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When the light interacts with matters in a lossy cavity, in the standardcavity quantum electrodynamics, the dissipation of cavity fields ischaracterized simply by the strengths of the two couplings: the light-matterinteraction and the system-environment coupling through the cavity mirror.However, in the ultrastrong light-matter interaction regime, the dissipationdepends also on whether the two couplings are mediated by the electric field orthe magnetic one (capacitive or inductive in superconducting circuits). Even ifwe know correctly the microscopic mechanism (Lagrangian) of thesystem-environment coupling, the coupling Hamiltonian itself is in principlemodified due to the ultrastrong interaction in the cavity. In this paper, weshow a recipe for deriving a general expression of the Hamiltonian of thesystem-environment coupling, which is applicable even in the ultrastronglight-matter interaction regime in the good-cavity and independent-transitionlimit.

A recipe for Hamiltonian of system-environment coupling applicable to ultrastrong light-matter interaction regime