波色凝聚体的不稳定性及量子混沌性质 Instability and Quantum Chaos of Bose-Einstein Condensate

1. 波色凝聚体的不稳定性及量子混沌性质Instability and Quantum Chaos of Bose-Einstein Condensate 刘杰(Jie Liu, IAPCM) Center for nonlinear studies and theoretical physics Institute of applied physics and computational mathematics, Beijing 北京应用物理与计算数学研究所 理论物理与非线性研究中心liu_jie@iapcm.ac.cn

2. Collaboration University of Texas, Austin Prof. Qian Niu Prof. M.Raizen Ph.D Chuanwei Zhang NUS, Singapore Prof. Baowen Li Dr. Wenge Wang Institute of Physics, China Prof. Biao Wu

3. Outline of my talk 1.Brief introduction to Bose-Einstein Condensates What is BEC? What is it good for? Instability is an important issue in applications of BECs 2. Linear response to perturbation: Bogoliubov excitations in BEC systems Physical Review Letters, 93, 074101(2004) 3. Nonlinear response to perturbation :Dynamical instability in BEC systems Physical Review Letters, Vol.92,054101 (2004) Physical Review A, (2006)

4. 4. Quantum essence of dynamical instability in mean field treatment of BECs cond-mat/0503036, to appear in Physical Review A, (2006) 5. How to control the instability of BECs ? New adiabatic theory for BEC Phys. Rev. Lett. 90, 170404 (2003) Phys. Rev. Lett. 94, 140402 (2005) 6. Conclusions

5. 1. Introduction

6. What is BEC? Liquid Gas Solid Matter 第五种 形态? BEC Coldest and Fragile！ Plasmas

7. Why Bosons can condensate

8. Criterion for Bose-Einstein condensate

9. How to realize BEC in lab

10. JILA group,Rubidium atoms, Science 269,198 (1995) MIT group, Sodium; Rice group, Lithium (1995) Phys. Rev. Lett. 75, 3969 (1995); ibid. 75, 1687(1995)

11. Property of BEC Atoms are identical and behaves in the same way, act collectively demonstrating macroscopic quantum fluid phenomena Typical parameters of BEC, Density Temperature nk Size The number of atom 10^2-10^8

12. Michael Albiez, et. al., Phys. Rev. Lett. 95, 010402(2005) W. Ketterle, Science 275, 637 (1997). Insulator - superfluidM. Greiner et al., Nature 415, 39 (2002).

13. What is BECs good for

15. applications (only to my knowledge) Atomic laser Atom clock Interferometer (microgravity) Waveguide BEC on chip Quantum tweezer Information storage Quantum computer Four-wave mixing Reducing light speed Superradiance Supernova S.L. Cornish et al., Phys. Rev. Lett. 85, 1795 (2000).

16. Instability is an important issue for BEC Perturbation Thermal Environment Manipulation or transportation process Instability leads to the collapse of BEC under perturbation BEC 1.Perturbation strength and atomic interaction effects ? 2.What happens after instability occurs? 3. How to control instability?

17. 2. Linear response to perturbation:Bogoliubov excitation of BEC in billiards

18. Ultra-cold atoms in optical trap of stadium billiards Intergrable: Possion distribution of the level spacings Localized eigenfunctions Nonintergrable: Wigner distribution of the level spacings Extended eigenfunctions (chaotic state) scars,

19. Numerical results with the phase shift method Results: The interaction does not change the distributions of level spacing C.W.Zhang, Jie Liu, M.Raizen and Qian Niu, in Physical Review Letters, 93, 074101(2004)

20. Predicting the distribution of the Bogoliubov excitation from

21. Predicting the distribution of the Bogoliubov excitation from the random Matrix theory:

22. 2. Nonlinear response to perturbation :Dynamical instability in BEC systems is interaction strength Time unit: Length unit: Energy unit: Normalization condition • Quasi-one dimensional ring trap • Gross-Pitaveskii equation Periodic Boundary condition:

23. Evolution of the mean energy of each particle Kick period Kick strength Initial state Anti-Resonance Quantum beating Instability

24. Normalization condition Population difference Relative Phase Quantum beating for weak interaction • Two-mode approximation • Total energy is very small • Total parity is conserved • Spin Hamiltonian • The Hamiltonian is similar to a kicked top model J. Liu, B. Wu, and Q. Niu, Phys. Rev. Lett. 90, 170404 (2003);

25. Analytic expressions for beat and oscillation frequencies Scatters: Numerical simulation Lines: Analytic expressions

26. is the chemical potential is condensate wave function is the projection operator Time dependent Bogoliubov Theory • Exponential sensitivity to initial condition: • ---Exponential growth of noncondensed atoms in unstable regime • Similar to the exponential divergence of nearby trajectories • Growth rate is similar to Lyapounov exponent • The mean number of noncondensed atoms at zero temperature describe the deviation of the wave function from the condensate wave function

27. Critical phenomena -- scaling law g=2.0 g=1.5 g=0.1 Noncondensed atoms number: Growth rate

28. Stable regime: 1. Condensate density oscillates regularly 2. Noncondensate density increases slowly and shows main peaks around and 0 ( ) mode 1. Condensate density oscillates irregularly 2. Noncondensate density increases exponentially and shows main peaks around ( ) mode Change of density distributions across critical point Unstable regime: C.Zhang, Jie Liu, M.Raizen and Qian Niu ,Physical Review Letters, Vol.92,054101 (2004) Jie Liu, C.Zhang, M.Raizen, and Qian Niu, Physical Review A (2006)

29. 4. Quantum essence of dynamical instability in BECs BECs Theoretical Meanfield Treatment Dynamical instability Quantum description No Dynamical instability dilemma Systematic instability of a quantum system

30. A simple Two-component BEC model Spinor BECs Rb 87, applied riadiation field, convert between two internal state BEC

31. Good correspondence for the initial coherent state BECs

33. No correspondence for the initial entangled state BECs • cond-mat/0503036 • Jie Liu, Wenge Wang, Chuanwei Zhang, Qian Niu, Baowen Li • Physical Review A (2006)

34. 4、How to control the instability of BECs ? Feasible way is the adiabatic process for BEC system

35. 3 Linear adiabatic theory Adiabatic Theorem and Berry phase in Quantum Mechanics E R Adiabaticity condition Occupation population is adiabatic invariant If parameter slowly moves round a circuit,

36. Dynamics of BEC is governed by Nonlinear Schrödinger equation (GP euqation) Problem: Linear adiabatic theory is not available superposition principle breaks down Our approach: combine classical action theory and Aharonov-Anandan phase theory. Set up the adiabatic theory for BEC system

37. Number of eigenstates >= dimension of Hilbert space Adiabaticity depends on the Bogoliubov spectrum AA phase is invariant Berry’s phase for eigenstates not complete in describing the non-eigenstates The Number of eigenstates = dimension of Hilbert space Adiabaticity depends on level spacings level population is invariant Berry’s phase for eigenstates complete in describing the non-eigenstates Main Results on Adiabatic Theory of BEC system Nonlinear case Linear case

38. Ref: Jie Liu, Biao Wu, and Qian Niu Phys. Rev. Lett. 90, 170404 (2003) Biao Wu, Jie Liu, and Qian Niu Phys. Rev. Lett. 94, 140402 (2005)

39. Conclusions 1) Instability is an important issue for BEC 2) Instability means the rapid production of Bogoliubov quasiparticle, leading to the collapse of BECs 3) Quantum essence of dynamical instability: sensitivity of quantum system on outer perturbation parameter 4) New Adiabatic theory of BECs to control instability

40. Thank you