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A cloud of cold atoms constitutes a new medium to study random lasing8,9,10,11,12,13,14, allowing a detailed microscopic understanding of gain and scattering. Multiple scattering of light in cold atoms has been extensively studied in the past16,17. Furthermore, quasi-continuous lasing with cold atoms as a gain medium, either placed inside optical cavities18,19,20,21 or based on distributed feedback22, has recently been demonstrated, illustrating the potential for a variety of gain mechanisms in a regime where optical coherence is limited by purely radiative decay channels. This is significantly different with respect to most random lasing devices, based on pulsed excitation of condensed-matter systems, where the relaxation rates of the optical coherence are several orders of magnitude faster than the decay of the excited state population. Long phase coherence times however allow for efficient feedback by resonant scattering, as expected in astrophysical lasers4.

To combine sufficient gain and scattering while using only one atomic species, we take advantage of the multilevel structure of rubidium atoms, shown in Fig. 1a (D2 line of 85Rb, wavelength λ = 780 nm). Two-photon Raman gain is obtained by a population inversion between the two hyperfine ground states |2〉 and |3〉 sustained by optical pumping. A Raman laser drives the |3〉→|2′〉 transition with a large detuning Δ, so that atoms can be transferred into the |2〉 state by stimulated emission. Scattering required for random lasing is provided by the |2〉→|1′〉 line, which is a closed transition efficient for multiple scattering. Both Raman gain and scattering can occur at the same frequency (that is, for the same photons) by an appropriate choice of the Raman detuning Δ = −4.8Γ, determined by the hyperfine splitting between the |1′〉 and |2′〉 states (Γ/2π = 6 MHz is the linewidth of the transition). We thus tune the Raman laser to the vicinity of this condition and define the detuning δ = Δ+4.8Γ as the relevant parameter (Fig. 1a). This scheme takes advantage of the selection rules, which forbid electric dipole transitions between the states |3〉 and |1′〉, so that the |1′〉 level does not affect the Raman gain.

Figure 1: Working principle of the random laser.
figure 1

a, Atomic transitions of the D2 line of 85Rb (at λ = 780.24 nm) used to create random lasing in cold atoms. The two hyperfine ground states are |2〉 = |F = 2〉 and |3′〉 = |F′ = 3〉, separated by 3 GHz. Similarly, the involved hyperfine excited states are the states |F′ = 1,2,3〉 denoted |1′〉, |2′〉 and |3′〉 with splittings of a few tens of megahertz. Optical pumping creates a population inversion between |2〉 and |3〉. This allows us to create Raman gain by applying a laser with a detuning Δ from the |3〉→|2′〉 transition. The gain frequency has a detuning δ from the closed |2〉→|1′〉 transition. Around δ∼0, this transition provides efficient scattering. Random lasing can thus occur around this frequency. b, Schematic of the experiment. The cold-atom cloud (yellow sphere) is exposed to two Raman-laser beams (green) and six optical-pumping beams (red). Its fluorescence is collected by a lens and detected by a photodiode. Zoom-in: light (in blue) is scattered by atoms in the |2〉 state (black) and amplified by atoms in the |3〉 state (yellow background).

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For a given amount of gain and scattering, the threshold of random lasing is determined by a critical minimum size of the sample1. In our case, gain and scattering are provided by the same atoms and depend on the atomic density n. We have shown23,24,25 that the critical parameter defining the random-laser threshold is the on-resonance optical thickness b0, defined for a homogeneous cloud of radius R as b0 = 2n σ34′R, with σ34′ being the on-resonance scattering cross-section for the |3〉→|4′〉 transition (used to measure b0; see Methods and Supplementary Fig. S1). Moreover, its critical value can be computed from the atomic polarizability alone (see Supplementary Information).

Our sample consists of a cloud of cold 85Rb atoms collected in a magneto-optical trap. A controlled compression period provides a variable optical thickness b0 with a constant number of trapped atoms (see Methods). We then switch off all lasers and magnetic field gradients 1 ms before applying strong counterpropagating Raman beams (intensity IRa = 4.25 mW cm−2 per beam with crossed linear polarizations) tuned around δ∼0. In addition, we use an optical-pumping laser tuned slightly below the |2〉→|3′〉 transition to sustain a steady-state population inversion between the two hyperfine levels involved in our scheme. The relative intensity between the two external lasers allows us to adjust the relative populations, and thus to tune continuously from a sample with large gain and no scattering (with all atoms in the |3〉 state) to a situation without gain and with large scattering on the |2〉→|1′〉 line (with all atoms in the |2〉 state). Note that with the chosen detunings, these lasers operate in the single scattering regime. The data presented here have been obtained with an optical-pumping intensity IOP = 2.9 mW cm−2, ensuring a small population inversion.

In random lasers, there is not any privileged emission direction that allows one to spatially separate the random-laser light from amplified spontaneous emission and scattering from the pump beam, which is usually much stronger than the random-laser emission itself. In our system, however, the |2〉→|1′〉 transition does not scatter light from the two external lasers. The random-laser line thus has a strength comparable to the one of the other involved transition and the signature of random lasing can be obtained by the detection of the total emitted light from the sample, which we collect with a solid angle of 10−2 sr at an angle of 40° with respect to the Raman beam axes (Fig. 1b). Figure 2 shows the measured fluorescence as the Raman laser frequency is swept through the region of interest, for different values of the optical thickness b0 of the atomic cloud (see Methods). We stress that for these measurements we vary b0 while keeping the atom number constant. Variations in the fluorescence can thus only be related to collective features.

Figure 2: Fluorescence measurement.
figure 2

Total fluorescence PF as a function of the Raman laser detuning δ (in units of the linewidth Γ of the optical transition) recorded for optical thickness varying from b0 = 1.9 to b0 = 26. The number of atoms is kept constant at N = 7×108±12%. Two collective features are visible. In the wings (regions 1), the overall increase of the fluorescence with b0 is due to amplified stimulated emission. Around δ∼0 (region 2), an extra peak appears for large optical thickness. This enhanced light emission is due to the combination of Raman gain and multiple scattering provided by the |2〉→|1′〉 transition and is thus a signature of random lasing.

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The first signature of such a collective behaviour can be seen in a regime of negligible scattering, far from the |2〉→|1′〉 transition (regions 1 of Fig. 2): amplified spontaneous emission (ASE) induces an overall increase of the fluorescence as a function of b0. Photons from the Raman beam can indeed undergo a spontaneous Raman transition. The subsequent scattered light is then amplified by Raman gain produced by the surrounding atoms while leaving the sample with a ballistic path. The efficiency of this process is directly related to the optical thickness (see Supplementary Information). The ASE signal decreases as the Raman laser is detuned further away from the |3〉→|2′〉 transition (located at δ = +4.8Γ) because both the spontaneous (source contribution) and stimulated (gain contribution) Raman scattering rates decrease for larger detuning. Note that when tuning the Raman laser very close to the |3〉→|2′〉 line, single-photon scattering dominates. As detailed in ref. 26, population redistribution is then responsible for the increase of fluorescence. This effect is negligible for the detunings considered here, and only gain can explain the observed features.

When the Raman laser is tuned close to δ = 0 (region 2 of Fig. 2), the combination of gain and scattering gives rise to a random laser. It appears as an enhanced fluorescence bump that emerges as the optical thickness b0 is increased. To better extract this signal, we fit the wings of the curves (regions 1) by adjustable slope and curvature and remove this ASE background. The remaining random-laser signal is a Gaussian peak, well centred at δ = 0 (Fig. 3a), which thus comes from the scattering due to the |2〉→|1′〉 transition. Therefore, the observed peak is due to the combination of gain and scattering. Moreover, although the signal consists of different emission lines, a threshold of the peak amplitude is clearly visible, with a change of slope at b0 = 6±1 (Fig. 3b). This threshold is the signature of the occurrence of random lasing in our sample when the Raman beams are tuned around δ∼0 and when b0>6. We stress that varying the optical thickness acts simultaneously on the amount of gain and feedback provided by the medium. This is unusual in laser physics, where the threshold is most commonly defined as a critical pump power. In our case, increasing the optical-pumping intensity indeed increases the population inversion that provides gain, but simultaneously decreases the feedback, so that random lasing needs a fine tuning of the laser parameters.

Figure 3: Random laser emission around δ = 0.
figure 3

a, Measured supplementary fluorescence δ PF due to random lasing as a function of the Raman laser detuning δ, measured for optical thickness varying from b0 = 1.9 to b0 = 26. The source data and the colour scale are the same as in Fig. 2, but the wings (regions 1 in Fig. 2) have been subtracted. The inset illustrates the fitting procedure. For clarity we also smoothed the data, corresponding to a detuning resolution of 0.3Γ. b, Measured amplitude of the peak due to random lasing. A threshold optical thickness is clearly visible at b0∼6. Vertical error bars correspond to the root mean squared noise on the data of Fig. 2 and horizontal error bars to shot-to-shot fluctuations of b0. c, Computed supplementary fluorescence δ PF as a function of the detuning δ, using our self-consistent model for random lasing, neglecting the contribution from ASE, for four different optical thicknesses, b0 = 26 (blue), b0 = 30 (green), b0 = 34 (orange) and b0 = 38 (red). d, Maximum supplementary fluorescence δ PF computed from the self-consistent model for random lasing. Inset: principle of the model. The atomic response allows the computation of the threshold optical thickness following Letokhov’s criterion1,23. Owing to saturation effects, this threshold depends on the random-laser intensity. Therefore, for each b0, we find the random-laser intensity such that the computed threshold equals b0, corresponding to a steady state (see Supplementary Information).

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Finally, we have exploited the possibility to perform ab initio theory by developing two simple models, one for ASE and the other for random lasing. Both are detailed in the Supplementary Information. Here, we describe briefly the random-lasing model. It consists of self-consistently coupling the atomic response, based on optical Bloch equations with additional scattering on the |2〉→|1′〉 line, to a diffusion equation for the light scattered on the |2〉→|1′〉 resonance. The optical Bloch equations allow us to compute the atomic polarizability and, including the additional scattering on the |2〉→|1′〉 line, the mean-free path ℓsc and the gain length ℓg, including saturation effects due to the random laser intensity IRL inside the sample. As in conventional laser theory, we look for a steady-state solution where gain exactly compensates losses. In the diffusive regime and taking into account only the diffuse mode with the longest lifetime, this condition is equivalent to Letokhov’s result on the random-lasing threshold1,23,

where Rcr is the critical sample size. As a consequence, for a given b0, we find the value of IRL such that equation (1) is fulfilled. Considering the absence of any free fitting parameter and the simplicity of our model, which neglects for instance the contribution of ASE, the experimental data and the computed values for the supplementary fluorescence show satisfactory qualitative agreement (Fig. 3). The quantitative discrepancies suggest the need for more involved models. Many ingredients could play a role in our experiment and have been neglected in our models, such as the inelastic spectrum of the emitted light, interference effects on light transport, light polarization, the Zeeman degeneracy of the involved atomic levels, the finite temperature, the inhomogeneous atomic density distribution, and cooperative effects21,27. The comparison between the experiment and new models including some of these effects will allow one to identify the most relevant ones and thus to better understand random-laser physics.

We have presented experimental evidence of combined gain and scattering of light in a cloud of cold atoms, demonstrating random lasing in a dilute vapour. This type of experiment, based on well-controlled atomic systems, with the possibility of ab initio calculations, will allow studying the role of interferences and cooperativity in random lasing27,28,29. The combined theoretical and experimental approach described in this work can also be applied to realistic atomic structures encountered in astrophysical systems and to test new detection schemes, based for instance on high-order photon correlations30.

Methods

Sample preparation.

In our experiment 6 counter-propagating trapping beams with a waist of 3.4 cm (1/e2 radius of the intensity distribution) are used to load 85Rb atoms from a background atomic vapour into a magneto-optical trap (MOT). The trapping beams are detuned by −3Γ from the |3〉→|4′〉 hyperfine transition. To maintain the atomic populations in |3〉, we add 6 repumper beams tuned slightly below the |2〉→|3′〉 transition. We can load between 108 and 1011 atoms by changing the background vapour pressure and the duration of the trap loading (from 10 to 500 ms). Once the atoms are trapped in the MOT, we perform a temporal dark-MOT stage by increasing the detuning of the trapping beams to −6Γ and by reducing the intensity of the repumper beams to a few per cent of their initial value. This leads to an increase of the spatial density and thus of the optical thickness b0 of the cloud, without loss of atoms. By changing the duration of this compression stage, we are able to tune b0 from 1.9 to 27, while keeping almost constant the total number of atoms, which, for the measurements presented here, is set to 7×108±12%. The temperature is T∼50 μK. The optical thickness b0 is measured by a transmission spectrum with a small and weak probe beam on the |3〉→|4′〉 transition24. The shot-to-shot fluctuations of b0 (horizontal error bars in Fig. 3b) are evaluated by repeating the measurement five times.

Data acquisition.

After the sample preparation, we switch off magnetic field gradients and trapping lasers and we expose the sample to two counterpropagating Raman beams with waists of 2.4 cm, intensities of 4.25 mW cm−2 each and linear orthogonal polarizations, and 3 pairs of counterpropagating optical pumping beams with waists of 3.4 cm, intensities of 0.48 mW cm−2 and σ+/σ− polarizations, each at a detuning of −3Γ from the |2〉→|3′〉 transition. Note that the diameters of these lasers are large enough to ensure that their effective intensities on the atom cloud are independent from the chosen optical thickness. The Raman laser is obtained from a distributed-Bragg-reflector laser diode and is frequency-tuned by a double-pass acousto-optic modulator before it is amplified by two stages of saturated slave lasers. This system allows us to scan the frequency in a range up to 16Γ with intensity variations of only 0.1%.

The measuring procedure consists of scanning in 2 ms the Raman beam detuning δ from −3,2Γ to 4,8Γ while a high-gain photodiode gathers the fluorescent emission of the cloud in a solid angle of ∼10−2 sr. The detected power is of the order of 0.4 nW. We checked that the direction of the sweep does not change the detected fluorescence, and that the duration of the sweep is short enough to avoid significant variations of b0 during the measurements (<5%) and long enough to probe a quasi-steady state (the sweep rate is 4Γ ms−1). We averaged over 4,000 subsequent measurements to increase the signal-to-noise ratio, thus performing also an averaging over the disorder configurations.