Pump quantization changes optimal measurements for quantum sensing
Standard analyses of quantum phase sensing often neglect the overhead of generating squeezed-light by treating the pump as an undepleted classical resource. Realistically, the pump is a quantum object that undergoes significant depletion and entangles with the produced squeezed light, raising the question of how the crossover from a classical to a quantum pump limits the quantum advantage in sensing. Contrary to prior analyses, we show that quantum advantage persists despite pump quantization, but depends critically on the nonlinear $(χ^{(2)})$ source dynamics. In particular, we demonstrate that Heisenberg scaling emerges in the regime of strong pump depletion, rather than maximum signal squeezing. At maximum squeezing, the quantum Fisher information (QFI) for interferometric sensing of an unknown phase scales as $N^{3/2}$, where $N$ is the pump photon number, surpassing the shot-noise limit but establishing that squeezing itself is insufficient to attain Heisenberg scaling of $N^2$. At maximal pump depletion, where the signal flux is maximized, the QFI scales as $N^2$. In this regime, standard photon counting is suboptimal, while a joint nonlinear measurement of the entire pump-signal system is shown to achieve Heisenberg scaling. Our work shows that pump quantization changes both the optimal operating point and the measurement strategies required to preserve a realistic quantum advantage in phase estimation and, more broadly, many-body quantum dynamics.