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No. 5 Symmetry Breaking Ground States of Bose–Einstein Condensates in 1D Double Square Well and Optical Lattice Well 815

where Φ(r, t) is the condensate wave function, m is the

mass of a single bosonic atom and N the number of

atoms in the condensate. g = 4πh2a/m is the strength

of interatomic interaction, with a the atomic scattering

length. The normalization condition of the wave function

is dr

|Φ(r, t)

|2 = 1. In steady state, we can take the

equation form as

− h2

2m∇2 + V (r ) + gN |Φ(r )|2 − µ

Φ(r ) = 0 . (2)

In 1D case, without loss of generality we adopt the sta-

tionary NLSE with a potential as the form

[−∂ 2x + V (x) + η|Φ(x)|2]Φ(x) = µΦ(x) , (3)

where µ is the chemical potential, and η is the nonlinearity

parameter, which is proportional to the number of atoms

and the s-wave scattering length. All quantities in Eq. (3)

are dimensionless. Strict solutions of the GP equation are

not possible in general, so we have chosen to investigatethe infinite square well with symmetrically placed finite

rectangular potential barrier. The potential is of the form

V (x) =

∞ , |x| ≥ a ,

V 0 , |x| ≤ b ,

0 , b < |x| < a .

Double-well traps created in experiments usually have

Gaussian barrier, but the qualitative behavior of the sta-

tionary ground states of such wells will be the same as

discussed in this paper for a double square well.

We know the analytic standing-wave solutions of the

Schrodinger equation without the barrier and nonlinearterm

φn(x) = 1√

a sin

nπ

2a(x + a)

. (4)

When the barrier potential and the nonlinear term are

taken into account, the wave functions will be altered. We

can take the wave function Φ(x) as a linear combination

of φn(x)

Φ(x) =

nmaxn=1

C nφn(x) . (5)

The potential and nonlinear terms will take the matrix

form

V mn =

a−a

φ∗

mV φndx , (6)

(η|Φ(x)|2)mn = η

a−a

φ∗m|Φ(x)|2φndx . (7)

In order to accurately compute the numerical solutions,

the numerical calculation method we employed is as fol-

lows. Firstly, we take finite expansion terms of the barrier,

and solve the matrix H mn and obtain the ground state

wave function; Secondly, we add the nonlinear term and

use iterative method to get the ground state solution of

this GP equation (in calculation we take nmax = 201, and

make sure that energy convergence is reached).

By changing the barrier’s altitude V 0 and attractive

nonlinearity parameter η, we find that when the barrier is

raised, it can drive a transition between symmetric ground

state and asymmetric ground state wave functions, which

means inducing a macroscopic phase transition of BECs.

On the other hand, as strengthening the nonlinearity term

similar results appear too. Symmetry breaking solutions

as we observe in the following are expected in the attrac-

tive case, since an attractive condensate in the ground

state tends to localize in one well or the other.

Fig. 1 A barrier height of V 0 = 100, barrier width2b = 1/5, well width 2a = 1. (a) Changing η, thecondensate wave functions B(η = 0), C(η = −10.10),D(η = −10.15), E(η = −20); (b) Changing η from −9.70to −10.60 at the step of 0.05, and the correspondingground state energy E g.

Figures 1(a) and 1(b) describe the ground state wave

function and the energy of the Bose–Einstein condensates

in the double square well via varying attractive nonlin-

earity parameter η. Figure 1(a) reveals that stronger η

will lead more condensates to be located in one of the

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816 YUAN Qing-Xin and DING Guo-Hui Vol. 43

wells. In Fig. 1(b), when η jumps from −10.10 to −10.15

the ground state energy falls rapidly and the ground state

wave function’s parity shifts from symmetry preserving

state to symmetry breaking state.

Fig. 2 Barrier width 2b = 1/5, well width 2a = 1,η = −10.10. (a) Varying V 0 and the correspondingBECs ground wave function B(V 0 = 50), C(V 0 = 100),D(V 0 = 105), E(V 0 = 200); (b) Varying V 0 and the cor-responding ground state energy E g.

Then we will turn to reducing the barrier with re-

maining η constant. The numerical results are depicted

by Figs. 2(a) and 2(b). It is evident that the conden-sates transform from an asymmetric state to a symmetric

state, otherwise, from a symmetric state to an asymmetric

state as increasing the barrier. For an asymmetric ground

state, increasing the barrier height the condensates will

be localized more into the well, on the other hand, for a

symmetric ground state, reducing the barrier pushes the

peak of the condensate density more towards the center

of the well on top of the barrier. When V 0 goes from 100

to 105 the ground state energy of BECs falls rapidly, and

the ground state wave function changes from symmetry to

asymmetry.

3 Optical Lattice Well

Now considering BECs in a periodic optical lattice

well, specially we concern the following 1D equation

HΦ(x) = µΦ(x) , H = H 0 + H ,

H 0 = p2

2m + V lat(x) , H = η|Φ(x)|2 . (8)

V lat(x) indicates the optical lattice potential and m is the

mass of a single atom, with η the nonlinear parameter. In

experiment the optical lattice potential can be generated

by a standing wave of laser light.

We first briefly introduce the key points of the Fourier

grid Hamiltonian method. As an illustration, we take

Hamiltonian operator for H 0 = ˆ p2/2m + V lat(x). In the

coordinate representation, the basic vectors or kets of this

representation are denoted by |x and are eigenfunctions

of the position or coordinate operator x. The potential is

diagonal in the coordinate representation x|V lat(x)|x =

V lat(x)δ (x − x). The eigenfunctions of the momentum

operator are written as ˆ p|k = k h|k. The kinetic energy

operator is therefore diagonal in the momentum represen-

tation k|T |k = T kδ (k − k) ≡ (h2k2/2m)δ (k − k). The

transformation matrix elements between the coordinate

and momentum representations k|x = (1/√

2π) e−ikx.

Armed with these basic formula and definitions, we may

now consider the coordinate representation of the Hamil-

tonian operator

x|H 0

|x

=x|T |x

+ V lat(x)δ (x

−x)

= 1

2π

∞

−∞

e ik(x−x)T kdk + V lat(x)δ (x− x) .(9)

Now we try to get the discretized analog of the Hamil-

tonian operator matrix elements in Eq. (9). xi = i∆x,

i = 1, 2, · · · , N , where ∆x is the uniform spacing between

the grid point, and the normalization condition for the

wave function is ∆xN

i=1 |Φ(xi)|2 = 1. The relationship

between the grid spacings in momentum space and coordi-

nate space is ∆k = 2π/N ∆x. If we take the central point

in the momentum space grid to be zero, the grid points

are evenly distributed about it. Defining an integer n as2n = N − 1 (N must be an odd number), we can have

H 0ij = xi|H 0|xj

= 1

2π

nl=−n

e il∆k(xi−xj) h2

2m(l∆k)2

∆k +

V lat(xi)δ ij∆x

= 1

∆x

2

N

nl=1

cos l2π(i − j)

N

h2

2m(l∆k)2

+ V lat(xi)δ ij

. (10)

For convenience in calculations, we use H 0ij = ∆xH 0ij to

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No. 5 Symmetry Breaking Ground States of Bose–Einstein Condensates in 1D Double Square Well and Optical Lattice Well 817

replace H 0ij and the standard set of secular equations readj

( H 0ij −E λδ ij)Φλj = 0 , (11)

where Φλj = Φλ(xj). After finding the eigenvalues and

eigenfunctions of H 0ij, we take the nonlinearity term (i.e.

H ) as a perturbation. Using iterative method, we can getthe numerical solutions of NLSE.

Fig. 3 (a) Wave functions for the lowest energy state byaltering η: B(η = −0.10), C(η = −0.30), D(η = −0.61),E(η = −0.62), F(η = −0.80), G(η = −1.00) undermaintaining the amplitude of optical lattice potentialW = 0.05; (b) The ground state energy of BECs as todifferent η .

In our problems, Bose–Einstein condensates are lo-cated into an optical lattice which is approximatively rep-

resented by a periodic potential field. As a model for

quasi-1D confinement in this case we use the periodic po-

tential V lat(x) = W cos(x), where W denotes amplitude of

potential. Choosing 4 periods from the optical lattice well,

i.e. taking x ranging from −4π to 4π, periodic boundary

conditions are assumed. The grid points in coordinate

space are given by xi = i∆x, 0 ≤ i ≤ N − 1, where

∆x = L/N and L is the length of the range of x val-

ues sampled. We calculate the results with the number

of sampled grid points N equals to 125, 251, 501, respec-

tively, and find the relative error is less than 10−7. For

simplicity, in the following calculations we set N = 251.

Fig. 4 (a) Wave functions for the lowest energy state byaltering W : B(W = 0.05), C(W = 0.15), D(W = 0.29),E(W = 0.30), F(W = 0.50), G(W = 1.00). For all,η = −0.20; (b) The ground state energy of BECs as todifferent W .

The condensate wave function corresponding to the

lowest energy will be solved on the conditions of varying

nonlinear term parameter η and the amplitude of optical

lattice potential W , respectively. Figure 3(a) shows that

the ground state wave functions with respect to different

atomic interactions while keeping W = 0.05 fixed. When

the attractive nonlinearity is relatively weaker, BECs will

take up all over the optical lattice well and in the groundstate the nonlinear attractive potential is not stronger

enough to break the original periodic spatial structure of

optical lattice. This case is shown in the inset of Fig. 3(a).

But as η reaches to −0.62 the existence is turned over,

namely, the wave function’s shape changes thoroughly.

Almost all of the condensates get into one of the optical

lattice well. Three wave functions are displayed when in-

tensifying their interactions, E (η = −0.62), F (η = −0.80)

and G(η = −1.00) respectively. In evidence, along with

strengthening the nonlinearity parameter more and more

BECs will be located in one of the optical lattice well.

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818 YUAN Qing-Xin and DING Guo-Hui Vol. 43

Along with this enhancing η the corresponding ground

state energy of condensates are shown in Fig. 3(b). The

ground state energy’s jump reflects the wavefunction’s

spatial transformation, and also means spatial symmetry

breaking, i.e., from symmetry preserving states to sym-

metry breaking states. Simultaneously a phase transition

happens.On the other hand, holding η = −0.20 and changing

the amplitude of the optical lattice well W , we get the

results which are illuminated by Figs. 4(a) and 4(b). This

gives the equivalent results. At first, we obtain BECs in

one of optical lattice wells with high amplitude. Decreas-

ing the height of barrier more and more condensates will

go away and spread throughout the lattice. Just as W

is less than 0.03 condensates will be distributed in every

optical lattice well uniformly. The spatial structure’s con-

version certainly reveals that a phase transition happens,

at the same time the ground state energy changes dramat-

ically. Figure 4(b) visualizes this process.According to the above results we know the ground

state of BECs undergoes a macroscopic phase transition

when the relative strength of two competing energy terms

in the Hamiltonian is varied across a critical value. This

transition changes its spatial symmetry and coherence.

As the potential depth of the lattice or the atomic at-

tractive interactions is decreased, a transition is induced

from a symmetry breaking (we call it Mott insulator) to

a symmetry preserving (superfluid) phase. In the super-

fluid phase, BECs are spread out over the entire lattice,

with long-range phase coherence. But in the insulator

phase, they are localized at a few individual lattice sites,

with weak phase coherence across the lattice; this phase

is characterized by a gap in the excitation spectrum.

4 Conclusions

In summary, we show the symmetry breaking ground

states of Bose–Einstein condensates trapped in both dou-

ble square well potential and optical lattice well potential

by varying the depth of the barrier and the attractive

nonlinear term. The methods we use are standing-wave

expansion, Fourier grid Hamiltonian, and iterative meth-ods based on the NLSE. Especially for the BECs trapped

in the optical lattice well potential, we display a structure

phase transition. This may reveal the transition between

superfluid and insulator phase.

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