Figure 1

Power dependence of the coherent and incoherent parts of two-photon interactions with down-converted light. (a) The relative magnitudes of the coherent signal $I^c$ (black line) vs the incoherent signals for TPA, SFG, and coincidence events ($I_{\mathrm{TPA}}^{\mathit{ic}}$, $I_{\mathrm{SFG}}^{\mathit{ic}}$, and $R{c}^{\mathit{ic}}$, respectively, gray lines), assuming typical physical parameters described in the text, represented on a log-log scale. The graph shows the $n^2+n$ dependence of $I^c$, with $n$ being the average spectral photon density, i.e., the total photon flux in each of the signal and idler modes, divided by their bandwidth $\Delta$. All the incoherent signals demonstrate a quadratic dependence on $n$. The dashed line represents a completely linear function, for comparison. (b) The dependence of $I^c$ on $n$ represented on a linear scale, demonstrating the nearly linear dependence at small values of $n$. The dash-dotted line depicts the quadratic response of $I^c$ assuming that down-converted light with $n=0.2$ is being attenuated by linear losses.Reuse & Permissions

Figure 2

The temporal behavior of the coherent and incoherent parts of two-photon interactions with down-converted light, as a function of a relative delay between the signal and the idler fields. (a) The relative magnitudes of the coherent signal $I^c$ (black line) vs the incoherent signals for TPA, SFG, and coincidence events ($I_{\mathrm{TPA}}^{\mathit{ic}}$, $I_{\mathrm{SFG}}^{\mathit{ic}}$, and $R{c}^{\mathit{ic}}$, respectively, gray lines), assuming typical physical parameters (described in the text), represented on a logarithmic scale. This graph depicts the instantaneous peak of the signals at $t={\tau }_i$ for one typical down-converted pulse, and shows the contrast between the sharp temporal behavior of $I^c$, and the behavior of the incoherent signals, which is on the same ns time scale as the pump pulse. Note that $I_{\mathrm{TPA}}^{\mathit{ic}}$ presents a smooth and symmetric graph, reflecting fact that the long-lived $(\sim 30\mathrm{ns})$ final atomic state integrates over the intensity variations of the pump pulse. Similarly, $R{c}^c$, which takes into account integration over the gating time $Tg\sim 1\mathrm{ns}$ is more smooth than $I_{\mathrm{SFG}}^c$, which represents an instantaneous parametric process. For the case assumed here of a nontransform-limited $3\text{−}\mathrm{ns}$ pump pulse, the ensemble average of all the incoherent signals becomes proportional to the normalized intensity correlation function of the pump ${\mathsf{g}}_p^{\left(2\right)}({\tau }_i-{\tau }_s)$, which exhibits a $\sim \times 2$ “bunching peak” at delays which are shorter than the pump’s coherence time $2\pi ∕{\delta }_p$. This is shown in (b) in black, together with the instantaneous incoherent TPA signal $I_{\mathrm{TPA}}^{\mathit{ic}}$ (gray), which can be approximated to be proportional to ${\mathsf{g}}_p^{\left(2\right)}$ even without averaging. (c) A zoomed-in presentation of the ultrashort-pulselike behavior of the coherent signal: $I^c({\tau }_i-{\tau }_s)\propto P_e({\tau }_i-{\tau }_s)$, assuming dispersion is compensated and no additional spectral filtering, resulting in a behavior that is identical to that of mixing two $35\mathrm{fs}$ transform-limited pulses. (e) $I^c$ at zero signal-idler delay as a function of the magnitude of a spectral phase filter ${\theta }_s\left(\omega \right)$ that is applied to the signal, for example by a pulse-shaper. ${\theta }_s\left(\omega \right)$ is drawn in (d) (black line) at a magnitude of $\pi$, together with the down-converted power spectrum (gray line). (f) The shaped temporal behavior of the coherent signal with ${\theta }_s\left(\omega \right)$ as depicted in (d) applied to the signal field.Reuse & Permissions

Figure 3

The spectrum of the coherent and incoherent parts of SFG light induced by high-power $(n⪢1)$ broadband down-converted light, with physical parameters as described in the text. The graphs are shown on a logarithmic scale in (a) and a linear scale in (b), and demonstrate how the coherent signal (black) replicates the narrow spectrum of the pump $({\delta }_p=0.01\mathrm{nm})$. In contrast, the spectrum of the incoherent signal (gray) is as wide as the phase-matching conditions at the crystal allow it to be $({\gamma }_{\mathrm{UC}}=0.3\mathrm{nm})$.Reuse & Permissions

Figure 4

The dependence of the coherent part of TPA ($I_{\mathrm{TPA}}^c$, black) induced by down-converted light, as a function of the center frequency of the pump. The graph shows how $I_{\mathrm{TPA}}^c$ behaves as if the $0.01\text{−}\mathrm{nm}$-wide pump itself is inducing the transition ($5S_{1∕2}\to 4D_{3∕2},4D_{5∕2}$ in atomic Rb, with the transition wavelengths drawn in gray). The incoherent part of the TPA is practically insensitive to the pump wavelength, and since for practical power levels $(n>1)$ it is smaller than the coherent part by roughly ${\delta }_p∕\Delta \approx 1∕2000$ (with ${\delta }_p$, $\Delta$ being the bandwidths of the pump and the down-converted light, respectively), it is too small to appear in this graph.Reuse & Permissions