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arXiv:1709.00116 · Optics Letters

Third-order nonlinearity OPO: Schmidt mode decomposition and tripartite entanglement

Carlos González-Arciniegas, Nicolas Treps, P. Nussenzveig

Published Nov 21, 2017• 11 citations
arXivHTMLPDF

Abstract

We investigate quantum properties of light in optical parametric oscillators (OPOs) based on four-wave mixing gain in media with third-order nonlinearities. In spite of other competing χ(3) effects such as phase modulation, bipartite and tripartite entanglement is predicted above threshold. These findings are relevant for recent implementations of complementary metal-oxide-semiconductor (CMOS)-compatible on-chip OPOs.

DOI
10.1364/ol.42.004865

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We investigate quantum properties of light in optical parametric oscillators (OPOs) based on four-wave mixing gain in media with third-order nonlinearities. In spite of other competing χ(3) effects such as phase modulation, bipartite and tripartite entanglement is predicted above threshold. These findings are relevant for recent implementations of complementary metal-oxide-semiconductor (CMOS)-compatible on-chip OPOs.
The Optical Parametric Oscillator (OPO) has been one of the most versatile source of non-classical states of light. Usual configurations of such devices are a macroscopic second order nonlinear crystals inside an optical cavity. Recently the use of silicon photonics techniques allowed the implementation of high quality factor microcavities and OPOs which include several technological advantages over usual configuration as a small size, bigger bandwidth, CMOS compatibility, facility to engineer the dispersion properties and compatibility with commercial optical fiber communications. Nevertheless the nonlinearity present within these systems is a third order nonlinearity for which theoretical calculations lack in the literature. Here we describe theoretically the quantum properties of the light generated in an OPO with a third order nonlinearity. We showed that the effects of phase modulation (which are not present in the second order nonlinearity) and dispersion are determinant in the way that oscillation and entanglement is produced in the system. Despite of these effects, bipartite and tripartite entanglement is predicted with the use of the Schmidt modes formalism. We also describe the system when there are more modes exited within the cavity and a frequency comb is formed. In such a situation, using again the Schmidt modes formalism, multipartite entanglement was predicted as well.
The Optical Parametric Oscillator (OPO) has been one of the most versatile source of non-classical states of light.Usual configurations of such devices are a macroscopic second order nonlinear crystals inside an optical cavity.Recently the use of silicon photonics techniques allowed the implementation of high quality factor microcavities and OPOs which include several technological advantages over usual configuration as a small size, bigger bandwidth, CMOS compatibility, facility to engineer the dispersion properties and compatibility with commercial optical fiber communications.Nevertheless Contents vii 4.3.2.2 Three Modes Correlations . . . . . . . . . . . . . . . . . .4.3.2.
The quantum features of ultrashort-pulse optical parametric oscillators (OPOs) are investigated theoretically in the nonlinear regime near and above threshold. Viewing the pulsed OPO as a multimode open quantum system, we rigorously derive a general input-output model that features nonlinear coupling among many cavity (i.e., system) signal modes and a broadband single-pass (i.e., reservoir) pump field. Under appropriate assumptions, our model produces a Lindblad master equation with multimode nonlinear Lindblad operators describing two-photon dissipation and a multimode four-wave-mixing Hamiltonian describing a broadband, dispersive optical cascade, which we show is required to preserve causality. To simplify the multimode complexity of the model, we employ a supermode decomposition to perform numerical simulations in the regime where the pulsed supermodes experience strong single-photon nonlinearity. We find that the quantum nonlinear dynamics induces pump depletion as well as corrections to the below-threshold squeezing spectrum predicted by linearized models. We also observe the formation of non-Gaussian states with Wigner-function negativity and show that the multimode interactions with the pump, both dissipative and dispersive, can act as effective decoherence channels. Finally, we briefly discuss some experimental considerations for potentially observing such quantum nonlinear phenomena with ultrashort-pulse OPOs on nonlinear nanophotonic platforms.
We analyze an optical parametric oscillator (OPO) in which cascaded down-conversion occurs inside a cavity resonant for all modes but the initial pump. Due to the resonant cascade design, the OPO present two χ^2 level oscillation thresholds that are therefore remarkably lower than for a χ^3 OPO. This is promising for reaching the regime of an effective third-order nonlinearity well above both thresholds. Such a χ^2 cascaded device also has potential applications in frequency conversion to far infra-red regimes. But, most importantly, it can generate novel multi-partite quantum correlations in the output radiation, which represent a step beyond squeezed or entangled light. The output can be highly non-Gaussian, and therefore not describable by any semi-classical model.
We analyze the three-mode correlation properties of the electromagnetic field in a optical parametric oscillator below-threshold. We employ a perturbative expansion of the It\^o equations derived from the positive-P representation of the density matrix. Using the generalized Cauchy-Schwarz inequality, we investigate the genuine quantum nature of the triple correlations between the interacting fields, since in this case continuous-variable entanglement is not detected by the van Loock-Furusawa criterion [Phys. Rev. A 67, 052315 (2003)]. Although not being a necessary condition, these triple correlations are sufficient evidence of tripartite entanglement. Of course, our characterization of the quantum correlations is applicable to non-Gaussian states, which we show to be the case of the optical parametric oscillator below-threshold, provided nonlinear quantum fluctuations are properly taken into account.