Radiative transport for two-photon light

The propagation of light in disordered media, including clouds, colloidal suspensions, and biological tissues, is generally considered within the framework of classical optics [1]. However, recent experiments have demonstrated the existence of novel effects in multiple light scattering, in which the quantized nature of the electromagnetic field is manifest. These include (i) the transport of quantum noise through random media [2], (ii) the observation of spatial correlations in multiply scattered squeezed light [3,4], (iii) the measurement of two-photon speckle patterns and the observation of nonexponential statistics for two-photon correlations [5,6], and (iv) the finding that interference survives averaging over disorder, as evidenced by photon correlations exhibiting both antibunching and anyonic symmetry [7,8]. Thus there is an interplay between quantum interference and interference due to multiple scattering that is of fundamental interest [9–15] and considerable applied importance. Indeed, applications to spectroscopy [16], two-photon imaging [17–26], and quantum communication [27–29] have been reported. In the multiple-scattering regime, the radiative transport equation (RTE) governs the propagation of light in random media [1]. The RTE is a conservation law that accounts for gains and losses of electromagnetic energy due to scattering and absorption. The physical quantity of interest is the specific intensity I (r,k̂), defined as the intensity at the position r in the direction k̂. The specific intensity obeys the RTE k̂ · ∇rI (r,k̂) + (μa + μs)I (r,k̂) = μs ∫ d2k′[p(k̂′,k̂)I (r,k̂′) − p(k̂,k̂′)I (r,k̂)], (1)