當絶熱地改變哈密頓量的參數時，波函數會獲得一個幾何相位，既 Berry相。它可以表示為參數空間內一個局域的規範場，叫作 Berry曲率。Berry曲率在凝聚態物理的許多領域的研究中起著至關重要的作用，例如量子霍爾效應以及拓撲絶緣體。因此它已成為固體的最基本的性質之一。在量子霍爾效應中，霍爾電導可以表示為 Berry曲率在布里淵區上的積分。這個積分是一個量子化的 Chern數，並且反映了系統的拓撲結構。然而由於時間反演對稱性，拓撲絶緣體的霍爾電導等於零。因此對時間反演不變絶緣體的 Berry曲率的直接以及非破壞性的測量已經成為凝聚態物理中的重要問題。 / 在這篇論文中，我們提出標準的非線性光譜學可以用來探測時間反演不變絶緣體的性質，而且非線性光譜學的實驗比直流實驗更可控。通過計算，我們發現當遠紅外光和 THz光入射到樣品上時，系統的三階光學響應不為零，這與輸運實驗的結果相比形成了鮮明的對比。更重要的是響應函數正比於能帶的非阿貝爾 Berry曲率。這個結果提供了直接測量時間反演不變系統的 Berry曲率的可能性。 / 對具有(近似的 )空間旋轉對稱性的時間反演不變絶緣體，我們發現三階光學響應與等能球面的 Berry曲率通量直接相關。由於 Berry曲率通量給出了能帶簡併點處的奇異單子的拓撲電荷，因此人們可以利用這種方法直接測量能帶的拓撲結構。作為一個例子，這個方法被應用於 III-V族化合物半導體的八帶模型，並給出了一個拓撲電荷為 3的非線性響應。 / Berry phase, a geometric phase acquired by a wave function by adiabatically varying the parameters of the Hamiltonian, can be expressed in terms of a local gauge field in parameter space, called Berry curvature. The Berry curvature plays an essential role in many fields of condensed matter physics, such as the quantum Hall eect and in the study of Topological insulators (TI) and hence it has become one of the most fundamental properties of solids. In Quantum Hall eect, the Hall conductance can be expressed as an integral of the Berry curvature over the Brillouin zone, which is a quantized Chern number and reflects the topology of the system. However in TI, the Hall conductance is equal to zero as a result of the time-reversal (TR) symmetry. Thus, the direct and nondestructive measurement of the Berry curvature of a TR invariant insulator is an important issue in condensed matter physics. / In this thesis, we show that the standard nonlinear optical spectroscopy, being more experimentally controllable than DC experiments, can be used to detect the bulk properties of TR invariant insulators. Through a general calculation, we nd that, when optical and terahertz light fields are employed, the third order optical eect is nonzero compared with the transport method. And the susceptibility is exactly proportional to the non-Abelian Berry curva-ture of the energy band, which provides the possibility of determining Berry curvature directly. / For the TR invariant insulator with (approximate) rotational symmetry, the third order optical susceptibility is related to the the Berry curvature flux through the iso-energy sphere, which gives the topological charge of the monopole at the degeneracy point. Hence it enables one to measure the topo¬logical property of the energy band explicitly. As an example, the method is applied to the eight-band model of III-V compound semiconductors and gives a quantized susceptibility with topological charge equal to 3. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yang, Fan = 時間反演不變絶緣體的Berry曲率的非線性光學測量 / 楊帆. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 77-[80]). / Abstracts also in Chinese. / Yang, Fan = Shi jian fan yan bu bian jue yuan ti de Berry qu lu de fei xian xing guang xue ce liang / Yang Fan. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Introduction of Berry phase --- p.1 / Chapter 1.1.1 --- Basic concepts of the Berry phase and Berry curvature --- p.2 / Chapter 1.1.2 --- Degeneracy and monopole --- p.5 / Chapter 1.1.3 --- Berry phase in Bloch bands --- p.7 / Chapter 1.1.4 --- Non-Abelian Berry curvature --- p.8 / Chapter 1.2 --- Quantum Hall effect and topological insulator --- p.10 / Chapter 1.2.1 --- Anomalous velocity and Quantum Hall effect --- p.11 / Chapter 1.2.2 --- Topological insulator --- p.14 / Chapter 1.3 --- Introduction of nonlinear optics --- p.16 / Chapter 1.3.1 --- Nonlinear optical susceptibilities --- p.16 / Chapter 1.3.2 --- Density matrix formalism --- p.19 / Chapter 1.3.3 --- Diagrammatic analysis of nonlinear optical processes --- p.21 / Chapter 1.4 --- Polarization operator of band electrons --- p.24 / Chapter 1.5 --- Outline of this thesis --- p.26 / Chapter 2 --- Third-order Optical Response of a General Insulator --- p.28 / Chapter 2.1 --- Introduction --- p.28 / Chapter 2.2 --- Microscopic mechanism --- p.30 / Chapter 2.3 --- Third-order nonlinear susceptibility --- p.31 / Chapter 2.3.1 --- A general model --- p.31 / Chapter 2.3.2 --- Perturbative calculation I --- p.35 / Chapter 2.3.3 --- Perturbative calculation II --- p.40 / Chapter 2.3.4 --- Total response --- p.43 / Chapter 2.4 --- Diagrammatic calculation of the third-order response --- p.45 / Chapter 2.5 --- Application to topological insulators --- p.56 / Chapter 2.6 --- Summary --- p.59 / Chapter 3 --- Nonlinear Optical Measurement of Topological Charge --- p.61 / Chapter 3.1 --- Introduction --- p.61 / Chapter 3.2 --- Third-order response with resonant interband transitions --- p.62 / Chapter 3.3 --- Third-order response and topological charge in a rotationally symmetric insulator --- p.66 / Chapter 3.4 --- Quantized susceptibility of III-V compound semiconductors --- p.70 / Chapter 3.5 --- Summary --- p.74 / Chapter 4 --- Summary and Conclusions --- p.75 / Bibliography --- p.77 / Chapter A --- Calculation of equation (2.32) --- p.81 / Chapter B --- Proof of formula (3.20) --- p.89 / Chapter C --- Third-order response with multiple conduction and valence bands --- p.92

Wong Hon Man = 量子力學中之幾何相位 / 黃漢文. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 75-78). / Text in English; abstracts in English and Chinese. / Wong Hon Man = Liang zi li xue zhong zhi ji he xiang wei / Huang Hanwen. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Geometric Phase in Orthogonal States --- p.2 / Chapter 1.2 --- Projective Phase --- p.3 / Chapter 1.3 --- Bose-Einstein Condensate --- p.3 / Chapter 1.4 --- Arrangement of the Thesis --- p.3 / Chapter 2 --- Review of Geometric Phase --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.2 --- Berry's Phase --- p.6 / Chapter 2.3 --- Aharonov-Anadan Phase --- p.9 / Chapter 2.4 --- Pancharatnam Phase --- p.10 / Chapter 2.5 --- Example of a Two-State System --- p.13 / Chapter 2.6 --- Remark on Phase Factor --- p.15 / Chapter 3 --- Review of Geometric Phase near Orthogonal States --- p.16 / Chapter 3.1 --- π Phase Change --- p.17 / Chapter 3.2 --- Off-diagonal Geometric Phase --- p.18 / Chapter 4 --- Projective Phase --- p.21 / Chapter 4.1 --- Projective Measurement --- p.21 / Chapter 4.2 --- Definition in Differential Geometry --- p.22 / Chapter 4.3 --- Gauge Transformation --- p.24 / Chapter 4.4 --- Two-state System and Monopole --- p.26 / Chapter 5 --- π Phase Change --- p.30 / Chapter 5.1 --- Projective Phase near Orthogonal States --- p.30 / Chapter 5.2 --- Two-state system --- p.32 / Chapter 5.3 --- Three-state systems --- p.33 / Chapter 5.4 --- Topological restriction for Spin-1 System --- p.36 / Chapter 5.5 --- Remark on Continuity of Curves --- p.37 / Chapter 6 --- Topological Number Associated with Projective Phase --- p.38 / Chapter 6.1 --- Curves Separated by Zero Dot Products --- p.38 / Chapter 6.2 --- Definition of Phase without 2π arbitrary phase --- p.41 / Chapter 6.3 --- 1st Chern Number Associated with Closed Loops --- p.42 / Chapter 6.4 --- Example with a Spin-m System --- p.43 / Chapter 6.5 --- πphase change --- p.45 / Chapter 6.6 --- Variation of Projection States and Curves --- p.46 / Chapter 7 --- Off-diagonal geometric phase --- p.50 / Chapter 7.1 --- Experiment to Measure Projective Phase --- p.52 / Chapter 8 --- Summary on Geometric Phase near Orthogonal States --- p.53 / Chapter 9 --- Review of Bose-Einstein Condensate --- p.55 / Chapter 9.1 --- Gross-Pitaevskii Equation --- p.55 / Chapter 9.2 --- One-Dimensional GP equation --- p.57 / Chapter 9.3 --- Elementary Excitations --- p.57 / Chapter 10 --- Geometric Phase in Bose-Einstein Condensate --- p.59 / Chapter 10.1 --- Two-state Geometric Phase --- p.59 / Chapter 10.2 --- Geometric Phase in Perturabation --- p.60 / Chapter 10.3 --- Numerical Results --- p.63 / Chapter 10.4 --- Discussions --- p.64 / Chapter A --- Monopole --- p.69 / Chapter B --- Definition of Fibre Bundle --- p.71 / Chapter C --- Evaluation of Bargmann invariant by Pancharatnam Phase Formula --- p.73 / Bibliography --- p.75

published_or_final_version / abstract / Physics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

Not available / text

Not available / text

published_or_final_version / abstract / toc / Physics / Master / Master of Philosophy

No description available.

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.