12)

The explicit formulas for∆(Q, 0)N=L/2+1 and∆(Q, 0)N=1

show on one hand the similarity between these two cases andon the other that the filled sub-band does have an effect onthe extra particle. Actually as can be seen in Fig. 2 and 3, the

negative fringes survive even for more than one atom in thesecond band.

Eq.(11) also shows that the absence or the presence of aninterference peak atQa = π can be used to distinguish thefractional and Mott insulating phases: the peak atQa = πdisappears when the system is a unit filled insulator but con-tinues to exist ataQ = π when the system is a half-filledinsulator. On the other hand, for both insulating phases theintensity of the central peak isν(ν + 1).

IV. SOFT CORE BOSONS

In this section we relax the hard-core constraint and ex-plore the interplay between the finite interaction effects andthe competing periodicity induced by the superlattice poten-tial. The key questions that we address are: (1) how a fi-nite value ofU affects the various phases observed in theHCB limit, (2) how generic are the characteristic signals ofthe phase transitions observed in the HCB noise correlationsasU becomes finite and (3) what novel characteristics of theinterference pattern emerge as we explore the(J, U, λ, ν) pa-rameter space.

Before we describe the details of HBTI, we will first usemean field theory to gain some insight about the phase dia-gram as(U/J, λ/J, ν) vary. Although previous studies basedon numerical Monte carlo simulations have calculated the var-ious phases in the soft core regime [14, 15, 16, 17, 18], herewe obtain the phase diagram analytically by using second or-der perturbation theory. It is well known that the mean-fieldcalculations provide a good characterization of the phase di-agram and gives qualitative prediction of the different phasetransitions even though the exact transition thresholds may notbe quite correct. We would also like to point out that earliermean field studies [14, 15, 16, 17] of the phase diagram weredone in a different case, namely when the hopping parameteris site dependent.

A. Mean-Field Phase Diagram

To calculate the phase diagram we use the well knowGutzwiller approximation [27] that decouples the kinetic en-ergy term of the Hamiltonian by introducing a superfluid or-der parameter. The details of the mean field calculations arediscussed in Appendix A. Mean field theory predicts three in-sulating phases in addition to the superfluid phase. The insu-lating phases can be classified as: (1) a MI phase withν = 1where all sites have unit occupancy, (2) a staggered MI phasewith ν = 1 characterized by every alternate site doubly oc-cupied and (3) a fractional MI state withν = 1/2 where al-ternate sites are singly occupied. At other filling factors,thesystem is always a superfluid.

Fig. 4 summarizes the mean-field calculations. The top leftpanel shows the critical values ofU ≡ U/J and λ ≡ λ/Jfor the onset to the three insulating phases. They correspondto the tips of the insulating lobes ( multicritical points) that

5

10 20 30 40 50 60UJ

01020304050

ΜJ

0 10 20 30 40 50 60UJ

-100

10203040

ΜJ

0 2 4 6 8ΛJ

10

20

30

40

Uc

J

10

20

30

10 20 30 40 50 60UJ

01020304050

ΜJ

FIG. 4: (color online) Mean field phase diagram for the periodtwo superlattice. The top left panel shows the critical curves,Uc/J −λc/J forthe onset to various transitions to insulating phases. The dash dotted blue line is for the transition to the band half-filled MI The the long dashedand short dashed asymptotes corresponds to2.3/(λ/J − 1) and6.1/(λ/J) respectively. The red dotted line corresponds to the transition tothe unit filled MI and the black solid line to the staggered unit filled insulator. The grid lines are atλ = J and 3.75. The top right, bottom leftand bottom right panels show the phase diagram as a function of the chemical potential for systems withλ = 0.5J, 4J and 8 J

are shown for selected values ofλ in the various panels of thefigure.

One of the interesting results of the mean-field theory isthe prediction of a reentrance to the superfluid phase from thestaggered MI phase. In contrast, to the critical curves describ-ing the threshold to the fractional MI and integer MI phases,which vary monotonically in the(U , λ) space, the curve forthe staggered MI atν = 1 bends (see left panel in Fig. 4 ) andbecomes a double-valued function. As a consequence, for afixed λ > λc, with the threshold value being approximatelyλc = 3.75, as we increase the interactionU , the system goesfrom superfluid to staggered MI, reenters the superfluid phaseand ends in the unit filled MI phase. Note that this reentranceto the superfluid phase is absent forλ < λc, where one onlysees a direct transition from superfluid to unit MI. This behav-ior is explicitly shown in the other panels of the figure wherethe chemical potential is plotted as a function of the interac-tion energy, for a few selected values ofλ. In the top rightpanelλ < λc and so only the unit Mott and fractional Mottphases are present. Forλ > λc, the staggered phase appearsas an isolated island in theµ−U plane whose size grows withλ.

The numerical Monte-Carlo calculations reported in Ref.[18] are in agreement with the intermediate superfluid phasefound at mean-field level. However, according with their cal-

culations the superfluid region is only present forU < 12. ForU > 12 the system is either a unit filled MI or a staggered MI.Our mean field calculations, on the other hand, predict thatthe intermediate superfluid phase exists for all values ofU .As we will discuss later, this result is in qualitative agreementwith our numerical calculations performed by exact diagonal-ization of the BH Hamiltonian for finite size systems.

The staggered phase describes an interesting manifestationof the interplay between the onsite interaction energy,U , andthe energy modulation introduced by the superlattice poten-tial, λ. This phase is absent in the HCB limit and only existsfor values ofλ ≡ λ/J beyond a threshold value and for afinite window ofU ≡ U/J values. The analytic calculationspredict that the threshold value isλc = 3.75, which is an over-estimation of the corresponding value,λc ≈ 1.75, found withthe Monte Carlo simulations [18].

For the half filled case, the mean-field critical curve has1/λas an asymptote (see top left panel in Fig. 4). This scaling isconsistent with Eq. (5) (for largeλ the effective tunneling ratebetween consecutive odd sites goes likeJ2/λ). Therefore, forlargeλ, a fractional MI state can exist for rather small valuesof U . On the other hand Fig. 4 shows that for moderate valuesof λ the transition to an insulating phase requires very largeinteractions. We would like to point out that the existence ofa fractional Mott phase only for values ofλ > 1 seems to be

6

an artifact of the mean-field theory. In fact, in the HCB limit(U → ∞) the system is a band insulator for any infinitesimalvalue ofλ.

B. Noise Correlations and Phase Diagram

To establish a correspondence between the various phasetransitions and their signatures in noise-spectroscopy, we re-sort to exact diagonalization procedures. We first numericallydiagonalize the BH system to obtain the ground state of thesystem. This is then used to obtain two and four point corre-lations and their Fourier transform. Here we will describe ourresults for8 wells (L = 8).

1 2 3 4 5 6 7 8N

-1

-0.5

0

0.5

1

1.5

2

Vis

ibili

ty

FIG. 5: For fixedλ/J = 1 andU/J = 20, we plot the visibilityof the central (dotted-dashed line) and superlattice induced (dashedlines) peaks as the filling factor varies. The solid and dotted lines arethe correspondent HCB curves

Fig. 5 describes the visibility [Eq. (8)] of the second orderfringes for various filling factors. The important point to benoticed here is that the peak to dip transition, a key signatureof the fractional Mott transition in the HCB limit, is preservedfor finiteU . The figure also shows the corresponding result forthe HCB case. A comparison between theL = 80 ( Fig. 3)andL = 8 results for HCB illustrates the finite size effect onthe peak to dip transition. As expected, the finite size of thesystem broadens the transition and hence reduces the visibilityof the interference peaks. Fig. 6 further demonstrates the be-havior of the peak to dip effect as the interactionU is varied.In Fig.6 we plot the normalized HBTI pattern as a functionof U for a system just above half filling. For large but finiteU & 10, the superlattice induced peak in∆(Qa = π) changessign and becomes a dip. Our numerical analysis confirms thatthe peak to dip transition is a generic signature of the stronglyinteracting regime where bosons exhibit fermion-type charac-teristics.

We next show that the second order interference patternnot only complements the characteristics of the various phasetransitions as observed in the first order Bragg spectroscopy,but it also provides new definitive signatures of variousphases. For clarity, we will describe our results forν = 1 and

FIG. 6: Visibility plot (see text for definition) as a function of U/Jfor a system just beyond half filling (L = 8, N = 5) and forλ = 2.

ν = 1/2 separately.

(B.I) Unit-Filled Case

Fig. 7 shows the variation in the central as well as in thesuperlattice induced peak for both the first and the second or-der interference patterns as the on-site repulsive interaction isvaried. An important aspect of the figure is the appearanceof various local maxima in∆(0). The single maximum ob-served for smallλ splits asλ increases. A comparative studybetweenN = 6 and8 shows that the height at the variousmaxima increase withN , suggesting that it may be divergentin the thermodynamic limit. This observation along with thefact that the peak for smallλ occurs at a value ofU very closeto the well known MI transition, suggests that the maximamay be associated with the onset of the various superfluid toMI phase transitions.

By monitoring the locations of the maxima, we obtain thephase diagram in the(U , λ) plane. Our results are shown inFig. 8. The rather remarkable qualitative agreement betweenthis phase diagram and that obtained by mean-field and alsoby earlier Monte carlo studies, supports the validity of ourconjecture regarding the relationship between a peak in∆(0)and the onset of a phase transition, and suggests that noisecorrelations can be used as a practical tool to obtain phase di-agrams of many body quantum systems. It should be noted(Fig. 7) that in contrast to∆(0), the corresponding zeroquasimomentum component does not provide any sharp sig-natures of the different critical points asU is varied.

As shown in the figure forλ = 3, for moderate on-siteinteraction, the system is in the superfluid phase. AsU in-creases, correlations begin to build up and whenU becomescomparable toJ , for λ > 1.5, the system enters the staggeredMott insulating phase with two atoms in the low energy wellsand none in the high energy ones. The onset to this transitionis signaled by a peak in∆(0) (as seen in the figure) and bythe development of a positive intensity peak atQa = π. It

7

0 5 10 15 20 25 30UJ

1

2

3

4

5

6

Inte

nsity

ΛJ=5

0 5 10 15 20 25 30UJ

-1

0

1

2

3

Inte

nsity

ΛJ=5

0 5 10 15 20 25 30UJ

1

2

3

4

5

6

Inte

nsity

ΛJ=3

0 5 10 15 20 25 30UJ

-1

0

1

2

Inte

nsity

ΛJ=3

0 5 10 15 20 25 30UJ

0

2

4

6

8

Inte

nsity

ΛJ=0.5

0 5 10 15 20 25 30UJ

-1

-0.5

0

0.5

1

Inte

nsity

ΛJ=0.5

FIG. 7: The figure shows the noise correlations (solid line) andquasimomentum fridges (dashed line) for variousλ values. The leftand right panel respectively describe the intensities of the centralpeak (Q = 0) and the superlattice induced peak (Qa = π).

0 1 2 3 4 5ΛJ

0

5

10

15

20

25

Uc

J

FIG. 8: Phase diagram for the period two superlattice calculated byreading the maxima of the central peak in the∆(0) vs. U plot fordifferent λ . The dotted, dashed and solid lines show the criticalvalues in theU vs λ plane for the transition to the half-filled MI, theunit filled MI and staggered filled insulators respectively.

should be noted that the critical value ofλ = 1.5 predicted bythis method for the appearance of the staggered phase is veryclose to the value of 1.75 found by the numerical Monte-Carlocalculations [18].

The system continues to be in the staggered insulatingphase asU increases until the point when the competition be-tween U and the energy offset induced by the superlattice,4λ, drives the system back to a superfluid phase. A simple

understanding of this reentrance can be obtained by realiz-ing that whenU is of the order of4λ the state|2020 . . . 〉 isdegenerated with the states|20112011 . . . 〉, |10211021 . . . 〉,|21102110 . . .〉, and |11201120 . . . 〉 which have superfluidcharacter as their average densities are3/2 and1/2 in the lowand high energy wells respectively. The reentrance to the su-perfluid phase is signaled by the second peak in∆(0). Fur-thermore, it is accompanied by the disappearance of superlat-tice induced peak atQa = π. For values ofλ close to 1.5 thestaggered phase exist only for a very narrow range ofU valuesand the first and second order peak can not be resolved.

As U increases beyond4λ, it is energetically costly to havetwo atoms in the same well and hence the system enters theMI phase where sites are singly occupied. This transition isalso signaled by the third peak in∆(0). As λ is increasedfrom 1.5 up toλ ≈ 5, the separation between the last twomaxima (which determines the range ofU values where theintermediate superfluid phase exists) decreases. Beyond thisvalue, asλ is increased further the position of the two peaks isshifted to larger values ofU . However, the relative separationbetween the peaks was found to remain constant. This findingis in disagreement with the results of Monte Carlo simulationswhich find points inU − λ space where staggered and Mottphases coexist. One may argue that the absence of coexis-tence between the staggered and Mott phase in our analysismay be due to the finite size of our system. Nevertheless, thecoexistence of such phases seems to imply the existence ofa first order transition. This makes such a coexistence pointsomewhat subtle and needs further investigation. Besides thisdifference, the phase diagram calculated by monitoring themaxima in∆(0) is in very good qualitative agreement withthe Monte carlo as well as the mean field calculations. Thequantitative differences are due to finite size effects.

We would like to point out that∆(Q = π/a) also con-tains valuable information about the different phases. Forexample the existence of a staggered phase is indicated bythe development of a positive interference peak atQa = π.This peak is a second order effect and it is not present inthe quasimomentum distribution. The quasimomentumdistribution does not give definite signatures of the transitionpoints but it gives information about the phase coherence ofthe various phases. As shown in Fig. 7, inside the superfluidphases then(0) vs U develops a maximum which disappearsin the insulating phases where the curve tends to become flat.

(B.II) Half filled Case

We next discuss theν = 1/2 case, where one sees a transi-tion from superfluid to fractional Mott phase. The formationof this phase is also signaled by the development of a sharppeak in the intensity of the central noise correlation peak as Uis varied across the transition. This is shown in Fig. 9. In ourfinite size system, the peak was found to exist only for valuesof λ & 0.205. However, we expect this value to decrease asthe size of the system increases. Fig. 8 shows the position ofthe peak as a function ofλ. For largeλ the criticalU valuedecreases as1/λ in consistence with the mean field results.

The formation of the half filled band insulator is also

8

0 5 10 15 20 25 30UJ

1

1.5

2

2.5

Inte

nsity

ΛJ=2

0 5 10 15 20 25 30UJ

-1

-0.5

0

0.5

1

Inte

nsity

ΛJ=2

0 5 10 15 20 25 30UJ

11.5

22.5

3

Inte

nsity

ΛJ=1.2

0 5 10 15 20 25 30UJ

-1

-0.5

0

0.5

1

Inte

nsity

ΛJ=1.2

0 5 10 15 20 25 30UJ

0

1

2

3

4In

tens

ity

ΛJ=0.1

0 5 10 15 20 25 30UJ

-1

-0.5

0

0.5

1

Inte

nsity

ΛJ=0.1

FIG. 9: The figure shows the noise correlations (solid line) andquasimomentum distribution (dashed line) forν = 1/2 and variousλ values. The left and right panel respectively describe the inten-sities of the central peakQ = 0 and the superlattice induced peak(Qa = π).

indicated by the superlattice induced peak at∆(π/a). Itshould be noted that∆(π/a) has negative intensity for thenon-interacting system. Precisely atU = 0 ∆(π/a) =−N/4λ2/(λ2 +J2). Its amplitude increases asU is increasedand becomes positive for the values ofλ that allow the forma-tion of a band-insulator. ForU > U

ν=1/2c , with Uν=1/2

c theMott band insulator critical point, the peak intensity almostreaches its hard-core value.

V. TRAPPED SYSTEM

In this section we study the experimentally relevant case ofa superlattice in the presence of an additional parabolic po-tential. The two key results of this study are following: First,to find that in contrast to the homogeneous system where theMI phases only exist at specific critical fillings, the trappingpotential stabilizes the MI phases and allows for their exis-tence over a large range of filling factors. Second, to showthat noise-spectroscopy provides a detailed information of theformation of the insulating domains and the shell structurethatarises in presence of a parabolic confinement.

Understanding the effects of the trap on period-2 superlat-tice follows along the lines of the earlier studies of the trappedsystem in the absence of any additional lattice modulation.Therefore, we first briefly review the key aspects of theλ = 0case which has been studied in great detail [28, 29, 30, 31].The combined lattice plus harmonic confinement system pos-

sesses two distinct classes of eigenstates: low energy statesthat extend symmetrically around the trap center and high en-ergy states that are localized on the sides of the potential.Theorigin of these two distinct classes is related to the two energyscales in the system, namely the tunneling (J) and the trappingenergy (Ω). Modes with excitation energy below4J , whichcorrespond to the band width of the translationally invariantsystem, are extended and can be thought of as harmonic os-cillator like modes with effective frequencyω∗ =

√4JΩ and

effective massm∗ = ~/(2Ja2). Modes with excitation ener-gies above4J are close to position eigenstates since for thesestates the kinetic energy required for an atom to hop from onesite to the next becomes insufficient to overcome the potentialenergy cost. The high energy eigenstates are almost two-folddegenerate with energy spacing mostly determined byΩ. Thelocalization of these modes can be understood by means of asemiclassical analysis [29]. Within WKB scheme, the local-ization of the higher energy modes can be linked to the appear-ance of new turning points related to umklapp processes. Incontrast to the turning points of the classical harmonic oscil-lator that appear at zero quasimomentum, the Bragg turningpoints emerge when the quasimomentum reaches the end ofthe Brillouin zone and can therefore be associated with Braggscattering induced by the lattice.

The period-2 superlattice splits a band into two subbandsand hence the nature of modes can be now classified by theenergy scales associated with these two bands and the bandgap which is equal to4λ.

In the top panel of Fig.10 we plot the various wave func-tions. Each eigenstate has been offset along they axis by itsenergy in units of

√JΩ. In the lower panel we also show the

energy spectrum. As shown in the figure, we now see two setsof extended and also two sets of localized states. Again, thelocalized states are related to umklapp processes and to em-phasize this idea, the classical and the Bragg turning pointsof both bands are explicitly displayed. In Appendix B, weprovide additional details of the trapped superlattice model.

Figs. 11 and 12 describe the density and number fluctua-tions that reflect the band structure of the trapped superlatticesystem discussed above. To simplify the description, we in-troduce three different quantities:NLE

1 , N1 andNLE2 which

respectively denote the number of low energy extended statesin the first band, the number of states below the first mode ofthe second band and the total number of modes below the firstlocalized mode of the second band. In Fig. 10, these threenumber are explicitly indicated with grid lines.

For filling factors belowNLE1 atoms tend to spread over

the central sites populating mostly the low energy wells. Theextended character of the modes induce large number fluctu-ations. For filling factors betweenNLE

1 < N ≤ N1, thelocalized modes in the first band become occupied. Becausethese modes are localized at the edges of the cloud, as thenumber of atoms is increased the occupation of the centralsite remains constant and instead sites farther away from thetrap center become populated. Thus the presence of single-particle localized modes in the Fermi sea leads to the forma-tion of fractionally filled insulating domains at the trap centerin the many-body system. Alternatively, as soon as localized

9

0 10 20 30 40 50 60mode

0

20

40

60

80

100

Ene

rgy

-40 -20 0 20 40Site

20

40

60

80

100E

nerg

y

FIG. 10: (color online) The top panel shows the single-particle eigen-states of a system withλ/J = 1.1 andΩ/J = 0.005. The heightof each mode is proportional to its energy in units of

√

ΩJ . Theparabola correspond to the classical and Bragg turning points thatappear in the first and second band respectively (see text). In thelower panel we show the spectrum of the modes in units of

√

ΩJ .The dots, triangles, boxes and stars indicate the low energymodesin the first band, the high energy localized modes in the first band,the low energy modes in the second band and high energy localizedmodes in the second respectively. Note the degeneracy that appearsbetween the localized high energy modes of the two bands. Thegridlines are atNLE

1 = 9, N1 = 32 andNLE

2 = 42.

modes are populated, number fluctuations become only rele-vant at the edges. The insulator character of the atoms at thesecentral sites and the reduced number fluctuations can be seenin Fig.12.

ForN > N1 the trapping energy cost of placing an atomat the trap edge is higher than the energy needed to place theatom at the center. For1 < N < NLE

2 , atoms occupy the ex-tended modes of the second band and tend to spread over thedifferent empty sites at the trap center. When the number ofatomsN = NLE

2 , the first localized mode of the band is pop-ulated and the site at the trap center acquires filling factorone.As N is further increased, the width of the unit-filled centralcore grows. ForN > NLE

2 the density profile alternates fromunit filled MI, superfluid, fractional MI and superfluid as onemoves from the center towards the edge of the cloud. Thisshell structure in the HCB limit resembles in many aspects

the shell structure observed in theλ = 0 case at moderatevalues ofU . In the former case, the gap is induced by theexternal modulation and the shells consist of fractional andunit filled insulator domains surrounded by superfluid regions.In the later case, the gap is induced by the onsite interactionenergyU and the shells consist of integer filled MI domainssurrounded by superfluid regions.

FIG. 11: (color online) Density distribution for a system with λ =1.1 andΩ/J = 0.005 as a function of the total number of atoms.Thegrid lines are atNLE

1 = 9, N1 = 32 andNLE

2 = 42.

FIG. 12: (color online) Number fluctuations in the presence of trapfor a system withλ = 1.1 andΩ/J = 0.005 as a function of thetotal number of atoms. The grid lines are atNLE

1 = 9, N1 = 32 andNLE

2 = 42.

We now describe the first and second order interferencepatterns in the trapped system. The quasimomentum distri-bution is shown in Fig. 13. Consistent with the behavior ob-served in the density and number fluctuations, for filling fac-torsN < NLE

1 andN1 < N < NLE2 the quasimomentum

distribution is sharply peaked atQ = 0 andQa = π, signalingthe superfluid character of the system. The insulating phasesthat appear for filling factorNLE

1 < N < N1 andN > NLE2

are signaled in the quasimomentum distribution by the drop ofthe peak intensities and the flattening of the quasimomentumdistribution profile.

Fig. 14 shows the HBTI pattern for a different total number

10

FIG. 13: (color online) Quasimomentum distribution for a systemwith λ/J = 1.1 andΩ/J = 0.005 as a function of the filling factor.The grid lines are atNLE

1 = 9, N1 = 32 andNLE

2 = 42.

FIG. 14: (color online) Noise interference pattern as a function of thefilling factor or a system withλ/J = 1.1 andΩ/J = 0.005. Thegrid lines are atNLE

1 = 9, N1 = 32 andNLE

2 = 42.

of atoms. ForN < NLE1 , only extended modes are occupied

and the intensity of the central peak grows monotonically withN . Similarly the peak atQa = π increases from the neg-ative value it takes forN = 1 and becomes a peak with apositive intensity. As the localized modes enter the Fermi sea(N > NLE

1 ) the rate of growth of the central peak slows downand this change in the slope generates a maximum in the inter-ference pattern. Note that in contrast to the strong reductionobserved in theΩ = 0 HCB case, in the presence of the traponly a decrease in the rate of growth is observed. The reasonof this behavior is the fact that in the trapped case there is al-ways a superfluid component at the edges of the atomic cloud.For NLE

1 > N ≥ N1 the interference peak atqa = π isclearly visible, reflecting the staggered character of the phase.

At N = N1 +1, we see a rather sharp peak to dip transitionatQa = π. Similar to the homogeneous system, the transi-tion signals the beginning of the population of an empty band.The dip becomes a peak when the number of atoms increasesbeyond a certain value. ForN1 < N ≤ NLE

2 , one also ob-serves an increase in the growth rate of the central peak with

N, consistent with the superfluid properties of the system atthese fillings. Finally forN > NLE

2 , a unit filled insulatingdomain at appears at the trap center. This leads to a decreasein the amplitude of the central peak until it reaches a constantvalue. The peak atQa = π disappears as the unit filled coreat the trap center grows. In the regimeN > NLE

2 , Fig. 10shows some additional small modulations of the central peakamplitude at certain fillings. The oscillations take place whena high energy mode of the first band, degenerated with anotherhigh energy mode of the second band, enters the Fermi sea.

In analogy with the homogeneous case, the transition to in-sulating state can best be illustrated in the normalized inten-sity of the noise correlations, namely the visibility. In Fig. 15,we plot the visibility of the central and the superlattice peaks.As mentioned above, the normalized pattern is perhaps thebest experimental observable as the normalization procedurefilters some of the technical noise introduced in the measure-ment procedure. The normalization procedure maps the cen-tral peak maxima that appear before the formation of the insu-lator domains to minima. Furthermore, when normalized, theamplitude of the superlattice induced peak becomes compara-ble in intensity to that of the central peak.

0 10 20 30 40 50 60N

-0.2

0

0.2

0.4

0.6

0.8

Viss

ibilit

y

0 10 20 30 40 50 60N

1.5

2

2.5

3

3.5

Viss

ibilit

y

FIG. 15: Visibility of the second-order interference peaksat Q =0 (top panel) andQa = π (bottom panel) as the total number oftrapped atoms is varied. The grid lines are atNLE

1 = 9, N1 = 32andNLE

2 = 42.

VI. CONCLUSIONS

In this paper, we have studied shot noise correlations forinteracting bosons in the presence of an additional lattice. Al-though we have focused on a period-2 superlattice, this simplecase exhibits all the key aspects of the phase diagram of moregeneral superlattice potentials.

One of the central results of this paper is that noise corre-lations may provide a practical tool to construct the phase di-agram of many body quantum systems exhibiting transitionsfrom superfluid to fractional and integer Mott phases whichalso include the staggered insulating phase. Our study sug-

11

gests that the intensity of the second order Bragg peaks pro-vides a new order parameter to characterize these phase tran-sitions. Although our calculations in the soft core regime weredone for systems with a reduced number of atoms and wellsand further studies for larger systems are needed to confirmour conjecture, we believe that this result is important in thetheory of quantum phase transitions. Furthermore, our studyof the trapped system demonstrates the possible realization ofthese phases in the laboratory.

Shot noise spectroscopy has already proven to be a usefulexperimental tool to identify Mott insulating states. We con-sider our analysis of the shot noise and its possible relevancein identifying various phase transitions in interacting bosonsin superlattices is pertinent for atom optics experiments.Upto date all the experiments in super-lattice has been restrictedto the Bose-Einstein-condensate regime. However, ongoingexperimental efforts are trying to reach the strongly corre-lated regime in superlattices. The motivation of these effortsis not only to gain understanding of the many-body physicsbut also because atoms selectively loaded in super-lattices canhave wider separation and stronger confinement, useful prop-erties for the implementations of lattice based quantum com-puting proposals. We hope that our study will stimulate fur-ther experimental probing of one dimensional bosonic sys-tems loaded in superlattices via HBTI.

VII. APPENDIX A

To study the phase diagram for the homogeneous system weuse the well know decoupling approximation [27] and substi-tute:

a†i aj = ψiaj + ψj a†i − ψiψj (13)

into the BH Hamiltonian. Hereψj = 〈aj〉 ≈ √nj is the

superfluid order parameter andnj is the expectation value ofthe number of particles on sitei. We will assume the orderparameter to be real. This substitution leads to :

Hef = −J∑

j

(ψj+1 + ψj−1)(a†j + aj − ψj) (14)

+∑

j

(2λ cos[πj] − µ)nj +U

2

∑

j

nj(nj − 1),

where we have also introduced the chemical potentialµ. As-suming periodic boundary conditions, one can reduced theproblem to a two mode system due the fact that the Hamil-tonian is exactly the same for all even and all odd sites. Wedefineψ2 = ψ2i, ψ1 = ψ2i+1, a2 = a2i anda1 = a2i+1, so

Hef =N

2(Hef

1 + Hef1 ), (15)

Hef1 = −2Jψ2(a

†1 + a1 − ψ1) (16)

−(2λ+ µ)n1 +U

2n1(n1 − 1),

Hef2 = −2Jψ1(a

†2 + a2 − ψ2) (17)

+(2λ− µ)n2 +U

2n2(n2 − 1).

The above Hamiltonian can be diagonalized using a Fock statebasis truncated below a certain occupation value. However,second order perturbation theory usingψ as an expansionparameter can provide a good analytic approximation of thephase diagram. At zero order inψ the Hamiltonian is diago-nal in the Fock basis. The integer occupation numbersn1, n2

that minimized the energy are given by the conditions

n1 − 1 <µ+ 2λ

U< n1, (18)

n2 − 1 <µ− 2λ

U< n2. (19)

We consider three different cases:

• −2λ < µ < min(U − 2λ, 2λ)

In this case the ground state of the system is the Fock State|010101 . . . 〉 and the unperturbed energy isE(0)

g = −2λ− µ.The first order correction to the energy vanishes and to secondorder one gets:

E(2)g = 4J

(

Jψ21

µ− 2λ− Jψ2

2

µ+ 2λ+

2Jψ22

µ+ 2λ− U+ ψ1ψ2

)

.(20)

Minimizing the energy respect toψ1,2 yields the followingequation:

(

1 2Jµ−2λ

2J(µ+2λ+U)(µ+2λ−U)(µ+2λ) 1

)

(

ψ2

ψ1

)

=

(

00

)

. (21)

Eq. (21) has nontrivial solution only when the determinant ofthe matrix vanishes. Therefore the surface where the determi-nant vanishes determines the insulating phase:

4J2(µ+ 2λ+ U) = (µ+ 2λ− U)(µ2 − 4λ2), (22)

where the bar denotes the dimensionless variablesU = U/J ,λ = λ/J andµ = µ/J . In Fig. 4 we show, with a crossedblue line, the solutions of Eq. (22) in theµ vs. U plane, forλ = 0.5, 4 and8 (top right, bottom left panel and bottom rightpanels respectively). The region inside the blue loops corre-sponds to the1/2 filled insulator. In the top left panel we also

show with a blue dash-dotted line the critical value ofUν=1/2c

as a function ofλ. Uν=1/2c corresponds to the smallestU in

the lobe, below which the system is always a superfluid. Ex-actly atUν=1/2

c the upper and lower branches of the chemical

12

potential merge and the energy gap closes up. It is impor-tant to point out that in general mean field calculations do notaccurately predict critical values but, in general, they capturevery well the physics of the phase transition.

• 2λ < µ < U − 2λ

In this case the ground state of the system is the unit filled statewith exactly one atom per site. The zero order ground stateenergy isE(0)

g = −2µ. For this case we setψ1 = ψ2 = ψ. Tosecond order the ground state energy is given by:

E(2)g =

(

8J2ψ2

µ− 2λ− U+

8J2ψ2

µ+ 2λ− U

)

−(

4J2ψ2

µ+ 2λ+

4J2ψ2

µ− 2λ

)

+ 4Jψ2. (23)

Minimizing with respect toψ one gets an algebraic equationthat determines the boundary between the superfluid and insu-lating phases. The solution is displayed in Fig.4 forλ = 0.5, 4and8. The critical value ofUν=1

c is also shown in the top leftpanel with a dotted red line. Note the critical value increaseswith λ. This increase is in agreement with the idea that forweak interactions disorder helps to delocalize the atoms.

• U − 2λ < µ < 2 min(U − λ, λ)

In this case the on-site repulsion is not large enough to avoiddouble occupancies and the ground state of the system is thestate with two atoms in the low energy sites and zero in theothers. This situation can not be described in the hard coreregime and it is only present at moderate values ofU . To zeroorder in J, the ground state energy isE(0)

g = −2µ− 4λ+ U .To second order it is given by

E(2)g =

4J2ψ21

µ− 2λ− U− 4J2ψ2

1

µ+ 2λ+

4J2ψ22

2λ− µ+

8J2ψ22

µ+ 2λ− U

+4Jψ1ψ2 (24)

By first minimizing with respect toψ1,2, and finding the so-lutions for which the determinant vanishes one obtains theboundary for this phase. The solution is displayed in Fig.4for λ = 0.5, 4 and8 with a black line. At mean field levelthe minimum value ofλ required for the existence of this in-sulating phase isλ = 3.75. For values ofλ > 3.75 the unitfilled system is a superfluid forU < Uν=1

c1 , a staggered insu-lator for Uν=1

c1 < U < Uν=1c2 and a unit filled Mott insulator

for U > Uν=1c . In the top right panel, the lower and up-

per branches of the solid black curve correspond toUν=1c1 and

Uν=1c2 respectively.

VIII. APPENDIX B

The super-lattice potential splits the main band into twodifferent sub-bands. In this ”multi”-band picture, it is sim-ple to understand the modification introduced by the trap.For excitation energies below the band width of the first sub-band:E(n) − E(0) < E

(1)with ≡ 2

√J2 + λ2 − 2λ, the modes

are delocalized states that spread every other site symmet-rically around the potential minimum. In the semiclassicalpicture these modes only see the classical turning point at

x(1)cla = ±a

√

(E(n) + 2√λ2 + J2)/Ω. For higher energies,

E(n) − E(0) ≥ E(1)with, the eigenstates become localized on

both sites of the potential. These modes see besides the clas-sical turning point, the Bragg turning point of the first bandx

(1)B = ±a

√

(E(n) + 2λ)/Ω.

At excitation energies higher than4λ (the energy gap be-tween the two subbands), the potential energy cost of local-izing a state at the edge is larger than the energy requiredto populate the high energy wells at the trap center. As aconsequence, states with quantum numbern >

√

4λ/Ω ap-pear centered again around the trap minima. These secondgroup of extended states see the classical turning point at

x(2)cla = ±a

√

(E(n) − 4λ+ 2√λ2 − J2)/Ω. Finally, for exci-

tation energies larger than the band width of the second bandE(n) − E(0) ≥ E

(2)with = E

(1)with, the modes feel the Bragg

turning point of the bandx(2)B = ±a

√

(E(n) − 2λ)/Ω andbecome again localized at the edge. Sometimes a high energymode in the second band becomes degenerated with some ofthe very high energy modes of the first band. This effect canbe seen in Fig. 10.

AcknowledgmentsWe would like to thank J.V. Porto, I.B.Spielman and S. Sachdev for their suggestions, commentsand useful input. This work is supported in part by the Ad-vanced Research and Development Activity (ARDA) contractand the U.S. National Science Foundation through a grantPHY-0100767. A.M.R. acknowledges support by a grant fromthe Institute of Theoretical, Atomic, Molecular and OpticalPhysics at Harvard University and Smithsonian Astrophysicalobservatory.

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