Temporal coherence and correlation of counterpropagating twin photons

Abstract

This work analyzes the temporal coherence and correlation of counterpropagating twin photons generated in a\ud quasiphase matched nonlinear crystal by spontaneous parametric down-conversion.We find out different pictures\ud depending on the pump pulse duration relative to two characteristic temporal scales, determined, respectively, by\ud the temporal separation between the counterpropagating and the co-propagating wave packets. When the pump\ud duration is intermediate between the two scales, we show a transition from a highly entangled state to an almost\ud separable state, with strongly asymmetric spectral properties of the photons

Heralding pure single photons: A comparison between counterpropagating and copropagating twin photons

“…At first order, it requires Λ on the same order as the pump wavelength in the medium. The central frequencies of the emitted signal and idler fields, ω s and ω i = ω p − ω s , are thus determined by the poling period Λ and the pump central frequency ω p according to k s − k i = k p − k G (a)counter-propagating case (1) where k j = ωj c n j (ω j ), j = s, i, p are the wave-numbers at the corresponding central frequencies ω j . In the co-propagating geometry ( Fig.1b) all the three fields propagate along the positive z direction.…”

“…(9) and (15) is valid only for small bandwidths, and corresponds to neglecting the temporal dispersion. This is is well justified in the counter-propagating configuration, which involves narrow down-conversion spectra [1,17,22], while it is less justified in the co-propagating case, because of the larger bandwidths in play. In particular, it is not justified when τ s τ − i ( e.g.…”

“…(17) without resorting to any approximation, by using the complete Sellmeier dispersion formula in [26][27][28] for evaluating the wave-numbers. If the linear approximations (9) and (15) are instead employed, the Fourier transform (17) can be explicitly calculated [1], obtaining…”

“…with analogous definitions in the spectral domain. In terms of its temporal G (1) , the reduced state of the heralded photon takes the formρ…”

“…which in general represents a mixed state, unless the coherence function G j (t, t ) is factorable in its arguments. For equal times, G (1) j (t, t) = Â † out j (t)Â out j (t) = I j (t) gives the temporal intensity profile of the j-th wave, i.e. the probabilty distribution of detecting the photon at time t at the crystal exit face.…”