Photon-Pair Advantage for Dissipative Parameter Estimation in Whispering-Gallery-Mode Interferometers

Abstract

We study the estimation of detuning and intrinsic loss in a lossy whispering-gallery-mode (WGM) resonator located in a balanced Mach--Zehnder interferometer. We use indistinguishable photon pairs and coherent-state probes. We show that using indistinguishable photon pairs can provide higher Quantum Fisher Information (QFI) of dissipative parameters, such as intrinsic loss, at resonance away from critical coupling. Taking into account all photon-loss outcomes, we obtain analytical QFI expressions and compare photon-pair and coherent-state probe results when they have the same mean photon number. For estimating the round-trip amplitude transmission at resonance and away from critical coupling, the photon-pair probe gives a higher QFI than the coherent-state probe, and photon-number-resolved detection saturates this QFI. This advantage is due to the photon-loss statistics and it is still accessible for a typical overcoupled condition after averaging over fluctuations of the unresolved detuning. We identify the limits of this advantage: phase doubling does not improve resonant detuning estimation, and coincidence-only detection is inefficient close to a zero transmission. These results identify the cases in which photon pairs can be useful for absorption and pump--probe WGM sensing.

The paper asks when indistinguishable photon pairs improve estimation of intrinsic loss or detuning in a lossy whispering-gallery-mode resonator placed in one arm of a balanced Mach–Zehnder interferometer. It compares a two-photon N00N probe with a coherent input of mean photon number two, giving both probes one mean photon incident on the resonator. A vacuum-loss dilation retains every photon-loss outcome before tracing out the environment. For complex transmission $t=\sqrt{T}e^{i\phi}$, the resulting analytical QFIs are $F_Q^{\rm pair}=(\partial_\vartheta T)^2/[T(1-T)]+8T^2(\partial_\vartheta\phi)^2/(1+T^2)$ and $F_Q^{\rm coh}=(\partial_\vartheta T)^2/T+4T(\partial_\vartheta\phi)^2$. Pure transmission encoding therefore gives a pair-to-coherent ratio $1/(1-T)>1$, whereas pure phase encoding gives $2T/(1+T^2)\leq1$. On resonance, intrinsic-loss estimation realizes the former and detuning estimation the latter, away from critical coupling. Photon-number-resolved detection saturates the resonant loss-estimation QFI; coincidence-only detection retains the fraction $4T/(T+3)$. The advantage is at fixed coupling: optimizing coupling separately gives the same coherent maximum and regular-point photon-pair supremum. At the exact critical-coupling transmission zero, the pair’s SLD QFI vanishes despite finite nearby limits. Numerical averaging over unresolved Gaussian detuning fluctuations preserves an accessible advantage at one representative overcoupled point: QFI and photon-counting information ratios are approximately 1.410 and 1.372. Absorption and pump–probe implications follow by reparametrization for predominantly dissipative signals. Coherent finite-bandwidth wavepackets, detector imperfections, imperfect indistinguishability, and source-heralding efficiency remain untreated; exploiting vacuum outcomes requires known preparation trials.