# 量子退相干

## 理論概述

### 馮諾伊曼量子測量綱要

${\displaystyle |\psi _{1}\rangle |E_{i}\rangle \to |\psi _{1}\rangle |E_{1}\rangle }$
${\displaystyle |\psi _{2}\rangle |E_{i}\rangle \to |\psi _{2}\rangle |E_{2}\rangle }$

${\displaystyle |\psi _{i}\rangle =c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle }$

${\displaystyle |\Psi _{i}\rangle =|\psi _{i}\rangle |E_{i}\rangle \to |\Psi _{f}\rangle =c_{1}|\psi _{1}\rangle |E_{1}\rangle +c_{2}|\psi _{2}\rangle |E_{2}\rangle }$

### 約化密度算符

${\displaystyle {\hat {O}}={\hat {O}}_{s}\otimes {\hat {I}}_{e}}$

${\displaystyle \langle O\rangle =Tr({\hat {\rho }}{\hat {O}})=Tr_{s}({\hat {\rho }}_{s}{\hat {O}}_{s})}$

${\displaystyle {\hat {\rho }}_{s}{\stackrel {def}{=}}Tr_{e}\left(|\Psi _{f}\rangle \langle \Psi _{f}|\right)}$

${\displaystyle {\hat {\rho }}_{s}={\begin{pmatrix}|c_{1}|^{2}&c_{1}c_{2}^{*}\langle E_{2}|E_{1}\rangle \\c_{1}^{*}c_{2}\langle E_{1}|E_{2}\rangle &|c_{2}|^{2}\end{pmatrix}}}$

### 分辨性

${\displaystyle |\psi _{f}\rangle =(c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle )|E_{1}\rangle }$

### 退相干機制

${\displaystyle {\hat {\rho }}_{s}={\begin{pmatrix}|c_{1}|^{2}&c_{1}c_{2}^{*}\langle E_{2}|E_{1}\rangle \\c_{1}^{*}c_{2}\langle E_{1}|E_{2}\rangle &|c_{2}|^{2}\end{pmatrix}}}$

{\displaystyle {\begin{aligned}D(x)&=\langle x|{\hat {\rho }}_{s}|x\rangle \\&=|c_{1}|^{2}|\psi _{1}(x)|^{2}+|c_{2}|^{2}|\psi _{2}(x)|^{2}+c_{1}c_{2}^{*}\psi _{1}(x)\psi _{2}^{*}(x)\langle E_{2}|E_{1}\rangle +c_{1}^{*}c_{2}\psi _{1}^{*}(x)\psi _{2}(x)\langle E_{1}|E_{2}\rangle \\&=|c_{1}|^{2}|\psi _{1}(x)|^{2}+|c_{2}|^{2}|\psi _{2}(x)|^{2}+2{\mathfrak {Re}}(c_{1}c_{2}^{*}\psi _{1}(x)\psi _{2}^{*}(x)\langle E_{2}|E_{1}\rangle )\\\end{aligned}}}

${\displaystyle D(x)=|c_{1}|^{2}|\psi _{1}(x)|^{2}+|c_{2}|^{2}|\psi _{2}(x)|^{2}}$

${\displaystyle {\hat {\rho }}_{s}={\begin{pmatrix}|c_{1}|^{2}&0\\0&|c_{2}|^{2}\end{pmatrix}}}$

### 退相干時間尺度

${\displaystyle \langle E_{i}(t)|E_{j}(t)\rangle \propto e^{-t/\tau _{d}}}$

${\displaystyle \langle E_{x}(t)|E_{y}(t)\rangle \propto e^{-\Lambda |x-y|^{2}t}}$

${\displaystyle \tau _{d}=1/(\Lambda \Delta ^{2})}$

${\displaystyle \langle E_{x}(t)|E_{y}(t)\rangle \propto e^{-\Gamma _{tot}t}}$

${\displaystyle \Lambda \approx 10^{20}a^{6}T^{9}\qquad [cm^{-2}s^{-1}]}$

${\displaystyle \Lambda \approx 10^{39}a^{2}T^{3/2}\qquad [cm^{-2}s^{-1}]}$

## 實驗觀察

• 製備出可分辨的幾個宏觀態或介觀態的量子疊加態。
• 設計一套證實量子疊加的方法。
• 量子退相干時間尺度必須足夠長久，這樣才能正確地觀測量子退相干。
• 設計一套監督量子退相干的方法。

### 腔量子電動力學實驗

1996年，在法國巴黎高等師範學校，物理學者塞爾日·阿羅什實驗團隊在腔量子電動力學實驗中，首先定量觀測到輻射場的介觀疊加態的相位相干性逐漸地因量子退相干而被摧毀。[8]

### 量子干涉學實驗

2002年，奧地利維也納大學物理學者安東·蔡林格研究團隊發表論文報告觀察C70富勒烯干涉行為的結果。C70富勒烯的質量為840amu，直徑約為1nm，是由超過1000個微觀粒子所組成的相當複雜的物體，因此很不容易觀察到量子干涉效應，必須特別使用一種應用塔爾博特效應英語Talbot effect的干涉儀，稱為。碰撞退相干、熱力學退相干、振動微擾引起的退相位[註 1]，這幾種效應會促使干涉圖案的可視性會逐漸衰減。量子退相干可以用可視性的衰減來量度，因此可視性的衰減表徵量子退相干效應。[7]:225-226

## 歷史

1935年，在普林斯頓高等研究院阿爾伯特·愛因斯坦、博士後納森·羅森、研究員鮑里斯·波多爾斯基合作完成論文《物理實在的量子力學描述能否被認為是完備的？》，並且將這篇論文發表於5月份的《物理評論[10]:303。這是最早探討量子糾纏的一篇論文。在這篇論文裏，他們詳細表述愛因斯坦-波多爾斯基-羅森佯謬，試圖藉著一個思想實驗來論述量子力學的不完備性質[11]。他們並沒有更進一步研究量子糾纏的特性。

## 註釋

1. ^ 碰撞退相干指的是C70富勒烯與環境氣體分子之間的碰撞而發生的量子退相干。熱力學退相干指的是C70富勒烯因發射熱力學輻射而發生的量子退相干。干涉儀的衍射光柵會振動，因此造成經典的振動微擾。[7]:225-226

## 參考文獻

1. Maximilian A. Schlosshauer. Decoherence And the Quantum-To-Classical Transition. Springer Science & Business Media. 1 January 2007. ISBN 978-3-540-35773-5.
2. Schlosshauer, Maximilian. Decoherence, the measurement problem, and interpretations of quantum mechanics. Reviews of Modern Physics. 2005, 76 (4): 1267–1305. Bibcode:2004RvMP...76.1267S. arXiv:quant-ph/0312059. doi:10.1103/RevModPhys.76.1267.
3. ^ Lidar, Daniel A.; Whaley, K. Birgitta. Decoherence-Free Subspaces and Subsystems. (編) Benatti, F.; Floreanini, R. Irreversible Quantum Dynamics. Springer Lecture Notes in Physics 622. Berlin. 2003: 83–120. arXiv:quant-ph/0301032. Decoherence is the phenomenon of non-unitary dynamics that arises as a consequence of coupling between a system and its environment.
4. ^ Bernard d' Espagnat. Conceptual Foundations of Quantum Mechanics. Advanced Book Program, Perseus Books. 1999. ISBN 978-0-7382-0104-7.
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6. Zurek, Wojciech. Decoherence and the Transition from Quantum to Classical. Physics Today. October 1991, 44 (10): 36. doi:10.1063/1.881293.
7. Daniel Greenberger; Klaus Hentschel; Friedel Weinert. Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy. Springer Science & Business Media. 25 July 2009. ISBN 978-3-540-70626-7.
8. ^ Serge Haroche; 等. Observing the Progressive Decoherence of the 「Meter」 in a Quantum Measurement. Phys. Rev. Lett. 9 December 1996, 77 (24): 4887. doi:10.1103/PhysRevLett.77.4887.
9. ^ nobelpress. Press release - Particle control in a quantum world. Royal Swedish Academy of Sciences. [9 October 2012].
10. ^ Kumar, Manjit. Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality Reprint edition. W. W. Norton & Company. 2011. ISBN 978-0393339888.
11. ^ Einstein, A; B Podolsky; N Rosen. Can Quantum-Mechanical Description of Physical Reality be Considered Complete? (PDF). Physical Review. 15 May 1935, 47 (10): 777–780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777.
12. Schrödinger, Erwin. Die gegenwärtige Situation in der Quantenmechanik (The present situation in quantum mechanics). Naturwissenschaften. November 1935.
13. ^ Trimmer, John. The Present Situation in Quantum Mechanics: A Translation of Schrödinger's "Cat Paradox" Paper. Proceedings of the American Philosophical Society (American Philosophical Society). 10 October 1980, 124 (5): pp. 323–338. JSTOR 986572.
14. ^ Hans, Zeh. On the interpretation of measurement in quantum theory. Foundations of Physics. March 1970, 1 (1): 69–76. doi:10.1007/BF00708656.
15. ^ Zurek, Wojciech. Pointer Basis of Quantum Apparatus: Into What Mixture Does the Wave Packet Collapse?. Physics Review D. 15 September 1981, 24 (6): 1516–1525. doi:10.1103/PhysRevD.24.1516.
16. ^ Zurek, Wojciech. Environment-Induced Superselection Rules. Physics Review D. 15 October 1982, 26 (8): 1862–1880. doi:10.1103/PhysRevD.26.1862.
17. ^ Zurek, Wojciech. Reduction of the Wavepacket: How Long Does it Take?. 2003. arXiv:quant-ph/0302044v1 [quant-ph].

## 延伸閱讀

• Mario Castagnino, Sebastian Fortin, Roberto Laura and Olimpia Lombardi, A general theoretical framework for decoherence in open and closed systems, Classical and Quantum Gravity, 25, pp. 154002–154013, (2008).