在量子資訊的領域中，有一部份的人致力於研究如何延長光子的
壽命(Lifetime) 和光子的量子相干性(Quantum coherence)，延長光子的
壽命的目的是希望光子在長途傳輸下，光子仍然可以存活，而延長光
子的量子相干性是因為無論是在量子計算(Quantum computing) 或量
子傳輸(Quantum teleportation) 上都需要仰賴光子的量子相干性才有辦
法進行。而目前已經發展出很多延長光子的壽命和量子相干性的方
法，如EIT 系統、次輻射陣列(Subradiant array) 系統或量子芝諾效應
(Quantum Zeno effect) 等等。
我們主要關注在量子芝諾效應下，一個系統的壽命和量子相干性
可不可以被延長。根據量子電動力學(Quantum Electrodynamics)，一
個處在激發態(Excited state) 的電子一定會隨時間衰退到基態(Ground
state)。在量子芝諾效應下，如果對一個不穩定的系統作快速連續測
量，那麼這個系統便不會隨時間演化，也就是說這個系統不會隨時間
衰退，因達到延長系統的壽命的效果。此外，如果把一個系統放在開
放系統(Open system)，則系統因為環境的去相干化(Decoherence)，而
使得系統的量子相干性隨時間衰退，但量子芝諾效應亦可以藉由連續
測量去破壞環境的去相干化，以達到降低系統的量子相干性隨時間衰
退的效果。
我們選擇要測量的系統是由N 個二能階發射子(N Two-level emitters
N-TLE) 構成的系統，並讓該系統處在超幅射態(Superradiance state)。
除了利用量子芝諾效應去看該系統的狀態的衰退率外，我們主要關注
於量子相干性的時間是否能夠被延長，而選擇超幅射態的原因是超
幅射態是一種疊加態，而疊加態的形成是因為量子相干性。而在看
N-TLE 系統的量子相干性和考慮量子芝諾效應前，我們會先歸納出處
在超幅射態得N-TLE 在各種參數下的現象，以預測之後的結果。
有很多方法可以讓我們知道系統的量子相干性，如：LG 不等式，
Bell 不等式或CHSH 不等式等等，而我們主要以量子見證(Quantum
witnessing) 來看N-TLE 系統的量子相干性如何隨時間演化，而量子見
證跟其他不等式相比，不論在理論上及實驗上都較容易觀察到N-TLE
系統的量子相干性。
本文會先從處在超幅射態的N-TLE 系統出發，去看超幅射態在不
同實驗參數下，超幅射態隨時間的演化。接著，考慮超幅射態在不同
實驗參數下，系統的量子相干性隨時間的變化。最後，在量子芝諾效
應下且不同實驗

In the region of Quantum information, some scientists focus on how to
extend the lifetime and the Quantum coherence of the photon. Because of
the dissipation during the teleportation,we have to extend the lifetime of the
photon in order to make it propagate much longer. On the other hand, Quantum
computing and Quantum teleportation strongly depend on the Quantum
coherence of the photon. As long as the Quantum coherence of the photon
dissipates, Quantum computing and Quantum teleportation are invalid. However,
the scientists have found some methods to extend the lifetime and the
Quantum coherence of the photon such as EIT, Subradiant array or Quantum
zeno effect .etc.
In our research,we work on the Quantum zeno effect which is the suppression
of Hamiltonian evolution of the system by continuous measurements to
extend the lifetime and the Quantum coherence of the system . According to
QED, an electron in excited state must decay to its ground state as time passes
by. However, if we apply rapid and continuous measurements to the system,
the system stop evolving with time. That’s , the system never decays because
of the Quantum zeno effect. Besides, if our system stays in the open quantum
system, the Quantum coherence of our system dissipates due to the decoherence
from the environment. By means of the Quantum zeno effect , we can
still slow down the decay rate of Quantum coherence , because continuous
measurements suppress the decoherence from the environment.
We purpose to measure the N Two-level emitters(N-LTE) system and suppose that the N-LTE system is in the Superradiance state. As mentioned
above, the extending of the coherence time of the system is mainly focused ,
so we intend to choose the Superradiance state which contains strong Quantum
coherence. By classifying several special phenomenons of the Superradiance
state under different parameters, the Quantum coherence of the N-TLE
system and the results of the Quantum zeno effect can be predicted.
There are many ways to observe the Quantum coherence , such as LG inequality,
Bell inequality,CHSH inequality and Quantum witness. We observe
the time evolution of the Quantum coherence in our system mainly by Quantum
witness . Compared with other inequalities, Quantum witness provides
us a simpler way to observe the Quantum coherence not only in experimental
realization but in theoretical analysis.
In this dissertation, we start from studying the time evolution of N-TLE
system in Superradiance state in different parameters. Then observing its time
evolution of Quantum coherence by Quantum witness in different parameters.
Finally, demonstrating the final result of variance in Quantum coherence and
the decay rate of N-TLE system under Quantum zeno effect.

誌謝iii
摘要v
Abstract vii
1 單光子超輻射(Single-photon Superradiacne) 1
1.1 緒論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 理論模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 總結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 見證量子相干性(Witnessing Quantum coherence) 17
2.1 緒論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 理論模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 總結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 量子芝諾效應(Quantum Zeno Effect) 27
3.1 緒論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 理論模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
ix
3.4 結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 總結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 總結與未來工作47
x

[1] R.H.Dicke. Coherence in spontaneous radiation processes. Phys. Rev., 93, 99(1954).
[2] Jun-Tao Chang Anatoly A. Svidzinsky and Marlan O. Scully. Dynamical evolution
of correlated spontaneous emission of a single photon from a uniformly excited cloud
of n atoms. Phys. Rev. Lett., 100, 160504 (2008).
[3] H. Eleuch Z. Yang R.D.Nevels E.A. Sete, A.A. Svidzinsky and M.O.Scully. Correlated
spontaneous emission on the danube. J. Mod. Opt, 57, 1311 (2010).
[4] C. Eichler A. Wallraff J. A. Mlynek, A. A. Abdumalikov. Observation of dicke
superradiance for two artificial atoms in a cavity with high decay rate. Nat. Commun,
5:5186 doi: 10.1038/ncomms6186 (2014).
[5] M.D. Havey I.M. Sokolov S.J. Roof, K.J. Kemp. Observation of single-photon superradiance
and the cooperative lamb shift in an extended sample of cold atoms.
Phys. Rev. Lett., 117.073003(2016).
[6] Robin Kaiser William Guerin, Michelle O. Ara´ujo. Subradiance in a large cloud of
cold atoms. Phys. Rev. Lett., 116.083601(2016).
[7] J Marek. Observation of superradiance in rb vapour. J. Phys. B: At. Mol. Phys, 12
L229(1979).
[8] Lambert N. Li C.M. Chen Y.N. Nori F Chen, G.Y. Delocalized single-photon
dicke states and the leggett-garg inequality in solid state systems. Sci. Rep, 2, 869;
DOI:10.1038/srep00869 (2012).
[9] J. S. Bell. On the einstein podolsky rosen paradox. Physics 1, 195 (1964).
[10] A. Shimony J. F. Clauser, M. A. Horne and R. A. Holt. Proposed experiment to test
local hidden-variable theories. Phys. Rev. Lett., 23, 880 (1969).
[11] A. J. Leggett and Anupam Garg. Quantum mechanics versus macroscopic realism:
is the flux there when nobody looks? Phys. Rev. Lett., 54, 857 (1985).
[12] B Podolsky Einstein A and N Rosen. Can quantum-mechanical description of physical
reality be considered complete? Physical Review, 47 (10): 777–780(1935).
[13] Lambert N. Chen Y.N. Chen G.Y.. Nori F Li, C.M. Witnessing.quantum coherence:
from solid-state to biological systems. Sci. Rep., 2, 885; DOI:10.1038/srep00885
(2012).
[14] B Sudarshan, E. C. G.; Misra. The zeno’s paradox in quantum theory. Journal of
Mathematical Physics, 18 (4): 756–763. (1977).
[15] D. J. Twitchen M. Markham R. Hanson T. H. Taminiau N. Kalb, J. Cramer. Experimental
creation of quantum zeno subspaces by repeated multi-spin projections
in diamond. Nat. Commun, 7, 13111 doi: 10.1038/ncomms13111 (2016).
[16] J. J. Bollinger Wayne M. Itano, D. J. Heinzen and D. J. Wineland. Quantum zeno
effect. Phys. Rev. A, Phys. Rev. A 41, 2295(1990).
[17] S. D. Jenkins S.-Y. Lan T. A. B. Kennedy T. Chanelière, D. N. Matsukevich and
A. Kuzmich. Storage and retrieval of single photons transmitted between remote
quantum memories. Nature, 438, 833 (2005).
[18] S. D. Jenkins G. Facchinetti and J. Ruostekoski. Storing light with subradiant correlations
in arrays of atoms. Phys. Rev. Lett., 117, 243601(2016).
[19] R. L. Zaffino N. Lo Gullo F. Plastina. S. Maniscalco, F. Francica. Protecting entanglement
via the quantum zeno effect. Phys. Rev. Lett., 100, 090503 (2008).