  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
pulse-like 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 narrowband 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 travelling-wave down-conversion with continuous or pulsed pumping, both for
high- and low-power 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 high power down-converted light and with entangled photons [Dayan
et al., Phys. Rev. Lett. 93, 023005 (2004), Dayan et al., Phys. Rev. Lett. 94,
043602, (2005), Pe'er et al., Phys. Rev. Lett. 94, 073601 (2005)].
