Nonlocalized Generation of Correlated Photon Pairs in Degenerate Down-Conversion

The achievement of optimum conversion efficiency in conventional spontaneous parametric down-conversion requires consideration of quantum processes that entail multisite electrodynamic coupling, actively taking place within the conversion material. The physical mechanism, which operates through virtual photon propagation, provides for photon pairs to be emitted from spatially separated sites of photon interaction; occasionally pairs are produced in which each photon emerges from a different point in space. The extent of such nonlocalized generation is influenced by individual variations in both distance and phase correlation. Mathematical analysis of the global contributions from this mechanism provides a quantitative measure for a degree of positional uncertainty in the origin of down-converted emission.

Figure 1

Schematic for two SPDC mechanisms. Above: Photon annihilation and both photon creation events are colocated at $A$. Below: The annihilation and one creation occur at $A$, the other creation event at another location $B$. Points $A$ and $B$ are coupled via virtual photon exchange, indicated by the double-headed arrow. In the full calculation, all time orderings are taken into account, and the positional geometry is arbitrary.

Figure 2

State sequence diagram accommodating 120 path contributions for two-center down-conversion, with the initial state of the radiation field and matter on the far left and the corresponding final state on the right; intervening entries represent intermediate states; $k$ and $k^{\prime }$ denote input and output mode photons, $p$ a virtual photon. Subscripts on $A$ and $B$ denote electronic states, 0 the ground state, and $r$, $s$ virtual states.

Figure 3

Plot of the nonlocalized contribution to SPDC from the phase- and distance-weighted factor given by Eq. (9), normalized against ${ϵ}_0$, as a function of the cutoff distance $C$. The inset shows the derivative of the main graph.