The Impact of the Spectral Filter Bandwidth on the Spectral Entanglement and Indistinguishability of Photon Pairs of SPDC Process

In this paper, we have investigated the dependence of the spectral entanglement and indistinguishability of photon pairs produced by the spontaneous parametric down-conversion (SPDC) procedure on the bandwidth of spectral filters used in the detection setup. The SPDC is a three-wave mixing process which occurs in a nonlinear crystal and generates entangled photon pairs and utilizes as one of the most useful resources in a variety of fields such as quantum computation, quantum cryptography, and quantum communication. The amount of the spectral entanglement and the indistinguishability of photon pairs are the two critical characteristics of the photon pairs determining their potential applications. The degree of the spectral entanglement of a quantum system, i.e. photon pairs, is determined by the entanglement entropy which is a measure of the system disorder. First, we derive the eigenvalue equation of reduced density operator of the signal and the idler photons in terms of the bandwidth of spectral filters. Then, by numerically solving the eigenvalue equation, we calculate the Schmidt coefficients for different values of the bandwidth of spectral filters. Finally, by calculating the entropy operator one can obtain the dependence of spectral entanglement of the photon pairs on filter bandwidth. The amount of indistinguishability of the photon pairs is measured by the visibility of the HOM diagram. Since the whole spectral information of the photon pairs exists in their two-photon mode function, using this function and the presented model we calculate the amount of indistinguishability by obtaining the visibility of the HOM diagram for different values of the bandwidth of the spectral filters. In this way, the dependence of the indistinguishability of the photon pairs on this quantity is reported.