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-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 limits.

Figure 1

In both circuits, a LC resonator with capacitance $C_{\mathrm{R}}$ and inductance $L_{\mathrm{R}}$ couples with a semi-infinite transmission line (environment) with capacitance $C_{\mathrm{T}}$ and inductance $L_{\mathrm{T}}$ per unit length. They are coupled by a capacitance $C_{\mathrm{c}}$. The resonator also couples with a Cooper pair box (CPB) with capacitance $C_{\mathrm{J}}$ and Josephson energy $E_{\mathrm{J}}$. Whereas the resonator-CPB coupled system itself is equivalent in both circuits and the resonator-environment coupling is also equivalent, the two circuits are not equivalent as a whole due to the difference of the position of the CPB, which corresponds to the difference of the inductive resonator-CPB interaction (circuit A) and the capacitive one (circuit B).

Figure 2

The resonator-CPB interaction strength is supposed as $\tilde{g}=0.01$ for (a), (c), (e) and $\tilde{g}=0.1$ for (b), (d), (f). (a), (b) The eigenfrequencies of lower and upper modes (dashed and solid curves) in the resonator-CPB system of circuits A and B are plotted versus the bare resonator frequency ${\omega }_0$ (the results are equivalent for the two circuits). (c), (d) The loss rates ${\kappa }_{L/U}^{\mathrm{A}}$ of the two modes are plotted for circuit A in Fig 1. The dashed-dotted curve represents ${\kappa }_{\mathrm{LC}}$ of the bare resonator mode. (e), (f) The loss rates ${\kappa }_{L/U}^{\mathrm{B}}$ for circuit B. The loss rates are normalized to ${\kappa }_{\mathrm{LC}}^0$ at ${\omega }_0={\omega }_x$. Although the two circuits have almost the same loss rates for $\tilde{g}=0.01$, they are significantly different in the ultrastrong-interaction regime $\tilde{g}=0.1$.

Figure 3

Sketch of a Fabry-Perot cavity. A perfect mirror is placed at $z=d$ and a nonperfect mirror is at $z=0$.

Figure 4

A resonator is made by a transmission line with a finite length $d$. It couples with a semi-infinite transmission line by capacitance $C_{\mathrm{c}}$. Artificial atoms can exist inside the resonator except at the contact with the external line.