Unity-Efficiency Parametric Down-Conversion via Amplitude Amplification

We propose an optical scheme, employing optical parametric down-converters interlaced with nonlinear sign gates (NSGs), that completely converts an $n$-photon Fock-state pump to $n$ signal-idler photon pairs when the down-converters’ crystal lengths are chosen appropriately. The proof of this assertion relies on amplitude amplification, analogous to that employed in Grover search, applied to the full quantum dynamics of single-mode parametric down-conversion. When we require that all Grover iterations use the same crystal, and account for potential experimental limitations on crystal-length precision, our optimized conversion efficiencies reach unity for $1\le n\le 5$, after which they decrease monotonically for $n$ values up to 50, which is the upper limit of our numerical dynamics evaluations. Nevertheless, our conversion efficiencies remain higher than those for a conventional (no NSGs) down-converter.

Figure 1

Down-conversion efficiencies for $n$-photon Fock-state pumps optimized over nonlinear-crystal lengths cut to a precision of ${10}^{-3}v/\kappa$. Lower (red) curve: maximum conversion efficiencies for a ${\chi }^{(2)}$ crystal without Grover-search amplitude amplification. Upper (blue) curve: maximum UPDC conversion efficiencies, where the $n=1$ point did not employ a NSG.

Figure 2

UPDC runtime (defined to be $M_n{L}_1/v$ with $M_n$ being the number of Grover iterations used in Fig. 1 to achieve maximum efficiency for an $n$-photon pump) versus $\sqrt{n}$.