非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表

1、非线性混沌现象及其在激光原子相互作用和玻色爱因斯坦凝聚中的表现刘杰北京应用物理与计算数学研究所非线性研究中心/zhaoshengjj/boshizy/lilunwl/lilunwl.htmliu_jie扬州大学物理系 2005年三月概要1、非线性动力学混沌现象简介 保守系统的混沌 耗散系统的混沌 量子混沌2、应用于强激光的原子电离3、应用于波色-爱因斯坦凝聚体4、应用于纳米材料（如有时间）一、非线性系统的混沌现象保守系统与耗散系统 区别点 相体积收缩与否 相同点混沌轨道有李雅谱诺夫指数描述保守系统的标准映象耗散系统的洛仑兹系统保守哈密顿系统的混沌根源及KAM 定理1、正则变换与作用量角变量pq

2、Area=2piIItheta2、可积系统与不可积系统Integrability: if K is independence of angular variable,i.e.all motions are either periodic or quasiperiodic不可积的解析根源-小分母共振现象 几何根源-分界线的复杂化（稳定和不稳定流形的 横截相交） bad news good news不可积系统的普遍性 KAM定理3、统计物理的基本假设-各态历经KAM定理：考虑条件：1）排除共振 2）非兑化结论：存在不变环面（积分）Arnold 扩散 4、经典混沌的量子表象能谱统计波函数统计可信度（

3、Fidelity)量子运动是周期的 或准周期的无混沌不可积性的表现能谱间距统计Fidelity of quantum evolution二、强激光场中的原子非微扰现象及理论强场物理概况超短超强激光技术已经提供激光功率密度达到10*21Wcm*2时，相应的局域电场将高达10*12V/cm，已是氢原子中束缚基态电子的库仑场强的170倍。激光与原子、分子、离子、电子、团簇以及等离子体等各种形态物质之间的相互作用研究，将进入到了一个前所未有的高度非线性和相对论性的强场范围。由此而开创出了超强激光场物理这样一个全新的现代科学技术的前沿学科领域。强场物理概况超短、超强激光与物质相互作用产生的主要物理现象研

4、究。1）激光与原子、分子、离子、电子、团簇各种形态物质之间的相互作用研究。2）与等离子体相互作用。反常吸收和加热效率，相对论效应和有质动力引起的临界面现象，强激光在等离子体中传播和对光传播的影响，带电粒子加速，自生磁场产生和对光束传播的影响等。强场物理概况相关高技术及基础学科中潜在应用的研究。1）快点火研究的突破可能降低激光聚变点火能量约一个量级，使激光聚变能源研究产生革命性影响。2）超快、高相干性、高亮度的X光源的应用在推进深层次物质结构的认识、发展生命科学和21世纪材料科学以及信息科学等都有重要意义。3) 强场超强梯度加速器的研制导致加速器超小型化，并导致加速器技术革命性变化。激光技术的发

5、展Intensity(watts per square centimeter)Q_switchingMode lockingChirped pulse amplificationNonlinear QEDNonlinear relativistic opticsatoms, plasmas, ICF强激光场中的原子-引言近些年来，强场与原子相互作用问题引起人们的广泛关注。这主要由于在过去几十年里激光技术和实验技术的飞速发展使得电磁场强度与库仑场(原子玻尔半径处)在同一量级(甚至超过)；可以非常精确地测量光电子的能量及角分布等物理量。由于场强很强，传统的微扰理论失效，因此需要发展新的非微扰理论。

6、这在理论上是具有挑战性的。主要的非微扰现象A. 多光子电离、阈上电离原子可以吸收超过阈值数的光子。1999年Agostini首先在实验上观察到这种现象，引起了人们的广泛兴趣与研究热潮。B原子稳定化现象 原子稳定化现象就是指在一定的参数条件下，电离率随着场强的增大而减小。主要物理机制是：在超强场中快速抖动（QUIVER）的电子有较少的机会与核交换动量。 Dynamic稳定化现象 Adiabatic稳定化现象 C. 高次谐波原子和激光相互作用时可发射光子。当场很强时有高次谐波。由于高次谐波发射可能成为有效的全相干短波光源，因此引起人们的重视。一、单电子电离实验 G.G.Paulus et al P

7、RL 72 (1994) 2851B.Yang, et al experiment, PRL 71 (1993) 3770H atoms in the intense laser fieldsPhys.Letter A, 236, 533-542 (1997) H atoms in the intense laser fieldsH atoms in the intense laser fieldsHe atom in intense laser fieldsHe atom in intense laser fieldsExplain the “knee” structure and the

8、ratio between the He1+ and He2+ ions observed in experimentsExplain the momentum distribution of the He ions recorded in experiments.Phys. Rev. A,Vol.63, 011404(R)(2001) Phys. Rev. A, Vol 65 (2): art. no. 021406 FEB (2002) Phys.Rev.A, Vol.63, 043416 (2001) He atom in intense laser fieldsHe atom in i

9、ntense laser fieldsHe atom in intense laser fieldsThe Momentum distribution of He2+ion原子在激光场中的稳定化现象稳定化现象对应于在超强超高频的激光场作用下，原子的电离随场强和频率的增加反而下降的现象。这种下降可区分为以下两类：动力学稳定化和渐进稳定化。这里主要介绍渐进稳定化。在这方面比较成功的理论是由Gavrila等人发展起来的基于对激光场频率倒数进行展开高频Floquet(HFFT)理论。三、Bose-Einstein Condensates (BECs) and Bogoliubov Excitation

10、1. Basic concept of BECsJILA group,Rubidium atoms, Science 269,198 (1995)MIT group, Sodium; Rice group, Lithium (1995) Cornell and Wieman cooled a small sample of atoms down to only a few billionths (0.000,000,001) of a degree above Absolute Zero! That was what they needed to do to see Bose-Einstein

11、 condensation. How cold of BECWhat it isTypical parameters of BEC, Density Temperature nkSize The number of atom 102-108 Atoms are identical and behaves in the same way, act collectivelydemonstrating macroscopic quantum fluid phenomenaWhat is BECs good forToo new and we know too littlePotential appl

12、ication: Sensitive measurement, tiny instrument, atom laser, quantum information, etc.We should understand BECs dynamics firstUltra cold atoms in a pulsed laser beam: Periodically kicked rotator model Laser beamWhat happens if we replace the ultra-cold atom with the BEC?1.Kicked Bose-Einstein conden

13、sate on a ringQuasi-one dimensional ring trapis the healing length Gross-Pitaveskii equationis interaction strengthLength unit:Energy unit:Time unit:Normalization conditionPeriodic Boundary condition:Evolution of the mean energy of each particleAnti-ResonanceQuantum beatingInstabilityKick strengthKi

14、ck periodInitial state Two-mode approximationQuantum beating for weak interaction Total energy is very small Total parity is conservedSpin HamiltonianPopulation differenceRelative PhaseNormalization condition The Hamiltonian is similar to a kicked top modelJ. Liu, B. Wu, and Q. Niu, Phys. Rev. Lett.

15、 90, 170404 (2003); Spin Dynamics Noninteraction: A new explanation of anti-resonance Free evolution: an angle rotation about the z axis Kick: an angle rotation about -x axis Weak interaction: Quantum beating Free evolution: an angle rotation about the z axis Kick: an angle rotation about -x axisRed

16、 arrows: quantum beatingBlue arrows: anti-resonanceEvolution of relative phase and population difference Population difference oscillates with time, similar as the energy Relative phase increases almost linearly and reaches 2 in onebeat periodBeat FrequencyThe slope:To the first order in gIn one bea

17、t period, it reachesAnalytic expressions for beat and oscillation frequencies Scatters: Numerical simulationLines: Analytic expressionsPhase space portraits in two-mode approximationQuantum Anti-ResonanceQuantum BeatingRed dots and line: the trajectories with initial conditionsBlack dots and lines:

18、the trajectories with other initial conditionsInstability for strong interactionDynamics in two-mode approximation As instability occurs for strong interaction, two-mode is not a good approximationRandom evolution in the temporal domain: -Irregular pattern of energy evolutionEvidences for instabilit

19、y in a BECBogoliubov Theory The mean number of noncondensed atoms at zero temperatureis the projection operatoris condensate wave functionis the chemical potentialdescribe the deviation of the wave function from the condensate wave function Exponential sensitivity to initial condition: -Exponential

20、growth of noncondensed atoms in unstable regime Similar to the exponential divergence of nearby trajectories Growth rate is similar to Lyapounov exponentNoncondensed atoms number:Growth rateCritical phenomena - scaling lawg=0.1g=1.5g=2.0Change of density distributions across critical pointUnstable r

21、egime: Stable regime: 1. Condensate density oscillates regularly2. Noncondensate density increases slowly and shows main peaks around and 0 ( )mode1. Condensate density oscillates irregularly2. Noncondensate density increases exponentially and shows main peaks around ( )modeCan be used to identify t

22、he transition to instability58 The beating phenomenon can be used to measure atom interaction strength.Applications and future directions Dynamical localization and quantum resonance in a kicked BEC? - the effects of nonlinearity The density distribution can be used to observe the transition to inst

23、ability in experimentThe article published in Physical Review Letters, Vol.92,054101 (2004)4. BEC in billiards Phys. Rev. Lett. 93, 074101 (2004)Ultra-cold atoms in optical trap of stadium billiardsIntergrable: Possion distribution of the level spacingsLocalized eigenfunctionsNonintergrable: Wigner

24、distribution of the level spacingsExtended eigenfunctions (chaotic state) scars,Question: What happens if we replace the ultra-cold atoms with the BECThe distribution of level spacings of Bogoliubov excitation?Eigenstate of Bogoliubov excitation?The properties of the Bogoliubov wave?Geometric optics approximationssupposeConsider slowly-varying media is small The bogoliubov equation can be transformed into the Eikonal equationIn the noninteraction and strong interaction, ray trajectories are