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Physical Review A

Heisenberg treatment of pair generation in lossy coupled-cavity systems

Mohsen Kamandar Dezfouli, Marc M. Dignam, Michael J. Steel, John E. Sipe

Published Oct 17, 2014• 14 citations
Publisher

Abstract

We employ a quasimode representation of the adjoint master equation to study nonlinear pair-generation dynamics via spontaneous four-wave mixing in coupled-cavity systems in the presence of scattering loss. Perturbative analytic solutions for signal and idler photon-number operators are derived, which enable us to examine the effect of scattering loss on pair-generation dynamics. In particular, for a simple three-mode system, we show that any loss mismatch for the signal and idler photons increases the probability of finding unpaired photons. While decreasing the loss rates for signal and idler photons increases the pair signal brightness, it also results in higher generation rates for multiple pairs in the system, which can result in lower operational signal-to-noise ratios.

DOI
10.1103/physreva.90.043832

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We employ a quasimode representation of the adjoint master equation to study nonlinear pair-generation dynamics via spontaneous four-wave mixing in coupled-cavity systems in the presence of scattering loss. Perturbative analytic solutions for signal and idler photon-number operators are derived, which enable us to examine the effect of scattering loss on pair-generation dynamics. In particular, for a simple three-mode system, we show that any loss mismatch for the signal and idler photons increases the probability of finding unpaired photons. While decreasing the loss rates for signal and idler photons increases the pair signal brightness, it also results in higher generation rates for multiple pairs in the system, which can result in lower operational signal-to-noise ratios.
We theoretically model pair generation and evolution via the nonlinear process of spontaneous four-wave mixing in a coupled-resonator optical waveguide in a photonic crystal slab. Using the adjoint master equation for a system of lossy coupled cavities, we calculate a symmetrized second-order coherence function to determine pair detection probability. We find that the scattering loss can have as large an effect on pair generation as waveguide dispersion. In particular, the wave-vector dependence of the loss can shift the frequency of the maximum detection probability and therefore cannot, in general, be ignored or treated via a simple overall loss factor.
We present a detailed study of the generation of photon pairs by spontaneous four-wave mixing in a bicavity system with a single atom inside. Cavity modes lead to larger coupling strength than that of the normally assumed thermal vacuum, and thus higher generation rates are expected. Unlike approaches that have emerged so far, this paper presents an approach based on the solution of the master equation. This method allows us to investigate the photon statistics more strictly, and we show that the choices of pumping detuning and leaking rate of the cavities significantly affect the degree of the autocorrelation, and a higher-purity heralded single photon source is possible. We also derive a concise expression for the output state and the generation rate for the extreme cases where the cavity leaking rate is much smaller or larger than the rate of spontaneous emission.
We develop a general Hamiltonian treatment of spontaneous four-wave mixing in a microring resonator side-coupled to a channel waveguide. The effect of scattering losses in the ring is included, as well as parasitic nonlinear effects including self- and cross-phase modulation. A procedure for computing the output of such a system for arbitrary parameters and pump states is presented. For the limit of weak pumping an expression for the joint spectral intensity of generated photon pairs, as well as the singles-to-coincidences ratio, is derived.
We investigate entangled photon-pair generation in a lossy microring resonator using an input-output formalism based on the work of M. G. Raymer and C. J. McKinstrie [Phys. Rev. A 88, 043819 (2013)] and P. M. Alsing et al. [Phys. Rev. A 95, 053828 (2017)] that incorporates circulation factors that account for the multiple round-trips of the fields within the cavity. We consider the nonlinear processes of spontaneous parametric down-conversion and spontaneous four-wave mixing, and we compute the generated biphoton signal-idler state from a single-bus microring resonator, along with the generation, coincidence-to-accidental, and heralding efficiency rates. We compare these generalized results to those obtained by previous works employing the standard Langevin input-output formalism.
We show that the light generated via spontaneous four-wave mixing or parametric down conversion in multiple, coupled, lossy cavities is a multimode squeezed thermal state. Requiring this state to be the solution of the Lindblad master equation results in a set of coupled first-order differential equations for the time-dependent squeezing parameters and thermal photon numbers of the state. The benefit of this semianalytic approach is that the number of coupled equations scales linearly with the number of modes and is independent of the number of photons generated. With this analytic form of the state, correlation variances are easily expressed as analytic functions of the time-dependent mode parameters. Thus, our solution makes it computationally tractable and relatively straightforward to calculate the generation and evolution of multimode entangled states in multiple coupled, lossy cavities, even when there are a large number of modes and/or photons.
We present two complementary strategies for modeling nonlinear quantum optics in realistic resonant integrated optical devices, where scattering loss is present. In the first strategy, we model scattering loss as an effective absorption; in the second, we employ a Hamiltonian treatment, with the loss modelled as a coupling to a 'phantom' channel. As an example, we use these two approaches to model spontaneous four-wave mixing in (i) a ring-channel system and (ii) an add-drop system. We present the rates of photon pairs, broken pairs, and lost pairs for both systems, as well as the full biphoton wavefunction (BWF) including the effects of scattering. We find that for a high-finesse resonator coupled to an arbitrary number of channels, the full BWF can be extracted from the BWF associated with photon pairs in an experimentally accessible output channel.