Two-photon spectral coherency matrix and multi-parameter optical entanglement

Abstract

We introduce the concept of two-photon spectral coherency matrix and the spectral two-photon Stokes parameters as a counterpart to the classical coherency matrix of broadband polarized light. We discuss its use for characterizing frequency-polarization optical entanglement.The detailed characterization of quantum optical states is important both in fundamental studies of quantum mechanics and in practical quantum information and quantum measurement applications. Techniques for reconstructing polarization density matrix have been developed in the literature for the case of polarization qubits carried by monochromatic single-photon and twophoton entangled states [1]. In the case of two-photon entangled states the two-photon Stokes parameters can be introduced [2].In classical optics the polarization state of broadband light has been considered on several occasion and the spectral coherency matrix and spectral Stokes parameters have been considered [3,4]. We extended these concepts to the quantum optical domain, introducing the spectral twophoton coherency matrix, and the spectral two-photon Stokes parameters.The polarization state tomography for monochromatic photons became a useful tool for characterizing properties of polarization state after it undergoes some transformation. The quantum version of spectral polarization tomography will allow retrieving the spectral and polarization entanglement properties of a broadband radiation after it has been transformed by some optical modulation such as reflection from a surface or propagation through a dispersive material. This technique can find application in quantifying multi-parameter entanglement (hyper-entanglement) when both polarization and frequency degrees of freedom are considered simultaneously. This can help in developing spectroscopic entangled-photon ellipsometry [5] and two-photon coherent control techniques [6]. Moreover, it could be used to characterize quantum states for wavelength division multiplexed QKD when different polarization states are encoded in different spectral channels.