Bath

The simplest Hamiltonian for a bosonic bath is

$$H_B= \sum_Q \omega_Q a_Q^{\dagger} a_Q.$$ (111)

• Example: free photons, $\omega_Q=c \vert Q\vert$.
• Example: phonons in crystals. However, more realistic Hamiltonians would contain phonon-phonon interaction. In particular, in order to explain the thermal expansion of materials one needs boson-boson interaction ($\leadsto$ `Grüneisen parameter'), cf. N. W. Ashcroft and N. D. Mermin, `Solid State Physics', Saunders College (Philadelphia, 1976).
• Photonic or phononic crystals or cavities have more complicated bandstructures and nontrivial dispersion relations $\omega=\omega_Q$. Example: Phonon cavities in 2-dimensional thin elastic plate, Rayleigh-Lamb waves; cf. L. D. Landau and E. M. Lifshitz, `Theory of Elasticity', Vol. 7 of Landau and Lifshitz, Course of Theoretical Physics (Pergamon Press, 1970); B. Auld, `Acousic Fields and Waves', (Wiley, New York, 1973).