Photon–quantum-dot dynamics in coupled-cavity photonic crystal slabs

Abstract

We derive a master equation for the total-system density matrix to treat the evolution of quantum dots and photons in a photonic crystal slab with multiple, coupled, lossy defect cavities. In such systems, when the resonant lossy quasimodes of the system overlap in frequency and space, they are generally nonorthogonal. We find that this nonorthogonality can have important consequences for the system evolution. Our formalism allows for the calculation of multiple-photon effects in the presence of quasimode nonorthogonality. It can be applied to the calculation of strong-and weak-coupling dynamics and to first-and second-order photon correlation functions in a wide variety of complicated coupled-cavity systems. As an example, we apply the formalism to an asymmetric two-cavity photonic crystal slab and demonstrate how quasimode nonorthogonality results in oscillations in the decay of the photons out of the system.

Heisenberg treatment of pair generation in lossy coupled-cavity systems

“…If needed, for complicated structures made of several defects, the tight-binding approach may be employed to obtain the QMs of the coupled-cavity system [72] based on the modes of the individual cavities calculated numerically. In general, QMs overlap in both frequency and space, so they form a nonorthogonal basis; while workable [73], this does bring some complexity to a typical quantum optical calculation of photon dynamics. However, for the purpose of this work, we assume the existence of an orthogonal QM basis, which could be a consequence of the underlying symmetry of the structure of interest.…”

“…As the detailed derivation of our QM approach is rather involved and is presented elsewhere [73], in this work we only outline the general procedure and key results. Using the fact that the QMs can, in principle, be expanded in the basis of true modes, we define a non-Hermitian quantum projector operator to project the system Hamiltonian from the basis of the true modes onto the basis of the QMs.…”

“…As shown earlier [73], the projected Hamiltonian can then be used to construct the master equation by using the Heisenberg equations of motion in the QM basis and then including the proper norm-conserving terms for the density matrix. From this, the corresponding adjoint master equation for a given Heisenberg operator,Â(t), can be derived as [73] dÂ…”

“…Using the fact that the QMs can, in principle, be expanded in the basis of true modes, we define a non-Hermitian quantum projector operator to project the system Hamiltonian from the basis of the true modes onto the basis of the QMs. Following [73], the projected linear Hamiltonian in the QM basis is given byH…”

“…Following previous work by two of us [73], which established a general QM formalism for nonlinear quantum optics, we first project the full Hamiltonian of the system onto the QM basis. We then use the QM representation of the adjoint quantum master equation [73,74] to calculate the time evolution for different operators. Alternatively, one could solve the quantum master equation itself to obtain the time evolution for the photon density matrix.…”