Fig. 1
From: Heisenberg-scaling measurement of the single-photon Kerr non-linearity using mixed states
Strategies for precision measurements. a The scheme: a single-photon goes through an interferometer, where in one arm it interacts with a coherent state and a phase is acquired, while in the other nothing happens. At the exit port the two paths interfere so the probability of the photon coming out there depends on the phase it acquired by the interaction, and thus, also on the number of photons in the probe. Post-selecting the probe pulses accordingly induces a shift in the average photon number, from which the interaction strength can be estimated. b The principle of our method: a statistical ensemble of \(\left| \alpha \right\rangle \) (only a few are drawn), showing the two paths for each member: blue with the phase and magenta without. Due to the probability of postselection, Eq. (3), the ensemble is shifted in the radial direction. The shift in photon number, Eq. (5), δn ∝ (Δn)2 ∝ N2 so the shift in \(\left| \alpha \right|\) is ∝N3/2. The uncertainty in n can be fluctuated to be ∝N. The ratio between the uncertainty and the shift is ∝1/N, resulting in the Heisenberg scaling
