Interference of photons from independent hot atoms

TL;DR

The paper shows that photons scattered from independent, room-temperature atomic ensembles can interfere via a Doppler-selective, forward-backward scheme, even as individual first-order coherence is washed out by thermal motion. By employing a retro-reflected standing wave and velocity-selective scattering, photons from two opposite velocity classes acquire a stable frequency difference of $2Δ$, enabling a measurable beating in the second-order correlation function $g^{(2)}(τ)$ with period $1/(2Δ)$. The authors derive a two-source Siegert-like relation and demonstrate experimentally in $^{87}$Rb vapor that $g^{(2)}(τ)$ exhibits clear beating with $f_{mod}=210.8\pm1.2$ MHz, along with strong photon bunching ($g^{(2)}(0)\approx1.96$) and a small residual coherent contribution ($r\approx0.02$). The technique enables Doppler-free, sub-Doppler spectroscopy in small, dilute hot-atom samples and is robust to phase drifts, offering a practical path to precise absolute-detuning measurements and potential studies of rare isotopes or molecules using photon-correlation readouts.

Abstract

The coherence of light from independent ensembles of elementary atomic emitters plays a paramount role in diverse areas of modern optics. We demonstrate the interference of photons scattered from independent ensembles of warm atoms in atomic vapor. It relies on the feasibility of the preservation of coherence of light scattered elastically in the forward and backward directions from Doppler-broadened atomic ensembles, such that photons with chaotic photon statistics from two opposite atomic velocity groups contribute to the same detection mode. While the random phase fluctuations of the scattered light caused by a large thermal motion prevent direct observability of the interference in the detected photon rate, the stable frequency difference between photons collected from scattering off counter-propagating laser beams provides strong periodic modulation of the photon coincidence rate with the period given by the detuning of the excitation laser from the atomic resonance. The presented interferometry represents a sensitive and robust methodology for Doppler-free optical atomic and molecular spectroscopy based on photon correlation measurements on scattered light.

Figures (5)

• Figure 1: The principle of interference of light from independent warm atomic ensembles. The laser at the frequency $\omega_\mathrm{L}$ is detuned by $\Delta$ from the atomic transition $\omega_\mathrm{A}$ and scatters off the particular velocity class of atoms which follow the thermal velocity distribution $f(v)$ along the laser propagation axis. In the forward scattering (F), the corresponding Doppler shift effectively compensates. The retroreflected laser scatters off atoms possessing the opposite direction of motion but with the same velocity magnitude. The corresponding backward-scattered photons (B) are frequency-shifted by $\approx \Delta$ concerning an observer in the frame moving with the atomic scatterer and by approximately $2\Delta$ to the forward-scattered photons.
• Figure 2: Experimental scheme and example measurement of interference of light from warm atomic vapors. The scattering of two counter-propagating laser beams off the particular velocity classes of atoms results in forward (F) and backward (B) scattered photons with a frequency difference of $\approx 2\Delta$ collected in the same optical mode defined by the single mode optical fiber (SMF). The measurement of the second-order correlations $g^{(2)}(\tau)$ provides coherent frequency beating with a period of $1/(2\Delta)$. The additional optical filtering (OF) in the detection path suppresses various noise contributions. Its detailed specification can be found in Supplementary materials Supplement.
• Figure 3: a) shows the measurements of the individual $g^{(2)}_\mathrm{ F(B)}(\tau)$ for photons emitted from two different velocity classes of ensembles of warm atoms. The corresponding $g^{(2)}_\mathrm{ F+B}(\tau)$ emerging from their interference is shown in b). The fit of the interference pattern uses Eq. (\ref{['eq.g2']}), where $\bar{g}^{(1)}(\tau)$ of atoms from the two atomic ensembles are taken from their independent measurements. The evaluation of particular $g^{(2)}(\tau)$ data points using the Siegert relation from the independently measured $|g^{(1)}(\tau)|^2+1$ is shown with black triangles. The error bars depict a single standard deviation. Graph in c) summarizes measurements of $g^{(2)}(\tau)$ for different laser detunings $\Delta_\mathrm{L}$. The evaluated mean modulation frequencies $f_\mathrm{ mod}$ from a direct Fourier analysis are shown in d). Here, green dotted lines illustrate its linear dependence on independently measured detuning $\Delta_\mathrm{ L}$. The stability of such frequency estimation is practically limited by detected photon rates shown as orange data points. The filled area depicts the corresponding simulation considering the atomic populations at different velocity classes with a Gaussian uncertainty of the Doppler broadened spectra $\sigma_\mathrm{ DB}$, including also the residual modification due to the Fabry-Pérot spectral filter. The marked data point corresponds to the measurement presented in b).
• Figure S1: Experimental scheme for observation of interference of light from warm atomic vapors. a) The excitation laser beam generated from an external-cavity diode laser at 780 nm is detuned by $\Delta$ from the $5\mathrm{S}_{1/2}(F=2) \leftrightarrow 5\mathrm{P}_{3/2}(F'=3)$ transition of $^{87}$Rb. The laser alignment is set in a counter-propagating standing wave configuration. The scattered photons are observed in the single optical mode at angle $\theta$ defined by the composite optical filter, including the spatial mode filter implemented by the combination of collecting lens (L) and single-mode optical fiber (SMF). The polarization mode coincident with the excitation laser polarization is set by the combination of the quarter waveplate (QWP) and Glan-Thompson polarizer (GT). The Fabry-Pérot (F-P) resonator is set to suppress the contributions from residual Raman scattering. The optical attenuator (A) corresponding to the optical neutral density filter was employed to suppress the detected count rate below the saturation limit of employed single photon detectors. The control of the position of the observed interaction area of lasers with atoms within the employed long atomic vapor cell allows for additional tunability and reduction of losses experienced by the scattered light upon traveling towards the detector. b) The collected photons are analyzed in the setup including two single-photon counting modules (D) for measurement of the second-order correlations $g^{(2)}(\tau)$. Additional measurement of the degree of the modulus of the first-order coherence $|g^{(1)}(\tau)|$ is implemented in the Michelson interferometer with feasible relative arm length difference corresponding to up to $\tau=12$ ns.
• Figure S2: Second order correlation functions $g^{(2)}(\tau)$ for various detunings $\Delta_\mathrm{L}$ of the laser from the employed transition. For better visualisation, the values of $g^{(2)}(\tau)$ are shifted. The error bars for the noisiest dataset (-40 MHz) reach $\pm0.03$ and the main noise contribution comes from shot noise.