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We investigate the transient coherent transmission of light through an optically thick cold Strontium gas. We observe a coherent superflash just after an abrupt probe extinction, with peak intensity more than 3 times the incident one (see Fig. (a) below). We show that this coherent superflash is a direct signature of the cooperative forward emission of the atoms. By engineering fast transient phenomena on the incident field, we give a clear and simple picture of the physical mechanisms at play. Finally, taking advantage of the fast decay time of the cooperative emission, we create a pulse train of light with a repetition time shorter than the atomic lifetime (see Fig. (b) below). In this regime single atom spontaneous emission can be quenched.

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**Related publications:**

[1] M. Chalony et *al*, Phys. Rev. A **84**, 011401(R) (2011).

[2] C.C. Kwong et *al*, Phys. Rev. Lett. **113**, 223601 (2014).

[3] C.C. Kwong et *al*, Phys. Rev. Lett. **115**, 223601 (2015).

**1. Experimental setup**

We consider the simple situation depicted in the figure below.

A resonant probe is send through a cold cloud of 88-strontium atoms. The coherent transmitted intensity is recorded on a detector. An acousto-optic modulator (AOM) and an electro-optic modulator (EOM) are used to change the amplitude and the phase of the incident electromagnetic field respectively. The probe is tuned across the ^{1}S_{0} -> ^{3}P_{1} intercombination line at 689 nm. The ^{3}P_{1} excited state lifetime is *τ* = 21 μs which can be much longer that the AOM and EOM characteristic time constant.

**2. Forward scattered field**

In the stationary regime, the coherent transmission intensity is given by the Beer-Lambert law: * I*_{t} = *I*_{0}e* ^{-b}*, where

*b*is the optical thickness. For the transmitted field, the Beer-Lambert law becomes, in the frequency domain:

(1) *E*_{t}(ω) = *E*_{0}(ω)exp[i*n*(ω)ω*L*/c]

where *n*(ω) is the medium complex index of refraction, and *L* the medium thickness. Moreover, one has *b* = 2ωIm[*n*(ω)]*L*/c and *θ *= ωRe[*n*(ω)-1]*L*/c, where *θ* is the relative phase shift of the transmitted field.

We know also that the coherent transmitted field results from the interference of the incident field and the field scattered by the atoms in the forward direction:

(2) *E*_{t} = *E*_{0}+ *E*_{s}

According to Eq. (2), the forward scattered intensity *I*_{s} can be measured switching off abruptly the incident field. Eq. (2) allows us also to reconstruct the relative phase of *E*_{s} performing measurements of the transmitted field before and after the incident field switch off. The result of this reconstruction is shown on the graph below (dots) as well as a direct measurement using interferometric means (stars). The plain curve is the prediction using Eqs. (1) and (2).

**3. Coherent superflash**

One the graph above, the forbidden region corresponds to a violation of the energy conservation law. The normal flash light gray region corresponds to *I*_{s} <= *I*_{0}. Finally the superflash region is associated to *I*_{0} < *I*_{s} <= 4*I*_{0}.

We call this phenomena a coherent flash, since the transmitted field rapidly decay once the incident field is switch off. Using a naive picture, we may expect that the peak intensity of this flash, corresponding to *I*_{s}, is smaller than the incident intensity. This actually always the case, if the laser is at resonance. However, at finite detuning, the real part of the index of refraction induces a phase shift of the transmitted field which may finally increases *I*_{s} such as *I*_{0} < *I*_{s} <= 4 *I*_{0}. If the latter inequality turns out to be true, the flash becomes a superflash.

**4. Flash duration and cooperative effect**

We observe that the decay time of the (super)flash is much faster than the excited state lifetime. To understand this effect, one has to note that the atoms are emitting cooperatively, i.e. with a well-defined phase relation, in the forward direction. The decay time is then:

(3) *τ*_{c} = *τ* /*μN*

Here *τ* is the excited state lifetime of a single atom, *N* the number of atoms and *μ* a geometrical factor corresponding to the numerical aperture of the forward scattered mode. In the Dicke superradiance regime *μ* = 1. Here *μ* ~ λ^{2}/*L*^{2}. Thus *μN* ~ *b*. In our experiment we find *τ*_{c} ≈ 0.02 *τ*.

**5. Cooperative pulse**

We take advantage of the short decay rate of the flash to generate a cooperative pulse train. To perform this experiment, we bring the probe laser to resonance and periodically change by π the phase of the incident field. In the Fig. (a) below, we give a schematic representation of the electric fields at play. Before the phase change, the transmitted intensity is almost zero, because of the large optical thickness of the cold cloud. After the phase change, the transmitted intensity results from the constructive interference between the incident field and the forward scattered field. Hence *I*_{t} <= 4 *I*_{0}. The previous scenario is valid if the system has reached the steady state regime before the change of the phase.

At long repetition time, i.e. *T*_{R} >> Γ^{-1}= *τ*, the system reaches indeed its steady state before the phase jump. Hence, we measure a pulse contrast *I*_{c} ≈ 4*I*_{0} [see red circles and solid curve in Fig. (b) above]. We note that transfer efficiency, defined as the mean transmitted intensity, is <*I*_{t}> ≈ 0 [see blue open circles and dashed curve in Fig. (b) above]. Thus most of the incident power is scattered out by single atom fluorescence events.

In the *τ*_{c}< *T*_{R} < Γ^{-1} intermediate regime, *I*_{c} oscillates and can reach a larger value. Moreover, the mean intensity <*I*_{t}> rapidly increases to its maximal value, *I*_{0}. Here, the incident power is almost perfectly transferred to the pulse train. This interesting result can be understood considering cooperativity in forward scattering. Indeed, its characteristic relaxation time scales like *τ*_{c}. Therefore, if b>>1, coherent processes relax much faster than single atom fluorescence events. The latter are quenched leading to the good figure of merit at repetition time shorter than Γ^{-1}.

For *T*_{R} < *τ*, the repetition rate is faster than any time scale of the atomic ensemble. Even though the probe power is fully transmitted, the contrast *I*_{c} tends to zero.

The plain curves on the figure above, are the theoretical predictions, obtained using Eq. (1) and (2).