We demonstrate coherent Doppler-free three-photon excitation of the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ optical clock transition and the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ intercombination transition in free-space thermal clouds of
88Sr atoms. By appropriate orientation of the wave vectors of three lasers incident on the atoms, the first-order Doppler shift can be eliminated for all velocity classes. Three-photon excitation of the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ transition enables high-contrast Ramsey spectroscopy with interrogation times comparable to the $21\phantom{\rule{0.16em}{0ex}}\textmu{}\mathrm{s}$ natural lifetime using a single near-resonant laser source. Three-photon spectroscopy on the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ clock transition, using only laser frequencies nearly resonant with the $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{0}$ and $^{1}S_{0}\phantom{\rule{0.16em}{0ex}}\ensuremath{\leftrightarrow}\phantom{\rule{0.16em}{0ex}}^{3}P_{1}$ transitions, enables a reduction in Doppler broadening by two orders of magnitude and a corresponding $\ensuremath{\sim}470\phantom{\rule{0.16em}{0ex}}\mathrm{Hz}$ linewidth without a confining potential.
Sub-Doppler spectroscopy in alkali-metal vapor cells using two counterpropagating dual-frequency laser beams allows the detection of high-contrast sign-reversed natural-linewidth sub-Doppler resonances. Previously, a qualitative theory based on a simplified \ensuremath{\Lambda}-scheme model has been reported to explain underlying physics of this phenomenon. In this paper, an extended theoretical model of dual-frequency sub-Doppler spectroscopy (DFSDS) for the Cs
D1 line is reported. Taking into account the real atomic energy structure, main relaxation processes, and various nonlinear effects, this model describes quantitatively the respective contributions of involved physical processes and predicts main properties (height and linewidth) of the sub-Doppler resonances. Experimental tests are performed with a Cs vapor microfabricated cell and results are found to be in correct agreement with theoretical predictions. Spatial oscillations of the sub-Doppler resonance amplitude with translation of the reflection mirror are highlighted. A beat note between two laser systems, including one stabilized with DFSDS on a Cs vapor microcell, yields a fractional frequency stability of $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$ until 10-s averaging time. These results demonstrate that DFSDS could be an interesting approach for the development of a high-performance microcell-based optical frequency reference, with applications in various compact quantum devices.
The sub-Doppler resonances linewidths and amplitudes depend on the laser beam intensity. The effect of laser beam intensity on the resonance linewidths and amplitudes obtained from different energy transitions of atoms varies from resonance to resonance. The effect of laser beam intensity on resonance linewidths and amplitudes is of great importance for diode laser frequency stability applications. It needs to be determined by measuring. The effect of the laser beam intensity on the linewidths and amplitudes of sub-Doppler resonances were measured by laser heterodyne spectroscopy using the linearly polarized frequency stabilized extended cavity diode lasers. The measurements are compatible with the theory and the uncertainty of the measurements are fewer than 1.6 MHz and 0.3 mV for linewidths and amplitudes, respectively.