Optimal single-photon spectroscopy of a single emitter.

Light-matter interaction at the single-photon and single-emitter level is a cornerstone of quantum technologies, including quantum network, key distribution, and metrology. Assessing the true potential of single photons in these protocols is limited by the current difficulty in characterizing the full quantum state of the electromagnetic field after the interaction. This is essential for accurately calculating entropic quantities to quantify the information content of a single-photon.

In this work, we solve this problem for a pulsed single-photon that weakly couples to a single-emitter where we characterize the full quantum state of the scattered pulse. We use the Quantum Fisher Information (QFI) to quantify the pulse’s maximum information content about a parameter, a critical measure for quantum spectroscopy and sensing.

We apply our framework to two emitter systems. First, for an isolated single-molecule, we estimate the emission linewidth and site energies. We demonstrate that the maximum QFI for the linewidth is independent of the emitter's bare Hamiltonian details, and we characterize the optimal single-photon probes that achieve this precision limit. Second, for a two-level emitter coupled to a phononic environment, we show that the QFI for linewidth estimation decays exponentially with phononic coupling at zero temperature, with even faster decay at finite temperatures. Finally, we quantify the performance of two experimentally feasible measurements that can saturate the Cramér-Rao bound.