Recipe for the Hamiltonian of system-environment coupling applicable to the ultrastrong-light-matter-interaction regime

When the light interacts with matters in a lossy cavity, in the standard cavity quantum electrodynamics, the dissipation of cavity fields is characterized simply by the strengths of the two couplings: the light-matter interaction and the system-environment coupling through the cavity mirror. However, in the ultrastrong-lightmatter- interaction regime, the dissipation depends also on whether the two couplings are mediated by the electric field or the magnetic one (capacitive or inductive in superconducting circuits). Even if we know correctly the microscopic mechanism (Lagrangian) of the system-environment coupling, the coupling Hamiltonian itself is in principle modified due to the ultrastrong interaction in the cavity. In this paper, we show a recipe for deriving a general expression of the Hamiltonian of the system-environment coupling, which is applicable even in the ultrastrong-light-matter-interaction regime in the good-cavity and independent-transition limits.