Pulsed parametric down-conversion (PDC) processes generate photon pairs with a rich spectral-temporal structure, which offer an attractive potential for quantum information and communication applications. We investigate the four-dimensional chronocyclic Wigner function ${W}_{\mathrm{PDC}}({\ensuremath{\omega}}_{\mathrm{s}},{\ensuremath{\omega}}_{\mathrm{i}},{\ensuremath{\tau}}_{\mathrm{s}},{\ensuremath{\tau}}_{\mathrm{i}})$ of the PDC state, which naturally lends itself to the pulsed characteristics of these states. From this function we derive the conditioned time-bandwidth product of one of the pair photons, a quantity which not only is a valid measure of entanglement between the PDC photons but also allows us to highlight a remarkable link between the discrete- and continuous-variable descriptions of PDC. We numerically analyze PDC processes with different conditions to demonstrate the versatility of our approach, which is applicable to a large number of current PDC sources.
We construct a projection measurement process for the maximally entangled
N-photon state [the NOON state $(\ensuremath{\mid}N,0⟩+\ensuremath{\mid}0,N⟩)∕\sqrt{2}$] with only linear optical elements and photodetectors. This measurement process will give null result for any
N-photon state that is orthogonal to the NOON state. We examine the projection process in more detail for
N=4 by applying it to a four-photon state from type-II parametric down-conversion. This demonstrates an orthogonal projection measurement with a null result. This null result corresponds to a dip in a generalized Hong-Ou-Mandel interferometer for four photons. We find that the depth of the dip in this arrangement can be used to distinguish a genuine entangled four-photon state from two separate pairs of photons. We next apply the NOON state projection measurement to a four-photon superposition state from two perpendicularly oriented type-I parametric down-conversion processes. A successful NOON state projection is demonstrated with the appearance of the four-photon de Broglie wavelength in the interference fringe pattern.
Quantum-state conversion among different information-carrying media is essential in a quantum network. It is found that the many schemes of quantum-state conversion can be unified under one simple process---that is, the parametric process of three-wave mixing. These schemes include quantum-state storage in atomic media, where a photonic state is converted to an atomic Raman state and vice versa, the frequency up- and down-conversion in photonic states, and photonic-state conversion to Brillouin phononic states and vice versa. The efficiency of conversion can in principle reach 100%, and the conversion process is noiseless. When the same idea is applied to a four-wave mixing process, unit conversion efficiency between a two-photon state and a single-photon state can be achieved. When applied to a two-photon entangled state in parametric down-conversion, entangled states of multiple photons can be generated.