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arXiv:quant-ph/0505076 · Physical Review A

Bloch theory of entangled photon generation in nonlinear photonic crystals

William T. M. Irvine, M. J. A. de Dood, Dirk Bouwmeester

Published Oct 20, 2005• 20 citations
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Abstract

We present a quantum-mechanical description of parametric down conversion and phase matching of Bloch waves in nonlinear photonic crystals. We discuss the theory in one-dimensional Bragg structures giving a recipe for calculating the down-converted emission strength and direction. We exemplify the discussion by making explicit analytical predictions for the emission amplitude and direction from a one-dimensional structure that consists of alternating layers of ${\mathrm{Al}}_{0.4}{\mathrm{Ga}}_{0.6}\mathrm{As}$ and air. We show that the emission is suitable for the extraction of polarization-entangled photons.

DOI
10.1103/physreva.72.043815

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We present a quantum-mechanical description of parametric down conversion and phase matching of Bloch waves in nonlinear photonic crystals. We discuss the theory in one-dimensional Bragg structures giving a recipe for calculating the down-converted emission strength and direction. We exemplify the discussion by making explicit analytical predictions for the emission amplitude and direction from a one-dimensional structure that consists of alternating layers of Al0.4Ga0.6As{\mathrm{Al}}_{0.4}{\mathrm{Ga}}_{0.6}\mathrm{As}Al0.4​Ga0.6​As and air. We show that the emission is suitable for the extraction of polarization-entangled photons.
Nonlinear photonic crystals can be used to provide phase matching for frequency conversion in optically isotropic materials. The phase-matching mechanism proposed here is a combination of form birefringence and phase velocity dispersion in a periodic structure. Since the phase matching relies on the geometry of the photonic crystal, it becomes possible to use highly nonlinear materials. This is illustrated considering a one-dimensional periodic Al0.4Ga0.6As/air structure for the generation of 1.5 microm light. We show that phase-matching conditions used in schemes to create entangled photon pairs can be achieved in photonic crystals.
A description of spontaneous parametric down-conversion in finite-length one-dimensional nonlinear photonic crystals is developed using semiclassical and quantum approaches. It is shown that if a suitable averaging is added to the semiclassical model, its results are in very good agreement with the quantum approach. We propose two structures made with GaN∕AlN\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}GaN∕AlN that generate both degenerate and nondegenerate entangled photon pairs. Both structures are designed so as to achieve a high efficiency of the nonlinear process.
Spontaneous parametric down-conversion in two-dimensional photonic crystal made of semiconductor material with large quadratic nonlinear susceptibility is proposed to generate polarization entangled photon pairs. On one hand, the large quadratic nonlinear susceptibility of AlGaAs crystal insures the high nonlinear conversion efficiency; on the other hand, the abnormal dispersion property of two-dimensional photonic crystal causes the satisfaction of the phase-matching condition. In particular, the dispersion property sensitively depends on the structure of photonic crystal. Then the walk-off between the down-converted photons with orthogonal polarizations can be minimized through appropriate structure parameter design; hence compensation measures to mitigate labeling effect on polarization entangled photon pairs can be eliminated.
We have developed a rigorous quantum model of spontaneous parametric down-conversion in a nonlinear 1D photonic-band-gap structure based upon expansion of the field into monochromatic plane waves. The model provides a two-photon amplitude of a created photon pair. The spectra of the signal and idler fields, their intensity profiles in the time domain, as well as the coincidence-count interference pattern in a Hong-Ou-Mandel interferometer are determined both for cw and pulsed pumping regimes in terms of the two-photon amplitude. A broad range of parameters characterizing the emitted down-converted fields can be used. As an example, a structure composed of 49 layers of GaN∕AlN\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{N}GaN∕AlN is analyzed as a suitable source of photon pairs having high efficiency.
Nonlinear metal-dielectric layered structures are shown to be able to efficiently generate entangled photon pairs using spontaneous parametric down-conversion. Increase of electric-field amplitudes in these structures enhanced by the presence of metal layers is sufficient to compensate for losses inside thin metal layers. As an example, photon pairs emitted from a structure composed of alternating nonlinear dielectric GaN layers and metal Ag layers are analyzed in spectral, temporal, as well as spatial domains. Also, correlations and entanglement between two photons in a pair are determined. Very narrow photon-pair spectra together with strong directionality of photon-pair emission are observed making the photons suitable for photon-atom interactions. Highly enhanced electric-field amplitudes provide high photon-pair generation efficiencies.
Type-II parametric down conversion in nonlinear optical crystals has turned out to be one of the most powerful methods to generate entangled photon pairs, and even allowed a large set of experiments on foundations of quantum mechanics and recent applications in quantum communication. In this paper, we present an experimental investigation of the emission characteristics for down-converted photons in order to investigate the possibility of high brightness sources for correlated photon pairs. Furthermore, we discuss possible ways to increase the collection efficiency for polarization entangled photon pairs.