Dual-frequency sub-Doppler spectroscopy: Extended theoretical model and microcell-based experiments

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 Λ-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 $D_1$ 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 $2\times {10}^{-12}{\tau }^{-1/2}$ 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.

Figure 1

Sketch of the proposed optical configuration: M, movable mirror; λ/4, quarter-wave plate; PD, photodiode.

Figure 2

Relevant energy levels of the $D_1$ line of an alkali-metal atom. Solid arrows denote optical transitions induced by the waves propagating along the $z$ axis, while dashed arrows stand for backward waves. Wavy arrows are for spontaneous relaxation. $\hslash {\mathrm{\Delta }}_{\mathrm{g}}$ is the energy hyperfine splitting of the atom ground state. The case depicted here corresponds to the atoms at rest under the null Raman detuning (${\delta }_{\mathrm{R}}$) and the null one-photon optical frequency detuning (δ). For Cs, we have $F_1=3, F_2=F_3=4$.

Figure 3

Sub-Doppler resonances calculated for the single- (solid and dashed curves at the bottom) and dual-frequency (solid red spike) regimes of light-field excitation. Here, $A$ is the height of the sub-Doppler resonance under the DF regime, while $A_{\mathrm{D}}$ is the height of the broad Doppler background. The total laser beam power at the entrance of the cell is 50 μW. The magnetic field is switched off. Other parameters are written in the text.

Figure 4

Analysis of different contributions to the excited-state population. (a) Influence of linear polarizations orientation at mutual backward waves phase ${\varphi }_{12}=0$. (b) Influence of the mutual phase ${\varphi }_{12}$ for orthogonal linear polarizations of the counterpropagating waves ($\alpha =\pi /2$). The total laser beam power is 50 μW; the static magnetic field $B$ is switched off.

Figure 5

Calculated contribution $W_0$ in the single-frequency regime, when only the transition $F_2\to F_3$ is excited. β is the branching ratio for the transition. The total laser beam power is 50 μW. The static magnetic field is null.

Figure 6

Influence of a difference in light wave intensities on $W_0$ and $W_z$. Parameters: ${\varphi }_{12}=0, \alpha =\pi /2$. The total laser beam power is 50 μW, $B=0$. Intensities $I_3$ and $I_4$ are assumed to be smaller than $I_1$ and $I_2$ by 30%.

Figure 7

Influence of the static magnetic field on $W_0$ and $W_z$. The field is applied along the light wave vectors ($z$ axis) at (a) parallel and (b) orthogonal linear polarizations of counterpropagating laser beams. Parameters are $P=10\mu \mathrm{W}$ and ${\varphi }_{12}=0$. Ω is the Larmor frequency.

Figure 8

(a): Photograph of a Cs microfabricated vapor cell. (b) Experimental setup for DFSDS measurements in the Cs microcell. ECDL, external-cavity diode laser; EOM, electro-optic modulator; LO, 4.596-GHz microwave synthesizer; HWP, half-wave plate; PBS, polarizing beam splitter; QWP, quarter-wave plate; M, mirror; PD, photodiode. The 4.596-GHz signal is applied to the EOM for the dual-frequency measurements.

Figure 9

Measurements of single-frequency (SF) and dual-frequency (DF) sub-Doppler spectra through the microcell at 60 °C. The laser power is about 200 μW after a single pass. The SF spectrum is obtained from the $F_1=3$ state. The spectra are fitted by two-Doppler plus two-Lorentzian functions with a quadratic background. Fit parameters for similar scans were used to deduce the height and width of the Doppler-broadened and sub-Doppler profiles.

Figure 10

Height of the sub-Doppler absorption spike normalized to the Doppler background height vs the distance between the retroreflection mirror and the microcell (Fig. 8). The solid line is the result of numerical calculations. The cell temperature is 42 °C. The laser is connected to the $F_3=4$ excited state and the laser power is 45 μW.

Figure 11

(a) Linewidth and (b) normalized height of the sub-Doppler resonance as a function of the laser power under the single-frequency (SF) and dual-frequency (DF) regimes. Theory results are shown as solid curves. In the SF regime, the transition $F_1=3\to F_3=4$ is excited. Other parameters are $\alpha =\pi /2, {\varphi }_{12}=0, B=0$.

Figure 12

Non-normalized height of the resonance divided by its linewidth vs laser power under the single-frequency (SF) and dual-frequency (DF) regimes.

Figure 13

Allan deviation of the laser beat note for free-running (black squares), single-frequency (red triangles), and dual-frequency (blue circles) regimes. In the SF case, the transition $F_1=3\to F_3=4$ is excited.