arX

iv:1

504.

0819

2v2

[co

nd-m

at.q

uant

-gas

] 2

6 Ju

n 20

15

Cluster Gutzwiller study of Bose-Hubbard ladder: ground-state phase diagram and

many-body Landau-Zener dynamics

Haiming Deng1,2,3, Hui Dai1, Jiahao Huang1,2, Xizhou Qin1,2, Jun

Xu1,2,4, Honghua Zhong1,2,5, Chunshan He1, and Chaohong Lee1,2∗1State Key Laboratory of Optoelectronic Materials and Technologies,

School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China2Institute of Astronomy and Space Science, Sun Yat-Sen University, Guangzhou 510275, China

3College of Electronic Information and Electrical Engineering, Xiangnan University, Chenzhou 423000, China4Center of Experimental Teaching for Common Basic Courses,

South China Agriculture University, Guangzhou 510642,China and5Department of Physics, Jishou University, Jishou 416000, China

(Dated: July 24, 2018)

We present a cluster Gutzwiller mean-field study for ground states and time-evolution dynamics inthe Bose-Hubbard ladder (BHL), which can be realized by loading Bose atoms in double-well opticallattices. In our cluster mean-field approach, we treat each double-well unit of two lattice sites asa coherent whole for composing the cluster Gutzwiller ansatz, which may remain some residualcorrelations in each two-site unit. For a unbiased BHL, in addition to conventional superfluid phaseand integer Mott insulator phases, we find that there are exotic fractional insulator phases if theinter-chain tunneling is much stronger than the intra-chain one. The fractional insulator phases cannot be found by using a conventional mean-field treatment based upon the single-site Gutzwilleransatz. For a biased BHL, we find there appear single-atom tunneling and interaction blockade ifthe system is dominated by the interplay between the on-site interaction and the inter-chain bias. Inthe many-body Landau-Zener process, in which the inter-chain bias is linearly swept from negativeto positive or vice versa, our numerical results are qualitatively consistent with the experimentalobservation [Nat. Phys. 7, 61 (2011)]. Our cluster bosonic Gutzwiller treatment is of promisingperspectives in exploring exotic quantum phases and time-evolution dynamics of bosonic particlesin superlattices.

PACS numbers: 03.75.Lm, 05.70.Fh, 67.25.dj, 02.70.-c

I. INTRODUCTION

The unprecedented experimental techniques of ma-nipulating and detecting ultracold atoms in optical lat-tices [1, 2] provide an ideal testing ground to investi-gate Bose-Hubbard (BH) models [1–8]. The remark-able cleanness and high tunability of ultracold atomicsystems allow one to explore various many-body quan-tum phenomena in BH models [9–11]. For an example,the experimental realization of the one-dimensional (1D)atomic Hubbard model [12] provides new opportunitiesto exploring quantum statistical effects and strong cor-relation effects in low-dimensional quantum many-bodysystems [13]. Quantum dynamics as well as quantumphase transition between superfluid (SF) phase and Mottinsulator (MI) phase in BH models are of great interestsand have been widely investigated [14–21].In recent, by loading ultracold Bose atoms into a

double-well optical lattice potential, the Bose-Hubbardladder (BHL) had been realized and the many-bodyLandau-Zener (LZ) dynamics has been explored [22].Different from the single-particle LZ process, the break-down of adiabaticity in the inverse sweeping from thehighest excited state had been observed in the many-

∗ Corresponding author. Email: chleecn@gmail.com

body LZ process of the BH ladder. This experiment hasstimulated extensive investigation of both stationary anddynamic behaviors in the BHL via different theoreticalmethods, such as, full diagonalization method [23] andtime-dependent density-matrix renormalization group (t -DMRG) technique [24–26]. However, the full diagonal-ization and t -DMRG methods should cost a huge numberof computational resources.

To simulate the BHL with less computational re-sources, the bosonic Gutzwiller method [27, 28] is an al-ternative option. Although the Gutzwiller method hascommon restrictions of the mean-field methods, it hasprovided versatile applications in qualitative calculationsof both stationary states and time-evolution dynam-ics. In recent, cluster bosonic Gutzwiller methods [29–34] have been developed by coupling multi-site clustersrather than single sites with the mean field. By em-ploying the cluster Gutzwiller method, some propertiesof many-body LZ phenomena in repulsive BHL [35] andsome quantum phases in attractive BHL [36] have beenexplored. In Ref. [35], the phase diagram and LZ dy-namics for fixed average number of particles per site havebeen shown. However, it does not discuss how the phasediagram and LZ dynamics depend on the chemical poten-tial. In the large-size multi-site Gutzwiller method [36],the three-body constraint has been imposed to each lat-tice site. In a realistic experimental system, the num-ber of particles in each lattice site may break this con-

2

straint. In addition, the ground-state phase diagrams ofBH systems have been obtained by the analytical mean-field approach [38], the cell strong-coupling perturbationtechnique [39] and the composite boson mean-field the-ory [40, 41] etc.

In this article, we present a cluster Gutzwiller mean-field study for the ground-state phase diagram and many-body LZ dynamics of a BHL. In our mean-field treat-ment, we regard each double-well unit of two lattice sitesas a coherent whole for composing the cluster Gutzwilleransatz, which remains some residual inter-site correla-tions in each double-well unit. For a unbiased BHL, inaddition to superfluid (SF) and integer Mott insulator(MI) phases which may be found by single-site Gutzwillertreatment, we find that there exist exotic fractional insu-lator phases if the inter-chain tunneling is much strongerthan the intra-chain one. The exotic fractional insulatorphase in BHL is similar to the rung-Mott phase in hard-core BH system [36, 37]. We also obtain the phase dia-gram for the asymmetry BHL, which have not yet beenreported. In further, by linearly sweeping the inter-chainbias from negative to positive or vice versa, we analyzemany-body LZ dynamics in the system and confirm theexistence of adiabaticity breakdown.

This article is organized as follows. In Sec. II, we givethe physical model and discuss its realization. In Sec. III,we present the cluster Gutzwiller mean-field method andobtain the ground-state phase diagram for both sym-metric and asymmetric BHLs. In Sec. IV, we studythe many-body LZ dynamics and show the adiabaticitybreakdown. At last, in Sec. V, we briefly summarize anddiscuss our results.

II. MODEL

We consider an ensemble of Bose atoms confined withina double-well superlattice potential,

V (x, z) = Vxl sin2(2πx/λxl) + Vxs sin

2(2πx/λxs)

+ Vz sin2(2πz/λz), (1)

where the first and second terms are generated by super-imposing two standing-wave lasers along the x-directionwith wavelengths λxl and λxs. The two potential depthsVxs and Vxl are determined by the laser intensities. Toform the double-well lattice potential, the wavelengthsare set to be λxl = 2λxs. The last term describes a lat-tice potential along the z-direction with the wavelengthλz and the depth Vz. During the experiment, the energydifference between the lattices in each double-well unitcan be ramped up or down with time. The schematicdiagram for the double-well lattices is shown in Fig. 1.

If the barriers between neighboring double-well unitsalong the x-direction is sufficiently high, the system canbe described by several parallel BHLs with ignorableinter-ladder couplings. The Hamiltonian for a single BHL

FIG. 1. Schematic diagram of the superlattice potential (1)projected onto the x-z plane. The time-dependent bias ∆(t)can be achieved by ramping up the lattices to obtain the tiltedpotential along the x-axis. Here, the parameters are set asλz = λxl = 2λxs and Vz = 1.5Vxl = 1.5Vxs. The poten-tial along the x-axis within a period of λxl is an asymmetricdouble-well potential and the potential along the z-axis is astanding-wave potential. The two lattice sites in each double-well potential are packed as a cluster, which are depicted bythe gray rectangles. The clusters are decoupled by apply-ing the Gutzwiller mean-field treatment, in which the crossesstand for the decoupling between neighboring clusters and thegray dashed and solid lines respectively denote the intra- andinter-chain tunneling.

reads as,

H(t) = −J‖∑

〈jk〉σ

b†jσ bkσ − J⊥∑

j

(

b†jLbjR + h.c.)

+U

2

∑

jσ

njσ

(

njσ − 1)

− ∆(t)

2

∑

j

(njR − njL)

−µ∑

jσ

njσ, (2)

where, 〈jk〉 indicates the summation comprising all near-est neighboring sites in the same chain and the indexσ = (L,R) denotes the left or right chain. The symbols

b†jσ (bjσ) creates (annihilates) a Bose atom on the j-th

lattice of the σ-chain, and njσ = b†jσ bjσ stands for theatomic number. The parameters J‖ and J⊥ are the intra-and inter-chain nearest-neighbor hopping strengths, re-spectively. The on-site interaction U is determined bythe s-wave scattering lengthes and the chemical poten-tial µ determines the particle filling.

To characterize different regimes of the BHL, we intro-duce the ratio between the intra- and inter-chain hoppingstrengths β = J‖/J⊥. If β ≪ 1, the intra-chain tunnel-ing is ignorable and the ladder system can be regarded asisolated double-wells. However, if β ≫ 1, the intra-chaintunneling becomes dominant, the system can be treatedas two decoupled single BH chains. The parameter ∆(t)represents the inter-chain energy bias.

3

III. GROUND-STATE PHASE DIAGRAM

In this section, we show how to obtain the ground-state phase diagram via the cluster Gutzwiller mean-fieldtreatment. In the first subsection, we describe the clus-ter Gutzwiller mean-field approach for the BHL. Thenin the second subsection, we present the self-consistentprocedure for determining ground states. In the last sub-section, we give the ground-state phase diagram.

A. Cluster Gutzwiller mean-field approach

The standard Gutzwiller method assumes the wave-function of the whole system as a product state ofsingle-site wavefunctions. By implementing the standardGutzwiller procedure, the BH model is decoupled as sin-gle sites which couple with surround sites via their aver-age mean fields. In further, attribute to the equivalenceof all lattice sites in the model, one can replace the meanfields of surround sites with the mean field of the siteitself and so that the mean-field version for the originalHamiltonian can be written as a sum of single-site terms.In the following, the cluster Gutzwiller mean-field ap-

proach is an extension of the single-site Gutzwiller mean-field approach. As shown in Fig. 1, the decoupling holdsfor each double-well cluster which includes one lattice sitein the left chain and one lattice site in right chain. There-fore all clusters are equivalent and the state for the wholesystem is written as a product state of the single-clusterstates which remains the correlations between lattice sitesin the same cluster. Unlike the single-site Gutzwiller ap-proach, in which all tunneling terms are decoupled, thecluster Gutzwiller approach keeps the intra-cluster tun-neling terms and only decouple the inter-cluster tunnel-ing terms. By using the mean-field treatment, the inter-cluster tunneling terms are decoupled as

b†jσ bkσ ≈ b†jσ〈bkσ〉+ 〈b†jσ〉bkσ − 〈b†jσ〉〈bkσ〉= b†jσϕkσ + ϕ∗

jσ bkσ − ϕ∗jσϕkσ , (3)

where ϕkσ = 〈bkσ〉 and the high-order fluctuations

δb†jσ

δbkσ

= (b†jσ−〈b†jσ〉)(bkσ−〈bkσ〉) are neglected. There-fore the original Hamiltonian (2) is decoupled as

HMF =∑

j

HMFj , (4)

with the single-cluster mean-field Hamiltonian

HMFj = −J‖

∑

σ,k=j±1

(

ϕkσ b†jσ + ϕ∗

kσ bjσ − Re[

ϕ∗jσϕkσ

]

)

−J⊥

(

b†jL bjR + h.c.)

+U

2

∑

σ

njσ

(

njσ − 1)

−∆

2(njR − njL)− µ

∑

σ

njσ. (5)

Making use of the Gutzwiller ansatz, the state for thewhole system can be expressed as a product state ofsingle-cluster states,

∣

∣ΨGA⟩

=∏

j

|Ψ〉j , (6)

where the state for the j-th cluster |Ψ〉j can be expandedas

|Ψ〉j =Nmax∑

N=0

N∑

m=−N

f(j)N,m |N,m〉j . (7)

with |N,m〉j denoting the state basis for the j-th cluster.

Here, N = NL + NR, m = NL − NR, NL (NR) standsfor the number of particles in the left (right) chain, the

probability amplitudes f(j)N,m are complex numbers, and

Nmax is the truncation of the maximum particle number.Obviously, it is easy to find that the eigenequation

HMF∣

∣ΨGA⟩

= E∣

∣ΨGA⟩

for the whole system is equiv-alent to the single-cluster eigenequation

HMFj |Ψ〉j = Ej |Ψ〉j (8)

with E =∑

j Ej . By substituting the single-cluster

state (7) into the single-cluster eigenequation (8), we ob-tain

Ejf(j)N,m = − J‖√

2φjL

√N +mf

(j)N−1,m−1

− J‖√2φjR

√N −mf

(j)N−1,m+1

− J‖√2φ∗jL

√N +m+ 2f

(j)N+1,m+1

− J‖√2φ∗jR

√N −m+ 2f

(j)N+1,m−1

+ J‖Re[

ϕ∗jLφjL + ϕ∗

jRφjR

]

f(j)N,m

− J⊥2

√N +m

√N −m+ 2f

(j)N,m−2

− J⊥2

√N +m+ 2

√N −mf

(j)N,m+2

+

[

U

4

(

N2 +m2 − 2N)

+∆

2m− µN

]

f(j)N,m.

(9)

Here, the order parameters are quantum expectationvalues of bosonic annihilation operators, i.e. ϕjL =

〈ΨGA|bjL|ΨGA〉 and ϕjR = 〈ΨGA|bjR|ΨGA〉. After somemathematical calculation, we have

ϕjL =∑

N,m

√

N +m+ 2

2f(j)∗N,mf

(j)N+1,m+1, (10)

ϕjR =∑

N,m

√

N −m+ 2

2f(j)∗N,mf

(j)N+1,m−1. (11)

For convenience, we define φjL = ϕ(j+1)L + ϕ(j−1)L andφjR = ϕ(j+1)R + ϕ(j−1)R.

4

B. Self-consistent procedure for determining

ground states

As the single-cluster Hamiltonian (5) depends on themean fields, one has to implement self-consistent pro-cedure for determining the mean fields and the groundstates. Given the parameters U , J‖, J⊥, µ and ∆, one canobtain the ground state from the single-cluster eigenequa-

tion (8) and the self-consistent relations ϕσ = 〈bσ〉. InFig. 2, we show the key steps of the self-consistent pro-cedure for determining ground states.(i) Initialize ϕσ = 0, ϕ′

σ = ϕσ and set the trial groundstate energy Emin

GS an arbitrary value.(ii) Substitute ϕσ into the single-cluster Hamiltonian,

and diagonalize the Hamiltonian to obtain its groundstate with eigen-energy EGS .(iii) If EGS < Emin

GS , replace EminGS and ϕ′

σ with EGS

and ϕσ, respectively. Otherwise, let ϕσ = ϕσ +∆ϕσ andimplement step (ii) again.(iv) Repeat Steps (ii) and (iii) until ϕσ ≥

√Nmax.

(v) Set ϕIσ = ϕ′

σ and calculate the ground state |GSI〉from the single-cluster eigenequation.(vi) Calculate the order parameters ϕII

σ for the groundstate |GSI〉.(vii) Compare ϕI

σ and ϕIIσ , if

∣

∣ϕIσ − ϕII

σ

∣

∣ < ǫ (where ǫis pre-given tolerance), then output |GSI〉 as the groundstate. Otherwise, set ϕI

σ = ϕIIσ and return to step (v).

Through the procedure from step (i) to step (iv), onecan numerically minimize the system energy with re-spect to the order parameters in the interval of ϕσ ∈[0,

√Nmax]. Usually, the steps (v-vii) are the so-called

self-consistent procedure.

C. Phase diagram

In this subsection, we show the ground-state phase di-agram for the BHL. Our cluster mean-field approach canbe applied to both symmetric and asymmetric systems.Below, we first consider the symmetric system with nointer-chain bias (i.e. ∆(t) = 0), then consider the asym-metric cases with nonzero inter-chain bias ∆.In Fig. 3, we show the ground-state phase diagram for

symmetric BHL with ∆(t) = 0 and different ratios β =J‖/J⊥. Due to the absence of asymmetry, the two chainsare completely equivalent and the order parameters ofboth chains are always equal ϕjL = ϕjR = ϕj . Therefore,it is enough to give the phase diagram via analyzing theorder parameter for one of the two chains. The groundstates sensitively depend on the chemical potential µ, theon-site interaction U , the intra-chain hopping J‖ and theinter-chain hopping J⊥.Usually, determined by the superfluid order parame-

ter, the BH systems have two typical phases: (i) thesuperfluid (SF) phase of nonzero order parameter and(ii) the Mott insulator (MI) phase of zero order param-eter and integer filling number. For our atomic BHLsystem, when the inter-cluster hopping J‖ is sufficiently

maxFor parameters , , , and, tolerance: J J U N

min

GS GSE E

YN

1/2

maxN

min

GS0, , and set initi ,al E

min

GS GS, E E

MFdiagonal )(H

GSfind the ground energy: E

YN

I

MF Idiagonal ( )H Ifind the ground state: GS

II

I IˆGS GSb

II I

I II

I

I, GS

Y

N

FIG. 2. The numerical self-consistent procedure for solvingthe single-cluster eigenequation.

strong, the atoms can move freely between neighboringdouble-well clusters and there appears a SF along thechain direction. In contrast, when the on-site interactionU becomes sufficiently strong, the atoms are localized ineach cluster and there is no SF along the chain direction.The chemical potential µ controls the filling number, i.e.the average atomic number per site.Under the condition of β = J‖/J⊥ ≫ 1, i.e. the intra-

chain tunneling is much stronger than the inter-chaintunneling, the BHL can be regarded as two decoupledchains and the corresponding phase diagram is almost assame as the one for a single BH chain. In Fig. 3 (c),we show the phase diagram for β = 10. At the side ofstrong intra-chain tunneling, J‖/U → +∞, the groundstates are SF phases of nonzero order parameter. At theside of strong interaction, J‖/U → 0, there appear sev-eral integer MI lobes which has integer filling numbers perlattice site and zero order parameter. The blue region inthe bottom corresponds to the vacuum state with no anyatoms. The biggest lobe corresponds to the MI phasewith definitely one atom (n = 1) in each site and thesmaller one stands for the MI phase of n = 2. This phasediagram reminds us the one for the one-dimensional BHmodel [44].

5

0.02 0.0401

0

1

2

0n

1n

1 2n

3 2n

2n( ) 0.1a ( ) 1b

1n

0n

| |J U

0n

1n

( ) 10c

0 0.2 0.4 0.6 0.8 1.0 1.2

2n 2n

5 2n

0.050.030.01 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05

FIG. 3. (Color online) The ground-state phase diagram for the symmetric Bose-Hubbard ladder (∆ = 0) with the on-siteinteraction U = 1 and different values of β = J‖/J⊥: (a) β = 0.1, (b) β = 1, and (c) β = 10. In our calculation, we setthe truncation of maximum particle number Nmax = 6 whose validity has been numerically verified. The blue areas are theinsulator phases with zero order parameter ϕj = 0 with n denoting the filling number (the average atomic number per latticesite). The Mott insulator (MI) lobes have integer filling numbers, while the loophole insulator (LI) phases have half-integerfilling numbers. The regions outside the blue areas are the superfluid (SF) phases with nonzero order parameters ϕj 6= 0.

The areas of MI lobes shrink if the ratio β decreases,see Fig. 3 (a-c). Qualitatively, the shrinking of MI lobescan be understood by the intra-chain tunneling assistedby the inter-chain tunneling. When the ratio β becomesvery small, the inter-chain hopping J⊥ are much strongerthan the intra-chain hopping J‖, the areas of MI lobesshrink dramatically, and, interestingly, several loopholeinsulator (LI) phases of zero order parameters appearbetween the conventional MI lobes, see Fig. 3 (a).

To distinguish the LI and MI phases, we calculate thefilling numbers (the average atomic numbers per site)and find that the LI phases have half-integer filling num-bers and while the MI phases have integer filling num-bers. The half-integer filling numbers mean that the to-tal atomic numbers per cluster are odd integer numbersand the residual atom in each double-well cluster canfreely move between the two wells of each cluster. Infurther, we calculate the intra-cluster first-order correla-

tion Cor(1)⊥ =

∣

∣

∣〈b†jLbjR〉

∣

∣

∣and find that the LI phases have

nonzero Cor(1)⊥ .

The appearance of the LI phases is a direct result ofU ≫ J‖ and J⊥ ≫ J‖. As U ≫ J‖, the tunneling alongthe chain direction is suppressed and the order param-eter vanishes. However, the atoms in each double-wellcluster may still freely move between the two wells andso that the total atomic numbers per cluster may beodd integer numbers. The insulator phases of fractionalfilling numbers have also been found in one-dimensionalsuperlattice BH models [16, 29] via mean-field method,quantum Monte Carlo simulation and numerical densitymatrix renormalization group simulation. Different fromthe one-dimensional superlattice BH chains [16, 29], our

ladder system includes two coupled one-dimensional BHchains and the coupling between different clusters aremore complex.

Now, we discuss the ground-state phase diagrams forBHL with nonzero bias ∆. Due to nonzero bias ∆, thetwo chains are no longer equivalent and so that the orderparameters ϕjL and ϕjR for the left and right chains mayhave different values. In Fig. 4, we show the two orderparameters (ϕjL, ϕjR) (the second and third rows) and

the intra-cluster first-order correlation Cor(1)⊥ (the first

row).

In our numerical simulation, by employing the clustermean-field method presented in Subsection A, we con-sistently obtain the ground-state phase diagram in the(∆/U, µ/U)-plane for the intra-chain hopping strengthJ‖ = 0.01, the on-site interaction U = 1, and differentvalues of the hopping ratio β = J‖/J⊥ = (0.1, 1, 10). Ourresults show that, for given J‖, β and µ, the order param-eter for the left chain ϕjL with bias ∆ equals to the orderparameter for the right chain ϕjR with bias −∆. There-fore, the ground-state phase diagrams of ϕjL and ϕjR

are symmetric with each other about the axis ∆ = 0 forgiven J‖ and β. Moreover, the intra-cluster correlation

Cor(1)⊥ is also symmetric about the axis ∆ = 0.

If the inter-chain tunneling is very weak, that is thehopping ratio β ≫ 1, the BHL can almost be treated astwo independent chains. In the third column of Fig. 4,

we show the inter-chain coherence Cor(1)⊥ and the order

parameters (ϕjL, ϕjR) for β = 10. In phase diagramsof (ϕjL, ϕjR), there appear several parallel and equal-spaced SF stripes with nonzero order parameters ϕjL orϕjR, see Fig. 4 (f) and (i). The SF stripes of ϕjL become

6

FIG. 4. (Color online) The ground-state phase diagrams for the biased Bose-Hubbard ladder with different values of β = J‖/J⊥

versus the bias ∆. In our calculation, we fix the on-site interaction U = 1, the intra-chain hopping J‖ = 0.01, and set the

truncation of maximum particle number Nmax = 6. The first row shows the intra-cluster first-order correlation Cor(1)⊥ , and the

second and third rows respectively correspond to the order parameters ϕL,R for the left and right chains. The hopping ratiosare chosen as β = 0.1 (the first column), β = 1 (the second column) and β = 10 (the third column), respectively.

narrower and its corresponding values become smaller asthe bias ∆ increases from the negative to the positiveside, see Fig. 4 (f). While for the order parameter ϕjR,its SF stripes change oppositely as they are symmetricwith the ones of the order parameter ϕjL about the axis∆ = 0, see Fig. 4 (i). The blue regions correspond to MIphases with zero order parameters ϕjL or ϕjR. For the

inter-chain coherence Cor(1)⊥ , nonzero values only appear

in the vicinity surrounding some specific points, whichform an inverted triangle lattice structure, see Fig. 4 (c).

The interplay between the inter-chain bias and the on-site interaction under weak hopping leads to the exoticsingle-atom tunneling [the bright spots in Fig. 4 (c)] andthe interaction blockade [the blue regions in Fig. 4 (c)].

Under strong on-site interactions, that is U ≫ J‖ andU ≫ J⊥, the system behavior can be understood by aperturbation picture. As U ≫ J‖ and U ≫ J⊥, thehopping terms can be regarded as perturbations and thedominant part of the BHL Hamiltonian reads as

Hd =U

2

∑

jσ

njσ

(

njσ − 1)

− ∆

2

∑

j

(njR − njL)

−µ∑

jσ

njσ. (12)

From the Fock states for a cluster, a single Fock state|nL, nR〉 corresponds to a MI phase of zero order param-eters, the quasi-degeneracy of |nL, nR〉 and |nL + 1, nR〉will result a nonzero order parameter ϕjL, and the quasi-

7

degeneracy of |nL, nR〉 and |nL, nR + 1〉 will induce anonzero order parameter ϕjR.In the MI regions of ϕjL (i.e. ϕjL = 0), the clus-

ter state is a single Fock state |nL, nR〉. The quasi-degeneracy of |nL, nR〉 and |nL+1, nR〉 under weak intra-chain hopping, which means that one atom can freelymove in the left chain, will result a nonzero order pa-rameter ϕL. From the energy quasi-degeneracy relationE(nL, nR) = E(nL + 1, nR), we have the system param-eters obeying

µ =∆

2+ UnL, (nL = 0, 1, 2, ...). (13)

Obviously, the above relation (13) between the chem-ical potential µ and the bias ∆ well agree with the SFstripes shown Fig. 4 (f). Similarly, from the energy quasi-degeneracy relation E(nL, nR) = E(nL, nR+1), we have

µ = −∆

2+ UnR, (nR = 0, 1, 2, ...), (14)

for the SF stripes of nonzero order parameter ϕjR shownin Fig. 4 (i).In addition to the single-atom tunneling along a spe-

cific chain, there exists single-atom tunneling betweentwo chains, which corresponds to a nonzero intra-cluster

first-order correlation Cor(1)⊥ . The inter-chain single-

atom tunneling is caused by the quasi-degeneracy of|nL, nR〉 and |nL+1, nR−1〉. Therefore, from the energyquasi-degeneracyE(nL, nR) = E(nL+1, nR−1), we have

∆ = U(nR − nL), (nL,R = 0, 1, 2, ...) (15)

As nonzero Cor(1)⊥ appears in the region of both ϕjL 6= 0

and ϕjR 6= 0, the corresponding chemical potential µ isgiven as,

µ =U

2(nL + nR), (16)

with nL,R = (0, 1, 2, ...). This means that nonzero Cor(1)⊥

appears in the vicinity surrounding (∆∗/U, µ∗/U) =(

nR − nL,12 (nL + nR)

)

with nL,R = (0, 1, 2, ...), see thebright spots in Fig. 4 (b) and (c). For a given chemicalpotential µ = 1

2 (nL + nR)U , a sequence of single-atomtunneling and interaction blockade takes place when thebias ∆ increases from negative infinity to positive infin-ity. The single-atom tunneling and interaction blockadein the BHL with fixed chemical potential is reminiscentof that of the quantized Bose-Josephson junction withstrong interaction [42, 43]. To the best of our knowledge,the single-atom tunneling and interaction blockade in theBHL have never been reported before.In the second column of Fig. 4, we show the phase

diagrams for the case of β = 1. The parallel SF stripes ofϕjL or ϕjR still appear but blur at the quasi-degenerateregions in the vicinity of (∆∗/U, µ∗/U). Different fromthe case of large β, ϕjL 6= 0 and ϕjR 6= 0 may coexistin some specific regions. Correspondingly, due to the

increase of J⊥, the area of nonzero Cor(1)⊥ surrounding

(∆∗/U, µ∗/U) extend.

In the first column of Fig. 4, we show the phase di-agrams for the case of β = 0.1. The strong intra-chain tunneling makes the occurrence of inter-chain intra-cluster single-atom tunneling more easy, the regions of

Cor(1)⊥ 6= 0 extend and merge into an entire area. Cor-

respondingly, due to the strong inter-chain hopping, theproperties of ϕjL and ϕjR change dramatically. The par-allel SF stripes are tailored and several avoided crossingsappear in the vicinity of (∆∗/U, µ∗/U). The avoided

crossings, which have Cor(1)⊥ 6= 0 and zero order parame-

ters (ϕjL = ϕjR = 0), correspond to the LI phases shownin Fig. 3 (a). This means that atoms may move freelybetween the two chains although there is no superfluidalong the chain direction.

IV. LANDAU-ZENER DYNAMICS

In this section, we analyze the many-body LZ dynam-ics in the BHL. In the many-body LZ process, the inter-chain energy bias ∆(t) is linearly swept from negative topositive or vice versa. The linear sweep of bias is de-scribed by ∆(t) = ∆0 +αt with ∆0 being the initial biasand α denoting the sweeping rate. In the first subsec-tion, we show how to apply the Gutzwiller mean-field tothe time-evolution problem of our BHL system. In thesecond subsection, we present the population dynamicsin the ground-state sweep and the inverse sweep, respec-tively. In the ground-state sweep, the initial state is theground state, while in the inverse sweep, the initial stateis the highest excited state.

ini ini finalinitial state , set , , t t t

ini ini,t t

ˆ ˆ, ( )j j jR jRb n t n

MF MF

MF

exp ( )

( ) ( )jU iH t

t t U t

finalt tY

N

( )jRn t

t t t

FIG. 5. The numerical simulation procedure for the time-evolution of Bose-Hubbard ladder system via our clusterGutzwiller mean-field method.

8

A. Time-evolution problem

The time-evolution obeys the Schrodinger equation

i~d

dt|Ψ(t)〉 = H(t) |Ψ(t)〉 , (17)

where H(t) is the original time-dependent Hamilto-nian (2). By applying the dynamical Gutzwiller mean-field method, the time-evolution is described by the dy-namical Gutzwiller equations

i~d

dt

∣

∣ΨGA(t)⟩

= HMF(t)∣

∣ΨGA(t)⟩

, (18)

where HMF(t) is the time-dependent mean-field Hamil-tonian and

∣

∣ΨGA(t)⟩

=∏

j |Ψ(t)〉j denotes the time-dependent Gutzwiller ansatz. Here, the single-clusterstate reads as

|Ψ(t)〉j =Nmax∑

N=0

N∑

m=−N

f(j)N,m(t) |N,m〉j . (19)

Similar to determining the ground states, the time-dependent mean-field Hamiltonian HMF(t) can be decou-pled as a sum of single-cluster Hamiltonians. Therefore,the dynamical Gutzwiller equations (18) can be simpli-fied to the single-cluster equations

i~d

dt|Ψj(t)〉 = HMF

j (t) |Ψj(t)〉 , (20)

with the time-dependent single-cluster Hamiltonian

HMFj (t) = −J‖

∑

σ,k=j±1

(

ϕkσ b†jσ + ϕ∗

kσ bjσ − Re[

ϕ∗jσϕkσ

]

)

−J⊥

(

b†jLbjR + h.c.)

+U

2

∑

σ

njσ

(

njσ − 1)

−∆(t)

2(njR − njL)− µ

∑

σ

njσ , (21)

in which the time-dependent order parameters are given

as ϕjσ(t) = 〈Ψj(t)|bjσ|Ψj(t)〉. Substituting Eq. (19) andEq. (21) into Eq. (20), one can obtain the following dif-ferential equations for the expansion coefficients

i~d

dtfN,m(t) =− J‖√

2φjL(t)

√N +mfN−1,m−1(t)−

J‖√2φjR(t)

√N −mfN−1,m+1(t)

− J‖√2φjL(t)

∗√N +m+ 2fN+1,m+1(t)−

J‖√2φjR(t)

∗√N −m+ 2fN+1,m−1(t)

− J⊥2

√N +m

√N −m+ 2fN,m−2(t)−

J⊥2

√N +m+ 2

√N −mfN,m+2(t)

+

[

U

4

(

N2 +m2 − 2N)

+∆(t)

2m− µN + J‖Re [ϕjL(t)

∗φjL(t) + ϕjR(t)∗φjR(t)]

]

fN,m(t), (22)

with the time-dependent order parameters

φjσ(t) = ϕj+1,σ(t) + ϕj−1,σ(t), (23)

ϕjL(t) =∑

N,m

√

N +m+ 2

2f(j)∗N,m(t)f

(j)N+1,m+1(t), (24)

ϕjR(t) =∑

N,m

√

N −m+ 2

2f(j)∗N,m(t)f

(j)N+1,m−1(t). (25)

By using the fourth-order Ronge-Kutta method, we sim-ulate the dynamics obeying Eq. (22). The flow chart forthe numerical procedure is shown in Fig. 5. Given theparameters J‖, J⊥, U , the initial bias ∆0, the sweepingrate α, and the initial state, the time-dependent order pa-rameters should be estimated by the instantaneous statesstep by step. That is, for a specific time step, based uponthe current state and the current order parameters, weneed estimate not only the time-dependent state but alsothe time-dependent order parameters for the next timestep.

B. Population dynamics

We consider two typical sweep processes: the groundstate sweep and the inverse sweep. In the ground-statesweep, the system is prepared in the ground state of allparticles in the lower chain, and the initial bias betweenleft and right chains is set as ∆0 = −50 and then thebias ∆(t) is linearly swept from ∆0 to −∆0 with thesweep rate α = −2∆0/T > 0. In the inverse sweep,the system is prepared in the highest excited state of allparticles in the higher chain, and the initial bias betweenleft and right chains is set as ∆0 = 50 and then thebias ∆(t) is linearly swept from ∆0 to −∆0 with thesweep rate α = −2∆0/T < 0. Here T is the total sweeptime. For convenience, we assume that the initial statefor both two sweep processes is the state of all atoms inthe left chain. To show the many-body LZ dynamics, wecalculate the transfer fraction nR(t) = NR(t)/N , whichis the fraction of the particles in the right chain at a giventime t. Obviously, the bias ∆(t) vanishes at time t = T/2,which corresponds to an instantaneous symmetric BHL.

9

0 0.2 0.4 0.6 0.8 1

1

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

/t T

( ) 25a ( ) 5b ( ) 1c

Ground-state Sweep Inverse Sweep

FIG. 6. (Color online) The many-body Landau-Zener dynamics in the Bose-Hubbard ladder for different sweeping rates |α|.Here, nR(t) stands for the transfer fraction, the cutoff of the maximum particle number is fixed as Nmax=6. The otherparameters are chosen as: |∆0| = 50, µ = 3, U = 0.5, J‖ = 0.25, J⊥ = 1. Red-dashed lines and blue-solid lines correspond tothe ground-state sweep and the inverse sweep, respectively. (a) For a large sweep rate, |α| = 25, the transfer fractions nR(T )for both the ground-state sweep and the inverse sweep are much smaller than 1. The difference between the the two sweeps issmall. (b) For an intermediate sweep rate, |α| = 5, the transfer fractions nR(T ) for the ground-state sweep increases to almost1, and nR(T ) for the inverse sweep is still far below 1. (c) For a small sweep rate, |α| = 1, the transfer fractions nR(T ) forthe ground-state sweep reaches 1, while nR(T ) for the inverse sweep is still below 1. This means that the ground-state sweepevolves adiabatically, while the inverse sweep evolves non-adiabatically.

If there are no intra-chain hopping and no on-site in-teraction, i.e. J‖ = 0 and U = 0, the physical picture forthe many-body LZ dynamics is as same as the one for theconventional two-level LZ problem. This means, the finaltransfer efficiency is given by the conventional LZ formulanR(+∞) = 1 − exp(−2πJ2

‖/~|α|) and there is no sig-

nificant difference between the ground-state and inversesweeps. However, taking into account the on-site inter-action and the intra-chain hopping, the many-body LZdynamics becomes very different from the conventionaltwo-level LZ problem. Below, we analyze the many-body LZ dynamics for the on-site interaction U = 0.5,the inter-chain hopping J⊥ = 1, the intra-chain hoppingJ‖ = 0.25, and different sweep rates α.

Independent on the sweep rate α, significant popula-tion transfers from the left chain to the right chain appeararound the time t = T/2. This significant populationtransfer between the two chains is caused by the avoidedlevel crossing in the vicinity of the bias ∆(t) = 0. How-ever, the transfer fraction nR(t) = NR(t)/N sensitivelydepends on the sweep rates, the physical parameters andthe initial states. In particular, for slow sweep rates,there appear significant difference of the final transferfraction for the ground-state sweep and the inverse sweep,see Fig. 6.

For a large sweep rate, |α| = 25, both the ground-state sweep (α = +25) and the inverse sweep (α = −25)are non-adiabatic, see Fig. 6 (a). The dynamics of theground-state sweep and the inverse sweep is very simi-lar. The transfer fraction nR(t) rapidly increase around∆(t) = 0 and then keep oscillates around a specific value.The final transfer fraction nR(T ) is much below 1 because

of the non-adiabatic evolution under fast sweeps.

For an intermediate sweep rate, |α| = 5, the non-adiabatic excitation in the ground-state sweep is not verysignificant, while the non-adiabatic excitation in the in-verse sweep is very significant, see Fig. 6 (b). After thesystem goes through the avoided level crossing regionaround ∆(t) = 0, the transfer fraction for the ground-state sweep (α = +5) is very close to 1 and its oscilla-tion amplitude is very small. While in the inverse sweep(α = −5), the final transfer fraction is much below 1 andthe corresponding oscillation amplitude is much largerthan the one for the ground-state sweep.

For a small sweep rate, |α| = 1, the ground-state sweepundergoes adiabatic evolution and but the inverse sweepstill show significant non-adiabatic excitations, see Fig. 6(c). In the ground-state sweep (α = +1), there is no sig-nificant oscillations in the transfer fraction and the finaltransfer fraction is almost the perfect limit nR(T ) = 1,which means that all particles in the left chain can becompletely transferred into the right chain. However, inthe inverse sweep (α = −1), the final transfer fraction isstill much below 1 and the oscillation amplitude is stillvery significant, which indicates that there still exist sig-nificant non-adiabatic excitations.

The adiabaticity breakdown in the inverse sweep quali-tatively explains the recent experimental observation [22,24]. The observed adiabaticity breakdown, which cannot be found in the conventional two-level LZ problem,is a result of the inter-particle interaction. Due to theinter-particle interaction, swallow-tail-shaped loop struc-tures [45, 46], which correspond to the macroscopic quan-tum self-trapping in mean-field models [47–55], may ap-

10

pear in the energy spectrum for our BHL system. Unlikethe conventional two-level LZ problem, whose energy-level structures for the ground state and the highest-excited state are similar, the energy-level structures forthe ground state and the highest-excited state of ourBHL system are very different. Because of their differ-ent energy-level structures, the ground-state sweep andthe inverse sweep show different adiabatic/non-adiabaticdynamics.

V. CONCLUSION AND DISCUSSION

In summary, we present a cluster Gutzwiller mean-fieldapproach to explore the static and dynamical behavior ofthe BHL, which can be experimentally realized by loadingBose atoms into a double-well optical superlattice poten-tial. In our mean-field treatment, the wavefunction of thewhole system is assumed in form of the Gutzwiller ansatz,the two sites in each double-well unit are packed as acluster and the inter-cluster hopping is decoupled by us-ing the conventional mean-field approximation. Throughimplementing the numerical self-consistent procedure, forboth unbiased and biased BHLs, we obtain the groundstates and give the phase diagram by calculating the or-der parameters.For an unbiased BHL, if the intra-chain hopping is

much stronger than the inter-chain hopping (i.e. β <<1), there appear several exotic loophole-shaped insula-tor regions of the half-integer filling numbers, which liebetween the conventional MI lobes of integer filling num-bers. As β increases, the loophole-shaped insulator re-gions gradually shrink and disappear. Differently, if theinter-chain hopping is much stronger than the intra-chainhopping (i.e. β >> 1), the unbiased BHL system can beregarded as two single BH chains and the correspondingphase diagram is almost as same as the one for a singleBH chain.For a biased BHL, single-atom tunneling and inter-

action blockade appear if the hopping terms are weakenough to be treated as perturbations. We present an an-alytical interpretation for the single-atom tunneling andinteraction blockade based upon the quasi-degeneracy ofdifferent Fock states for the considered cluster. If theinter-chain hopping is much stronger than the intra-chainone, there appear exotic LI phases with no superfluids

along the chain direction but nonzero inter-chain coher-ence.In further, we analyze the many-body LZ process of

the BHL, in which the inter-chain bias is linearly sweptfrom positive to negative or vice versa. We consider twodifferent sweeps: the ground-state sweep and the inversesweep. In the ground-state sweep, the initial state isthe ground state and the final transfer fraction can reach1 if the sweep rate is small enough. While in the in-verse sweep, whose initial state is the highest excitedstate, there still exist significant non-adiabatic excita-tions when the corresponding ground-state sweep obeysadiabatic evolution. The breakdown of adiabaticity inthe inverse sweep, which are well consistent with therecent experimental observations [22, 24], is a result ofthe swallow-tail-shaped loop structures induced by inter-particle interaction [45, 46].In recent, for ultracold atoms in optical lattices, arti-

ficial gauge fields have been realized by lattice shakingtechnique [56] or laser-induced tunneling [57]. The ar-tificial gauge fields, which allow one to generate spin-orbit couplings and effective magnetic fields, opens anew path to explore quantum Hall effect and topolog-ical phases of matters. Our cluster Gutzwiller mean-field approach can also be extended to investigate thebosonic ladders in the presence of an artificial magneticfield [26, 57–63], such as the observation of chiral cur-rents [57], the measurement of Chern number in Hofs-tadter bands [58, 63], and the two-leg Bose-Hubbard lad-der under a magnetic flux [26, 61]. In addition, our clus-ter Gutzwiller mean-field approach may also use to ex-plore the non-equilibrium dynamics of two coupled one-dimensional Luttinger liquids [64] and the dynamical in-stability of interacting bosons in disordered lattices [65].

ACKNOWLEDGEMENTS

H. Deng, H. Dai and J. Huang made equal contri-butions for this work. This work is supported by theNational Basic Research Program of China (NBRPC)under Grant No. 2012CB821305, the National Natu-ral Science Foundation of China (NNSFC) under GrantsNo. 11374375 and No. 11465008, and the PhD Pro-grams Foundation of Ministry of Education of China un-der Grant No. 20120171110022.

[1] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, andI. Bloch, Quantum phase transition from a superfluidto a Mott insulator in a gas of ultracold atoms, Nature(London) 415, 39 (2002).

[2] I. Bloch, Ultracold quantum gases in optical lattices, Nat.Phys. 1, 23 (2005).

[3] O. Morsch and M. Oberthaler, Dynamics of Bose-Einstein condensates in optical lattices, Rev. Mod. Phys.78, 179 (2006).

[4] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski,A. Sen (De), and U. Sen, Ultracold atomic gases in op-tical lattices: mimicking condensed matter physics andbeyond, Adv. Phys. 56, 243 (2007).

[5] I. Bloch, J. Dalibard, and W. Zwerger,Many-bodyphysics with ultracold gases, Rev. Mod. Phys. 80, 885(2008).

[6] S. Will, T. Best, U. Schneider, L. Hackermuller, D.Luhmann, and I. Bloch, Time-resolved observation of

11

coherent multi-body interactions in quantum phase re-vivals, Nature (London) 465, 197 (2010).

[7] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat-tore, Colloquium: Nonequilibrium dynamics of closed in-teracting quantum systems, Rev. Mod. Phys. 83, 863(2011).

[8] D. Chen, M. White, C. Borries, and B. DeMarco, Quan-tum Quench of an Atomic Mott Insulator, Phys. Rev.Lett. 106, 235304 (2011).

[9] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling,I. Cirac, G. V. Shlyapnikov, T. W. Hansch, and I. Bloch,TonksCGirardeau gas of ultracold atoms in an opticallattice, Nature (London) 429, 277 (2004).

[10] T. Kinoshita, T. Wenger, and D. S. Weiss, Observation ofa One-Dimensional Tonks-Girardeau Gas, Science 305,1125 (2004).

[11] S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm,and J. Schmiedmayer, Non-equilibrium coherence dy-namics in one-dimensional Bose gases, Nature (London)449, 324 (2007).

[12] T. Stoferle, H. Moritz, C. Schori. M. Kohl, and T.Esslinger, Transition from a Strongly Interacting 1D Su-perfluid to a Mott Insulator, Phys. Rev. Lett. 92, 130403(2004).

[13] X.-W. Guan, M. T. Batchelor, C. Lee, Fermi gases inone dimension: From Bethe ansatz to experiments, Rev.Mod. Phys. 85, 1633 (2013).

[14] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, andP. Zoller, Cold Bosonic Atoms in Optical Lattices, Phys.Rev. Lett. 81, 3108 (1998).

[15] J. K. Freericks, and H. Monien, Phase diagram of theBose-Hubbard Model, Europhys. Lett. 26, 545 (1994).

[16] F. Grusdt, M. Honing, and M. Fleischhauer, TopologicalEdge States in the One-Dimensional Superlattice Bose-Hubbard Model, Phys. Rev. Lett. 110, 260405 (2013).

[17] D. Muth, A. Mering, and M. Fleischhauer, Ultracoldbosons in disordered superlattices: Mott insulators in-duced by tunneling, Phys. Rev. A 77, 043618 (2008).

[18] B.-L Chen, S.-P Kou, Y. Zhang, and S. Chen, Quantumphases of the Bose-Hubbard model in optical superlat-tices, Phys. Rev. A 81, 053608 (2010).

[19] I. Danshita, J. E. Williams, C. A. R. Sa de Melo, andC. W. Clark, Quantum phases of bosons in double-welloptical lattices, Phys. Rev. A 76, 043606 (2007).

[20] C. Kollath, A. M. Lauchli, and E. Altman, Quench Dy-namics and Nonequilibrium Phase Diagram of the Bose-Hubbard Model, Phys. Rev. Lett. 98, 180601 (2007).

[21] C. Lee, Bose-Einstein Condensation of Particle-HolePairs in Ultracold Fermionic Atoms Trapped within Op-tical Lattices, Phys. Rev. Lett. 93, 120406 (2004).

[22] Y. Chen, S. D. Huber, S. Trotzky, I. Bloch, and E. Alt-man, Many-body LandauCZener dynamics in coupledone-dimensional Bose liquids, Nature Phys. 7, 61 (2011).

[23] W. Tschischik, M. Haque, and R. Moessner, Nonequilib-rium dynamics in Bose-Hubbard ladders, Phys. Rev. A86, 063633 (2012).

[24] C. Kasztelan, S. Trotzky, Y.-A. Chen, I. Bloch, I. P.McCulloch, U. Schollwock, and G. Orso, Landau-ZenerSweeps and Sudden Quenches in Coupled Bose-HubbardChains, Phys. Rev. Lett. 106, 155302 (2011).

[25] B. Pandey, S. Sinha, and S. K. Pati, Quantum phasesof two coupled XXZ spin chains: A DMRG study,arXiv:1406.5849v1 (2014).

[26] A. Keles and M. O. Oktel, Mott transition in a two-legBose-Hubbard ladder under an artificial magnetic field,Phys. Rev. A 91, 013629 (2015).

[27] D. S. Rokhsar and B. G. Kotliar, Gutzwiller projectionfor bosons, Phys. Rev. B 44, 10328 (1991).

[28] W. Krauth, M. Caffarel, and J.-P. Bouchaud, Gutzwillerwave function for a model of strongly interacting bosons,Phys. Rev. B 45, 3137 (1992).

[29] P. Buonsante, V. Penna, and A. Vezzani, Fractional-filling loophole insulator domains for ultracold bosons inoptical superlattices, Phys. Rev. A 70, 061603(R) (2004).

[30] P. Pisarski, R. M. Jones, and R. J. Gooding, Applicationof a multisite mean-field theory to the disordered Bose-Hubbard model, Phys. Rev. A 83, 053608 (2011).

[31] T. Mclntosh, P. Pisarski, R. J. Gooding, and E. Zaremba,Multisite mean-field theory for cold bosonic atoms in op-tical lattices, Phys. Rev. A 86, 013623 (2012).

[32] D. Yamamoto, I. Danshita, and C. A. R. S de Melo,Dipolar bosons in triangular optical lattices: Quantumphase transitions and anomalous hysteresis, Phys. Rev.A 85, 021601(R) (2012).

[33] D. Yamamoto, A. Masaki, and I. Danshita, Quantumphases of hardcore bosons with long-range interactionson a square lattice, Phys. Rev. B 86, 054516 (2012).

[34] D.-S. Luhmann, Cluster Gutzwiller method for bosoniclattice systems, Phys. Rev. A 87, 043619 (2013).

[35] S. F. Caballero-Benıtez, and R. Paredes, Phase diagramof Landau-Zener phenomena in coupled one-dimensionalBose quantum fluids, Phys. Rev. A 85, 023605 (2012).

[36] M. Singh, T. Mishra, R. V. Pai, and B. P. Das, Quan-tum phases of attractive bosons on a Bose-Hubbard lad-der with three-body constraint, Phys. Rev. A 90 013625(2014).

[37] J. Carrasquilla, F. Becca, and M. Fabrizio, Bose-glass,superfuid, and rung-Mott phases of hard-core bosons indisorderd two-leg ladders, Phys. Rev. B 83,245101(2011).

[38] P. Buonsante, V. Penna, and A. Vezzani, Analyticalmean-field approach to the phase diagram of ultracoldbosons in optical superlattices, Laser Phys. 15, 361(2005).

[39] P. Buonsante, V. Penna, and A. Vezzani, Fractional-filling Mott domains in two-dimensional optical super-lattices, Phys. Rev. A 72, 031602(R) (2005).

[40] D. Huerga, J. Dukelsky, and G. E. Scuseria, CompositeBoson Mapping for Lattice Boson Systems, Phys. Rev.Lett. 111, 045701(2013).

[41] J. Zhao, C. A. Jimenez-Hoyos, G. E. Scuseria, D. Huerga,J. Dukelsky, S. M. A. Rombouts and G. Ortiz, Compositefermion-boson mapping for fermionic lattice models, J.Phys.: Condens. Matter 26, 455601 (2014).

[42] C. Lee, L.-B. Fu and Y. S. Kivshar, Many-body quantumcoherence and interaction blockade in Josephson-linkedBose-Einstein condensates, EPL 81, 60006 (2008).

[43] P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger, M.Moreno-Cardoner, S. Folling, and I. Bloch, CountingAtoms Using Interaction Blockade in an Optical Super-lattice, Phys. Rev. Lett. 101, 090404 (2008).

[44] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D.S. Fisher, Boson localization and the superfluid-insulatortransition, Phys. Rev. B 40, 546 (1989).

[45] B. Wu and Q. Niu, Nonlinear Landau-Zener tunneling,Phys. Rev. A 61, 023402 (2000).

[46] J. Liu, L. B. Fu, B. Y. Ou, S. G. Chen, D. I. Choi, B. Wu,and Q. Niu, Theory of nonlinear Landau-Zener tunneling,

12

Phys. Rev. A 66, 023404 (2002).[47] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy,

Quantum Coherent Atomic Tunneling between TwoTrapped Bose-Einstein Condensates, Phys. Rev. Lett.79, 4950 (1997).

[48] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy,Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, oscillations,and macroscopic quantum self-trapping, Phys. Rev. A59, 620 (1999).

[49] L. M. Kuang and Z. W. Ouyang, Macroscopic quantumself-trapping and atomic tunneling in two-species Bose-Einstein condensates, Phys. Rev. A 61, 023604 (2000).

[50] C. Lee, W. H. Hai, L. Shi, X. W. Zhu, and K. L. Gao,Chaotic and frequency-locked atomic population oscil-lations between two coupled Bose-Einstein condensates,Phys. Rev. A 64, 053604 (2001).

[51] C. Lee, W. H. Hai, L. Shi, and K. L. Gao, Phase-dependent spontaneous spin polarization and bifurcationdelay in coupled two-component Bose-Einstein conden-sates, Phys. Rev. A 69, 033611 (2004).

[52] M. Albiez, R. Gati, J. Folling, S. Hunsmann, M. Cris-tiani, and M. K. Oberthaler, Direct Observation of Tun-neling and Nonlinear Self-Trapping in a Single BosonicJosephson Junction, Phys. Rev. Lett. 95, 010402 (2005).

[53] R. Gati and M. K. Oberthaler, A bosonic Josephson junc-tion, J. Phys. B: At. Mol. Opt. Phys. 40, R61 (2007).

[54] C. Lee, Universality and Anomalous Mean-Field Break-down of Symmetry-Breaking Transitions in a CoupledTwo-Component Bose-Einstein Condensate, Phys. Rev.Lett. 102, 070401 (2009).

[55] C. Lee, J. Huang, H. Deng, H. Dai, and J. Xu, Nonlin-ear quantum interferometry with Bose condensed atoms,Front. Phys. 7, 109 (2012).

[56] J. Struck, M. Weinberg, C. Olschlager, P. Windpassinger,J. Simonet, K. Sengstock, R. Hoppner, P. Hauke, A.Eckardt, M. Lewenstein, and L. Mathey, Engineering

Ising-XY spin-models in a triangular lattice using tun-able artificial gauge fields, Nat. Phys. 9, 738 (2013).

[57] M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B.Paredes, and I. Bloch, Observation of chiral currents withultracold atoms in bosonic ladders, Nat. Phys. 10, 588(2014).

[58] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J.T. Barreiro, S. Nascimbene, N. R. Cooper, I. Bloch, andN. Goldman, Measuring the Chern number of Hofstadterbands with ultracold bosonic atoms, Nat. Phys. 11, 162(2014).

[59] M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky,Y. -A. Chen, and I. Bloch, Experimental Realization ofStrong Effective Magnetic Fields in an Optical Lattice,Phys. Rev. Lett. 107, 255301 (2011).

[60] S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P.McCulloch, U. Schollwock, and T. Vekua, Sponta-neous increase of magnetic flux and chiral-current re-versal in bosonic ladders: Swimming against the tide,arXiv:1504.0656v1 (2015).

[61] A. Tokuno and A. Georges, Ground states of aBoseCHubbard ladder in an artificial magnetic field:field-theoretical approach, New J. Phys. 16, 073005(2014).

[62] X. Li, A. Paramekanti, A. Hemmerich and W. V. Liu,Proposed formation and dynamical signature of a chiralBose liquid in an optical lattice, Nat. Commun. 5, 3205(2014).

[63] D. Hugel, and B. Paredes, Chiral ladders and the edgesof quantum Hall insulators, Phys. Rev. A 89, 023619(2014).

[64] L. Foini, and T. Giamarchi, Nonequilibrium dynamicsof coupled Luttinger liquids, Phys. Rev. A 91, 023627(2015).

[65] P. Buonsante, L. Pezze, and A. Smerzi, Interactingbosons in a disordered lattice: Dynamical characteriza-tion of the quantum phase diagram, Phys. Rev. A 91,031601(R) (2015).