Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

High-order terahertz sideband generation (HSG), recently discovered experimentally in semiconductors, is an extreme nonlinear optical phenomenon with physics similar to high-order harmonic generation (HHG) but in a much lower frequency regime. A key concept in understanding the HSG and HHG is the quantum trajectories, where the quantum evolution of particles under strong fields can be essentially captured by a small number of quantum trajectories that satisfy the stationary phase condition of the Dirac-Feynmann path integral. However, in contrast to HHG in atoms and molecules, HSG in semiconductors can have interesting effects due to nontrivial “vacuum” states of band materials. A rich structure of the Bloch states in condensed matter systems would lead to a variety of phase effects in extreme nonlinear optics. / In this thesis, we show that in semiconductors with nontrivial gauge structures in the energy bands, the curved quantum trajectory of an electron-hole pair under a strong elliptically polarized terahertz field can accumulate a geometric phase. In particular, the geometric phase becomes the famous gauge invariant Berry phase for a cyclic trajectory. Taking monolayer MoS₂ as a model system, we show that the Berry phase appears as the Faraday rotation angle in the pulse emission from the material under short-pulse excitation. This finding reveals the Berry phase effect in the extreme nonlinear optics regime for the first time. / We further apply the Berry phase dependent quantum trajectory theory to biased bilayer graphene under strong elliptically polarized terahertz fields. The biased bilayer graphene with Bernal stacking has similar Bloch band features and optical properties to the monolayer MoS₂, such as the time-reversal related valleys and valley contrasting optical selection rule. However, the biased bilayer graphene has much larger Berry curvature than that in monolayer MoS₂, which leads to a large Berry phase of the quantum trajectory and in turn a giant Faraday rotation of the optical emission (∼ 1 rad for a THz field with frequency 1 THz and strength 8 kV/cm). This surprisingly big angle shows that the Faraday rotation can be induced more efficiently by the Berry curvature in momentum space than by the magnetic field in real space. It provides opportunities to use bilayer graphene and THz lasers for ultrafast electro-optical devices. / Finally, we study the geometric phase of a quantum wavepacket driven adiabatically along a trajectory in a parameterized state space. Inherent to quantum evolutions, the wavepacket can not only accumulate a quantum phase but may also experience dephasing, or quantum diffusion. We show that the diffusion of quantum trajectories can also be of geometric nature as characterized by the imaginary part of the geometric phase. Such an imaginary geometric phase results from the interference of geometric phase dependent fluctuations around the quantum trajectory. As a specific example, we again study the quantum trajectories of HSG in monolayer MoS₂. We find that while the real part of the geometric phase leads to the Faraday rotation of the linearly polarized light that excites the electron-hole pair, the imaginary part manifests itself as the polarization ellipticity of the terahertz sidebands which can be measured experimentally. The discovery of the geometric quantum diffusion extends the concept of geometric phases. / 最近，在實驗上發現了半導體中的一個極端非線性光學現象，即高次太赫茲邊帶產生(HSG)。它是原子与分子系统里的高次谐波产生(HHG)在太赫茲頻域的一個推广。HSG与HHG的關鍵物理過程均可用量子轨道理论解释，其中粒子的路徑積分描述的量子演化由若干滿足穩相近似條件的量子軌道主導。但是HHG与HSG之間存在着本質區別，即半導體的“真空態”可以具備一些非平凡的拓撲結構，從而給極端非線性光學领域帶來許多有趣的物理效應。 / 在這篇論文中，我們發現在強橢圓偏振太赫茲場作用下的具有非平凡规范結構的半導體中，電子空穴對的量子軌道可以積累一個非零的幾何相。特別地，如果我們考慮週期量子軌道，這個幾何相便成為著名的規範不變的Berry相。我們取單層MoS₂為模型系統，發現在光脉衝激勵下的材料中的光信號經歷一個法拉第旋轉，而且轉角由量子軌道的Berry相給出。這個發現首次揭示了極端非線性光學領域內的Berry相效應。 / 我們進一步將含Berry相效應的量子軌道理論應用于強橢圓偏振太赫茲場作用下的雙層石墨烯中。Bernal堆疊的雙層石墨烯与單層MoS₂具有某些相似的能帶結構与光學性質，例如兩者都具有兩個時間反演對稱的谷，且兩個谷內具有不同的躍遷選擇定則。但是雙層石墨烯有遠遠大於單層MoS₂的Berry曲率，從而其內的量子軌道也會積累一個遠遠大於單層MoS₂的Berry相。這個Berry相可以導致光信號巨大的法拉第旋轉（在頻率1THz以及場強8kV/cm的太赫茲場下約為1rad）。這個傳統方法下所無法產生的巨大法拉第旋轉說明比起實空間內的磁場，動量空間內的Berry曲率可以更加有效地誘發光信號的法拉第旋轉。我們的結果可以促使雙層石墨烯以及太赫茲激光在超快光電設備中的應用。 / 最後，我們考慮具有非平凡規範結構的參數空間內的量子波包在絕熱驅動下的量子演化。在演化過程中，這個波包不僅可以獲得一個量子相位，而且會經歷退相干（即量子擴散）。我們發現波包的一部分量子擴散具有幾何性質，而且這部分量子擴散可以表示為一個复幾何相的虛部。這個复幾何相可以通過量子軌道附近的帶有幾何相的量子路徑的相干來解釋。作為例子，我們研究了強橢圓偏振太赫茲場作用下的單層MoS₂中的量子軌道的复幾何相。我們發現此幾何相的實部誘發光的法拉第旋轉，而虛部則表現為邊帶光信號的橢圓偏振度，並且進而可以從實驗上進行測量。我們關於虛幾何相的研究拓展了幾何相這一概念的新領域。 / Yang, Fan = 強太赫茲場下半導體中的量子軌道的Berry相 / 楊帆. / Thesis Ph.D. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 71-75). / Abstracts also in Chinese. / Title from PDF title page (viewed on 13, September, 2016). / Yang, Fan = Qiang tai he zi chang xia ban dao ti zhong de liang zi gui dao de Berry xiang / Yang Fan. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

In this thesis, the critical phenomenon in Berry curvature of nonlinear systems that occurs at phase boundaries is described by using the Bogoliubov excitation of the semiquantal dynamics. Its is shown that when the critical boundary in the parameter space is crossed, the nonlinear geometric phase of the Bogloubov excitations surrounding the elliptic fixed points experiences non-analytic behavior. / 在本論文，我們利用半古典動力學的博戈留波夫激發研究非線性系統的貝里曲率在相邊界上出現的臨界現象。結果顯示，當參數空間中的臨界曲面被越過，環繞橢圓不動點的博戈留波夫激發的非線性幾何相位發生非解析行為。 / Kam, Chon Fai = 非線性系統的貝里曲率 / 甘駿暉. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 49-56). / Abstracts also in Chinese. / Title from PDF title page (viewed on 18, October, 2016). / Kam, Chon Fai = Fei xian xing xi tong de Beili qu lu / Gan Junhui. / Detailed summary in vernacular field only.

Thesis (Ph. D.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 86-94).

published_or_final_version / abstract / toc / Physics / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 134-138).

<p>Boundary quantum phases ---a special type of quantum phenomena--- occur in the boundary part of the system. The boundary part can be a surface of a bulk material, an interface between two distinct system, and even it can be a single impurity or a impurity cluster embedded into a bulk system. The properties of the boundary degree of freedom can be affected by many strong electron correlation effects, mesoscopic effects, and topological effects, which, therefore, induce a vast variety of exotic boundary quantum phases. Many techniques for precise fabrication and measurement in nanostructures had been developed,</p><p>which can provide ways to prob, understand, and control those boundary quantum phases.</p><p>In this thesis, we focus on three types of the boundary quantum phases : Kondo effects, boundary quantum phase transitions, and Majorana fermions. Our motivation is to design and prob those effects by using a important type of nanostructures, i.e. quantum dots. A vast variety of models related to quantum dots (QDs) are studied theoretically, which includes a QD coupled to a mesoscopic bath, a quadruple QD system with metallic leads, a QD with dissipative environments, and a QD coupled to a Majorana fermion zero mode.</p><p>Quantum dots provide a way to study the interplay of Kondo effects and mesoscopic fuctuations. In chapter 5, we consider a model including an Anderson impurity (small QD) coupled to a mesoscopic bath (large QD). Both the weak and strong coupling Anderson impurity problems are characterized by Fermi-liquid theories with weakly interacting quasiparticles. We find that the fluctuations of single particle properties in the two limits are highly correlated and universal : The distributions of the spectrum within the Kondo temperature collapse to universal forms; and the strong coupling impurity changes the wave functions corresponding to the spectrum within the Kondo temperature. </p><p>Quantum dots also bring the possibility to study more complex quantum impurities (multi-QDs) and the competition among dierent interactions, which may induce exotic effects: boundary quantum phase transitions and novel Kondo effects. In chapter 7, we design a quadruple quantum dot system to study the competition among three types of interactions: Kondo, Heisenberg, and Ising. We find a rich phase diagram containing two sharp features : a Berezinsky-Kosterlitz-Thouless type quantum phase transition between a charge-ordered phase and a charge liquid phase and a U(1)XU(1) Kondo state with emergent symmetry from Z2 to U(1). In chapter 8, we study a dissipative resonant level model in which the coupling of a fermionc bath competes with a dissipation-induced bosonic bath. we establish an exact mapping from this dissipative resonant level model to a model of a quantum dot embedded into a Luttinger liquid wire, and we also find two kinds of boundary quantum phase transitions (a Berezinsky-Kosterlitz-Thouless type and a second order type).</p><p>Finally, in chapter 9, we propose an experimental system to detect Majorana fermion zero modes. This system consists of a spinless quantum do coupled to a Majorana fermion which exists in the end of a p-wave superconductor wire. The Majorana Fermion strongly infuence the transport properties of the quantum dot. The zero temperature conductance peak value (when the dot is on resonance and symmetrically coupled to the leads) is e^2/2h. In contrast, if the wire is in its topological trivial phase, the result is e^2/h; if the side-coupled mode is a regular fermionic zero mode, the result is zero. Driving the wire through the topological phase transition causes a sharp jump in the conductance by a factor of 1/2. This result can be used to detect the existence of Majorana fermions.</p> / Dissertation

Matter exists in many different phases, for example in solid state or in liquid phase. There are also phases in which the ordering of atoms is the same, but which differ in some other respect, for example ferromagnetic and paramagnetic states. According to Landau's symmetry breaking theory every phase transition is connected to a symmetry breaking process. A solid material has discrete translational symmetry, while liquid phase has continuous translational symmetry. But it has turned out that there also exist phase transitions that can occur without a symmetry breaking. This phenomenon is called topological order. In this thesis we consider one example of a theoretical model constructed on a two dimensional lattice in which one obtains topological order.

The discovery of topological phases of quantum matter has brought about a new paradigm in the understanding of rich and exotic phases which fall outside the conventional classification of phases using Landau’s theory of broken symmetries. The thesis addresses various aspects of nonequilibrium and semiclassical dynamics in systems hosting such topological phases. While the study of nonequilibrium closed quantum systems is an exciting field in itself, it has gained a lot of importance in the context topological systems. Much of this has been fuelled by the immense progress in the experimental realisation of such topological systems with ultracold atoms in optical lattices. As measurements of real-time responses are natural in such experiments, they have served as ideal platforms to study the nonequilibrium responses of topological systems. The studies presented in this thesis can be brought under the umbrella of two broad questions, first, how non-equilibrium dynamics can be used to characterize topological phases or locate topological critical points, and second, what new topological phases can be realized out of equilibrium. Generally, non-trivial topology of a system manifests itself via quantised responses at the edges of a system or via appropriate non-local string order parameters which are rather difficult to measure in experiments. Local measurements in the bulk are more conducive to experiments. We address this question by showing that within a non-equilibrium setup obtained via a quantum quench, local bulk observables can show sharp signatures of topological quantum criticality via a non-analyticity in parameter space at the critical point. Although via non-local basis transformations, topological phase transitions can often be mapped onto conventional phase transitions, a remarkable aspect of this result is that within the non-equilibrium setup, the local bulk observables can locate the critical point in the natural basis where the phase transition is topological and not described by a local order parameter. The next question that the thesis explores is how nonequilibrium and semiclassical dynamics, more precisely wavepacket dynamics, can be used to probe topological phases with an emphasis on Chern insulators in two dimensions. Chern insulators are essentially similar to quantum Hall systems except that they show quantised Hall responses in the absence of external magnetic fields due to intrinsically broken time-reversal symmetry. The Hall conductance in these systems is related to an integer-valued topological invariant characterising the energy bands, known as the Chern number, which is the net flux of Berry curvature through the entire two-dimensional Brillouin zone. The Berry curvature modifies the semiclassical equations of motion describing the dynamics of a wavepacket. Hence, the real-time motion of a wavepacket is used to map out the Berry curvature and thence the topology of the band. Complementary to these bulk responses, spatially local quenches in Chern insulators are also shown as probes for the presence or absence of chiral edge modes. The idea of semiclassical equations of motion can be extended to the case of a three-dimensional Weyl semimetal. Weyl semimetals are a new class of gapless topological systems in three dimensions, elementary fermionic excitations of which are described by the Weyl equation. Since in cold atom experiments, magnetic fields are realized synthetically via phases in complex hoppings, exploring the Hofstadter limit is a natural scenario. When the magnetic field penetrating a two-dimensional system becomes so large that the associated magnetic length becomes comparable to the lattice spacing, the energy spectrum of the system is described by fractal known as the Hofstadter butterfly. We introduce the Weyl butterfly, a set of fractals which describes the spectrum of a Weyl semimetal subjected to a magnetic field, and we characterize the fractal set of Weyl nodes in the spectrum using wavepacket dynamics to reveal their chirality and location. Moreover, we show that the chiral anomaly -- a hallmark of the topological Weyl semimetal -- does not remain proportional to the magnetic field at large fields, but rather inherits a fractal structure of linear regimes as a function of external field. Finally, the thesis addresses the question of novel nonequilibrium topological phases of matter. In the context of phase structures of nonequilibrium systems, periodically driven, also known as Floquet systems, has received a lot of attention. Moreover, the role of disorder has been shown to be rather crucial as generically such Floquet systems heat up to featureless infinite temperature states. Also, in the context of topological systems like Chern insulators, disorder is expected to play an interesting role given that it is important in localising the bulk cyclotron orbits in an integer quantum Hall system. With this motivation, the phase diagram of the disordered Chern insulator with a Floquet drive is explored in the thesis. In the model considered there are indeed topological Floquet edge modes which are exclusive to Floquet systems, for instance, the edge modes in gaps of the quasienergy spectrum around ±pi. There are also disorder-induced topological transitions between different Floquet topological phases, due to a mechanism shown to be of levitation-annihilation type.

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