我的研究主軸是相空間中的物理特性與現象，其中又特別選擇了具有眾多優點的維格納函數作為探討的重點。因為維格納函數帶有負數的量子特性又同時可以表現出同調態的古典性質，使其在眾多相空間中的分布函數顯得特別優異。本研究受到2011年O. Steuernagel博士等人在維格納函數流研究啟發，進而探討各種系統在相空間中的表現。
首先我們在模擬以及實踐頻域分辨光學開關的實驗，相空間的現象不再只是圖面上的模擬，應證了維格納函數是可以模擬對應到光學系統上。重現了W.H. Zurek教授所提出的羅盤態，也證明了其中棋盤結構的干涉條紋在良好的控制下是可以超越測不準原理面積的極限。
再者我們利用維格納函數流探討非埃爾米特哈密頓算符的系統，其中又特別著重於鏡像及時間反轉對稱的系統，實數本徵值跟複數本徵值共存是該系統最大的特色。我們利用維格納函數流中的散度定理證明了在實數與複數能量臨界點是二階相變的現象。
最後我們將工作延伸到原子分子系統，其中又特別著重於SU(2)的系統。在這樣的系統下相空間不再是投影到平面上，而是投影在布洛赫球面，我們遵循了G.S. Agarwal教授建構維格納函數的工作，將一次及二次哈密頓算符的隨時變系統做了完整的建構。相信這對未來原子分子系統中的相空間研究而言是相當重要的基礎工作。

We present phase space studies in this thesis, especially focusing on Wigner distribution function. Because of the negativity reveals quantum phenomenon and coherent state shows classical behaviour, Wigner distribution function is an excellent choice to study in phase space. This work is inspired by the publication of Dr. O. Steuernagel et al. We made several Wigner function flow studies in different systems.
First, we show that frequency resolve optical gating maps in a chronocyclic phase space and the Wigner distribution function. To contract the compass state which proposed by Prof. W.H. Zurek, we have demonstrated the existence of the interference structure that changes on areas smaller than minima uncertainty limit.
Second, we study on non-Hermitian Hamiltonian, especially focusing on the system with parity and time reversal symmetry. Real and complex eigenvalues could exist at the same time in this kind of Hamiltonian. By the analogy of divergence theorem in Wigner function flow, we show that at the energy exception point between real and complex energy is a second order phase transition.
At last, we made the extension to atomic systems, especially on SU(2) systems. Phase space is no longer projecting on x-p plane but on Bloch sphere. We followed the works done by Prof. G.S. Agarwal, who constructed Wigner distribution function for atomic systems. We constructed time dependent Wigner function flow for linear and quadratic Hamiltonians. These are fundamental works for atomic systems in phase space.

誌謝v
Acknowledgements vii
摘要ix
Abstract xi
1 Introduction 1
2 Direct Measurement of Time-Frequency Analogs of Sub-Planck Structures 5
2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Wigner Distribution Function Flow 17
3.1 Continuity Equation and Wigner Flow in Generalized Hamiltonian . . . 17
3.2 Phase-Space Representation of a non-Hermitian System withPT Symmetry
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 Model Hamiltonian forPT -Symmetry Systems . . . . . . . . 20
3.2.2 PT -Symmetry in Wigner Function Representation . . . . . . 24
3.2.3 Wigner Function Flow at the Exceptional Points . . . . . . . . 27
3.3 Wigner Flow in Generalized Quantum Harmonic Oscillator . . . . . . . 29
4 The Wigner Flow on the Sphere 31
4.1 Wigner Function on the Sphere . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Dynamics and Wigner Flow on the Sphere . . . . . . . . . . . . . . . . 34
4.2.1 Linear Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.2 Kerr Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Conclusion 41
A Lauricella Hypergeometric Function 43
References 47

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