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Demystifying decoherence and the master equation of quantum Brownian motion
2.Wojciech H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44(10), 36–44 (1991).
3.J. J. Halliwell, “Two derivations of the master equation of quantum Brownian motion,” J. Phys. A 40, 3067–3080 (2007).
6.E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I.-O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed. (Springer, New York, 2003), pp. 66–81.
7.George Greenstein and Arthur G. Zajonc, The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, 2nd ed. (Jones and Bartlett, Sudbury, MA, 2006), 2nd ed., pp. 193–199.
8.Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, 2nd ed. (Cambridge U. P., Cambridge, 2000), pp. 71–75, 98–108.
9.Roland Omnès, Understanding Quantum Mechanics (Princeton U. P., Princeton, 1999), pp. 200–203, 235–256.
10.Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë, Quantum Mechanics (Hermann, Paris, France, 1977), pp. 114–116 501, 646.
11.David J. Griffiths, Introduction to Quantum Mechanics (Pearson Education, Upper Saddle River, NJ, 2005), pp. 56–57, 65, 160–161.
12.Leslie E. Ballentine, Quantum Mechanics: A Modern Development (World Scientific, Singapore., 1998), pp. 42–52, 85–87, 160–162.
13.States that are prepared statistically, called mixtures or mixed states, have diagonal state operators. Such states are inherently non-quantum-mechanical and have no corresponding state vector. To convince oneself of this remarkable fact, try to represent the mixture in Eq. (18) as the product .
14.Jamshid Sabbaghzadeh and Ali Dalafi, “The role of the density operator in the statistical description of quantum systems,” Am. J. Phys. 75(12), 1162–1165 (2007).
15.Background image from Bretislav Friedrich and Dudley Herschbach, “Stern and Gerlach: How a bad cigar helped reorient atomic physics,” Phys. Today 56(12), 53–59 (2007).
16.Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics: Quantum Mechanics (Addison-Wesley, Reading, MA, 1965), pp. 3–13–4, (18–7).
17.John Townsend, A Modern Approach to Quantum Mechanics (University Science Books, New York, 2000), Sec. 6.4.
18.Some authors represent by and refer to it as the probability density.
19.L. V. Tarasov, Basic Concepts of Quantum Mechanics, translated by Ram S. Wadhwa (MIR, Moscow, 1980), pp. 79–84.
20.David Halliday, Robert Resnick, and Jearl Walker, Fundamentals of Physics, 7th ed. (Wiley, Hoboken, NJ, 2005), p. 222.
21.Recall that for an arbitrary matrix , we calculate the trace by setting and summing from 1 to . Similarly, in the continuous case, we set and sum over all possible position values. Hence, we have an integral over all real numbers.
22.Charles Kittel and Herbert Kroemer, Thermal Physics, 2nd ed. (Freeman, San Francisco, 1980), p. 164.
23.Equation (67) is missing the dissipation term, which damps the motion of the system. This damping is not that important in the regime in which we are looking. To develop a master equation with dissipation we would use Wigner’s quasi-probability function in phase-space, improving the accuracy of our approximations (Ref. 29). Further background on the Wigner function is discussed in Refs. 27 and 28.
24.For example, Ref. 6 calculates the decoherence time of a dust particle with radius in air to be about .
25.F. Laloë, “Do we really understand quantum mechanics? Strange correlations, paradoxes, and theorems,” Am. J. Phys. 69(6), 655–701 (2001).
26.C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland, “Decoherence of quantum superpositions through coupling to engineered reservoirs,” Nature (London) 403, 269–273 (2000).
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