  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 light-matter 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
limit.
