Abstract
This chapter describes the density operator, which is used to calculate light-matter interaction and the generation of coherent phonons in Chap. 6. The density operator can represent both a pure quantum state and a mixed state. The von Neumann equation for the time evolution of the density operator is derived. The perturbative expansion of its time evolution is explained and represented with double-sided Feynman diagrams. Time evolution in a two-level system is shown as an example.
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Notes
- 1.
This equation corresponds to the Liouville equation for classical mechanics. Then, this is also called the quantum Liouville equation or Liouville–von Neumann equation [2].
- 2.
This interaction corresponds to an dipole interaction between the two-level system and light with an angular frequency of \(\omega \). When we use \(\mu \) as a dipole moment and \(E_0\) as an amplitude of electric field, \(\gamma = \mu E_0\). \(\gamma e^{-i \omega t} {\left| {2}\right\rangle }{\langle {1}\vert }\) and \(\gamma e^{i \omega t} {\left| {1}\right\rangle }{\langle {2}\vert }\) correspond to the excitation process with light absorption and the de-excitation process with emission. In addition, the rotating wave approximation is assumed.
References
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I wrote this chapter referencing Refs. [1–3] and following books: Yoshio Kuramoto and Junichi Ezawa, Ryoushi Rikigaku (in Japanese, Quantum Mechanics), Asakura Shyoten (2008); Keiji Igi and Hikari Kawai, Kiso Ryousi Rikigaku (in Japanese, Fundamental Quantum Mechanics), Kodansya (2007); Takahiro Sunagawa and Masahiro Ueda, Ryoshi Sokutei to Ryoshi Seigyo (in Japanese, Quantum Measurement and Quantum Control), Science Sya (2016), Masahito Ueda Gendai Ryoushi Buturigaku (in Japanese, Modern Quantum Physics), Baifukan (2004); Akira Shimizu, Shin-han Rryoushi Ron no Kiso (in Japanese, New edition Fundamental of Quantum Physics), Science Shya (2004); Kyo Inoue, Kogaku Kei no Tameno Ryoushi Kogaku (in Japanese, Quantum Optics for Engineer), Morikita Shyoten (2015); Masahiro Matsuoka, Ryoushi Kogaku (in Japanese, Quantum Optics), Shokabou (2000)
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Nakamura, K. (2019). Density Operator. In: Quantum Phononics. Springer Tracts in Modern Physics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-11924-9_2
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