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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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