超冷简并原子气体中的非线性量子特性研究
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摘要
非线性相互作用是超冷简并原子气体的一个重要特征。深入理解由于这种非线性相互作用导致的量子特性,不只是对完善与丰富线性量子理论有重要的意义,也对超冷简并原子气体的相干控制及其潜在的应用有现实的指导意义。本论文提出了一组用于非均匀玻色—爱因斯坦凝聚(Bose-Einstein condensates (BECs))描述的正交基；研究了非线性对其相干性的影响,以及利用谐振子势中多组份BECs理解和模拟一个重要的非线性问题的可能性进行了研究。具体地讲,可以概括为以下几个方面：
正交完备基是描述线性量子系统的重要基础,由于非线性相互作用和外势的影响,原则上不存在一组正交完备基。在一维无限深势阱中,我们发现一组与线性薛定谔方程的完备基一一对应的的非线性基底。利用这组正交基和波戈留波夫理论,我们对一维无限深势阱中的定态解的稳定性进行了研究,结果发现：对于基态解,无论是非线性参数为正或是为负均是稳定的；但对于第一激发态解,当取负的非线性大于某一个临界值时,它是不稳定的。
我们对具有非对称BECs的干涉现象进行了研究,发现由于初始波包的不对称性,所形成的干涉条纹也具有一定的不对称性。利用该干涉条纹的性质,结合BECs的特性,我们提出一种用于测量微弱类重力的量子干涉仪。如果对物质波的密度测量可以保证实现的话,则对微弱力的测量精度可以达到δg～10-6。此外,我们通过量子Loschmidt echo展示了非线性的增强效应,它对于测量微弱重力起到了积极的作用。
相干性是BECs的另一个引人注意的基本性质。由于环境或是其他人为因素,对其相干性的影响是实现原子激光中不得不考虑的问题。我们利用量子Loschmidt echo对囚禁在非谐势阱中、受到到环境扰动的BECs的相干性进行研究。结果表明：其相干性的减少呈现费米型的规律,且非简谐势阱、环境扰动和非线性相互作用三个条件必不可少。其中,非简谐势阱用来产生非简谐振荡；环境扰动则造成相干性损失；同时非线性相互作用增强了这种损失。基于这个费米型的规律,当对凝聚体进行相干性操作时,我们给出了延长其相干时间的建议。
最后,我们提议利用简谐阱中的多组份BECs来研究一个经典的非线性问题——Fermi-Pasta-Ulam (FPU)问题。简谐势阱中单组份BECs在不考虑外界扰动时,将做非线性的简谐振动,这已经被实验所证实。多组份凝聚体间的相互作用将不同的组份耦合在一起,实现FPU的基本模型,但是,模式之间耦合被非线性相互作用所代替。我们通过对其能量、以及所对应的Husimi分布函数的计算发现：当非线性从弱到强时,能量从一种转移过程转变为一种分享过程。在两组份时,这种能量分享几乎是均等的。推广到三组份时,在特定的条件下也可以看到类似的现象。
The nonlinear atomic interaction is an important character for the ultra-cold degenerate atomic gas. It is one of the hot points in current research to deeply understand the quantum behaviors induced by the nonlinear interaction, not only for extending the quantum theory, but also for coherent manipulation and potential application of the ultra-cold degenerate atomic gas. In this thesis, we have suggested one set of orthogonal basis for describing the nonuniform BECs, explored the coherence decay law enhanced by the nonlinear interaction and proposed one experiment to simulate one famous nonlinear problem.
It is important to exactly describe the quantum system based on one orthogonal and complete basis in the frame of linear quantum theory. Due to the nonlinearity and external trap, it is difficult to find this kind basis for nonlinear system. With the help of one exactly stationary solution for G-P with1-D infinite square well, a set of orthogonal basis was presented, which is one to one corresponding to the basis of the linear case. Combined the orthogonal basis and the Bogoliubov theory, we have investigated the stability of the stationary states for G-P with the one-dimensional infinite square well, and found that:Whether the positive or negative nonlinear interaction, the ground state is always stable; the first excited state is, however, instable for the negative large enough nonlinear interaction.
As one of the fundamental properties of BECs, the coherence has attracted more attentions both for theorists and experimenters. We have investigated the interference phenomena from spatially imbalanced BECs. An asymmetric interference pattern is reported by the asymmetry of the initial states. Considering the special character of the atom and this asymmetric interference pattern, we have suggested one novel scheme to detect weak gravity-like force, which benefits from the nonlinear interaction. The sensitivity can be around δg～10-6.
It is one of the crucial problems to be solved for the realization of the atom lasers, due to the negative factors from unavoidable environments and other factors. Using quantum Loschmidt echo, the evolution of the coherence of BECs with an anharmonic trap, and subjected to the surrounding perturbation is investigated. Our results have shown that the coherence loss can be described by a Fermi-decay law, in which there are three indispensable factors:the anharmonic trap, the external perturbation and the nonlinear interaction. The anharmonic trap creates the anharmonic oscillations, and the weak random perturbation causes the loss of coherence by disturbing their coherent oscillations, while the nonlinear interaction enhances the loss to the Fermi-decay law. Based on the Fermi-decay law, some suggestions are presented to prolong the coherent time during coherently manipulating condensates.
Finally, we have shown that the evolution of multicomponent BECs with a harmonic trap can be understood as the Fermi-Pasta-Ulam problem (FPU). The single-component BEC performs a nonlinear harmonic oscillation, proved by the experiment, and can be mode for FPU. The nonlinear couple between different modes in FPU, is realized by atomic nonlinear interaction. We have calculated the energy and the Husimi distribution of each component and found the energy recurrence and the energy sharing for different regimes of nonlinear interaction. For the two components case, the strong energy sharing is found, once nonlinear interaction is large enough. For the three components case, the situation is more complex but still energy sharing is observed.
正交完备基是描述线性量子系统的重要基础,由于非线性相互作用和外势的影响,原则上不存在一组正交完备基。在一维无限深势阱中,我们发现一组与线性薛定谔方程的完备基一一对应的的非线性基底。利用这组正交基和波戈留波夫理论,我们对一维无限深势阱中的定态解的稳定性进行了研究,结果发现：对于基态解,无论是非线性参数为正或是为负均是稳定的；但对于第一激发态解,当取负的非线性大于某一个临界值时,它是不稳定的。
我们对具有非对称BECs的干涉现象进行了研究,发现由于初始波包的不对称性,所形成的干涉条纹也具有一定的不对称性。利用该干涉条纹的性质,结合BECs的特性,我们提出一种用于测量微弱类重力的量子干涉仪。如果对物质波的密度测量可以保证实现的话,则对微弱力的测量精度可以达到δg～10-6。此外,我们通过量子Loschmidt echo展示了非线性的增强效应,它对于测量微弱重力起到了积极的作用。
相干性是BECs的另一个引人注意的基本性质。由于环境或是其他人为因素,对其相干性的影响是实现原子激光中不得不考虑的问题。我们利用量子Loschmidt echo对囚禁在非谐势阱中、受到到环境扰动的BECs的相干性进行研究。结果表明：其相干性的减少呈现费米型的规律,且非简谐势阱、环境扰动和非线性相互作用三个条件必不可少。其中,非简谐势阱用来产生非简谐振荡；环境扰动则造成相干性损失；同时非线性相互作用增强了这种损失。基于这个费米型的规律,当对凝聚体进行相干性操作时,我们给出了延长其相干时间的建议。
最后,我们提议利用简谐阱中的多组份BECs来研究一个经典的非线性问题——Fermi-Pasta-Ulam (FPU)问题。简谐势阱中单组份BECs在不考虑外界扰动时,将做非线性的简谐振动,这已经被实验所证实。多组份凝聚体间的相互作用将不同的组份耦合在一起,实现FPU的基本模型,但是,模式之间耦合被非线性相互作用所代替。我们通过对其能量、以及所对应的Husimi分布函数的计算发现：当非线性从弱到强时,能量从一种转移过程转变为一种分享过程。在两组份时,这种能量分享几乎是均等的。推广到三组份时,在特定的条件下也可以看到类似的现象。
The nonlinear atomic interaction is an important character for the ultra-cold degenerate atomic gas. It is one of the hot points in current research to deeply understand the quantum behaviors induced by the nonlinear interaction, not only for extending the quantum theory, but also for coherent manipulation and potential application of the ultra-cold degenerate atomic gas. In this thesis, we have suggested one set of orthogonal basis for describing the nonuniform BECs, explored the coherence decay law enhanced by the nonlinear interaction and proposed one experiment to simulate one famous nonlinear problem.
It is important to exactly describe the quantum system based on one orthogonal and complete basis in the frame of linear quantum theory. Due to the nonlinearity and external trap, it is difficult to find this kind basis for nonlinear system. With the help of one exactly stationary solution for G-P with1-D infinite square well, a set of orthogonal basis was presented, which is one to one corresponding to the basis of the linear case. Combined the orthogonal basis and the Bogoliubov theory, we have investigated the stability of the stationary states for G-P with the one-dimensional infinite square well, and found that:Whether the positive or negative nonlinear interaction, the ground state is always stable; the first excited state is, however, instable for the negative large enough nonlinear interaction.
As one of the fundamental properties of BECs, the coherence has attracted more attentions both for theorists and experimenters. We have investigated the interference phenomena from spatially imbalanced BECs. An asymmetric interference pattern is reported by the asymmetry of the initial states. Considering the special character of the atom and this asymmetric interference pattern, we have suggested one novel scheme to detect weak gravity-like force, which benefits from the nonlinear interaction. The sensitivity can be around δg～10-6.
It is one of the crucial problems to be solved for the realization of the atom lasers, due to the negative factors from unavoidable environments and other factors. Using quantum Loschmidt echo, the evolution of the coherence of BECs with an anharmonic trap, and subjected to the surrounding perturbation is investigated. Our results have shown that the coherence loss can be described by a Fermi-decay law, in which there are three indispensable factors:the anharmonic trap, the external perturbation and the nonlinear interaction. The anharmonic trap creates the anharmonic oscillations, and the weak random perturbation causes the loss of coherence by disturbing their coherent oscillations, while the nonlinear interaction enhances the loss to the Fermi-decay law. Based on the Fermi-decay law, some suggestions are presented to prolong the coherent time during coherently manipulating condensates.
Finally, we have shown that the evolution of multicomponent BECs with a harmonic trap can be understood as the Fermi-Pasta-Ulam problem (FPU). The single-component BEC performs a nonlinear harmonic oscillation, proved by the experiment, and can be mode for FPU. The nonlinear couple between different modes in FPU, is realized by atomic nonlinear interaction. We have calculated the energy and the Husimi distribution of each component and found the energy recurrence and the energy sharing for different regimes of nonlinear interaction. For the two components case, the strong energy sharing is found, once nonlinear interaction is large enough. For the three components case, the situation is more complex but still energy sharing is observed.
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