Light effects in the atomic-motion-induced Ramsey narrowing of dark resonances in wall-coated cells

We report on light shift and broadening in the atomic-motion-induced Ramsey narrowing of dark resonances prepared in alkali-metal vapors contained in wall-coated cells without buffer gas. The atomic-motion-induced Ramsey narrowing is due to the free motion of the polarized atomic spins in and out of the optical interaction region before spin relaxation. As a consequence of this effect, we observe a narrowing of the dark resonance linewidth as well as a reduction of the ground states’ light shift when the volume of the interaction region decreases at constant optical intensity. The results can be intuitively interpreted as a dilution of the intensity effect similar to a pulsed interrogation due to the atomic motion. Finally the influence of this effect on the performance of compact atomic clocks is discussed.

Figure 1

Scheme of the experimental apparatus. The light emitted by a distributed feedback (DFB) diode laser is separated in two parts. One part is used for optical frequency stabilization based on the saturated absorption signal in an evacuated uncoated reference cell. The second part is used for the main experiment: it is modulated by an electo-optical (EOM) at ${\nu }_{\mathrm{hf}}=6.834\hspace{0.5em}682\hspace{0.5em}610$ GHz and collected on a photodetector after passing through the different optic elements and the wall-coated cell.

Figure 2

Measurement of the typical dual structure [29] of dark resonances prepared in wall-coated cells. The resonant laser power is $12.1$ $\mu$W and the laser beam radius is $1.5$ mm (corresponding to a resonant laser intensity of $0.17$ mW cm${}^{-2}$). ${\delta }_R$ is the two-photon Raman detuning. The signal is recorded at zero magnetic field to avoid magnetic broadening and lineshape deformation of the broad resonance. The broad resonance has a linewidth (full width at half maximum (${\gamma }_b$) of the Lorentzian fit) of about $100$ kHz mainly determined by transit time and intensity broadening [33]. The linewidth of the narrow resonance is about $1000$ times narrower than the linewidth of the broad resonance.

Figure 3

Measurement of the narrow resonance corresponding to Fig. 2 when a longitudinal magnetic field $B_z=0.4$ $\mu$T is applied to remove the Zeeman degeneracy. The Zeeman structure of the CPT spectrum is shown in the inset. We focus our attention on the magnetic-insensitive dark resonance used in clock applications. The resonance linewidth (${\gamma }_n$), amplitude ($\Delta S$), and frequency shift ($\Delta \nu$) are $130$ Hz, $4$ mV, and $-316$ Hz, respectively.

Figure 4

Measurement of the linewidth (${\gamma }_n$) of the dark resonance as a function of the laser intensity contained in the carrier and one of the first-order sideband incidents on the cell, that is, the resonant laser intensity ($I$). The laser beam radius ($r$) is varied from $0.5$ to $4$ mm. The points are the experimental values while the lines are the linear fits with Eq. (1).

Figure 5

Evolution of the light broadening coefficient $g$ as a function of the square of the laser beam radius $r^2$. The values of $g$ were obtained by fitting Eq. (1) to the data of Fig. 4.

Figure 6

Evolution of the light-shift coefficient ($\mathrm{LS}=\partial y/\partial I$ with $y=\Delta \nu /{\nu }_{\mathrm{hf}}$) as a function of the square of the laser beam radius $r^2$.

Figure 7

Measurement of the dark resonance amplitude as a function of the laser intensity. The points are the experimental data, corresponding to the data reported in Fig. 4. Each line is a fit to the data using Eq. (2).

Figure 8

Sketch of the vapor cell used to explain the dynamics of the light-atom interaction. The trajectory of the atom is represented by a dotted (solid) line when the atom moves inside (outside) the laser beam. The $[t,I]$ plot summarizes the interaction sequence for a generic atom, where $I$ is the normalized laser intensity and $t$ is the time. The atomic spin evolves freely when $I=0$ while the atom interacts with the laser for $I\ne 0$, and after the time $t_A$ the atom is detected. The interaction time for each passage in our experiments is of the order of 10 $\mu$s.

Figure 9

Evolution of the Allan deviation ${\sigma }_y(1s)$ as a function of the laser intensity for different values of the laser beam radius. The values of ${\sigma }_y(1s)$ are calculated using Eq. (3) and the signal-to-(shot)noise measurements. In each plot the gray-dotted line corresponds to the value $3\times {10}^{-13}$, which is close to the optimal value obtained for each $r$.