Field quantization for open optical cavities

Abstract

We study the quantum properties of the electromagnetic field in optical cavities coupled to an arbitrary number of escape channels. We consider both inhomogeneous dielectric resonators with a scalar dielectric constant ǫ(r) and cavities defined by mirrors of arbitrary shape. Using the Feshbach projector technique we quantize the field in terms of a set of resonator and bath modes. We rigorously show that the field Hamiltonian reduces to the system-and-bath Hamiltonian of quantum optics. The field dynamics is investigated using the input-output theory of Gardiner and Collet. In the case of strong coupling to the external radiation field we find spectrally overlapping resonator modes. The mode dynamics is coupled due to the damping and noise inflicted by the external field. For wave chaotic resonators the mode dynamics is determined by a non-Hermitean random matrix. Upon including an amplifying medium, our dynamics of open-resonator modes may serve as a starting point for a quantum theory of random lasing.

Route from spontaneous decay to complex multimode dynamics in cavity QED

“…Note, that the form of this Hamiltonian is a bit simpler as compared to the one presented in [34] as we do not consider multiple scattering channels outside the cavity. The Hermitian resonator modes are described by a discrete set of operators a λ and corresponding eigenfrequencies ω λ , whereas the external radiation field corresponds to a continuous set of operators b(ω) and frequencies ω.…”

“…The resonator and external region communicate with each other via the coupling matrix elements W λ (ω) defined as the expectation value of the operator L P Q sandwiched between the resonator and external modes (see Eq. (52a) in [34]). This coupling operator is determined through the Feshbach projection formalism, which consists of separating space in two regions, the resonator Q and the external region P .…”

“…To verify the validity of the above results, we recalculated the temporal decay in all of the three regimes from above using a recently developed system-and-bath approach [34]. Under the rotating-wave and Born approximations this approach can, in principle, also be reduced to a single Volterra equation as in Eq.…”

“…The operators obey the usual canonical commutation relations (see Sec. IID in [34] for more details). The resonator and external region communicate with each other via the coupling matrix elements W λ (ω) defined as the expectation value of the operator L P Q sandwiched between the resonator and external modes (see Eq.…”

Theoretical Analysis of Mechanical Displacement Measurement Using a Multiple Cavity Mode Transducer

“…It will appear in the results as the degree of overcoupling η c = κ 0 /κ tot , with κ tot being the total cavity decay rate [30]. For multiple cavity modes, the output quadrature fluctuations are given by a generalized inputoutput relation [14] X out…”