Second-order correlations in directed emissions in sodium atoms

TL;DR

The study investigates second-order intensity correlations $g^{(2)}(\tau)$ in phase-matched forward- and backward-directed mid-infrared emission from sodium under bichromatic excitation. Using a 10 cm Na vapor cell and CW bichromatic pumping at $589.2$ nm and $569.0$ nm, the authors measure $g^{(2)}(\tau)$ for emissions at $2.21\,\mu$m and $2.34\,\mu$m, observing $g^{(2)}(0)$ values between 1 and ~1.14 and persistent oscillations in $g^{(2)}(\tau)$ whose frequencies are power-dependent. The forward and backward $2.21\,\mu$m channels are positively cross-correlated, signaling a phase-matched, cooperative CW emission with long-range dipole coherence and velocity-selective mode coupling. The results support a CW, phase-matched cooperative emission regime akin to steady-state superradiance, with AC Stark shifts and hyperfine structure shaping the dynamics and offering potential applications in laser guide stars and mesospheric remote sensing; future work may explore structured light and OAM-based quantum correlations in such cooperative atomic systems.

Abstract

We report on measurements of second-order intensity correlations $g^{(2)}(τ)$ of infrared emission under bichromatic excitation at 589.2\,nm and 569.0\,nm of sodium atoms contained in a buffer-gas-free and uncoated 10-cm-long vapor cell. Directional emissions at $2.34\,μ$m in the forward direction and $2.21\,μ$m in both forward and backward directions under different experimental parameters are considered for this study. The measured values of $g^{(2)}(0)$ in all cases are found to exceed unity, while remaining significantly below the thermal light limit of 2. Cross-correlation measurements reveal that forward- and backward-propagating $2.21\,μ$m radiations are correlated. Oscillatory features in $g^{(2)}(τ)$ are observed over a broad range of excitation powers, and the dependence of the oscillation frequency on laser power can be attributed to AC Stark shifts, with contributions from hyperfine atomic structure in selected atomic velocity groups even in the presence of Doppler broadening. Our study establishes that the observed mid-infrared emission arises from a phase-matched, continuous-wave cooperative process that combines features of lasing and collective amplified spontaneous emission. The results highlight the buildup of long-range dipole coherence and velocity-selective coupling of atomic groups, which together govern the observed photon correlations and forward-backward emission symmetry. The demonstrated backward emission is of particular interest for applications in laser guidestar generation and mesospheric remote sensing, where understanding the statistical properties of the emitted light is essential for optimizing sodium-based light sources.

Figures (5)

• Figure 1: Schematic top view of the experimental setup on the left. EOM: Electro-optic modulator driven at 1.713 GHz, PBS: Polarizing beam spliterr, BS: Non-polarizing beam splitter, M: Mirror, P: Polarizer, DM: Dichroic mirror, F: Infrared Filter, L: Lens, PD: Photodetector. Relevant Na energy levels and transitions are presented on the right.
• Figure 2: Autocorrelation functions vs. delay time $\tau$ for 589.2 nm laser (solid horizontal line), 569.0 nm laser (dotted horizontal line), (a) forward (solid) and backward (dashed) emission at $2.21\,\mu$m, and (b) forward emission at $2.34\,\mu$m. (c) Cross-correlation vs. delay time $\tau$ in solid line between forward and backward emitted lights at $2.21\,\mu$m. For reference, $2.21\,\mu$m forward (dash-dotted), and backward (dashed) auto-correlations are shown. The 569.0 nm laser power is 15 mW, the 589.2 nm laser power is 230 mW.
• Figure 3: Dependence of emitted radiation oscillation frequencies on 569.0 nm laser power for forward emitted (a) 2.21 $\mu$m and (b) 2.34 $\mu$m fields. 589.2 nm laser power is fixed at 230 mW.
• Figure 4: Coherence time (top), $g^{(2)}(0)$ (middle), normalized mean intensity (bottom) of $2.21\,\mu$m of forward (solid) and backward (dashed) emission (left pane); and $2.34\,\mu$m emitted light (right pane) dependence on 569.0 nm laser power. 589.2 nm laser power is 230 mW. Horizontal error bars indicate the uncertainty of the laser power ($\pm 0.7$ mW) due to power fluctuations.
• Figure 5: (a) Time dependence of the detected intensities at 2.21 $\mu$m in the forward (solid) and backward (dashed) directions, recorded with laser powers of 230 mW at 589.2 nm and 45 mW at 569.0 nm. (b) Corresponding second-order auto-correlation functions $g^{(2)}(\tau)$ calculated from the same data: forward emission (dash-dotted line), backward emission (dashed line), and forward-backward cross-correlation (solid).