Theory of two-photon interactions with broadband down-converted light and entangled photons

When two-photon interactions are induced by down-converted light with a bandwidth that exceeds the pump bandwidth, they can obtain a behavior that is pulselike temporally, yet spectrally narrow. At low photon fluxes this behavior reflects the time and energy entanglement between the down-converted photons. However, two-photon interactions such as two-photon absorption TPA and sum-frequency generation SFG can exhibit such a behavior even at high power levels, as long as the final state i.e., the atomic level in TPA, or the generated light in SFG is narrow-band enough. This behavior does not depend on the squeezing properties of the light, is insensitive to linear losses, and has potential applications. In this paper we describe analytically this behavior for traveling-wave down conversion with continuous or pulsed pumping, both for highand lowpower regimes. For this we derive a quantum-mechanical expression for the down-converted amplitude generated by an arbitrary pump, and formulate operators that represent various two-photon interactions induced by broadband light. This model is in excellent agreement with experimental results of TPA and SFG with highpower down-converted light and with entangled photons Dayan et al., Phys. Rev. Lett. 93, 023005 2004 ; 94, 043602 2005 ; Pe’er et al., ibid. 94, 073601 2005 .