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Quantum phase transition of Bose-Einstein condensates on a ring with
periodic scattering length
Dalian, Aug. 3, 2010
Zheng-Wei Zhou(周正威） Key Lab of Quantum Information , CAS, USTC
In collaboration with:
Univ. of Sci. & Tech. of ChinaS.-L. Zhang( 张少良）X.-F. Zhou ( 周祥发 ) X. Zhou ( 周幸祥 ) G.-C. Guo ( 郭光灿 )
Rice Univ.Han Pu ( 浦晗 )Lisa C. QianMichael L. Wall
Outline
Background: Bosons on a ring
Bosons on a ring with modulated interactionMany bosons: Mean field analysisA few bosons: Quantum mechanical analysis;
Entanglement and correlation
Conclusion
October, 2009 KITPC
Background: Ring potential for cold atoms
• Magnetic waveguides
Gupta, et al. PRL (2005)
4 coaxial circular electromagnets
BECs in a ring shaped magnetic waveguide.
• Optical dipole trap using
Laguerre-Gaussian beams
Background: Ring potential for cold atoms
Atom-Atom Interactions
• 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)
Feshbach resonance
Background: Bosons on a ring
Background: Bosons on a ring
r R
2 22
2 2( , ) ( , ) ( , )
2i t g t tt mR
Toroidal system with sufficient transverse confinement:
•Weakly interacting particles
•GP Equation
2
gN
L. D.Carr, et. al., PRA 62, 063211 (2000)
Background: Bosons on a ring
2
gN
Kanamoto, PRA 67,013608 (2003)
0.5 : uniform amplitute
0.5 : soliton state (symmetry breaking)
Phase transition at γ = -0.5
ground state
~ sin(2 )g
•Periodically modulated scattering length (2 periods)
2
2
2
( , )sin(2 )2 , ,
ti t t
t
2 22
2 2( , ) ( , ) ( , )
2i t g t tt mR
Bosons on a ring with modulated interaction---- Many bosons: Mean field analysis
MFT solutions
60.0
2-fold degeneracyin symmetry breaking regime
60.0
54.0
25.0
Symmetry breaking occurs at 0.52
The original symmetry manifest itself in the 2-fold degeneracy of GS.
den
sity
Energy vs. |γ|
Phase transition
0.52
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.
2. Decompose ψ into plane waves (Fourier decomposition)
1. Full many-body Hamiltonian
3. Rewrite Hamiltonian as
A kind of modified Bogoliubov method in the momentum space
关于玻色凝聚稳态的定义：（ a ）
(b) 经 Bogoliubov 变换之后，
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 回代入哈密顿量，
使得约束 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。
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
我们的发现 :
d ＝ 2
动力学非稳驱动量子相变！
动力学非稳点
d ＝ 3
d>=3，凝聚稳态的能级交叉导致量子相变。
动力学非稳点
Bosons on a ring with modulated interaction---- A few bosons: Quantum mechanical analysis
2. Decompose ψ into plane waves (Fourier decomposition)
1ˆ ( )
2
Lil
ll L
a e
22 † † †
20
ˆ ˆ ˆ ˆ ˆ ˆ ˆ(sin 2 )H dN
1. Full many-body Hamiltonian
3. Rewrite Hamiltonian as
22 24
ˆl l k l m k l m k l m k l mNi
l klm
H l a a a a a a a a a a
Bosons on a ring with modulated interaction---- A few bosons: Quantum mechanical analysis
4. Basis states are Fock states (angular momentum e-states)
1 1 0 1, ,..., , , ,....,L L Ln n n n n n
5. Diagonalize Hamiltonian in the span of this basis
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!
Energy and density profile of 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
ground state.
Entanglement of ground state for N=2
(N=2)
we calculate the overlap of the ground-state wave function defined as
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 ofthe quantum ground state and its first excited state.
Energy gap between the quantum mechanical ground state and the first excited state as a function of particle number N.
the mean-field states are “selected” states
另外一种求解该问题的途径 －－ Time evolving block decimation algorithm
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 in a local basis for qubit 2,
then we write each in terms of at most Schmidt vectors a and the corresponding Schmidt coefficients ,
finally we can obtain
A wave function for n-qubit system:
Repeat these steps, we can express state as:
coefficients
In a generic case grows exponentially with n. However, in one-dimensional 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 does not depend on time. After a time interval T, the evolved state is given by
The can be decomposed as
The Trotter expansion of order p for reads
where and where a for first and second order expansions.
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 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.
We studied the ground states of 1D BECs in a ring trap with d spatial periods of modulated scattering length, within and beyond the Gross-Pitaevskii 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.
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.
