臺灣博碩士論文加值系統

因為傳播速度的優勢與和環境交互作用力較弱的特性，光子，又名飛行位元成
為了量子通訊與量子演算中，科學家致力發展的資訊載體之一。然而，一個由光子
作為資訊載體所構成的量子網路中的各個量子節點(如: 邏輯閘、偵測器、光纖、記
憶體等等) 所響應的頻率，都是不相同的。為了讓所有節點都能在高效率下運行，以
增進運算或通訊的效率，如何實現一個高效率的量子轉頻器成為了全光學量子網路
中一個至關重要的問題。在我的博士論文中，我們提出並實現了一個基於共振量子
非線性光學的高效率量子頻率轉換器。有別於遠離共振型的非線性轉頻過程，這類
型的頻率轉換器利用共振引發的光與材料間的強交互作用力，有效降低了達到高轉
換效率所需要的介質密度與幫浦光強度，因此，這類型轉頻器在高轉換效率的條件
下，不會誘發額外的非線性噪音，這使得這類型的轉頻器具有相當高的保真度。此
外，運用基於電磁波引發透明的四波混頻效應，本應由共振系統所引起的自發輻射
耗損，也能被有效的抑制。結合這兩個特性，使得我們可以在這類型的系統中享受
帶來的強交互作用(低噪音) 的優勢的同時，不需要付出高自發耗損的代價。藉由這
類型的系統，我們就可以完成一個高效率、高保真度的量子頻率轉換器了。在理論
分析上，我們使用一般儲層理論(general reservoir theory) 來研究基於共振四波混頻的
雙Λ 結構量子頻率轉換，根據我們的理論模型，我們發現這類型的系統確實可以藉
由電磁波引發透明來抑制自發輻射的現象。由分析的結果我們發現，當轉換效率接
近100% 的時候，入射頻率與轉換頻率的量子特性是幾乎完全一樣的，也就是保真
度接近1。此外，我們的理論也有給出多頻率模態的分析。根據我們的研究，我們發
現200 奈秒時間寬度的單光子波包可以在光學深度(OD) 300 的條件下，達到92.4%
的轉換效率與對應0.96 的保真度。在實驗上，我們展示了基於電磁波引發透明之反
向四波混頻的實驗結果。結果指出，91.2% 的轉換效率可以在光學深度130 的情況
下以87Rb 的原子完成。我的博士論文主要是在理論與實驗上完成了深入分析一套基
於共振型量子非線性光學且低噪音高轉換效率的量子轉頻器。這個研究提供了未來
一套有潛力被當作高效率量子轉頻器且用來完成量子通訊，與量子運算的實驗方法。

Due to the advantages of fast propagation and weak interaction with the environment, photons or flying qubits have become one of the excellent quantum carriers that scientists are committed to developing in quantum communication and quantum computing. However, in a quantum network connected by photons, the wavelengths of photons applied to various quantum nodes are quite different. To enable all nodes to work under optimal conditions, the realization of efficient quantum frequency conversion (QFC) becomes a key issue in the construction of optical quantum network. In this doctoral dissertation, we propose and demonstrate a high-efficiency QFC based on resonance-type quantum nonlinear optics. This resonant QFC uses a four-wave mixing (FWM) process based on the electromagnetically induced transparency (EIT) mechanism. Unlike far-resonant nonlinear systems, this resonant EIT-based FWM requires a relatively small pump light intensity and medium density. This feature makes the EIT-based QFC not cause additional nonlinear processes and accompanying noise photons, making it have an excellent performance in the fidelity of the quantum state of the converted photons. Furthermore, by leveraging the characteristics of EIT, the spontaneous emission loss, which is the shortage of the resonant system, can be considerably
suppressed. Thus, the EIT-based FWM system allows us to take advantage of the strong interaction strength without paying for the spontaneous emission loss, and a high-fidelity low-loss QFC can be implemented. In the theoretical model, we use the general reservoir theory to study a double-Λ QFC based on a resonant FWM. Due to the effect of EIT, this resonant QFC can considerably suppress vacuum field noise; consequently, the converted photon can inherit the quantum state of the input photon with high fidelity. Our research demonstrates that if the conversion efficiency (CE) of the EIT-based QFC is close to 100%, the wave function and quadrature variance of the converted photon are almost the same as
the input probe photon. In addition, the fidelity in the multi-mode case is also studied in our quantum model. Based on our theoretical calculation, a Fock state single photon with a duration of 200 ns can achieve a CE of 92.4% with a fidelity of 0.96 under an optical depth (OD) of 300. In the experiment, we demonstrate the use of EIT-based double-Λ FWM for efficient frequency conversion, and achieve a CE of 91.2% with an OD of 130 in cold 87Rb atoms. This doctoral dissertation has carried out a detailed research on the low-loss, high-fidelity QFC of resonant-type quantum nonlinear optics from both theoretical and experimental aspects.
The current work provides another promising scheme for the future development of efficient QFC applications in quantum communication, network and computing.

摘要 i
Abstract ii
誌謝 iii
Table of Contents vi
List of Tables viii
List of Figures ix
Chapter 1. Introduction 1
1.1 Why we need quantum technology 1
1.2 The breakthroughs of technology utilizing quantum mechanics 3
1.3 Why we need the quantum frequency conversion 7
1.4 How to realize the quantum frequency conversion 8
1.5 The arrangement of this thesis 12
Chapter 2. Semi-Classical Model of Electromagnetically Induced Transparency 13
2.1 Optical-Bloch equation and Maxwell-Schr¨odinger equation 14
2.2 Steady-state solution under the first-order perturbation expansion 35
2.3 Autler-Townes splitting (ATS) 40
2.4 Coherent population trapping (CPT) 49
Chapter 3. Semi-Classical Model of Double-Λ Four-Wave Mixing 55
3.1 Balance condition 56
3.2 All-resonance forward four-wave mixing (FFWM) 60
3.3 Detuned forward four-wave mixing (DFFWM) 75
3.4 Spatially-modulated four-wave mixing (SFWM) 81
3.5 Backward four-wave mixing (BFWM) 85
3.6 Phase-mismatch effect in backward four-wave-mixing system 95
Chapter 4. Experimental Detail 98
4.1 Information of Rubidium atom 98
4.2 Preparation of the 87Rb cold-atom cloud 106
4.3 Experimental data of the backward four-wave mixing 126
Chapter 5. Quantum Model of Backward Four-Wave Mixing 137
5.1 Heisenberg-Langevin equation and Maxwell-Schr¨odinger equation 138
5.2 Single-mode case 158
5.3 Multi-mode case 173
5.4 High conversion efficiency band in backward four-wave mixing 182
Chapter 6. Conclusion and Outlook 192
6.1 Conclusion 192
6.2 Outlook 194
References 197
Appendix A. Quantum Dissipation Theorem 207
A.1 Spontaneous emission of two-level system : Weisskopf-Wigner theory 208
A.2 Master equations : density matrix approach 212
A.3 Heisenberg-Langevin equation - Langevin approach 218
A.4 Comparing the master equation to the Heisenberg-Langevin equation: Damping light field problem 223
A.5 Master equation : Two-level system 225
A.6 Heisenberg-Langevin equation : Two-level system 228
Appendix B. Langevin Equation : Classical Fluctuation-Dissipation Theorem 234
B.1 Model-1 : Dissipation process only 235
B.2 Model-2 : Dissipation process accompanied by the noise process 238
Appendix C. Briefly Introduce of the Continuous Wave Packet 242
C.1 Expression of the fock state pulses 243
C.2 Expression of the Coherent State Pulses 249
Appendix D. Another Useful Description for the Light Pulse 251

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