Quantum Fisher information analysis for absorption measurements with undetected photons

TL;DR

This work addresses absorption estimation with undetected photons by benchmarking three architectures—SU(1,1) interferometry, induced coherence, and distributed loss—through quantum Fisher information (QFI). It develops a unified broadband twin-beam model using Heisenberg–Langevin equations, yielding analytic QFI expressions for lossless and lossy configurations and for both full and restricted mode access. The main finding is a regime map: SU(1,1) offers the highest QFI at moderate gain and losses below 99%, IC dominates at high gain with intermediate loss, and the DL scheme becomes optimal only under extreme attenuation; crossovers occur around 1–2% idler transmission. The results provide practical design criteria for optimizing quantum-sensor performance in absorption spectroscopy across wavelength ranges where undetected photons are advantageous.

Abstract

We theoretically compare the quantum Fisher information (QFI) for three configurations of absorption spectroscopy with undetected idler photons: an SU(1,1) interferometer with inter-source idler loss, an induced-coherence (IC) setup in which the idler partially seeds a second squeezer together with a vacuum ancilla, and a distributed-loss (DL) scheme with in-medium attenuation. We calculate the QFI as a function of parametric gain for both full and signal-only detection access. For losses below 99% and low to moderate gain, the SU(1,1) configuration provides the largest QFI. At high gain and intermediate loss, the IC scheme performs best, while under extreme attenuation (transmission $<$ 1%) the DL model becomes optimal. These results delineate the measurement regimes in which each architecture is optimal in terms of information theory.

Figures (4)

• Figure 1: Schematic representation of the different sensing configurations and the respective loss implementations. (a) SU(1,1) interferometer where the two nonlinear regions are taken to be identical and lossless. We treat losses via a beam splitter interaction (with transmission rates $\eta_{S}$ and $\eta_{I}$ for the signal and idler respectively) between both nonlinear regions. (b) IC system where in the second nonlinear region the idler is mixed with vacuum of an ancilla mode. (c) DL model where the signal and idler modes experience different decay rates ($\kappa_{S}$ and $\kappa_{I}$ respectively) as they propagate through the nonlinear region. To simulate optical path delays, we also allow the beam splitter interaction to induce additional dispersion for the signal and idler modes ($\Phi_{S}$ and $\Phi_{I}$ respectively) in the SU(1,1) and IC configuration. Both losses and transmission rates are taken to be frequency dependent. P: pump; S: signal; I: idler; A: ancilla.
• Figure 2: Comparing quantum fisher information for the SU(1,1), DL, and IC models. First row shows the full QFI with access to all modes (c.f. Eq. (\ref{['eq:qfi_twomode']})) for the (a) SU(1,1) and IC models, and (b) DL model. (c) shows the logarithm of the ratio of the DL and SU(1,1) QFI. (d) shows the two-mode QFI for the IC model after the idler is traced out (c.f. Eq (\ref{['eq:tm_ic_qfi']})). (e) and (f) show the logarithm of the ratio between the two-mode IC QFI and the single-mode QFI of the SU(1,1) and DL respectively. The third row shows the single-mode QFI for the (g) SU(1,1), (h) DL, and (i) IC models. The fourth row shows the logarithm of the ratio of the single-mode QFI between the (j) DL and SU(1,1) models, (k) IC and SU(1,1) models, (l) IC and DL models. Each figure includes several curves at different levels of gain. We plot the curves as a function of $\kappa_{I}$ and the chosen range corresponds to a transmission coefficient $\eta_{I}\in(99,0.1)\%$. Length of the nonlinear regions is set to $L=40$ mm.
• Figure 3: Inverse ratio $\left( N^{\text{Dist.}}_{S}(\omega)-N^{\text{Dist.}}_{I}(-\omega) \right)/H_{\epsilon}$ for different parametrizations for the DL model. The QFI, $H_{\epsilon}$, is evaluated numerically at a frequency where we are phase-matched. For (a) and (b), the estimation parameter is taken to be $\eta_{I}$ and we plot the inverse ratio as a function $\eta_{I}$ and $\kappa_{I}$ respectively. For (c) and (d), the estimation parameter is taken to be $\kappa_{I}$ and we plot the inverse ratio as a function $\eta_{I}$ and $\kappa_{I}$ respectively. For each plot we include curves at different levels of gain which is determined by $N^{P}_{S}$. Each plot also includes the approximate function, labelled as "Fit", with the fit parameter $\alpha=1.1$. Using a least-squared fit, we find an average value of $R^{2}=0.998$ when comparing the numerical solutions to our approximate form. Chosen range corresponds to a transmission coefficient $\eta_{I}\in(99,0.1)\%$. Length of the nonlinear regions is set to $L=40$ mm.
• Figure 4: Inverse error obtained from an intensity difference measurement (c.f. Eq. (\ref{['eq:inverse_error']})) for the (a) SU(1,1) and (b) DL model. (c) shows the logarithm of the ratio of the inverse errors. Each figure includes several curves at different levels of gain. We plot the curves as a function of $\kappa_{I}$ and the chosen range corresponds to a transmission coefficient $\eta_{I}\in(99,0.1)\%$. Length of the nonlinear regions is set to $L=40$ mm.