Multiphoton Hong-Ou-Mandel Interference Enables Superresolution of Bright Thermal Sources

TL;DR

This work tackles imaging of incoherent bright thermal sources beyond the diffraction limit by leveraging multiphoton interference at a balanced beamsplitter between a thermal two-mode image state and a reference single-photon Fock state. By deriving analytical $L$-photon coincidence probabilities and performing Fisher-information analysis, it shows that even-order coincidences yield sub-Rayleigh superresolution, while resolving transverse momenta in the Fourier plane enables finite imaging precision across sub- and large-Separations with coarse pixel sizes. For source brightness $N_s\sim1$, the scheme scales as $O(N_s)$ in precision and can saturate the ultimate quantum bound with a modest number of coincidence orders (up to $L\approx7$). The approach offers a practical, robust alternative to existing quantum imaging schemes, compatible with current large-format SPAD/SNSPD detector arrays, and has potential impact on non-invasive nanoscale biological and chemical sensing, as well as astronomical imaging.

Abstract

We present a quantum optical scheme for imaging transversely displaced thermal sources of arbitrary intensities by employing multiphoton interference with a reference single-photon Fock state at a beamsplitter. Obtaining an analytical form for transverse momenta-resolved $L$-photon probabilities in either output, we show via Fisher information analysis that separation estimators built using interference sampling of multiphoton events exhibit significantly enhanced precision vis-à-vis existing imaging schemes over a wide range of separations and brightness. Even-photon-number coincidences exhibit constant precision in the sub-Rayleigh regime, demonstrating quantum superresolution of our scheme beyond the diffraction limit. For sources emitting on average $N_s\sim1$ photon per frame (such as in IR emission of thermal sources), precision bounds for our scheme scale linearly in $N_s$, exemplifying an enhanced precision of estimators in relation to weak sources $N_s\ll1$, and matching the ultimate quantum scaling. Finally, transverse momenta resolution in the Fourier plane produces finite imaging precisions for intermediate and large source separations using coarse pixel sizes of order $δy\sim100\,μ\mathrm{m}$ for exemplary image spot sizes $σ_x \sim 0.1\, μ\mathrm{m}$, in contrast with existing schemes of diffraction-limited direct imaging and superresolved inversion interferometric imaging that are severely degraded by coarse pixel sizes and have limited use. Combining the relatively straightforward sensing operation of Hong-Ou-Mandel interferometers with multiphoton coincidence detection of arbitrarily bright thermal sources and inner variable resolution of transverse photonic momenta, our scheme offers a robust alternative to non-invasive single-particle tracking and imaging of bright sources in nanoscopic chemical and biological systems.

Figures (8)

• Figure 1: Imaging scheme setup: Image state $\rho_d(s)$ corresponding to thermal source distribution interferes at the balanced BS with a single-photon reference state whose wavepacket center coincides with the image centroid. After propagating to the far field, cameras $C_1$ and $C_2$ composed of single-photon detector arrays record in coincidence the spatial positions of the impinging photons, resolving the corresponding transverse momenta.
• Figure 2: Two-photon detection probability $P^{(2)}(\bar{K},\Delta k;X)$ as a function of difference $\Delta k$ and mean $\bar{K}$ of the detected transverse momenta for Gaussian PSF $\psi(x)$ with $\sigma_x=1$ for $N_s=1.5$, and (a) X=B, $s=5.0\sigma_x$; (b) X=A, $s=5.0\sigma_x$; (c) X=B, $s=20.0\sigma_x$; and (d) X=A, $s=20.0\sigma_x$. With increasing separation between sources, we see that the beating in $\Delta k$ becomes comparable to the width of the transverse momentum distribution of the photons, demonstrating the sensitivity of the spatial interference pattern to source separation $s$.
• Figure 3: Sub-Rayleigh FI $F(s/\sigma_x\rightarrow 0)$ (blue dashed line) as a function of source brightness $N_s$, and $\sigma_x=1.0$, from the closed expression in Eq. (\ref{['eq:FIsubrayleigh_allorders']}). Also displayed are sub-Rayleigh FIs for detection up to multiphoton coincidence orders $L=2$ (blue circles), $L=4$ (orange-rotated triangles), and $L=6$ (yellow triangles), demonstrating that for $N_s\sim 1$, lower order photon detection is sufficient to saturate almost all of the available imaging precision.
• Figure 4: Imaging FI $\sum_{i=1}^L F^{(i)}(s) \sim F(s)$ (blue dashed line) for coincident multiphoton sampling up to $i\leq L=7$ as a function of source separation $s/\sigma_x$, for (a) $N_s=0.01$ (feeble sources), and (b) $N_s=1.5$ (bright sources). For a meaningful comparative, we have also superposed lower bounds nair2016far for pixelated direct imaging FI for pixel sizes $\delta y=5\sigma_x$ (maroon solid line-triangle), and $\delta y=10\sigma_x$ (orange solid line-square); the lower bounds for pix-SLIVER FI (black dotted line-pentagrams) for pixel size $\delta x = 5\sigma_x$; and FI of only two-photon events $F^{(2)}$ (magenta dot-dashed) in (b). All multidimensional numerical integrations were performed using routines from the Cuba library hahn2005cuba.
• Figure 5: Comparative analysis of contributions to total multiphoton FI $F(s)$ for the imaging scheme from photon orders corresponding to $L=\{2,\dots,7\}$, as a function of source brightness $N_s$, and for (a) $s/\sigma_x = 0.01$, (b) $s/\sigma_x = 1.0$, and (c) $s/\sigma_x = 7.0$. For sub-Rayleigh separations, the analytically established ordering $F^{(2)}>F^{(3)}>F^{(4)}>\dots$ holds, but for intermediate and large separations, higher order FIs become increasingly important, and may even contribute more information than lower order FIs. All multidimensional numerical integrations were performed using the Cuba library hahn2005cuba.