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

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 tau. The function R(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-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.