SPDC Coincidence Analysis¶

Introduction

Photon-pair sources based on spontaneous parametric down-conversion (SPDC) are a standard platform for studying nonclassical light and polarization interference. In the degenerate regime, and under appropriate phase-matching and mode-overlap conditions, the emitted state is well described as a two-mode squeezed vacuum (TMSV) in the horizontal and vertical polarization modes. When this state is sent through polarization optics and measured with non-photon-number-resolving detectors, the experimentally relevant quantities are typically coincidence probabilities rather than resolved photon-number statistics. In many practical analyses, coincidence probabilities for such setups are estimated by truncating the TMSV expansion to the lowest few photon-number terms. While this can be sufficient in the weak-pumping regime, it does not provide an exact description of the full state. This limitation becomes especially relevant when higher-order pair contributions are non-negligible or when one wishes to retain a fully analytical treatment in the presence of loss. Here I present the derivation of an exact analytical expression for the coincidence probability of a type-II SPDC-generated TMSV state after a half-wave plate (HWP) and subsequent polarization-resolved detection with bucket detectors. My approach adds the Positive Operator-Valued Measure (POVM) description of lossy bucket detectors to the TMSV state transformed by the HWP. This way, all photon-number contributions are included exactly, and detector loss can be incorporated through mode-dependent efficiencies. The resulting formalism yields compact expressions for the vacuum probabilities entering the coincidence signal and therefore for the coincidence probability itself.