在本论文中我们研究光晶格中超冷玻色子系统的基本性质以及动力学行为，并对其进行一定的量子操控。光晶格中的量子气可以用哈伯德模型进行描述，由于光晶格系统的高度可控性，使得周期驱动的哈伯德模型可以在实验中实现。而理论上高频驱动的哈伯德模型可以利用 Floquet 的理论进行重整化，从而得到新奇的不含时有效哈密顿量。以此为基础我们主要关注周期驱动的哈伯德模型，包括均匀光晶格中的双驱动以及梯度势中的单驱动的模型，研究其中强关联粒子对的动力学行为并提出量子操控的方案。另一方面，利用激光对原子气的作用，可以在晶格体系中实现人造规范场。因此我们还关注了磁通晶格的模型，具体研究了一种磁通双链模型中的布洛赫振荡。此外，我们还研究了磁通 Creutz 模型中的量子相干性和相互作用引起的动力学效应。最后我们还探讨了一种泵浦过程对应的量子化输运性质。我们的希望这些结果可以为更加丰富的多体物理问题的研究提供理论基础。 在第一章中，我们首先对玻色-爱因斯坦凝聚和冷原子气作了介绍，包括其基本性质、实验实现和控制以及其理论描述。其次我们探讨了光晶格在冷原子气系统中的作用，并详细介绍了其理论模型以及一些扩展模型。然后我们回顾了冷原子系统在晶格体系中典型的动力学行为。最后我们以两种经典模型为例介绍了能带拓扑的概念。 第二章至第五章分别为我们的几个主要研究内容和成果，具体介绍如下： （一）我们分别在两种模型中研究强关联粒子对的量子操控。第一种为玻色-哈伯德模型理论框架下的双驱动模型。我们利用共振条件下的辅助隧穿效应，对哈密顿量进行重整化，得到的有效跃迁系数取决于粒子占据数。我们利用选择性隧穿的性质来控制强关联粒子对的运动，通过调节驱动场的振幅，分别在一维和二维均匀光晶格系统中实现粒子对的定向运动。第二种模型为梯度场中的振动晶格模型。我们提出特定的共振条件，同样利用辅助隧穿等量子效应，实现了粒子对的定向迁移。在此基础上，我们提出用粒子对与孤立原子的“碰撞”过程实现量子纠缠的方案。因此周期驱动的晶格模型为量子计算和量子信息提供了研究平台。 （二）我们研究了磁通双链晶格中的布洛赫振荡。在小的梯度势作用下，波包在晶格体系中做周期振荡。而在我们考虑的系统中，这种振荡表现出手征性。并且利用这种动力学行为可以区分迈斯纳相和涡旋相。当加入对角耦合时，其耦合系数大于某个临界值时，系统将经历拓扑相变，这一点同样可以通过动力学行为进行揭示。在拓扑非平庸相中，波包的自旋表示对应方向在一个布洛赫周期内转过一个圆周。而在临界点处，发生带隙闭合，动量态在两能带中交替演化，并导致布洛赫周期加倍。同时非平庸的拓扑性意味着边缘态的出现，而这里的边缘态可以不受手征对称性的保护。我们的研究为判断量子相变与拓扑相变提供了一种动力学的方式。 （三）我们以一种交错磁通的 Creutz 梯子模型为基础，研究了动量空间的约瑟夫森振荡和自俘获效应。其中的交错磁通由复数的跃迁系数给出。在不考虑同格点链间隧穿时，系统存在迈斯纳相和涡旋相。在涡旋相中，最低能带具有两个极小点，可将其看作动量空间的双阱结构。通过引入两个动量态之间的弱耦合，我们展示动量空间的约瑟夫森振荡。在相互作用超过某个临界值时，会出现宏观凝聚体局域在其中一个阱，此时的振荡对应于宏观量子自俘获效应。同时我们将时空间晶格和动量空间双阱各自的 on-site 相互作用进行了联系，这有助于我们定量的研究动量空间的约瑟夫森和自俘获的效应。 （四）我们用拓扑泵浦的量子化性质来解释表面声波引起的量子化电子输运过程。我们将一维量子通道中的门电压描述为一个类高斯型的势垒，并认为表面声波的作用为一维的周期势，并引起布洛赫能带的能谱结构。同时该周期势具有一个随时间绝热演化的相位因子，使得周期势呈一种行波的形式。泵浦过程引起的粒子流和对应填充能带的拓扑不变量陈数有直接联系。基于这样的理解，我们还提出用两束表面声波的叠加作用，可导致非单调的量子化电流平台。 最后在第六章中，我们对本论文的研究成果和意义进行详细的总结，并对后续的研究思路和方向做了展望。

In this thesis, we mainly focus on the basic properties, the dynamics and the quantum control of ultracold Boson gases in optical lattices which are typically described by the Hubbard model. With the high level of controllability of quantum gases in optical lattices, the periodically driven lattice systems are readily realized in cold atom experiments. A novel effective Hamiltonian can be obtained from the rapidly driven model by exploiting the Floquet theory. Thus we consider the Hubbard models with periodic modulations, including a double modulation scheme in a uniform lattice and a single driven model in a tilted lattice. The dynamical behaviors of a correlated particle pair are studied, which reveals the possibilities of controlling the migration of the pair. We also consider a lattice system with artificial gauge fields, in specific, we study the Bloch oscillations in a ladder lattice with magnetic flux. Besides, novel effects induced by quantum coherence and interactions are studied within the Creutz ladder lattice with flux. Finally, we investigate the quantized charge pump by surface acoustic waves in a one-dimensional channel. Our results may form the building blocks for investigating much richer physics in many particle systems. In chapter I, we first introduce the basics of Bose-Einstein condensates and cold atom gases, including their realizations, quantum controls and theoretical descriptions. The basic and extended models of quantum gases in optical lattices are discussed. We then make a brief review of the typical dynamical behaviors of cold atoms in lattice system. Finally we introduce the concept of band topology by taking two models as examples. Chapters from II to V are arranged to present our research and the main results in detail: Quantum controls for correlated particles are investigated within two driving models. Firstly, we study a double modulation scheme within the frame of Bose-Hubbard model. In a resonant condition, the time-dependent Hamiltonian is renormalized, leading to the density-dependent hopping coefficients. We can make use of this property to control the migration of a pair of strongly correlated particles in one- or two-dimensional uniform lattices via properly manipulating the amplitudes of the driven fields. The second model refers to a tilted optical lattice driven by an ac-field. By making use of both the photon-assisted tunneling and coherent destructive tunneling effects in specific conditions, the same migrating motion can be realized. We further propose a scheme of creating entanglement between the particle-pair and a single particle through interacting oscillations. Our model may provide a new building block for investigating quantum computing and quantum information processing with ultracold atoms in optical lattices. Bloch oscillations in a tilted ladder lattice are studied in the presence of artificial magnetic flux. Such oscillations exhibit chiral characteristics. The dynamical features may be used to distinguish the Meissner phase and the vortex phase. By incorporating the diagonal coupling, a topological phase transition occurs as the diagonal hopping rate exceeds a critical value, which can be revealed by the dynamical evolution. The topology of the system is implied by the rotation of the pseudospin polarization in a Bloch period. The pseudospin polarization rotates a full circle in the nontrivial regime. At the critical point, the band gap is closed and the wave packet propagates alternatively in the two bands that leads to a doubled Bloch period. Edge states arise in the region of nontrivial band topology even in the absence of chiral symmetry. Our study may provide a dynamical way to identify the quantum phases and topological transitions in condensed-matter physics. We study the Josephson oscillation and self-trapping in momentum space based on the Creutz ladder model with staggered flux. The unconventional flux is induced by complex tunneling rates along and between the two legs. In the vortex phase, the double-minima band structure is regarded as a double well. By introducing a tunable coupling between the two momentum minima, we demonstrate a phenomenon of Josephson oscillations in momentum space. The condensate density locked in one of the momentum valleys is referred to as macroscopic quantum self-trapping. The on-site interaction of the lattice provides an effective analogy to the double well model within the two-mode approximation which allows for a quantitative understanding of the Josephson effect and the self-trapping in momentum space. Quantized electron pumping by the surface acoustic wave across barriers created by a sequence of split metal gates is interpreted from the viewpoint of topology. The metal gates are described by Gaussian-like barriers. The surface acoustic wave serves as a one-dimensional periodical potential whose energy spectrum possesses the Bloch band structure. The time-dependent phase plays the role of an adiabatic parameter of the Hamiltonian which induces a geometrical phase. The pumping currents are related to the topological invariant Chern numbers of the filled bands. Based on this understanding, we predict a novel effect of quantized but non-monotonous current plateaus simultaneously pumped by two homodromous surface acoustic waves. Finally, a summary of the results in this thesis and an outlook for the future research are included in chapter VI.