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Quantum phase transition of BoseEinstein condensates on a ring with
periodic scattering length
Zheng-Wei Zhou(周正威）
Key Lab of Quantum Information , CAS, USTC
In collaboration with:
Univ. of Sci. & Tech. of China
Rice Univ.
S.-L. Zhang(张少良）
X.-F. Zhou (周祥发)
X. Zhou (周幸祥)
G.-C. Guo (郭光灿)
Han Pu (浦晗)
Lisa C. Qian
Michael L. Wall
Dalian, Aug. 3, 2010
Outline
 Background: Bosons on a ring
 Bosons on a ring with modulated interaction
Many bosons: Mean field analysis
A few bosons: Quantum mechanical analysis;
Entanglement and correlation
 Conclusion
October, 2009
KITPC
Background: Ring potential for cold atoms
• Magnetic waveguides
4 coaxial circular
electromagnets
BECs in a ring shaped magnetic
waveguide.
Gupta, et al. PRL (2005)
Background: Ring potential for cold atoms
• Optical dipole trap using
Laguerre-Gaussian beams
Background: Bosons on a ring
Atom-Atom Interactions
Feshbach resonance
• Ultracold collision
governed by s-wave
scattering length, a.
• a>0: repulsive interactions
• a<0: attractive interactions
• Control with external
magnetic or optical fields
Cornish, et al. PRL (2000)
Background: Bosons on a ring
Toroidal system with sufficient transverse confinement:
•Weakly interacting particles
•GP Equation
2


2
2
i
( , t )  
 g ( , t )   ( , t )
2
2
t
 2mR 

r
R

L. D.Carr, et. al., PRA 62, 063211 (2000)
gN
2
Background: Bosons on a ring
ground state
gN

2
  0.5 : uniform amplitute
  0.5 : soliton state (symmetry breaking)
Phase transition at γ = -0.5
Kanamoto, PRA 67,013608 (2003)
Bosons on a ring with modulated interaction
---- Many bosons: Mean field analysis
•Periodically modulated scattering length (2 periods)
2


2
2
i
( , t )  

g

(

,
t
)
 ( , t )

2
2
t
 2mR 

g ~ sin(2 )
2
 ( , t )   2
i
  2  sin(2 )2   , t    , t 
t
 

MFT solutions
2-fold degeneracy
in symmetry
breaking regime
  0.60
  0.54
density
  0.60
Symmetry breaking occurs at
  0.25

0.52
The original symmetry manifest itself in the 2-fold degeneracy of GS.
Phase
transition
0.52
Energy vs. |γ|
一个成功的经验：标准的Bogoliubov方法求解均匀调制
1. Full many-body Hamiltonian
2. Decompose ψ into plane waves (Fourier decomposition)
3. Rewrite Hamiltonian as
When γ<-0.5, ω_k can become complex for some k,
indicating instability of the condensate mode. This shows
that γ=-0.5 is a critical point.
A kind of modified Bogoliubov method in the
momentum space
1. Full many-body Hamiltonian
2. Decompose ψ into plane waves (Fourier decomposition)
3. Rewrite Hamiltonian as
关于玻色凝聚稳态的定义：
（a）
经Bogoliubov变换之后，
(b)
本征谱皆为非零实数。
如条件（a）（b)得以满足，则态
被称为玻色凝聚稳态。对于玻色凝聚稳态而言，系统的有
效哈密顿量为：
对于
最小的玻色凝聚稳态，我们称其为体系的玻色
凝聚基态（BEC）。
玻色凝聚稳态的约束条件：
将
使得约束
回代入哈密顿量，
成立的模式，即为玻色凝聚稳态的模式。
搜索最小能量找到基态能：
化学势：
Bogoliubov 激发谱：
矩阵M的正本征值即为Bogoliubov 激发谱能量。
我们的发现:
周期数
2
3
4
Bogoliubov方法
相变时粒
子间散
射长度
0.528
0.851
1.122
G-P方程虚时演化
相变时粒
子间散
射长度
0.525
0.85
1.07
d＝2
动力学非稳点
动力学非稳驱动量子相变！
d＝3
动力学非稳点
d>=3，凝聚稳态的能级交叉导致量子相变。
Bosons on a ring with modulated interaction
---- A few bosons: Quantum mechanical analysis
Bosons on a ring with modulated interaction
---- A few bosons: Quantum mechanical analysis
1. Full many-body Hamiltonian
2


2

†
ˆ
H   d  ˆ  
ˆ     (sin 2 )ˆ †  ˆ †  ˆ  ˆ  
2
0

N


2. Decompose ψ into plane waves (Fourier decomposition)
1
ˆ ( ) 
2
L
ae 
il
l  L
l
3. Rewrite Hamiltonian as
Hˆ   l 2 al al 
l

4 Ni
a

k
al am ak l  m  2  ak al am ak l m  2 
klm
4. Basis states are Fock states (angular momentum e-states)
n L , n L1 ,..., n1 , n0 , n1 ,...., nL
5. Diagonalize Hamiltonian in the span of this basis
Energy and density profile of ground states
ground-state energy per particles
Density profile of quantum mechanical
ground states with N=6.
No spontaneous symmetry breaking
happens in quantum mechanical
ground states!
Correlation and entanglement
Left-right spatial correlation function for N=2, 4, and 6.
This implies that the quantum ground state is a Schrödinger cat state for large
!
Correlation and entanglement
Entanglement of ground state for N=2
(N=2)
we calculate the overlap of the groundstate wave function defined as
ground state.
the mean-field states are “selected” states
Energy gap between the quantum mechanical ground state and the
first excited state as a function of particle number N.
The rapid vanishing of the energy gap for large
means that the ground
state and the first excited state essentially become degenerate, a result in
accordance with the MFT analysis. The two degenerate solitonlike states
found in MFT are just the symmetric and antisymmetric superpositions of
the quantum ground state and its first excited state.
另外一种求解该问题的途径
－－ Time evolving block decimation algorithm
A wave function for n-qubit system:
We first compute the SD of
according to the bipartite
splitting of the system into qubit 1 and the n-1 remaining qubits.
where
,we expand each Schmidt vector
local basis for qubit 2,
in a
then we write each
in terms of at most Schmidt vectors
a
and the corresponding Schmidt coefficients
,
finally we can obtain
Repeat these steps, we can express state
as:
coefficients
In a generic case
grows exponentially with n. However, in onedimensional settings it is sometimes possible to obtain a good
approximation to
by considering only the first terms, with
Problem: Numerical analysis shows that the Schmidt coefficients
of the state
of
decay exponentially with :
Initialization
We consider only Hamiltonians made of arbitrary single-body and
two-body terms. With the interactions restricted to nearest
neighbors,
The ground state can be obtained through one of the following
methods:
i) by extracting it from the solution of the DMRG method;
ii) by considering any product state,
and by using the present scheme to simulate an evolution in imaginary
time according to ,
The second method rely on simulating a Hamiltonian evolution from a
product state.
Evolution
For simplicity, we assume that
time interval T, the evolved state
The
does not depend on time. After a
is given by
can be decomposed as
The Trotter expansion of order p for
where
a
and where
for first and second order expansions.
reads
The simulation of the time evolution is then accomplished by
iteratively applying gates
and
to
a number
of
times, and by updating decomposition
at each step.
Errors and computational cost
The main source of errors in the algorithm are the truncation and
the Trotter expansion.
i) The truncation error is
Truncation errors accumulate additively with time during the
simulation of a unitary evolution.
ii) The order-p Trotter expansion error scale as
Lemma 2 implies that updating after a two-body gate requires
basic operations. Gates
and
are applied
times
and each of them decomposes into about n two body gates. Therefore
operations are required on
.
The finite-differerence discretization scheme
单粒子能量（d＝2）
单粒子能量（d＝3）
归一化的凝聚粒子数（d＝2）
归一化的凝聚粒子数（d＝3）
Conclusion
We studied the ground states of 1D BECs in a ring trap with d spatial
periods of modulated scattering length, within and beyond the GrossPitaevskii mean-field theory.
In the MFT, the ground state undergoes a quantum phase transition
between a sinusoidal state matching the spatial symmetry of the
modulated interaction strength and a bright solitonlike state that breaks
such a symmetry. the d－fold ground state degeneracy was found in the
symmetry-breaking regime.
We use the exact diagonalization and TEBD to study the behavior of few
particles systems, which reveals that the degeneracy found in the soliton
phase of the MFT is lifted. Instead, the ground state is comprised of a
strongly anti-correlated macroscopic superposition of solitons peaked at
different spatial locations, and can be regarded as a Schrödinger cat
state, which becomes increasingly fragile as the total number of atoms
increase.
Reference:
Lisa C. Qian, Michael L. Wall, Shaoliang Zhang, Zhengwei Zhou, and
Han Pu, Phys. Rev. A 77, 013611 (2008).
Zheng-Wei Zhou, Shao-Liang Zhang, Xiang-Fa Zhou, Xingxiang Zhou,
Guang-Can Guo, Han Pu, in preparation.