Second-order correlations in directed emissions in sodium atoms

We report second-order intensity correlations <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msup> <a:mi>g</a:mi> <a:mrow> <a:mo>(</a:mo> <a:mn>2</a:mn> <a:mo>)</a:mo> </a:mrow> </a:msup> <a:mrow> <a:mo>(</a:mo> <a:mi>τ</a:mi> <a:mo>)</a:mo> </a:mrow> </a:mrow> </a:math> of directional midinfrared emissions from sodium vapor under bichromatic continuous-wave excitation. Directional emission at <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mn>2.34</b:mn> <b:mspace width="0.16em"/> <b:mi>µ</b:mi> <b:mi mathvariant="normal">m</b:mi> </b:mrow> </b:math> in the forward direction and at <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mn>2.21</e:mn> <e:mspace width="0.16em"/> <e:mi>µ</e:mi> <e:mi mathvariant="normal">m</e:mi> </e:mrow> </e:math> in both forward and backward directions exhibits <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:msup> <h:mi>g</h:mi> <h:mrow> <h:mo>(</h:mo> <h:mn>2</h:mn> <h:mo>)</h:mo> </h:mrow> </h:msup> <h:mrow> <h:mo>(</h:mo> <h:mn>0</h:mn> <h:mo>)</h:mo> </h:mrow> </h:mrow> </h:math> values that exceed unity yet remain well below the thermal limit of 2, indicating subthermal photon bunching characteristics of a multimode collectively driven process rather than purely independent amplified spontaneous emission. Cross-correlation measurements reveal that the counterpropagating <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mn>2.21</i:mn> <i:mtext>−</i:mtext> <i:mi>µ</i:mi> <i:mi mathvariant="normal">m</i:mi> </i:mrow> </i:math> beams are positively correlated, demonstrating dynamic coupling between the two emission channels. Persistent oscillations in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:msup> <k:mi>g</k:mi> <k:mrow> <k:mo>(</k:mo> <k:mn>2</k:mn> <k:mo>)</k:mo> </k:mrow> </k:msup> <k:mrow> <k:mo>(</k:mo> <k:mi>τ</k:mi> <k:mo>)</k:mo> </k:mrow> </k:mrow> </k:math> , observed across a broad range of pump powers, shift systematically to higher frequencies with increasing excitation power, consistent with ac Stark shifts of hyperfine sublevels. Together, these observations establish that the emission arises from a phase-matched, continuous-wave collective process combining features of mirrorless lasing and amplified spontaneous emission, with long-range dipole coherence and velocity-selective coupling governing the photon statistics and the forward-backward emission symmetry. The experiment uses a 10-cm buffer-gas-free sodium vapor cell under bichromatic excitation at 589.2 and 569.0 nm; the emitted midinfrared light is detected in a Hanbury Brown–Twiss configuration. The demonstrated backward emission is of particular interest for applications in laser guide star generation and mesospheric remote sensing, where understanding the statistical properties of the emitted light is essential for optimizing sodium-based light sources.