Mollow triplet in cold atoms

The experimental setup is described in [26, 28]. A cold atomic cloud is provided by loading ⁸⁵Rb atoms in a MOT. A compression stage is added to increase its density and to obtain a smooth density profile. For this specific experiment, the cloud is made of typically 5 × 10⁷ atoms with a longitudinal rms radius σ_z = 0. 5 ± 0.1 mm, and a radial rms radius σ_r = 0. 7 ± 0.1 mm after a free expansion of 2 ms. The temperature measured by time of flight (TOF) technique is of the order of 100 μK before applying the strong laser beams to detect the Mollow spectrum.

To observe the Mollow triplet, the atomic cloud is illuminated by two counter-propagating laser beams A and B, as depicted in figure 1, applied along the longitudinal cloud axis z. Their power can be adjusted independently which allows, for instance, to avoid pushing the atoms due to imbalanced radiation pressure forces. The two beams come from the same distributed-feedback laser amplified by a tapered amplifier, and their frequency is locked close to the $F=3\to F^{\prime} =4$ hyperfine transition of the ⁸⁵Rb D₂ line, with linewidth Γ = 2π × 6.07 MHz. The locking system uses a master-slave configuration with an offset locking scheme [29], allowing us to adjust the laser frequency without changing the beam direction and power.

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We adjust the laser beam polarization, measured with a polarimeter just before the vacuum cell, with a λ/2 plate and a λ/4 plate. To probe a single closed transition, and thus consider the atoms as two-level systems, we use the same circular polarization for the two beams. For a σ⁺ polarization, the atoms are quickly pumped in (F = 3, m_F = 3) and we probe the $F=3\ ({m}_{F}=3)\to F^{\prime} =4\ ({m}_{F}^{{\prime} }=4)$ transition. Since both beams have the same polarization, if applied simultaneously, they would interfere and create a standing wave, leading to a strong spatial modulation of the intensity [30] with a corresponding convolution of the expected Mollow triplet. To avoid such a convolution, one need to apply a single laser beam with a well defined homogeneous Rabi frequency. We therefore illuminate the atoms alternating successively beam A and beam B, the switching on and off being done using two acousto-optic modulators.

On the one hand, achieving a homogeneous Rabi frequency in the transverse direction would require a uniform intensity distribution, i.e. a plane-wave. On the other hand, the maximum available laser power imposes to decrease the beam waist to get a Rabi frequency as high as possible and thus well-resolved Mollow sidebands. As a compromise between the laser intensity and its non-uniformity over the cloud, we have used a beam waist of w₀ = 2.5 mm on our cloud of radius 0.7 mm. The laser beams are centered on the atoms using absorption imaging. We also observe an increase of the transverse radius of the cloud of about 10% after the alternating laser beams have been applied. The observed increase in the longitudinal direction is much higher, with typically an increase by a factor of 2.

Finally, the power of the laser beams can be adjusted to vary the Rabi frequency, by means of a λ/2 and a polarizing beam splitter (PBS). The maximum available power is about 20 mW per beam, corresponding to a maximum on-resonance saturation parameter at the center of the laser beam of ${s}_{0,\max }=I/{I}_{\mathrm{sat}}\simeq 140$, with a saturation intensity I_sat = 1.67 mW cm⁻² for the transition considered in this experiment, and a maximum Rabi frequency of ${{\rm{\Omega }}}_{\max }={\rm{\Gamma }}\sqrt{{s}_{0}/2}\simeq 8\,{\rm{\Gamma }}$. We also use two photodiodes to monitor the power of each beam and their fluctuations during an experimental run which can last from a few hours to one day. We typically observe intensity fluctuations of 5%.

Our goal is to measure the spectrum ${\tilde{g}}_{\mathrm{sc}}{}^{(1)}(\omega )$ of the light scattered by the atomic cloud, or its Fourier transform which corresponds to the first order temporal correlation function ${g}_{\mathrm{sc}}{}^{(1)}(\tau )$. The latter quantity is obtained by recording the beat note between the scattered light and a local oscillator (LO). The light scattered while illuminating the atomic cloud with beam A or B is collected by a polarization-maintaining (PM) single-mode fiber. This allows us to select only one spatial mode. A λ/2 plate and a PBS placed in front of the collection fiber allows to select one polarization mode and to optimally couple the polarized scattered light into the fiber. Indeed, the angle between the laser beams and the axis of the fiber is close to 90°. As the polarization of beams A and B are circular, once the atoms are pumped into the stretched m_F = +3 Zeeman sublevel, the polarization of the fluorescence in the direction of the fiber will be mostly linear. The λ/2 plate is thus adjusted to align the linear polarization of the scattered photons in order to maximise the transmission of the PBS. The LO is derived from the same laser that provides beams A and B, but with an extra detuning ω_BN = 2π × 120 MHz. The LO is then injected in another PM single-mode fiber, with a polarization selected by a PBS parallel to the fluorescence one.

The two fibers that collect the LO and the light scattered by the atoms are then connected to a fiber beam splitter (FBS). We compute the ${g}_{{\rm{s}}{\rm{c}}}{}^{(1)}(\tau )$ function via the measurement of the intensity autocorrelation of the beat note ${g}_{\mathrm{BN}}{}^{(2)}(\tau )$, given by the following formula [27]:

$$\begin{eqnarray}&&{g}_{\mathrm{BN}}{}^{(2)}(\tau )=\displaystyle \frac{\langle {I}_{\mathrm{BN}}(t){I}_{\mathrm{BN}}(t+\tau )\rangle }{\langle {I}_{\mathrm{BN}}(t){\rangle }^{2}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&=\ 1+2\displaystyle \frac{\langle {I}_{\mathrm{sc}}(t)\rangle \langle {I}_{\mathrm{LO}}(t)\rangle }{{\left(\langle {I}_{\mathrm{sc}}(t)\rangle +\langle {I}_{\mathrm{LO}}(t)\rangle \right)}^{2}}{\mathfrak{R}}\left({g}_{\mathrm{sc}}{}^{(1)}(\tau ){e}^{-i{\omega }_{\mathrm{BN}}\tau }\right)+\displaystyle \frac{\langle {I}_{\mathrm{sc}}(t){\rangle }^{2}}{{\left(\langle {I}_{\mathrm{sc}}(t)\rangle +\langle {I}_{\mathrm{LO}}(t)\rangle \right)}^{2}}\left({g}_{\mathrm{sc}}{}^{(2)}(\tau )-1\right),\end{eqnarray} \tag{ 2 }$$

with I_BN, I_sc, I_LO the intensity of the beat note, the collected scattered light and the LO respectively, ${g}_{\mathrm{sc}}{}^{(2)}(\tau )$ the temporal intensity correlation of the scattered light, and with $\langle .\rangle$ corresponding to averaging over time t. It is important to note that this signal is not sensitive to the field correlation of the laser itself. This is due to the fact that the LO and laser beams A and B are derived from the same laser, a well known technique used to get rid of the laser linewidth, at least at the first order. This is an advantage compared to other kind of techniques often used to measure the spectrum, such as Fabry–Perot interferometers, which require a laser with a low spectral width as well as a high resolution spectrometer [9, 10, 13].

Experimentally, we have a LO intensity more than 10 times higher than the scattered intensity. In this case, one can neglect the last term in equation (2). In addition, it is shifted in frequency by ω_BN from the ${g}_{\mathrm{sc}}{}^{(1)}$ term under investigation. We are left with a signal oscillating at the frequency ω_BN and with an amplitude proportional to ${g}_{{\rm{s}}{\rm{c}}}{}^{(1)}(\tau )$. We then take the Fourier transform of ${g}_{\mathrm{BN}}{}^{(2)}(\tau )-1$ which contains the spectrum of the scattered light shifted by ω_BN.

The experimental setup to measure intensity correlations has been described in [26]. As shown in figure 1, the two outputs of the FBS are sent to two single-photon avalanche photodiodes (APDs, SPCM-AQRH from Excelitas Technologies). Their typical detection efficiency is 64% with an active area of 180 μm. This experimental scheme allows to overcome the photodiode dead time (a few tens of ns) and the afterpulsing (1% typically). The arrival time of each photon recorded on the APDs are processed by a time-to-digital converter (TDC, ID800 from IDQuantique), with a time resolution of 162 ps. The data are transferred in real time to a computer and processed by Matlab to calculate the histogram of the time intervals between the two channels. In this real time mode, the data transfer rate of the TDC is limited to about 2.5 × 10⁶ counts per second. We set the number of counts by adjusting the power of the LO and we typically work with 800 photons per 200 μs which corresponds to the total pulse duration of the laser beams.

To measure the Mollow triplet, the following time sequence is used. We first turn off the trapping laser beams and the magnetic field. The atoms are released from the MOT and a TOF of 2 ms is applied to ensure that the MOT magnetic field gradient is completely off while keeping a small cloud radius compared to the waist of the incident laser beam. Then, we apply the laser beams. To avoid any interference effect, the counter-propagating beams A and B are switched on successively during t_pulse = 20 μs for each pulse, with a waiting time of 20 μs between two consecutive pulses. We use for each run a train of 10 pulses, i.e. 5 pulses from beam A and 5 pulses from beam B. These parameters were chosen to minimize the effect of pushing and heating the atoms while keeping a reasonable pulse duration to perform a measurement.

The APDs are gated on the beam pulses. We calculate the histogram of the temporal coincidences for each pulse and after the first μs of the pulse once the laser intensity is stable. This also ensures that the stationary state is reached, as assumed in the calculations of the Mollow triplet [1, 31]. We then analyse the data coming from beams A and B separately. We first check on the histograms that the position of the sidebands as well as their shape do not depend on the number of the pulse, and we then sum the 5 histograms corresponding to each beam. The variation of temperature that could affect the histogram is averaged over the 5 pulses. Because of the finite time-window of the measurement, the histogram needs to be normalized to get the intensity correlation function, as explained in [26].

The spectrum ${\tilde{g}}_{\mathrm{sc}}{}^{(1)}(\omega )$ of light scattered by a two-level system driven by a strong incident field close to resonance is commonly known as the Mollow triplet. Its derivation can be found in [1] assuming a quantum two-level system driven by a classical field, or in [31] with a fully quantum-mechanical approach. It can be decomposed in two parts:

$$\begin{eqnarray}&&{\tilde{g}}_{\mathrm{sc}}{}^{(1)}(\omega )={\tilde{g}}_{\mathrm{el}}{}^{(1)}(\omega )+{\tilde{g}}_{\mathrm{inel}}{}^{(1)}(\omega ),\end{eqnarray} \tag{ 3 }$$

with ${\tilde{g}}_{\mathrm{el}}{}^{(1)}(\omega )$ the elastic part and ${\tilde{g}}_{\mathrm{inel}}{}^{(1)}(\omega )$ the inelastic part given by the following formula:

$$\begin{eqnarray}&&{\tilde{g}}_{\mathrm{el}}{}^{(1)}(\omega )=\displaystyle \frac{s}{2{\left(1+s\right)}^{2}}\delta (\omega ),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\tilde{g}}_{{\rm{i}}{\rm{n}}{\rm{e}}{\rm{l}}}{}^{(1)}(\omega )=\displaystyle \frac{{s}_{0}}{8\pi {\rm{\Gamma }}}\displaystyle \frac{s}{1+s}\times \displaystyle \frac{1+{\textstyle \tfrac{{s}_{0}}{4}}+{\left({\textstyle \tfrac{\omega }{{\rm{\Gamma }}}}\right)}^{2}}{{\left[{\textstyle \tfrac{1}{4}}+{\textstyle \tfrac{{s}_{0}}{4}}+{\left({\textstyle \tfrac{{\rm{\Delta }}}{{\rm{\Gamma }}}}\right)}^{2}-2{\left({\textstyle \tfrac{\omega }{{\rm{\Gamma }}}}\right)}^{2}\right]}^{2}+{\left({\textstyle \tfrac{\omega }{{\rm{\Gamma }}}}\right)}^{2}{\left[{\textstyle \tfrac{5}{4}}+{\textstyle \tfrac{{s}_{0}}{2}}+{\left({\textstyle \tfrac{{\rm{\Delta }}}{{\rm{\Gamma }}}}\right)}^{2}-{\left({\textstyle \tfrac{\omega }{{\rm{\Gamma }}}}\right)}^{2}\right]}^{2}},\end{eqnarray} \tag{ 5 }$$

where ω denotes the frequency relative to the laser frequency. The laser detuning is denoted by Δ, and the saturation parameter depends on the laser detuning as follows:

$$\begin{eqnarray}&&s({\rm{\Delta }})=\displaystyle \frac{\tfrac{I}{{I}_{\mathrm{sat}}}}{1+4{\left(\tfrac{{\rm{\Delta }}}{{\rm{\Gamma }}}\right)}^{2}}=\displaystyle \frac{{s}_{0}}{1+4{\left(\tfrac{{\rm{\Delta }}}{{\rm{\Gamma }}}\right)}^{2}}.\end{eqnarray} \tag{ 6 }$$

One can show that the ratio of the intensity of the inelastic part to the elastic part, obtained by integrating equations (4)–(5) over the emitted spectrum, is given by:

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{\mathrm{inel}}}{{I}_{\mathrm{el}}}=\displaystyle \frac{{s}^{2}/{\left(1+s\right)}^{2}}{s/{\left(1+s\right)}^{2}}=s.\end{eqnarray} \tag{ 7 }$$

For Ω ≫ Γ and Ω ≫ Δ, the inelastic part can be approximated by three Lorentzians, the carrier at the laser frequency and the sidebands separated from the carrier by the generalized Rabi frequency:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{G}}}=\sqrt{{{\rm{\Omega }}}^{2}+{{\rm{\Delta }}}^{2}},\end{eqnarray} \tag{ 8 }$$

with Ω the Rabi frequency given by:

$$\begin{eqnarray}&&{\rm{\Omega }}={\rm{\Gamma }}\sqrt{\displaystyle \frac{s}{2}\left[1+4{\left(\displaystyle \frac{{\rm{\Delta }}}{{\rm{\Gamma }}}\right)}^{2}\right]}={\rm{\Gamma }}\sqrt{\displaystyle \frac{{s}_{0}}{2}}.\end{eqnarray} \tag{ 9 }$$

At resonance, the height ratio is 1:3:1 and the width is 3Γ/2 for the two sidebands and Γ for the carrier, corresponding to an intensity (area) ratio of 1:2:1.

A typical experimental intensity correlation is plotted in the inset of figure 2. The laser frequency was set to resonance and with a saturation parameter of the order of 80. We first observe a fast oscillation at the frequency ω_BN corresponding to the frequency beat note between the LO and the light scattered by the atomic cloud. These oscillations decay on a time scale of the order of 1/Γ ≃ 26 ns. Finally, we can identify the beat note between different frequency components, namely between the Mollow sidebands and its carrier, giving rise to a revival of the amplitude of the oscillations. Considering the typical decay time of few tens of ns for the Mollow triplet at resonance, we adopted a window of ∼300 ns for the Fourier transform, which provides a good balance between the noise on the amplitude and the frequency resolution of the spectrum. This time window is then increased to 1 μs for off-resonant spectra, the elastic scattering part being more significant and thus corresponding to a much larger decay time. During our measurements, the imaginary part of the signal was always at the noise level, and no asymmetry in the spectra was observed. For this reason, we hereafter focus on the real part of the signal.

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In our experimental conditions, ${g}_{\mathrm{BN}}{}^{(2)}(\tau )-1$ is directly proportional to $| {g}_{\mathrm{sc}}{}^{(1)}(\tau )| \cos ({\omega }_{\mathrm{BN}}\tau )$. The Fourier transform, plotted in blue in figure 2, will thus directly give the spectrum of the scattered light. The height of this spectrum has been arbitrarily normalized to unity. The Mollow triplet is clearly visible with its carrier and the two sidebands.

When s₀ ≫ 1, the inelastic part is dominant and we thus neglect, at first, the elastic part. To extract s₀ from the experimental spectrum, and the corresponding Rabi frequency, we first fit the data using equation (5), using a least-squares procedure. The result corresponds to the dashed red line in figure 2. The first observation is that the full width at half maximum (FWHM) of the carrier is underestimated: the experimental data yield a FWHM of 1.4Γ instead of the expected value Γ. We can also notice that the height ratio between the sidebands and the carrier is not 1:3:1 as expected but of the order of 0.7:3:0.7 and that the sideband widths are larger than expected, with a small asymmetry for each sideband, the half width at half maximum being larger on the side of the carrier.

This discrepancy can be explained by different contributions. We will first focus on the sidebands. The first contribution comes from the laser linewidth ${\rm{\Delta }}{\nu }_{{\rm{L}}}=3\,\mathrm{MHz}$, measured with a beat-note technique between this laser and a laser with a much smaller linewidth, as explained in [28]. As said before, this linewidth has no effect on the carrier because we use the same laser for the LO and the laser beams A and B. On the other hand, different frequencies on the atoms lead to different generalized Rabi frequencies, and consequently to a broadening of the sidebands. However, if ${\rm{\Delta }}{\nu }_{{\rm{L}}}\ll {\rm{\Omega }}$, the change in the generalized Rabi frequency is of the order of ${\rm{\Delta }}{\nu }_{{\rm{L}}}^{2}/2{{\rm{\Omega }}}^{2}$. The minimum Rabi frequency used on this experiment is higher than 4Γ, corresponding to a change in Ω_G of less than 1% which can thus be neglected. The same argument holds if one considers the temperature of the cloud T. The distribution of the effective detunings, in the atomic rest frame, corresponds to a Gaussian with an rms width of $k\sqrt{{k}_{{\rm{B}}}T/M}$, where k is the wave vector, k_B is the Boltzmann constant and M is the atomic mass. We will see in the next section that, when the laser beams A and B are applied, T increases to a few tens of mK. This corresponds to a rms width of a few MHz, thus still with almost no effect on the generalized Rabi frequency and the sidebands broadening. The last contribution, which actually fully explains the sidebands broadening, comes from the finite waist of the laser beam. This results in a Rabi frequency not being the same for all the atoms. This does not affect the carrier width but leads to an inhomogeneous broadening of the sidebands: their height is thus decreased and their width increased.

Finally, the increase in the carrier width is due to temperature. In the previous paragraph, we have evaluated the impact of the frequency detuning of the incident photon in the atomic rest frame, but one also needs to take into account the frequency shift of the scattered photon. This effect has been characterized quantitatively in one of our previous experiments on the elastic spectrum [26], showing that the theoretical spectrum, given for the inelastic part by equation (5) for atoms at rest, should be convoluted by the Doppler-broadened spectrum. This spectrum is Gaussian with a rms width Δω_D that depends on the temperature as well as the angle θ between the incident photon and the scattered direction:

$$\begin{eqnarray}&&{\rm{\Delta }}{\omega }_{{\rm{D}}}=k\sqrt{2(1-\cos \theta ){k}_{{\rm{B}}}T/M}.\end{eqnarray} \tag{ 10 }$$

To take into account these two contributions, namely temperature and finite waist of the laser beam, we have implemented a new fitting procedure. First, the laser beam is considered homogeneous in the propagation direction z since the Rayleigh length is of several meters. Then, we model the cloud as a continuous Gaussian distribution which, after integration over the longitudinal axis z, reads $\rho ({r}_{\perp })=N\exp (-{r}_{\perp }^{2}/2{\sigma }_{r}^{2})/(2\pi {\sigma }_{r}^{2})$. The spectrum radiated by the cloud in the finite-waist beam, with Rabi frequency ${{\rm{\Omega }}}_{{{\bf{r}}}_{\perp }}={\rm{\Omega }}\exp (-{r}_{\perp }^{2}/{w}_{0}^{2})$, is given by:

$$\begin{eqnarray}&&{\tilde{g}}_{\mathrm{cloud}}{}^{(1)}=2\pi {\int }_{0}^{\infty }\rho ({{\bf{r}}}_{\perp }){\tilde{g}}_{{{\rm{\Omega }}}_{{{\bf{r}}}_{\perp }}}{}^{(1)}(\omega ){r}_{\perp }{{\rm{d}}{r}}_{\perp },\end{eqnarray} \tag{ 11 }$$

where the single-atom spectrum ${\tilde{g}}_{{{\rm{\Omega }}}_{{{\bf{r}}}_{\perp }}}{}^{(1)}$ is given by equation (5) for a Rabi frequency ${{\rm{\Omega }}}_{{{\bf{r}}}_{\perp }}$. Finally, to take into account the effect of the temperature, the spectrum is convoluted with the following Gaussian: $\exp (-{\omega }^{2}/2{\rm{\Delta }}{\omega }_{D}^{2})$ and the final height is renormalized to unity. The free parameters are the frequency beat note ω_BN (which is always in agreement with the experimental value, within fluctuations of ∼1%), the Rabi frequency at the center of the beam Ω, the ratio between the laser beam waist and the radial atomic cloud radius w₀/σ_r, and the temperature T. By incorporating these finite-beam and finite-temperature effects into the fitting procedure, the normalized least-square residuals are reduced by a factor of 4, from 5.6% for the single Rabi frequency model to 1.5%.

3.4.1. Resonant Mollow triplet

This fitting procedure is first applied on the data of figure 2. The result is plotted in dash–dotted black and is in very good agreement with the experimental data. The fitted ratio w₀/σ_r gives a radial atomic cloud radius of σ_r = 950 ± 30 μm, comparable to the radius measured by absorption imaging and when the laser beams are applied. The fitted broadening of the carrier width due to temperature is of the order of 0.3Γ, which, according to equation (10), corresponds to a temperature of 10 mK. This temperature is much higher than the one measured on the initial atomic cloud by TOF technique. However, the latter measurement has been realized without the laser beams, and the photons exchanged during the application of these beams significantly heat the atomic sample. When we repeat the TOF measurement after the application of the laser beams, we find a temperature between 10 and 40 mK. Hence, the temperature increases from 100 μK to a few tens of mK during the intensity correlations measurements, with a mean value of typically 20 mK over the 5 pulses, corresponding to the same order of magnitude extracted from the fit and thus validating the hypothesis that temperature is the main contribution to the carrier broadening.

We repeated the same kind of measurements, still at resonance, but with different powers. For each spectrum, we checked that the temperature and w₀/σ_r correspond to what is experimentally extracted by absorption imaging. We finally plot the Rabi frequencies measured in the spectrum as a function of the powers measured just before the vacuum cell (see figure 3). The vertical error bars are given by the uncertainty on the fit of the Rabi frequency. The horizontal ones correspond to the power fluctuations measured during each experimental sequence, which range from 1% to 10%, with an additional 3% accuracy due to the power-meter. The theoretical Rabi frequency can be deduced from equations (6) and (9), as well as the laser waist and peak power. This theoretical value corresponds to the curve in figure 3, and is in very good agreement with our measurements, the only free parameters being those of the fitting procedure. The error from the fitting procedure is of ∼6% for the Rabi frequency, ∼10% for Δω_D and ∼6% for w₀/σ_r, resulting in an overall normalized least-squares error of  8%.

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3.4.2. Off-resonance spectrum

We now turn to the analysis of the Mollow spectrum when the laser frequency is shifted from the atomic transition by a detuning Δ of a few linewidths. A typical spectrum is shown in figure 4, for ${\rm{\Delta }}=3\,{\rm{\Gamma }}$ and with s₀ = 140. Two sidebands due to the inelastic scattering are still observed. The sharp central part now corresponds to the inelastic contribution plus a significant elastic component. Because of the lower saturation parameter for a nonzero detuning, the elastic part can no longer be neglected in our analysis. We thus include equation (4) in the computation of the total spectrum given by (11) where, as before, the finite size of the laser waist and the temperature are accounted for. In particular, the temperature not only broadens the carrier of the inelastic spectrum, but also turns the homogeneous elastic component into a Doppler-broadened Gaussian peak. The resulting fit is plotted in figure 4 in dashed red, showing a good agreement with the experimental data, with a normalized least-squares residual of ∼5%. The temperature and the ratio w₀/σ_r are comparable to the values obtained at resonance, as expected from the large saturation parameters that are used.

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The amplitudes of the elastic component A_el and of the inelastic component A_inel are fitted independently as follows:

$$\begin{eqnarray}&&{\tilde{g}}_{\mathrm{sc}}{}^{(1)}(\omega )={A}_{\mathrm{el}}\times {\tilde{g}}_{\mathrm{el}}{}^{(1)}(\omega )+{A}_{\mathrm{inel}}\times {\tilde{g}}_{\mathrm{inel}}{}^{(1)}(\omega ).\end{eqnarray} \tag{ 12 }$$

The resulting ratio ${I}_{\mathrm{inel}}/{I}_{\mathrm{el}}$, corresponding to the ratio of the integral of the inelastic spectrum and the integral of the elastic one, is computed. It is theoretically equal to the saturation parameter, as shown in equation (7). This measurement has been done for different laser detunings and two different powers. On the other hand, the saturation parameter is calculated using the Rabi frequency Ω of the inelastic component, also extracted from the fit, thanks to equation (9). As shown in figure 5, the ratio I_inel/I_el is in good agreement with the computed saturation parameter. Here, the error on the saturation parameter, which comes from the fitting of the Rabi frequency, is of the order of 4%, whereas the one on the ratio I_inel/I_el comes from the fitting of both the inelastic and elastic components, and presents a much larger error bar, of 20%–50%. In particular, the elastic component is quite weak close to the atomic resonance, which corresponds to the largest saturation parameters in this figure, and thus results in the largest error bars.

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3.4.3. Simultaneous pulses and non-circular polarization

So far, we have paid attention to have the system as close as possible to an ideal two-level system, with a well defined Rabi frequency. To illustrate the importance of these points, we have performed two more tests. We first measured the spectrum of light scattered by the cold atomic cloud when it is illuminated by a laser beam with a linear polarization, at resonance and with ${s}_{0}\simeq 80$. The result is plotted in figure 6 (red dashed curve) where we compare the circular and linear polarization. As said before, for a left-handed or right-handed circular polarization, we only excite the transition $F=3\ ({m}_{F}=\pm 3)\to F^{\prime} =4\ ({m}_{F}^{{\prime} }=\pm 4)$. On the contrary, for a linear polarization, different transitions between different Zeeman substates contribute, that possess different dipole moments. The opening of the scattering process to different substates thus results not only in a decrease of the effective Rabi frequency, due to the lower values of the Clebsch–Gordon coefficients compared to the one associated to the $F=3\ ({m}_{F}=\pm 3)\to F^{\prime} =4\ ({m}_{F}^{{\prime} }=\pm 4)$ transition, but also in a broadening of the sidebands and a decrease of their height. In this case, a comparison with the theoretical Mollow triplet requires to compute the steady state populations, their coupling to the excited states with different Clebsch–Gordon coefficients and the scattering rates between different Zeeman sublevels. This last feature is not taken into account by a multiple two-level scheme and thus requires a more evolved modeling, which is out of the scope of this paper.

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In order to test the importance of a homogeneous longitudinal Rabi frequency, we illuminated the atomic cloud with a circular polarization, but with both beams with identical circular polarization, shined at the same time. Since the polarization is the same for the two beams, their interference generates a standing wave, that leads to a strong spatial modulation of the Rabi frequency along the beam propagation. Then, as shown in figure 6 (green dash–dotted curve), the presence of the Mollow sidebands manifests only in a pedestal around the carrier, differently from the sequence with successive pulses. The strong Rabi frequency modulation leads to the superposition of a large range of sidebands, and thus to a blurred signal. Hence, this effect is an extreme case of the broadening of the sidebands due to the beam finite waist, which is responsible for a weaker modulation of the driving amplitude.