Biphoton wave packets in parametric down-conversion: Spectral and temporal structure and degree of entanglement

Abstract

We investigate spectral and temporal features and entanglement of biphoton wave packets formed in spontaneous parametric down-conversion with a pulsed laser pump. The degree of entanglement is characterized by the experimentally measurable parameter $R$ defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration $\ensuremath{\tau}$. The function $R(\ensuremath{\tau})$ is shown to have a minimum and even in the minimum, at rather natural conditions, the degree of entanglement is found to be very high $({R}_{\mathrm{min}}=73)$. The Schmidt number $K$ is found analytically for both short and long pump pulses and interpolated for arbitrary pulse durations. All functional dependences of $R$ and $K$ are found to be identical and numerical coefficients are found to be rather close. Two-time temporal wave function of a biphoton state is investigated in detail, and a rather significant difference between the cases of short and long pump pulses is found to occur. In the case of long pulses, the temporal parameter ${R}_{t}$ (defined as the ratio of durations of the single-particle and coincidence signals) is shown to be very close to the Schmidt number $K$.