Supersensitive phase estimation by thermal light in a Kerr-nonlinear interferometric setup

Estimation of the phase delay between interferometer arms is the core of transmission phase microscopy. Such phase estimation may exhibit an error below the standard quantum (shot-noise) limit, if the input is an entangled two-mode state, e.g., a N00N state. We show, by contrast, that such supersensitive phase estimation (SSPE) is achievable by \textit{incoherent}, e.g., \textit{thermal}, light that is injected into a Mach-Zehnder interferometer via a Kerr-nonlinear two-mode coupler. Phase error is shown to be reduced below $1/\bar{n}$, $\bar{n}$ being the mean photon number, by thermal input in such interferometric setups, even for small nonlinear phase-shifts per photon pair or for significant photon loss. Remarkably, the phase accuracy achievable in such setups by thermal input surpasses that of coherent light with the same $\bar{n}$. Available mode couplers with giant Kerr nonlinearity that stems either from dipole-dipole interactions of Rydberg polaritons in a cold atomic gas, or from cavity-enhanced dispersive atom-field interactions, may exploit such effects to substantially advance interferometric phase microscopy using incoherent, faint light sources.