光晶格中玻色—爱因斯坦凝聚体隧穿动力学特性的研究
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摘要
光晶格中BEC的隧穿动力学性质已发展成为当前冷原子研究领域的一个热点课题。大量相关研究表明,原子间相互作用对光晶格中BEC的隧穿动力学性质,如BEC的能带特征、稳定性、局域化和自囚禁现象有着显著的影响。超冷原子在光晶格中的局域化和自囚禁现象是一类重要的非线性量子现象,与冷原子系统的稳定性以及量子相变密切相关。然而,目前已有的相关研究大部分仅局限于考虑原子两体相互作用下低维光晶格中BEC的动力学特性。众所周知,要研究真实的隧穿动力学过程,原子间三体相互作用和光晶格维数是不容忽视的。因此,为了更接近实际,本文一方面侧重研究当同时考虑原子间两体相互作用和三体相互作用时,三体相互作用对光晶格中BEC的能带特征、能量和动力学不稳定性、局域化和自囚禁现象的影响;另一方面,着重研究光晶格的维数对BEC发生局域化和自囚禁现象的影响;进一步研究了光晶格中超流费米气体的动力学性质,从理论上证明了超流费米气体在光晶格中也会出现自囚禁现象,且发现超流费米气体和玻色凝聚体在光晶格中有完全不同的动力学性质。论文包括六章:
第一章简单介绍了BEC的概念和实现、光晶格的产生和光晶格中BEC的研究进展。
第二章分别利用两模近似和紧束缚近似的方法,结合变分原理,研究了两体和三体相互作用下一维光晶格中BEC的动力学特性,系统地分析了三体相互作用对BEC的能带结构、能量和动力学不稳定性、以及局域化和自囚禁现象的影响。结果表明,三体相互作用对BEC能带结构和粒子流密度有很大影响,即粒子数存在一临界值,且此临界值与三体相互作用强度密切相关。只有当粒子数大于这一临界值时,在基态能带的第一布里渊区边界处能级才会出现loop结构。同时,粒子流密度不仅与粒子数、两体相互作用强度和光晶格的特性有关,还与原子之间三体相互作用强度有关;经过理论和数值分析比较发现,原子数和光晶格深度,特别是原子间三体相互作用对Bose系统发生能量和动力学不稳定性的边界条件有着显著的影响;由于原子间三体相互作用的出现,BEC发生扩散、自囚禁态、运动孤子态和呼吸子态的临界条件以及它们在相图中所对应的区域都发生了很大的变化。此外,在自囚禁现象发生以后,波包的最大宽度和最小宽度也与原子间三体相互作用密切相关。
第三章通过构造一个简单的五阱模型,解析和数值地分析了二维光晶格中BEC发生自囚禁现象的临界行为。结果表明,自囚禁现象在五阱模型中也可以发生。但是与两阱和三阱的情况相比,出现自囚禁态的临界参数值发生了改变;阱与阱之间的相位差对BEC的隧穿动力学性质有着显著的影响,尤其对三阱和五阱模型的影响更大。
第四章将一维深光晶格中BEC隧穿动力学性质的研究推广到三维情况,着重分析了光晶格的维数对BEC发生自囚禁和局域化现象的影响。通过求解控制三维深光晶格中BEC隧穿动力学性质的离散G-P方程,解析地得到了体系出现自囚禁态、稳定运动的孤子态和呼吸子态的临界条件,并给出了相应的数值验证。研究结果表明,自囚禁态在二维和三维光晶格中也存在,但是与一维情况相比,控制其发生的临界参数值发生了很人改变;并且发现稳定运动的孤子和呼吸子态只在一维光晶格中存在,在二维和三维光晶格中不存在;当自囚禁态出现以后,光晶格的维数对BEC原子云的最终波包宽度有着显著的影响。
第五章系统分析了超流费米气体在一维、二维和三维深光晶格中发生自囚禁和局域化现象的临界行为,系统讨论了超流费米气体在光晶格中的隧穿动力学性质。结果表明,费米波包的自囚禁态可存在于一维、二维和三维光晶格中,并且存在于BEC—BCS整个渡越区。其次,稳定运动的孤子态和呼吸子态在一维、二维和三维光晶格中都存在。但是,在一维和二维体系中,稳定运动的孤子态和呼吸子态可存在于BEC—BCS整个渡越区;而在三维体系中,稳定运动的孤子态和呼吸子态仅存在于BCS超流体中。进一步研究表明,当体系从分子BEC超流体向BCS超流体过渡时,发生自囚禁态、稳定运动的孤子态和呼吸子态的临界条件发生了很大的变化。相比而言,自囚禁态、孤子态和呼吸子态在BCS超流体中比在分子BEC超流体中更容易发生。此外,无论是在分子BEC超流体中还是在BCS超流体中,光晶格维数越高,发生自囚禁态、稳定运动的孤子态和呼吸子态所需要的临界参数值越大。
第六章给出总结和展望。
The research of the tunneling dynamics of Bose-Einstein Condensation (BEC) loaded into optical lattices have attracted enormous attention. One of the important discovery is that the energy band structure, the stabilities, the localized and self-trapped states of BEC are influenced dramatically by the atoms interactions. However, most previous studies are mainly focused on considering the two-body interaction in low dimension system. Indeed, the real behaviors of BEC will be influenced dramatically by the three-body interactions and the lattice dimension. Thus, one aim of this thesis is to investigate the tunneling dynamics of BEC in one dimensional optical lattices with two and three-body interactions. The influences of the three-body interaction on the energy band structure, the stabilities, the localized and self-trapped states are discussed analytically and numerically. Another aim of this thesis is to study the effect of lattice dimension on the localized and self-trapped states of the system. Furthermore, the tunneling dynamics of superfluid Fermi gas in optical lattices are studied analytically and numerically. We find that the phase diagrams vary greatly along the BEC-BCS crossover; the dynamics of Fermi wave packet are very different from that of Bose wave packet. The major investigations are listed as follows:
Chapter I: The conception, the realization of the BEC, and the progress of studing BEC trapped in optical lattices are reviewed briefly.
Chapter II: By using the two-mode approximation and tight-binding approximation, the dynamical properties, including energy band structure, energetic and dynamical instability, and the tunneling dynamics of a dilute BEC trapped in one-dimensional periodic optical lattices are discussed. It is demonstrated that the lowest energy band can show a loop structure at the boundary of the first Brillouin zone only when the number of atoms is larger than a critical value, which is influenced by the three-body interaction dramatically. The obtained atom fluid density depends on the number of atoms, two-body interaction, the optical lattice parameters, and, especially, the three-body interaction. It is also important to note that, the boundaries of dynamical instability and Landau instability are modified significantly due to the presence of the three-body interactions. In addition, after the three-body interactions are taken into account, the critical conditions for the BEC wavepacket to take place the self-trapping, diffusion, stable moving soliton and breather are all modified dramatically.
Chapter III: Developing a five-well model for describing the tunneling dynamics of BEC trapped in 2D optical lattices. The tunneling dynamics of BEC in this five-well model are investigated both analytically and numerically. We focus on the self-trapped states and the difference of the tunnelling dynamics among two-well, three-well and five-well systems. The criterions for the self-trapped states and the phase diagrams of the five trapped BEC in zero-phase mode andπ-phase mode are obtained. We find that, the criterions and the phase diagrams are largely modified by the dimension of the system and the phase difference between wells.
Chapter IV: The tunneling dynamics of a BEC loaded into 1D, 2D and 3D deep optical lattices are studied both analytically and numerically. We focus on the self-trapping state and the effect of the system dimension. Under the tight-binding approximation, we obtain an analytical criterion for the self-trapped state of BEC using time-dependent variational method. The phase diagram for self-trapping, soliton, breather, or diffusion of the BEC cloud is obtained accordingly and verified by directly solving the discrete Gross-Pitaevskii equation (GPE) numerically. Our results show that, the change of the lattice dimension leads to many interesting consequences: self-trapped state can also emerge in higher-dimension systems, however, compared to the 1D situation its parameter domain in the phase diagram is largely shrunken; the stable moving soliton and breather solutions of the wave packet that can exist in 1D system, no longer stand in 2D and 3D systems; In addition, when the self-trapping occurs, the ratio between the final BEC cloud width and the initial wave packet width is greatly modified in 2D and 3D systems compared to the 1D case.
Chapter V: We predict the existence of self-trapping, stable moving soliton and breather of Fermi wave packet along the BEC-BCS crossover in 1D, 2D, and 3D optical lattices. The dynamical phase diagrams for self-trapping, stable moving soliton and breather of the Fermi matter waves along the BEC-BCS crossover are presented analytically and verified numerically by directly solving a discrete nonlinear Gross-Pitaevskii equation. Our results show many new and interesting consequences: the self-trapped state of Fermi wave packet can exist in 1D, 2D, and 3D optical lattices along the BEC-BCS crossover; the stable moving soliton and breather solutions of Fermi wave packet along the BEC-BCS crossover in both 1D and 2D system can exist, while for 3D case, the stable moving soliton and breather solutions can exist only in the BCS superfluid. This point is totally different from the Bose system (in Bose system, the stable moving soliton and breather solutions can exist only in 1D optical lattices); the self-trapped, stable moving soliton, and breather states take place in the BCS superfluid always easier than that in the BEC superfluid. Moreover, the critical conditions for the occurrence of self-trapping, stable moving soliton, and breather solutions in both BEC and BCS superfluids increase sharply with lattice dimension.
Finally, in the sixth chapter of this paper, we summarize the main results and give some outlooks.
第一章简单介绍了BEC的概念和实现、光晶格的产生和光晶格中BEC的研究进展。
第二章分别利用两模近似和紧束缚近似的方法,结合变分原理,研究了两体和三体相互作用下一维光晶格中BEC的动力学特性,系统地分析了三体相互作用对BEC的能带结构、能量和动力学不稳定性、以及局域化和自囚禁现象的影响。结果表明,三体相互作用对BEC能带结构和粒子流密度有很大影响,即粒子数存在一临界值,且此临界值与三体相互作用强度密切相关。只有当粒子数大于这一临界值时,在基态能带的第一布里渊区边界处能级才会出现loop结构。同时,粒子流密度不仅与粒子数、两体相互作用强度和光晶格的特性有关,还与原子之间三体相互作用强度有关;经过理论和数值分析比较发现,原子数和光晶格深度,特别是原子间三体相互作用对Bose系统发生能量和动力学不稳定性的边界条件有着显著的影响;由于原子间三体相互作用的出现,BEC发生扩散、自囚禁态、运动孤子态和呼吸子态的临界条件以及它们在相图中所对应的区域都发生了很大的变化。此外,在自囚禁现象发生以后,波包的最大宽度和最小宽度也与原子间三体相互作用密切相关。
第三章通过构造一个简单的五阱模型,解析和数值地分析了二维光晶格中BEC发生自囚禁现象的临界行为。结果表明,自囚禁现象在五阱模型中也可以发生。但是与两阱和三阱的情况相比,出现自囚禁态的临界参数值发生了改变;阱与阱之间的相位差对BEC的隧穿动力学性质有着显著的影响,尤其对三阱和五阱模型的影响更大。
第四章将一维深光晶格中BEC隧穿动力学性质的研究推广到三维情况,着重分析了光晶格的维数对BEC发生自囚禁和局域化现象的影响。通过求解控制三维深光晶格中BEC隧穿动力学性质的离散G-P方程,解析地得到了体系出现自囚禁态、稳定运动的孤子态和呼吸子态的临界条件,并给出了相应的数值验证。研究结果表明,自囚禁态在二维和三维光晶格中也存在,但是与一维情况相比,控制其发生的临界参数值发生了很人改变;并且发现稳定运动的孤子和呼吸子态只在一维光晶格中存在,在二维和三维光晶格中不存在;当自囚禁态出现以后,光晶格的维数对BEC原子云的最终波包宽度有着显著的影响。
第五章系统分析了超流费米气体在一维、二维和三维深光晶格中发生自囚禁和局域化现象的临界行为,系统讨论了超流费米气体在光晶格中的隧穿动力学性质。结果表明,费米波包的自囚禁态可存在于一维、二维和三维光晶格中,并且存在于BEC—BCS整个渡越区。其次,稳定运动的孤子态和呼吸子态在一维、二维和三维光晶格中都存在。但是,在一维和二维体系中,稳定运动的孤子态和呼吸子态可存在于BEC—BCS整个渡越区;而在三维体系中,稳定运动的孤子态和呼吸子态仅存在于BCS超流体中。进一步研究表明,当体系从分子BEC超流体向BCS超流体过渡时,发生自囚禁态、稳定运动的孤子态和呼吸子态的临界条件发生了很大的变化。相比而言,自囚禁态、孤子态和呼吸子态在BCS超流体中比在分子BEC超流体中更容易发生。此外,无论是在分子BEC超流体中还是在BCS超流体中,光晶格维数越高,发生自囚禁态、稳定运动的孤子态和呼吸子态所需要的临界参数值越大。
第六章给出总结和展望。
The research of the tunneling dynamics of Bose-Einstein Condensation (BEC) loaded into optical lattices have attracted enormous attention. One of the important discovery is that the energy band structure, the stabilities, the localized and self-trapped states of BEC are influenced dramatically by the atoms interactions. However, most previous studies are mainly focused on considering the two-body interaction in low dimension system. Indeed, the real behaviors of BEC will be influenced dramatically by the three-body interactions and the lattice dimension. Thus, one aim of this thesis is to investigate the tunneling dynamics of BEC in one dimensional optical lattices with two and three-body interactions. The influences of the three-body interaction on the energy band structure, the stabilities, the localized and self-trapped states are discussed analytically and numerically. Another aim of this thesis is to study the effect of lattice dimension on the localized and self-trapped states of the system. Furthermore, the tunneling dynamics of superfluid Fermi gas in optical lattices are studied analytically and numerically. We find that the phase diagrams vary greatly along the BEC-BCS crossover; the dynamics of Fermi wave packet are very different from that of Bose wave packet. The major investigations are listed as follows:
Chapter I: The conception, the realization of the BEC, and the progress of studing BEC trapped in optical lattices are reviewed briefly.
Chapter II: By using the two-mode approximation and tight-binding approximation, the dynamical properties, including energy band structure, energetic and dynamical instability, and the tunneling dynamics of a dilute BEC trapped in one-dimensional periodic optical lattices are discussed. It is demonstrated that the lowest energy band can show a loop structure at the boundary of the first Brillouin zone only when the number of atoms is larger than a critical value, which is influenced by the three-body interaction dramatically. The obtained atom fluid density depends on the number of atoms, two-body interaction, the optical lattice parameters, and, especially, the three-body interaction. It is also important to note that, the boundaries of dynamical instability and Landau instability are modified significantly due to the presence of the three-body interactions. In addition, after the three-body interactions are taken into account, the critical conditions for the BEC wavepacket to take place the self-trapping, diffusion, stable moving soliton and breather are all modified dramatically.
Chapter III: Developing a five-well model for describing the tunneling dynamics of BEC trapped in 2D optical lattices. The tunneling dynamics of BEC in this five-well model are investigated both analytically and numerically. We focus on the self-trapped states and the difference of the tunnelling dynamics among two-well, three-well and five-well systems. The criterions for the self-trapped states and the phase diagrams of the five trapped BEC in zero-phase mode andπ-phase mode are obtained. We find that, the criterions and the phase diagrams are largely modified by the dimension of the system and the phase difference between wells.
Chapter IV: The tunneling dynamics of a BEC loaded into 1D, 2D and 3D deep optical lattices are studied both analytically and numerically. We focus on the self-trapping state and the effect of the system dimension. Under the tight-binding approximation, we obtain an analytical criterion for the self-trapped state of BEC using time-dependent variational method. The phase diagram for self-trapping, soliton, breather, or diffusion of the BEC cloud is obtained accordingly and verified by directly solving the discrete Gross-Pitaevskii equation (GPE) numerically. Our results show that, the change of the lattice dimension leads to many interesting consequences: self-trapped state can also emerge in higher-dimension systems, however, compared to the 1D situation its parameter domain in the phase diagram is largely shrunken; the stable moving soliton and breather solutions of the wave packet that can exist in 1D system, no longer stand in 2D and 3D systems; In addition, when the self-trapping occurs, the ratio between the final BEC cloud width and the initial wave packet width is greatly modified in 2D and 3D systems compared to the 1D case.
Chapter V: We predict the existence of self-trapping, stable moving soliton and breather of Fermi wave packet along the BEC-BCS crossover in 1D, 2D, and 3D optical lattices. The dynamical phase diagrams for self-trapping, stable moving soliton and breather of the Fermi matter waves along the BEC-BCS crossover are presented analytically and verified numerically by directly solving a discrete nonlinear Gross-Pitaevskii equation. Our results show many new and interesting consequences: the self-trapped state of Fermi wave packet can exist in 1D, 2D, and 3D optical lattices along the BEC-BCS crossover; the stable moving soliton and breather solutions of Fermi wave packet along the BEC-BCS crossover in both 1D and 2D system can exist, while for 3D case, the stable moving soliton and breather solutions can exist only in the BCS superfluid. This point is totally different from the Bose system (in Bose system, the stable moving soliton and breather solutions can exist only in 1D optical lattices); the self-trapped, stable moving soliton, and breather states take place in the BCS superfluid always easier than that in the BEC superfluid. Moreover, the critical conditions for the occurrence of self-trapping, stable moving soliton, and breather solutions in both BEC and BCS superfluids increase sharply with lattice dimension.
Finally, in the sixth chapter of this paper, we summarize the main results and give some outlooks.
引文
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