# 量子退相干

## 理论概述

### 冯诺伊曼量子测量纲要

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

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

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

### 约化密度算符

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

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

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

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

### 分辨性

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

### 退相干机制

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

{\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))}

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

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

### 退相干时间尺度

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

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

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

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

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

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

## 实验观察

• 制备出可分辨的几个宏观态或介观态的量子叠加态。
• 设计一套证实量子叠加的方法。
• 量子退相干时间尺度必须足够长久，这样才能正确地观测量子退相干。
• 设计一套监督量子退相干的方法。

### 腔量子电动力学实验

1996年，在法国巴黎高等师范学校，物理学者塞尔日·阿罗什实验团队在腔量子电动力学实验中，首先定量观测到辐射场的介观叠加态的相位相干性逐渐地因量子退相干而被摧毁。[8]

### 量子干涉学实验

2002年，奥地利维也纳大学物理学者安东·蔡林格研究团队发表论文报告观察C70富勒烯干涉行为的结果。C70富勒烯的质量为840amu，直径约为1nm，是由超过1000个微观粒子所组成的相当复杂的物体，因此很不容易观察到量子干涉效应，必须特别使用一种应用塔尔博特效应英语Talbot effect的干涉仪，称为塔尔博特-劳澳干涉仪。碰撞退相干、热力学退相干、振动微扰引起的退相位[注 1]，这几种效应会促使干涉图案的可视性会逐渐衰减。量子退相干可以用可视性的衰减来量度，因此可视性的衰减表征量子退相干效应。[7]:225-226

## 历史

1935年，在普林斯顿高等研究院阿尔伯特·爱因斯坦、博士后纳森·罗森、研究员鲍里斯·波多尔斯基合作完成论文《物理实在的量子力学描述能否被认为是完备的？》，并且将这篇论文发表于5月份的《物理评论[10]:303。这是最早探讨量子纠缠的一篇论文。在这篇论文里，他们详细表述爱因斯坦-波多尔斯基-罗森佯谬，试图藉著一个思想实验来论述量子力学的不完备性质[11]。他们并没有更进一步研究量子纠缠的特性。

## 注释

1. ^ 碰撞退相干指的是C70富勒烯与环境气体分子之间的碰撞而发生的量子退相干。热力学退相干指的是C70富勒烯因发射热力学辐射而发生的量子退相干。干涉仪的衍射光栅会振动，因此造成经典的振动微扰。[7]:225-226

## 参考文献

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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. (编) 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.
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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].
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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].

## 延伸阅读

• 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).