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 τ. The function R (τ) 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 (Rmin =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 Rt (defined as the ratio of durations of the single-particle and coincidence signals) is shown to be very close to the Schmidt number K.