我們利用Leggett-Garg不等式研究一銣(Rubidium)原子系綜系統的量子效應。我們用腔量子電動力學(Cavity quantum electrodynamics)來描述單光子與此原子系綜系統的交互作用，並考慮不同光學厚度(Optical depth)影響下所對應的擁有不同coupling strengths 以及 cavity loss rates的腔量子電動力學系統。我們在腔量子電動力學系統下考慮兩種情況：coupling strengths約等於或遠大於cavity loss rate，在這兩種情況中我們都有觀察到系統違反了Leggett-Garg不等式的現象，顯示了在原子系綜系統中出現了量子效應。在我們未來的研究中，溫度對於原子系綜系統的影響將會成為研究Leggett-Garg不等式的條件之一。

We utilize Leggett-Garge inequality to study the quantum phenomenon in an atomic ensemble of Rubidium atoms. The interaction between the ensemble and single photons is treated by cavity quantum electrodynamics (cavity QED). Ensembles of different optical depths, corresponding to cavity QED systems with different coupling strengths and cavity loss rates, are considered. In particular, we consider two cases where the coupling strengths are either comparable or much larger than the cavity loss rate. Violation of Leggett-Garge inequality are found in both cases, which demonstrates the presence of quantumness in an atomic ensemble. In future works, effect of temperature of atomic ensembles will be included to study the violation of Leggett-Garge inequality.

摘要……………………………………………………………………………………i
誌謝……………………………………………………………………………………iii
目錄……………………………………………………………………………………v
圖目錄…………………………………………………………………………………vi
第1章 簡介……………………………………………………………………………1
第2章 CHSH不等式v.s. Leggett–Garg不等式
2.1 EPRB experiment…………………………………………………3
2.2 Bell experiment……………………………………………………4
2.3 CHSH不等式：考慮CHSH experiment………………………………7
2.4 Leggett–Garg 不等式…………………………………………………8
2.5 總結………………………………………………………………………10
第3章 計算一維系統的Leggett-Garg 不等式
3.1 Dicke state…………………………………………………………11
3.2 多重量子井結構…………………………………………………………15
3.3 Dicke state–三維系統推廣……………………………………………17
3.4 Dicke state–圓柱形體積系統推廣………………………………………21
第4章 Ensemble原子系統
4.1 Markovian master equation………………………………………………22
4.2 Leggett–Garg 不等式計算………………………………………27
4.3考慮一「切片分段」系統………………………………………………28
第5章 Leggett-Garg不等式對時間作圖之比較 …………………………………30
第6章 討論與總結…………………………………………………………………36
參考文獻………………………………………………………………………………37

1. David Bohm, Quantum Theory (1951).
2. J.S.Bell, Physics 1,195–200 (1964).
3. J. F. Clauser, M. A. Horne, A. Shimony, and R .A. Holt, Phys. Rev. Lett. 23, 880 (1969).
4. Johan E. Mooij, Nature Physics 6, 401–402(2010).
5. A. J. Leggett and Anupam Garg, Phys. Rev. Lett. 54, 857, (1985).
6. Guang-Yin Chen, et al.,Scientific Reports 4, 3580 (2012).
7. R. H. Dicke, Phys. Rev. 93, 99, (1954)
8. Scheibner, M,et al., Nat. Phys. 3, 106–110 (2007).
9. Goldberg, D, et al., Nat. Photon. 3, 662–666 (2009).
10. Zhou, L, et al., Phys. Rev. Lett. 101, 100501 (2008).
11. Neill Lambert, et al., Phys. Rev. A 82, 063840 (2010).
12. Quantum Optics : An Introduction, Mark Fox, Oxford University (2006).
13. J. G. Rarity, Science 301, 604 (2003).
14. A. Aspect, et al., Phys. Rev. Lett. 49, 1804 (1982).
15. Johannes Kofler and Časlav Brukner, Phys. Rev. Lett. 99 (2007).
16. Anatoly A. Svidzinsky, et al., PRL 100, 160504 (2008).
17. Brahim Elattari and S. A. Gurvitz, Phys. Rev. A 62, 032102 (2000).