Quantum Nonlinear Optics in Lossy Coupled-Cavities in Photonic Crystal Slabs
Kamandar Dezfouli, Mohsen
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A general formalism is developed that can be used to obtain photon dynamics in coupled-cavity system in leaky photonic crystal slabs. This is accomplished using a non-Hermitian projection operator, where the coupled-cavity modes, known as quasimodes, are used as a basis. Because of this, intrinsic features of these quasimodes such as the leakage and the non-orthogonality are included in a self-consistent manner. The projection technique can be used to represent the Hamiltonian of a typical system in the basis of the quasimodes. In addition, the corresponding quantum Master equation and adjoint quantum Master equation are provided. By employing these, the time dependence of the density matrix and Heisenberg operators can be obtained. In particular, a multimode Jaynes-Cummings Hamiltonian is obtained for photonic crystal slabs interacting with multiple quantum dots. As a proof of principle, a simple system with two quasimodes is considered, where the mode non-orthogonality affects the photon dynamics in a non-trivial manner. It is shown that, while the number of photons in each quasimode decays off, it also oscillates due to the quasimode non-orthogonality. Using the same projection technique, the problem of nonlinear photon pair generation via spontaneous four-wave mixing in photonic crystal slabs is discussed. The main objective is to examine the effect of loss on pair generation in systems such as photonic molecules and coupled-resonator optical waveguides. Several conclusion are made. In addition to the overall loss rates of the pump, signal and idler photons, the loss difference between signal and idler channels plays an important role in minimizing the number of unpaired photon in the system. Also, there is a trade-off between source brightness and higher order generation depending on the losses in the system. This is important, because both the number of unpaired photons and the number of multiple photon pairs degrade device performance. Moreover, when slow light devices are considered, the probability of finding photon pairs at particular locations is affected both by the dispersive behavior of the waveguide and the lossy behavior. This is important as it opens up different possible design strategies that one might want to use in lossy systems.