Bidimensional measurements of photon statistics within a multimodal temporal framework

TL;DR

The paper addresses ultrafast, two-dimensional imaging of photon statistics by using difference-frequency generation in a BBO crystal to map temporal information into a detectable idler field. It introduces a temporal-mode decomposition framework, based on the joint spectral amplitude and Bloch-Messiah/Bogoliubov formalism, to model vacuum fluorescence and multimode amplification across many temporal modes, enabling quantitative predictions of photon-number statistics for both coherent and thermal inputs. The authors demonstrate 2D single-shot measurements, validate the multimodal model against vacuum, coherent, and thermal inputs, and show that deviations from ideal statistics arise from vacuum contamination and a multimodal response, with about 40 contributing modes in their setup. This framework provides a robust, predictive tool for bidimensional photon statistics imaging and suggests pathways to improve fidelity via engineered phase-matching, single-mode operation, or alternative nonlinear processes for high-efficiency, ultrafast statistics measurements. The work has implications for studying complex light-mmatter systems (e.g., exciton-polaritons) and for developing ultrafast, high-resolution photon-statistics probes in two dimensions.

Abstract

Ultrafast imaging of photon statistics in two dimensions is a powerful tool for probing non-equilibrium and transient optical phenomena, yet it remains experimentally challenging due to the simultaneous need for high temporal resolution and statistical fidelity. In this work, we demonstrate spatially resolved single-shot measurements of photon number distributions using difference-frequency generation (DFG) in a nonlinear BBO crystal. We show that our platform can discriminate between coherent and thermal photon statistics across two spatial dimensions with picosecond resolution. At the same time, we find that the retrieved distributions deviate from the ideal ones, a consequence of vacuum contamination and the multimodal response of the amplifier. To explain this, we develop a temporal mode decomposition framework that captures the essential physics of signal amplification and fluorescence, and quantitatively reproduces the experimental findings. This establishes a robust approach for measuring two-dimensional photon statistics while clarifying the fundamental factors that limit the fidelity of such measurements.

Figures (15)

• Figure 1: Experimental Setup. A 2 ps pump laser is sent to a BBO crystal where it is combined simultaneously with a coherent laser source (Ti:Sapph with typical power of 3-4 mW) and a thermal source (SLED with typical power of $40$$\mu$W after spectral filtering). Down-converted photons are filtered and recorded using a CMOS camera. The repetition rate of the camera, 1KHz, matches that of the pulsed laser, allowing for single shot image recording.
• Figure 2: (a) Measured average number of photons in a two-dimensional image using 10000 single-shot. (b) Variance of the photon count over the image.
• Figure 3: Center: Single shot of a two-dimensional spatial distribution and associated density distributions of the photon counts relative to two different positions in the image obtained from 10000 single-shot measurements. Dots are the experimental results, and in red and green lines are the theoretical curves calculated with $g_{\text{2D}}=5.963\times 10^{-13}$ s.m$^{-1}$ for both a coherent and a thermal state. The red and green dashed lines are coherent and thermal distribution, respectively, possessing the same average photon number as measured in the experiment.
• Figure 4: Schematic representation of the DFG process in the temporal eigenmode basis.
• Figure 5: (a) Phase-matching and (b) the Joint Spectra Amplitude (JSA) profile in the $(\Omega_i,\Omega_s)$ space.