- Letter
- Open Access
Quantum Lamb model
Phys. Rev. Research 7, L032004 – Published 7 July, 2025
DOI: https://doi.org/10.1103/9fxx-2x6n
Abstract
H. Lamb considered the classical dynamics of a vibrating particle embedded in an elastic medium before the development of quantum theory. Lamb was interested in how the back action of the elastic waves generated can damp the vibrations of the particle. We propose a quantum version of Lamb's model. We show that this model is exactly solvable by using a multimode Bogoliubov transformation. We find that the exact system ground state is a multimode squeezed-vacuum state, and we obtain the exact Bogoliubov frequencies by numerically solving a nonlinear integral equation. A closed-form expression for the damping rate of the particle is obtained, and it agrees with the result obtained by perturbation theory. The model provides a solvable example of the damped quantum harmonic oscillator.
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (20)
- A. Bachtold, J. Moser, and M. I. Dykman, Mesoscopic physics of nanomechanical systems, Rev. Mod. Phys. 94, 045005 (2022).
- A. N. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications (Springer-Verlag, Berlin Heidelberg, 2003).
- M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
- A. N. Cleland and M. R. Geller, Superconducting qubit storage and entanglement with nanomechanical resonators, Phys. Rev. Lett. 93, 070501 (2004).
- M. Blencowe, Nanoelectromechanical systems, Contemp. Phys. 46, 249 (2005).
- D. P. Clougherty, Quantum sticking of atoms on membranes, Phys. Rev. B 90, 245412 (2014).
- D. P. Clougherty, Infrared problem in quantum acoustodynamics at finite temperature, Phys. Rev. B 96, 235404 (2017).
- M.-H. Chou, H. Qiao, H. Yan, G. Andersson, C. R. Conner, J. Grebel, Y. J. Joshi, J. M. Miller, R. G. Povey, X. Wu, and A. N. Cleland, Deterministic multi-phonon entanglement between two mechanical resonators on separate substrates, Nat. Commun. 16, 1450 (2025).
- K. C. Schwab and M. L. Roukes, Putting mechanics into quantum mechanics, Phys. Today 58(7), 36 (2005).
- Y. Yang, I. Kladarić, M. Drimmer, U. von Lüpke, D. Lenterman, J. Bus, S. Marti, M. Fadel, and Y. Chu, A mechanical qubit, Science 386, 783 (2024).
- H. Lamb, On the peculiarity of the wave-system due to the free vibrations of a nucleus in an extended medium, Proc. London Math. Soc. s1-32, 208 (1900).
- H. Feshbach and Y. Tikochinsky, Quantization of the damped harmonic oscillator, Trans. NY Acad. Sci. 38, 44 (1977).
- A. Caldeira and A. Leggett, Quantum tunnelling in a dissipative system, Ann. Phys. 149, 374 (1983).
- B. Yurke, Quantizing the damped harmonic oscillator, Am. J. Phys. 54, 1133 (1986).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/9fxx-2x6n for derivation details.
- W.-M. Zhang, D. H. Feng, and R. Gilmore, Coherent states: Theory and some applications, Rev. Mod. Phys. 62, 867 (1990).
- A. G. D. Maestro and M. J. P. Gingras, Quantum spin fluctuations in the dipolar Heisenberg-like rare earth pyrochlores, J. Phys.: Condens. Matter 16, 3339 (2004).
- L. C. Biedenharn and H. van Dam, Quantum Theory of Angular Momentum: A Collection of Reprints and Original Papers (Academic Press, New York, 1965).
- X. Ma and W. Rhodes, Multimode squeeze operators and squeezed states, Phys. Rev. A 41, 4625 (1990).
- D. P. Clougherty, Variational approach to atom-membrane dynamics, APL Quantum 1, 046120 (2024).
