Time resolution at the quantum limit of two incoherent sources based on frequency resolved two-photon-interference

Abstract

The Rayleigh criterion is a widely known limit in the resolution of incoherent sources with classical measurements in the spatial domain. Unsurprisingly the estimation of the time delay between two weak incoherent signals is afflicted by an analogue problem. In this work, we show the emergence of two-photon quantum beats in the frequency domain from the interference at a beam splitter of a photon emitted by a reference source and one from the two incoherent weak signals. We demonstrate, based on this phenomena, that with a relatively low number of measurements of the frequencies of the interfering photons either bunching or antibunching at the beam splitter output one can achieve a precision amounting to half of the quantum limit, independently of both the temporal shape of the photonic wavepacket and the time delay to be estimated. The feasibility of the technique makes it applicable in astronomy, microscopy, remote clocks synchronization and radar ranging

Figures (4)

• Figure 1: Two-photon interferometer. Two incoherent photons are emitted with a time delay $\Delta t$ in the same line of sight, one of them interferes on a balanced beam splitter with a photon produced by a reference source. The possible outputs depend on the frequency of the detected photons and if they bunch together or hit different cameras. This scheme allows to retrieve the value of the time delay $\Delta t$ with quantum-enhanced sensitivity and a relatively low number of measurements.
• Figure 2: Quantum beat interference in the two-photon interferometer in Fig.\ref{['fig:Setup']}. Plots of the probability distribution $P(\Delta\omega, X; \Delta t)$ at the interferometer output given by the function in Eq. \ref{['Eq:deltafixed']} of the frequency difference $\Delta\omega$, in units of the frequency bandwidth $\sigma_{\omega}$, for a fixed time delay $\Delta t = 4/\sigma_{\omega}=8\sigma_{t}$. Plot a) corresponds to indistinguishable photons, $\nu = 1$, and plot b) corresponds to partially distinguishable photons, $\nu = 0.9$. The coincidence probability ($X = A$) in the dashed-dotted green line, and the bunching probability ($X = B$) in the dashed orange line. The probability clearly manifests quantum beats with a periodicity inversely proportional to $\Delta t$.
• Figure 3: Numerical simulations for Gaussian wavepackets of the rate of convergence of the variance of the maximum likelihood estimator $\widetilde{\Delta t}$ normalized to the Cramér-Rao bound in Eq. \ref{['eq:CRB']}, for different numbers $N$ of collected samples and for $\Delta t/\sigma_t=0.6,0.7,0.8$, in the case of (a) indistinguishable photons, $\nu=1$ (Fisher information $F_{\nu=1}$ in Eq. \ref{['eq:Fisher']}), and (b) partially distinguishable photons, $\nu=0.95$ (Fisher information in Eq. \ref{['eq:Fishnonres']}). We show that the data can be fitted with the curve $1+a/N$, where $a/N$ is a correction term of order $1/N$ of the variance of the estimator normalized to the Cramér-Rao bound in Eq. \ref{['eq:CRB']} for $N\gg 1$. Thus showing that our scheme approaches the Cramér-Rao bound already for $N\simeq 5000$. In the insets, we also plot the estimated expectation value of the maximum-likelihood estimator normalized by its real value, showing that the estimation is unbiased already again for $N\simeq 5000$.
• Figure 4: Plots of the Fisher information $F_{\nu}(\Delta t)$ in Eq. \ref{['eq:Fishnonres']}, for $\eta=1$, and Fisher information for time-resolved direct detection as a function of the time delay $\Delta t/\sigma_{t}$, normalized by the Quantum Fisher information in Eq. \ref{['eq:Qufi']}, considering as an example a Gaussian temporal wavepacket with unitary $\sigma_{t}$. Direct detection in solid black line and dashed green line for a direct detection with resolution $T=5\sigma_{t}$ and $T=10\sigma_{t}$ respectively, $F_{\nu}$ in Eq. \ref{['eq:Fishnonres']} in dotted red, $\nu=0.95$, dot-dashed blue, $\nu=0.98$, and thick solid orange, $\nu=1$. In the left inset we show the same curves (constant function $F_{\nu=1}/Q=1/2$ omitted here) for small delays, $0<\Delta t/\sigma_{t}<0.2$, while in the right inset we show them for large delays, $1<\Delta t/\sigma_{t}<6$, highlighting the clear advantage of our method in both regimes. In the right inset, it is also possible to observe the curves for $\nu\neq 1$ approaching the constant Fisher information value given in Eq. \ref{['eq:Fishlimit']}.