Atom-pair tunnelling-induced quantum phase transition and scaling behaviour of fidelity...

8/3/2019 Atom-pair tunnelling-induced quantum phase transition and scaling behaviour of fidelity susceptibility in the extended boson Josephson-junction model

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IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 025101 (8pp) doi:10.1088/0953-4075/44/2/025101

Atom-pair tunnelling-induced quantumphase transition and scaling behaviour offidelity susceptibility in the extendedboson Josephson-junction model

J-L Liu and J-Q Liang

Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, Peoples

Republic of China

E-mail: [email protected]

Received 14 September 2010, in final form 11 November 2010

Published 23 December 2010

Online at stacks.iop.org/JPhysB/44/025101

Abstract

In this paper we theoretically investigate the nonlinear tunnelling of two weakly linked

BoseEinstein condensates in a double-well trap in a strong interaction regime with the

two-body interaction extended to neighbouring lattice sites. As a consequence the boson

Josephson-junction tunnelling depends on the atomatom interaction and the total atom

number as well. A crucial atom-pair tunnelling term, obtained directly from the extension,

results in significant energy spectrum corrections and an abrupt change of the ground state

viewed as quantum phase transition. An atom-number oscillation state with a tunable relative

phase between two BoseEinstein condensates predicted for the first time is seen to be adegenerate ground state in an interaction region. The quantum phase transitions between

degenerate and non-degenerate ground states driven by the atom-pair tunnelling are analysed.

We show the scaling behaviours and the critical exponents of fidelity susceptibility, which can

classify the universality.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The BoseEinstein condensates (BECs) in a double-well trap

were previously suggested [1] for the investigation of themacroscopic quantum tunnelling analogous to the Josephsonjunction [2], where atoms oscillate through the central barrierwith coherent phase difference of macroscopic wavefunctionsof the two BECs. Unlike the single-particle Josephson-junction model some novel and unexpected quantum dynamicphenomena [36] are explored in the BEC system due to thetwo-body interaction of tunnelling atoms. It was demonstratedthat the tunnelling dynamics of BECs in a weak link can bedescribed by a nonrigid pendulum of momentum-dependentlength with the dynamic equations analogous to Josephsonoscillations only for the case of a weak two-body interaction.For particular initial conditions and larger interaction, the

nonzero population difference is approached, which iscalled macroscopic quantum self-trapping [46] and can be

described by a rotating pendulum [7]. It has become a hot

research topic to study the tunnelling dynamics in the strong

interaction regime both theoretically and experimentally

[8, 9] since the tunnelling theory of the many-body interactingsystem is still lacking.

The system of cold atoms in optical lattices has become

a test ground of quantum mechanical principles in many

extraordinary aspects and the new prospective regime since

the strength of the atomatom interaction can be manipulated

using modern experimental techniques such as Feshbach

resonance, which can change the scattering length almost

arbitrarily. To date, the ultracold atomgas clouds in optical

lattices are only studied based on the well-known and widely

applied BoseHubbard Hamiltonian, which can well describe

the Josephsonoscillations and nonlinear self-trapping of BECs

in a double-well trap [10] with weak atomatom interactions.However, the strong interaction may fundamentally alter the

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J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 025101 J L Liu and J-Q Liang

tunnel configuration and result in a correlated tunnelling,

which was explored most recently in ultracold atoms [8, 9]. It

was shown that the boson Josephson-junction model (BJJM)

should be modified by a nonlinear interaction-dependent

tunnelling term in the case of a large number of atoms [11],

which leads to a considerable contribution to the tunnelling

effect. The tunnelling dynamics of a few atoms loaded in adouble-well trap has been studied by varying the interaction

strength from a weak to strong limit and it is shown for the

two-atom case that the tunnelling character changes from

Rabi oscillation to an atom-pair co-tunnelling process with

increasing interaction [8, 9]. A direct observation of the

correlated tunnelling was reported recently [8] and theoretical

analysis hasalso been presented in terms of two-body quantum

mechanics [9]. In a previous paper we pointed out that the

well-known BoseHubbard Hamiltonian, which is valid in

a relatively weak interaction regime, is not able to describe

the dynamics of atom-pair tunnelling and should be extended

in the strong interacting regime to include the atomatom

interaction of neighbouring lattice sites [12]. A peculiar

atom-pair hopping term resulted from the common two-

body interaction explains very well the recently reported

experimental observation of correlated tunnelling [12].

In this paper, we adopt the modified BoseHubbard

model as in [12] to study the nonlinear tunnelling effect

on the energy spectrum and dynamics of cold atoms in the

strong interaction region. A peculiar but fundamentally

important phenomenon occurs which states that atoms in

optical lattices with strong repulsive interactions can tunnel

together as an atom pair through the inter-well barriers.This apparently contradictory effect, however, is a natural

result of two-body interaction in the second quantization

formulation of thequantum field theoryunlike theCooper pairs

in the superconductivity resulted from attractive interaction

between two electrons by the exchange of phonons. In

the double-well case the Josephson-junction model has to

be extended to include the atom-pair tunnelling, which we

called the extendedbosonJosephson-junction model(EBJJM).

New dynamic states induced by the nonlinear tunnelling

effect, particularly, a degenerate ground state of atom-number

oscillation with a tunable relative phase are found in the

strong interaction regime. Quantum phase transitions (QPTs)from non-degenerate to degenerate ground states driven by

the atom-pair tunnelling and scaling behaviours are analysed

based on the fidelity theory. We also show the analogy

between the EBJJM and the LipkinMeshkovGlick (LMG)

model.

This paper is organized as follows: in section 2, the

EBJJM in a symmetric double-well trap is derived in the strong

interaction regime. In section 3 we study the macroscopic

eigenstates and energy spectrum of the EBJJM by means of

the fixed points of phase-space portraits. Section 4 is the

main part of the paper devoted to the atom-pair tunnelling-

induced QPT and critical behaviours. Finally we summarize

and discuss our results in section 5.

2. EBJJM beyond the on-site approximation oftwo-body interaction

BECs in a double-well potential exhibit novel phenomena such

as atom-number self-trapping, the phase coherence oscillation

and so on, which were observed in recent experiments.

Generally, this system can be described by the BoseHubbardmodel, which, however, is valid in the weak interaction

region. For strong interaction we derive a Hamiltonian in the

frameworkof second quantization for a symmetric double-well

potential, which can be written as

H = H0 + Hintwhere

H0 =

(x)+ d

2

dx2+ vtrap(x)

(x) dx

Hint =1

2

(x1)

+(x2)+U (|x1 x2|)(x2)(x1) dx1 dx2,

and (x) denotes the field operator of the condensate. Inthe weak coupling case of two condensates, which can be

guaranteed by a high barrier making the energy-level space

between the first and second excited states much larger than

the energy splitting of the ground state, thus we can adopt

the two-mode approximation to expend the field operator in

terms of variational wavefunctions 1(x) and 2(x) localized

respectively in the first and second wells [4, 10, 11, 13]:

(x,t) = a1(t)1(x) + a2(t)2(x).Taking the atomatom interaction between neighbouring wells

into account besides the usual on-site one we obtain the

Hamiltonian for a symmetric double-well potential [12] with

a constant energy setting to zero:

H = JN

a1a2 + a1a

2

+

U0

2

2i=1

ni (ni 1)

+ 2U2n1n2 +U2

2

a

1a

1a2a2 + a1a1a

2a

2

(1)

where ai

a

i

denotes the boson annihilation (creation)

operator in the ith well and the total atom-number operator

is a conserved quantity n1 + n2 = N. The hopping couplingconstant

JN

=[J

U1(N

1)] (2)

depends on the total atom number N and modifies the

Josephson-junction-like single-atom tunnelling J.

The coupling parameters are defined as the usual overlap

integrals:

J =

dx1(x) d2

dx2+ vtrap(x)

2(x)

= dx2(x) d2dx2 + vtrap(x)1(x)U0 = g

dx41(x) = g

dx42(x)

U1 = g

dx31(x)2(x)

= g

dx32(x)1(x)

U2 = g dx21(x)22(x).

(3)

Obviously, here we consider only the -function interactionpotential for U (|x1 x2|) with g = 4 ash2/m, and as being

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the s-wave scattering length. U0 is the usual on-site interaction

strength and U1, U2 are the intersite interaction strengths. The

last part of the Hamiltonian (equation (1)) with the coupling

constant U2 describes a novel atom-pair tunnelling process.

The dynamics of condensate described by the field

operator (x,t) can be investigated by the Heisenberg

equation in Heisenberg picture. At zero temperature andfor a large atom number N we can use C-numbers a1(a2)

instead of the corresponding quantum operators (mean-field

approximation) and then the Heisenberg equations for the C-

numbers ai and a

i corresponding to the probability amplitudes

in the two wells can be written as a matrix equation [ 14]

i

t(t) = H(t) (4)

with the state (t) = [a1 a2]T and the effective matrix-Hamiltonian given by

H

= U0|a2|2 (JN U2a1 a2)

(JN U2a1a2 ) U

0|a1|2 , (5)

where we have normalized the parameters as U0 (U0

2U2)/N,U1 U1/N,U2 U2/N,JN = J U1(N 1)/N.

The nonlinear eigen-equation (4) can be solved

semiclassically. To this end we assume a1 = n1 ei1 ,a2 = n2 ei2 (1 and 2 being the phases of two condensates).The relative phase = 2 1 and the average of atom-number population difference = (n2 n1)/N are regardedas the canonical conjugate variables in EBJJM. Substituting

the conjugate variables and into equation (4), we obtain

the canonical equations

t= 2JN

1 2 sin() sin(2)U2(1 2),

t= JN 2

12cos() U0 + U2 cos(2).

(6)

The equivalent classical Hamiltonian corresponding to the

above canonical equations is seen to be

Hc = 2cos()JN

1 2 1

2cos(2)U2(1 2) 12 U

0

2, (7)

in which the second term is totally new compared with the

effective Hamiltonian of the BJJM.

3. Fixed points of the phase-space portraits andeigenvectors

It has been demonstrated that the eigenvectors of the nonlinear

equation (6) correspond to extreme energy or the fixed points

in the phase-space portraits of the classical Hamiltonian Hc[15]. In this paper we show that the EBJJM gives rise to more

eigenvectors than that of the BJJM [16].

Since Hc is a conserved quantity Hc(,) = E, thephase-space trajectories are constrained on the contours of

the constant energy E called phase-space portraits. The

intersite interactions result in a significant modification of the

phase-space trajectories, although parameters U1 and U2 are

much smaller compared with the on-site one U0, which asa matter of fact can be evaluated roughly with the Gaussian

(A) (E) (I)

(B) (F) (J)

(C) (G) (K)

(D) (H) (L)

Figure 1. Phase-space portraits for different values of the couplingconstant J = 0.08, J = 0.06, J = 0.04 and J = 0.02 (from top tobottom). Fixed points are marked respectively with p1, p2, p3, p4

and p5. Plots (A)(H) are for the EBBJM and plots (I)(L)correspond to the BBJM.

Wannier functions and function potential [12] such thatU2 2U0,U1 3/2U0, where = exp( 2s1/2/4), s =/Er with and Er being the well depth and lattice recoil

energy respectively. Using experimental values of and Erwe have evaluated the parameter values as U2 = 0.02, U1 =U

3/42 = 0.053 in the energy scale of U0 = 1. The variation

range of the coupling constant J is assumed from 1 to 0.01.

With the standard numerical procedure, phase-space portraits

of both the EBJJM (U2 = 0.02, U1 = 0.053) and the BJJM(U2

=U1

=0) in the strong atomatom interaction region

are shown in figures 1(A) (H) and (I)(L) respectively in the

coupling constant range 0 < J < U 1 + U2, where the fixed

points are marked respectively with p1, p2, p3, p4 and p5.

3.1. Fixed points and eigenstates of the BJJM

The fixed points of phase-space portraits are the points of

extreme energy corresponding to eigenvectors of the nonlinear

system [15]. Referring to the canonical equation (6), the fixed

points obtained by solving the extreme-value equation,

2JN1 2 sin() sin(2)U2(1

2)

=0,

JN 212

cos() U0 + U2 cos(2) = 0, (8)

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(a) (b)

(c)

Figure 2. Adiabatic energy eigenvalues of eigenvectorsp1, p2, p3, p4 and p5 as a function of the coupling constant J

varying from 0.01 to 0

.1 (a) and 0

.1 to 1 (b) for the EBJJM. Thelower panel (c) is the corresponding plot for the BJJM as a

comparison.

are markedwith p1, p2, p3, p4and p5 respectively in figure 1.

The phase-space portraits of the BJJM are depicted in

figures 1(I)(L) in which we find two elliptic fixed points p2

and p4 in thestrong interaction region (from J = 0.02to0.08).The elliptic fixed point is a true energy extremum with finite

frequency and is able to follow adiabatically with the variation

of the controlling parameter J. In our case p2 indicates the

zero average of atom-number population difference and zero

relative phase as well, while p4 corresponds to the -phase

self-trapping. In the eigenstate p2 atoms are equally populatedin the two wells showing the property of insulator because

of the strong interaction. The energy-eigenvalue variation

with respect to J is given in the lower panel of figure 2(c),

where we see that p2 (solid line) is the ground state in the

whole parameter region. The -phase self-trapping (p4) is an

excited state of highest eigenenergy existing only in the strong

interaction region when J < 0.5 (dashed line in figure 2(c)).

The hyperbolic fixed point p1, which is a saddle point of

energy corresponding to unstable eigenstate, indicates the zero

average of atom-number population difference with a relative

phase . The energy eigenvalue ofp1 coincides with that of

p2 at J = 0 and p4 at J 0.5 showing a two-fold degeneracyat these points. However, the ground state does not have abrupt

change and there is no QPT at all in the BJJM.

3.2. Eigenvectors of the EBJJM and abrupt change of the

ground state

The eigenvectors of the EBJJM are analytically solved

and compared with the BJJM showing a great difference.

Obviously, in the less strong interaction region when J >

U1 + U2 0.073, three eigenvectors p2 (ground state),p1(first excited state), p4 ( -phase self-trapping) and their

corresponding energies are seen to be almost the same

as in BJJM [16], as shown in figures 1(A), (E), (I) andfigures 2(b), (c). However in the stronger interaction region

when 0 < J < U 1 + U2, there are novel and significant

corrections of the energy spectrum and the structure of phase-

space portraits. In the interaction region (J = 0.060.04),we find a new elliptic fixed point p5 (besides p2, p1 and

p4 ) surrounded by periodic orbits, which being a ground

state in this parameter region (see figure 2(a) dashed line)

corresponds to a state of atom-number oscillation with zeroaverage of population difference. An interesting observation

is that the relative phase of the stationary state p5, which

does not exist in the BJJM, can be tuned by varying the

coupling constant J seen from figures 1(A)(C), where the

relative phase of p5 varies from 0 to . The ground

state p2 of the BJJM gradually changes to a hyperbolic

point in the EBJJM along with the increase of interaction

or decrease of J (see figures 1 (B)(D)). The unstable

hyperbolic point p1 in the BJJM becomes an elliptic fixed

point and finally coincides with p5 (see figures 1 (C), (D)).

A new stablestate(ellipticfixed point) p3 of the EBJJM,which

indicates the zero-phase self-trapping (figures 1 (G), (H)) has

no corresponding state in the BJJM. From figures 2(a), (c) it is

seen that in the super-strong interaction region ,when Jis less

than the critical value marked by T1, p1 becomes the ground

state (p2 is the first excited state) while between the critical

points T1 and T2 the ground state is p5. Above the critical

point T2 the ground state becomes p2. The QPT takes place

at the critical points T1(J = U1 U2) and T2(J = U1 + U2).Moreover the phase-space portraits of the EBJJM display

a number change of fixed points or the equivalent eigenvectors,

which increases from 3 (p1, p2, p4; figures 1(A), (E)), in

the region U1 + U2 < J < U 1 + (U0 U2)/2, to 4 (p1,

p2, p4 or p3 and p5; figures 1(B), (F), (C), (G)) in the region

U1 U2 < J < U 1 + U2. When the atomatom interactionbecomes super strong in the region 0 < J < U 1 U2 , thenumber of eigenvectors decreases again to 3, however with

the -phase self-trapping p4 replaced by the zero-phase self-

trapping p3 (see figures 1(D), (H)). Two new fixed points p3

and p5, which are stable states, are seen to be generated by

the atom-pair tunnelling.

In the BJJM the ground state p2 describes the atom-

number oscillation between two wells and the oscillation

amplitude is gradually squeezed with the increase of

interaction seen from figures 1(J)(L) attending to a stable

state of equal atom-number population in the two wells, since

the single-atom tunnelling is suppressed and the strong atomatom interaction forces atoms equally populated in the two

wells. However the situation is very different in the EBJJM,

where the elliptic fixed point p5, which is a gapless ground

state between critical pointsT1 and T2 (see thenext section) and

coincides with p1 at the critical point T1. p1 becomes a stable

atom-number oscillation state in the super-strong interaction

region (below T1) resulted by the atom-pair tunnelling. It is

found that the relative phase of the oscillation state can be

tuned by adiabatic variation of the parameterJin a wide range.

To verify the robust phase control we have numerically solved

equation (6) with the fourth- and fifth-order step-adaptive

RungeKutta algorithm to find the time evolution of conjugate

variables , near the elliptic point p5, which is expected tobe able to follow the slowly varying parameter J compared

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with the fundamental frequency of periodic orbits around the

fixed point. These states with the tunable phase by adiabatic

variation of the parameter J, can be used to realize the qubit in

quantum computation.

4. Quantum critical scaling of fidelity susceptibility

The atom-pair tunnelling in the EBJJM is the most essential

difference compared with the usual BJJM [16], which leads to

the structure change of the ground state spectrum viewed as

QPT. We now turn to the full quantum mechanical calculations

to demonstrate the QPT and critical behaviour at the critical

points. The critical points of QPT T1 and T2 in figure 2(a)

possess the same critical behaviour illustrated in terms of

the role of fidelity, which is a measure of similarity between

states useful in quantum-information theory [17] and quantum

critical phenomena [18]. A dramatic change in the ground-

state structure around the quantum critical point results in a

great fidelity difference of the two ground states on both sidesof the critical point. The fidelity denoted by F is defined as

the overlap between two ground states with an infinitessimal

variation of parameter strength (J here) such as |0(J ) and|0(J + J ) [19]:

F (J , J ) = |0(J )|0(J + J )|. (9)The Hamiltonian (equation (1)) of the quantum many-body

system can be divided into two parts

H = H0 + JNH

where H

= a

1a2 + a1a

2 is the driving Hamiltonian with anormalized coupling strengthJN and the rest partH0 describesthe atomatom interaction only. Assume that the Hamiltonian

Hhas orthogonal eigenstates

H|n(J ) = En|n(J ).With an infinitesimal variation of the parameter J J + J,the ground state |0(J + J evaluated by means of theperturbation method is obtained up to the second order as [19].

Substituting the state |0(J + J ) into the fidelity definition(equation (9)) and noting the orthogonality of the eigenstates

m(J )|n(J ) = mn we obtain up to the second order ofJ

F2(J,J) = 1 J2 n=0

|Hn0|2

(En E0)2, (10)

from which we see that the coefficient of J2 actually defines

a response of fidelity to the infinitesimal variation of J. From

this point of view, the concept of fidelity susceptibility (FS)

[19]

F

(J ) = limJ0

2 ln F (J + J )J2

is introduced and can be evaluated as [19]

F (J ) = n=0 |

n(J )

|H

|0(J )

|2

(En E0)2 . (11)

(a)

(b)

(c)

Figure 3. Energy gap E between ground and the first excitedstates as a function of the coupling constant J(a), relative-phase versus J(b) and F versus J(c) for Jvarying from 0 to 0.08 andatom number N

=60, 80, 100, respectively.

Fora finite atom numberN, theHamiltonian(equation (1))

can be exactly diagonalized with the basis vectors of the atom-

number state |m, Nm, in which m denotes the atom numberpopulated in the well-1, and thus the eigenstates

|n(J ) =N

m=0c(n)m |m, N m (12)

along with the eigen-energy En are obtained. So we are able

to evaluate numerically the FS in relation with the QPT.

Seen from section 3, the -phase-oscillation ground state(elliptic fixed point p1) adiabatically changes into the ground

state p5 with the increase of J at the critical point T1 and

continuously shifts to the ground state p2 at the critical point

T2. As a comparison with the FS the level space between the

ground- and first-excited states (E = E1 E0) is plottedin figure 3(a) for total atom number N = 60 (solid line),80 (dashed) and 100 (dotted) respectively. The new ground

state p5 between the two critical points T1 and T2 is doubly

degenerate in the limit N , whereas for the finite N,the level space is nonzero but decreases with the increase of

N (see figure 3(a)). The level space below T1 (EBJJM non-

degenerate phase) and above T2 (BJJM non-degenerate phase)

is almost symmetric however in the super-strong region theoscillation is induced by the atom-pair tunnelling while in the

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(a) (b)

Figure 4. Jm versus N(a), and F(Jm) versus Nwith various U2(b).

Table 1. Scaling exponent at various U2 obtained by sampling system size in different regions.

U2 0.01 0.02 0.03 0.04 0.05 0.06

(N [100, 200]) 1.232 11 1.258 65 1.270 95 1.277 69 1.2807 1.285 19(N [200, 300]) 1.269 81 1.285 25 1.291 47 1.295 31 1.299 88 1.299 78

BJJM non-degenerate phase it is resulted by the single-atom

tunnelling.

In the BJJM we have a non-degenerate ground state only.

The coherent phase difference shown in figure 3(b) is nothing

but the angle of the correlation function of the two wells

[8, 12]:

= angle0|a2a1|0

= angle

N1m=0

(m + 1)(N m)c(0)m c(0)m+1

in the ground state p5 obtained from the eigenstate |0(J )in equation (12), which continuously changes from to zero

in the interaction region between T1 and T2 with vanishing

level space. The phase-difference tends to the classical result

= arccos

JU1U2

obtained from equation (8) for N

and U1 U2 < J < U1 + U2.The FS curves as a function of J in figure 3(c) for atom

numbers 60 (solid), 80 (dashed) and 100 (dotted) show two

sharp peaks at the two critical points. The coupling constant

corresponding to the maximum value of FS denoted by Jm isdepicted as a function of the total atom numberNin figure 4(a),

where the Jm N curve ( solid line) to fit the numerical data(squares) can be described formally by the equation

Jm = ANB + C (13)with A = 0.116 72, B = 0.64 and C = 0.033 46. The criticalpoint

Jc limN

Jm = C = 0.033 46 (14)is obtained in the thermodynamic limit, N . In thevicinity of the critical point the scaling behaviour of Jm can

be found from equations (13) and (14), and is seen to be

characterized by the power-law decay in the large N:Jc Jm N (15)

with the exponent being = B = 0.64. With the variation ofU2 the exponent does not change, and neither does the criticalpointJc which follows the classical result Jc = U1 U2.

In theprevious analysis of critical phenomenaof fidelity, ithas been pointed out that the intensive FS scales in the vicinityof the critical point Jc generally like [20, 21]

F

(J )

1

(J Jc)

(16)

with being the corresponding exponent to be determined.The maximum value of FS at Jm for a finite system scalesfollowing the power-law

F(Jm) N. (17)Figure 4(b) is the plots of the maximal FS as a function of Nfor various U2 and the power-law exponent is extracted fromthe slope of the straight line, which fits exactly the numericaldata of F S(Jm) in the logarithmic scale. The correspondingvalues of the exponent for different U2 are obtained by theresults shown in table 1, which vary slightly in the finite regionofN. While in a larger scaling region the power-law exponent

can be estimated numerically as a universal constant value 1.31, which does not depend on the parameter U2. TheFS becomes divergent at the critical point as N ,demonstrating a Landau-type transition.

Following [20, 21] we introduce a compact functionto include the above two asymptotic behaviours given inequations (16) and (17):

F(J ) =G

Q(J Jm) + N(18)

where G 102 in this case is a constant, and Q is a nonzerofunction of J independent of the atom number N. From thecompact function (equation (18)), FS is seen to be a universalfunction of the driving parameter N (J

Jm), such that

F(Jm) F(J )F(J )

= f (N (J Jm)) (19)

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(a) (b)

Figure 5. The finite-size scaling analysis for system sizes N = 50, 100, 150, 200, 250, respectively. f(the rescaled FS) as a function ofN (J Jm) with = 0.64.

Figure 6. System-size analysis off as a function ofN (J Jm) with = 0.64 for U2 = 0.01, 0.03, 0.05 from the top to bottom panelsrespectively.

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where is a critical exponent and f(x) is an even deviation

function around x = 0. FS F(J ) as a function of J isplotted in figure 5(a) for atom numbers 60, 70, 80, 90, 100

respectively. Figure 5(b) is the plots of the universal function

f (N (J Jm)) for the same atom numbers as in figure 5(a)(i.e. N = 60, 70, 80, 90, 100), which fall into a single curvefor all the different atom numbers with the critical exponentdetermined as = 0.64. For different values of U2 =0.01, 0.03, 0.05, the plots of the universal function also fall

into a single curve with the atom number from 50 to 250,

as shown in figure 6. It is seen that the critical exponent

0.64 does not depend on the parameter U2. Combiningequation (18) and (19), a relation among the three exponents

can be found as

=

2. (20)In the usual study of scaling behaviour for the lattice model,

N is regarded as lattice size, i.e. the total number of lattice

sites, while in our system of the BEC in a double-well trap N

is the total atom number. Although the physical systems are

quite different the Hamiltonian is actually of the same type. To

this end we introduce the pseudoangular momentum operators

[12] defined by

Sx =1

2

a

1a2 + a1a

2

,

Sy =1

2i

a

1a2 a1a2

,

Sz =1

2

a

1a1 a2a2

with the total angular momentum S

2

=N

2N

2 + 1

; thus theHamiltonian (equation (1)) can be written as

H = 2JNSx + K1S2z + K2S2x (21)with the parameters given by K1 = U0 U2, and K2 = 2U2[12]. This is an LMG-typemodel Hamiltonian [11, 12, 22, 23],

in which theadditional K2-term compared with theBJJM is due

to atom-pair tunnelling and gives rise to the QPT in the strong

interaction region. From the viewpoint of the coupled spin

chain we regard Sk =N

i=1 i

k

2(k = x,y,z), as collective

spin with N being the total number of lattice sites. Thus the

finite-size scaling has obvious physical meaning. In our case

the FS is always an intensive quantity, which is different from

the LMG model considered in [22], with the relation amongthree exponents (equation (20)) being always satisfied around

the critical points of QPT from non-degenerate to degenerate

phases. Thus, the universality class can be characterized by

the critical exponents of the FS [21].

5. Conclusion

We show that the atom-pair tunnelling in the EBJJM for two

weakly coupled BECs in the strong atomatom interaction

regime leads to crucial modifications of the energy spectrum

and dynamics as well [4, 6], while in the weak interaction

case the effects of the two-body interaction of nearest-neighbouring sites are negligibly small and the results of

the EBJJM reduce to that of the well-known BJJM [16]. In

the exactly solvable system of BECs in a symmetric double-

well potential, the atom-pair tunnelling-induced QPTs from

non-degenerate to degenerate phases are demonstrated. We

have presented explicitly the finite-size scaling behaviours and

critical exponents of FS which can be used for classification of

the universality in the quantum critical phenomena. Moreover,the new degenerate ground state, i.e. atom-number oscillation

with tunable relative phase between two BECs may be used to

realize qubit in quantum computation.

Acknowledgments

This work was supported by National Nature Science

Foundation of China (grant no 10775091).

References

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