Fig. 1: Model of the proposed hybrid CEP cavity and the few-mode quantization for local density of states.

a Schematic diagram of the proposed hybrid CEP cavity based on the whispering-gallery mode (WGM) cavity, which supports the degenerate clockwise (CW) and counterclockwise (CCW) modes with unidirectional coupling provided by the mirror. Therefore, the hybrid CEP cavity is different from a usual hybrid cavity without CEP where two WGM modes are uncoupled. \({\kappa }_{c}\) is the decay of WGM modes induced by waveguide and \(L\) is the distance between the cavity-waveguide junction and the mirror. b Conceptual sketch of the mapping of the local density of states (LDOS) in a hybrid cavity into a quantized few-mode model, where the cavity QED system can be decomposed into a non-Markovian core and the Markovian bath. The gray dashed lines indicate the structured environment of hybrid cavity which is constituted of continuous electromagnetic modes. c Normalized LDOS of a hybrid CEP cavity for various \(\phi\) (solid lines). The inset shows the logarithmic plot of the normalized LDOS versus frequency and \(\phi\). Normalized LDOS of the usual hybrid cavity is shown for comparison (dashed black line). The parameters are \({\kappa }_{i}={\gamma }_{{{{{{\rm{p}}}}}}}=0\), \({g}_{1}=-20{{{{{\rm{meV}}}}}}\), \({g}_{a}=10{{{{{\rm{meV}}}}}}\), and the plasmon-photon detuning \({\Delta }_{{ac}}={\omega }_{a}-{\omega }_{c}=1{{{{{\rm{eV}}}}}}\). The quality factors of dipolar plasmonic antenna and WGM modes are \({Q}_{a}=18\) and \({Q}_{c}=2\times {10}^{3}\), respectively.