Cooperative Emission of a Pulse Train in an Optically Thick Scattering Medium

C. C. Kwong,1 T. Yang,2,* D. Delande,3 R. Pierrat,4 and D. Wilkowski1,2,5,†1School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore

2Centre for Quantum Technologies, National University of Singapore, 117543 Singapore, Singapore3Laboratoire Kastler Brossel, UPMC, CNRS, ENS, Collège de France, 4 Place Jussieu, F-75005 Paris, France4ESPCI ParisTech, PSL Research University, CNRS, Institut Langevin, 1 rue Jussieu, F-75005 Paris, France

5MajuLab, CNRS-University of Nice-NUS-NTU International Joint Research Unit UMI 3654, 117543 Singapore, Singapore(Received 8 April 2015; published 24 November 2015)

An optically thick cold atomic cloud emits a coherent flash of light in the forward direction when thephase of an incident probe field is abruptly changed. Because of cooperativity, the duration of thisphenomena can be much shorter than the excited lifetime of a single atom. Repeating periodically theabrupt phase jump, we generate a train of pulses with short repetition time, high intensity contrast, and highefficiency. In this regime, the emission is fully governed by cooperativity even if the cloud is dilute.

DOI: 10.1103/PhysRevLett.115.223601 PACS numbers: 42.50.Md, 42.25.Dd

The seminal work on superradiance of R. Dicke in 1954has opened up tremendous interest in studying cooperativeemission of electromagnetic radiation from an ensemble ofradiative dipoles (see [1] for the original proposal, [2,3] forreviews, and [4–10] for recent related works). In his originalproposal, R. Dicke considered an ensemble of N excitedtwo-level atoms confined inside a volume smaller than λ3,where λ is the transition wavelength. In this context, amacroscopic polarization is built up in the medium uponincoherent spontaneous emission. This Dicke superradiancemechanism leads to the coherent emission of an intensepulsewith a decay time, τD ¼ ðNΓÞ−1, that is shortened by afactor of N−1 with respect to the atomic excited statelifetime, Γ−1. For practical implementation in the opticaldomain, the Dicke model was extended to media withvolume larger than λ3 [2,11,12]. In those cases, the propa-gation of the electromagnetic field in the medium and thespatial mode density must be taken into account. If themedium is dense, i.e.,ρλ3 ≫ 1, where ρ is the radiator spatialdensity, it still exhibits the main feature of the Dickesuperradiance, namely, the emission of a short pulse aftersome delay [4,7,13,14]. It was, however, pointed out in [11],that the superradiant pulse decay time should be corrected asτ ¼ τD=μ. μ < 1 is a geometrical factor corresponding to thesolid angle subtended by the superradiant emission [2,12].For a dilute scattering medium, i.e., ρλ3 ≪ 1, the Dicke

superradiancemechanism does not occur [15]. Nevertheless,an optically thickmedium driven by a coherent incident fieldshares interesting similarities with Dicke superradiance;here, the cooperativity factor Nμ is replaced by the opticalthickness of the medium [18,19]. Once a driving coherentfield is abruptly switched off, like in a free induction decay(FID) experiment [20–27], a short coherent cooperative flashof light is emitted in the forward direction. The flash durationis inversely proportional to the optical thickness and thebare linewidth of the transition [26]. A similar phenomenon

occurs for the optical precursor, i.e., when the drivingcoherent field is abruptly switched on [28].In a coherently driven medium, the incident probe

frequency can be detuned with respect to the atomicresonance, leading to a nontrivial phase rotation of thecooperatively emitted field (see [27] in the optical domainand [29–31] for γ-ray pulses in Mössbauer spectroscopyexperiments). In this Letter, we report the generation of highrepetition rate and high intensity contrast pulse trains in anoptically thick cold dilute atomic ensemble using the setupschematically shown in Fig. 1(a). An example of a pulsetrain, generated in our experiment by periodically changingthe probe phase, is shown in Fig. 1(b). As a consequence ofcooperative emission, the repetition time TR of the pulsetrain can be shorter than the atomic excited state lifetime,Γ−1. Moreover, we show that at a high repetition rate, thesingle atom fluorescence is quenched. This constitutes arather counterintuitive result where the emission in freespace is fully governed by cooperativity, in contrast with theusual situationswhere it is enhanced by a cavity surroundingthe medium [32].The scattering medium is a cloud of laser-cooled 88Sr

atoms (see [33] and [27] for the details of the cold atomsproduction line). The ellipsoidal shape of the cold cloud hasan axial radius of 240ð10Þ μm and an equatorial radius of380ð30Þ μm, with peak density around 4.6 × 1011 cm−3 fora total of 2.5ð5Þ × 108 atoms. λ ¼ 689 nm is the wave-length associated to the 1S0 → 3P1 intercombination line(bare linewidth of Γ=2π ¼ 7.5 KHz) used in this experi-ment. ρλ3 ¼ 0.15, which puts us in the dilute regime. Thetemperature of the cold gas is T ¼ 3.3ð2Þ μK. We getkv ¼ 3.4Γ, indicating a significant Doppler broadening ofthe narrow intercombination line. k ¼ 2π=λ is the wavevector of the transition, and v is the rms velocity of the gas.The optical thickness depends strongly on the temperature.We measure 19(2) along the equatorial axis at resonance.

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A 150 μm diameter probe laser beam, tuned around theintercombination line, is sent through the cold atomic gasalong an equatorial axis. The probe power is 400(40) pW,corresponding to 0.45ð5ÞIsat (Isat ¼ 3 μW=cm2). We mea-sure the forward transmitted intensity of the probe using aphotodetector, integrating over the transverse dimensionsof the transmitted beam. We apply a bias 1.4 G magneticfield along the beam polarization during the probing phase,making the atom an effective two-level system on the 1S0,m ¼ 0 → 3P1, m ¼ 0 transition.The ellipsoidal shape of the cloud is modeled by a slab

geometry, so that the coherent transmitted electric field, inthe frequency domain, is given by

EtðωÞ ¼ E0ðωÞ exp�inðωÞωL

c

�: ð1Þ

In the above equation, nðωÞ, E0, c, and L are the complexeffective refractive index, the incident optical field, thespeed of light in vacuum, and the slab thickness along thelaser beam, respectively. For a dilute medium, nðωÞ ¼1þ ραðωÞ=2 [34], with the two-level atomic polarizability,

αðωÞ ¼ −3πΓc3

ω3

1ffiffiffiffiffiffi2π

pv

Z þ∞

−∞dv

exp ð−v2=2v2Þδ − kvþ iΓ=2

: ð2Þ

δ ¼ ω − ω0 is the detuning of the probe laser frequency ωwith respect to the bare atomic resonance frequency, ω0.The effect of Doppler broadening is included in thepolarizability by averaging over the thermal Gaussiandistribution of the atomic velocity v along the beampropagation direction. The transmitted intensity ItðtÞ iscomputed following [27], and by performing an inverseFourier transform. We define, for given δ and v, the opticalthickness bvðδÞ and the relative phase θvðδÞ between thetransmitted and the incident fields by

bvðδÞ ¼2ω

cIm½nðωÞ�L;

θvðδÞ ¼ω

cRe½nðωÞ − 1�L: ð3Þ

The transmitted field Et results from the interferencebetween the incident field E0 and the field scattered in theforward direction Es,

Et ¼ E0 þ Es: ð4ÞFor effective two-level atoms, we can drop the vectorialnature of the electric fields and represent them as scalarquantities. Because of the noninstantaneous response timeof the medium, the coherent scattered field in the forwarddirection is a continuous function across the abrupt changeof the incident field. In a FID experiment where the incidentfield is abruptly switched off at t ¼ 0, the intensity of thetransmitted field at t ¼ 0þ is a direct measurement of theforward scattered intensity in the stationary regime. Itsproperties are studied in detail in [26,27]. In particular, theintensity of the forward scattering is bounded by 4 times theincident intensity (“superflash effect”) [27]. The temporalevolution of the transmitted field, after the abrupt switch offof the incident field, is not a simple function having onlyone characteristic decay rate [26]. However, we get a clearphysical insight by considering only the initial decay time(at t ¼ 0þ), which takes a simple analytical expression (seeSupplemental Material [35]):

τvðδÞ ¼���� Itðt ¼ 0þÞ − Itðt ¼ ∞Þ

dIt=dtðt ¼ 0þÞ����

¼ 2

Γb0ð0Þ1þ expð−bÞ − 2 expð−b=2Þ cosðθÞ

1 − expð−b=2Þ cosðθÞ ; ð5Þ

where b≡ bvðδÞ and θ≡ θvðδÞ. In Eq. (5), b0ð0Þ isthe optical thickness at resonance and zero velocity. It islinked to bvð0Þ by bvð0Þ ¼ b0ð0Þgðkv=ΓÞ, where gðxÞ ¼ffiffiffiffiffiffiffiffiπ=8

pexp ð1=8x2Þerfcð1= ffiffiffi

8p

xÞ=x [26].For small optical thickness, Eq. (5) reduces to τvð0Þ ¼

gðkv=ΓÞ=Γ at resonance. It is shorter than τ0ð0Þ ¼ 1=Γ due

ProbeLaser

ColdAtoms Detector

0 0.5 1 1.5 20

1

2

3

40 10 20 30 40

t

I t(t

)/I 0

t (µs)

FIG. 1 (color online). (a) Experimental setup: a laser beam issent through an acousto-optic modulator (AOM), which may beused to switch on and off the incident beam, followed by anelectro-optic modulator (EOM), which abruptly changes thephase of the incident probe field, E0. (b) A pulse train generatedat a repetition time of TR ¼ 0.12Γ−1 by a periodic abrupt phasechange of π. Here, the probe laser is at resonance. (c) Electricfields just before and just after an abrupt phase jump of π arerepresented schematically in the complex plane. Before the phasejump, the forward scattered field Es destructively interferes withE0. After the phase jump, they constructively interfere. Thetransmitted field is denoted by Et.

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to the dephasing effect from the motion of the atoms. Thishas already been observed experimentally [see Fig. 3(b) of[26] where the transition is Doppler broadened, and Fig. 5of [24] where Doppler broadening can be ignored]. In ourexperiments, gðkv=ΓÞ ≈ 0.16; thus, for bvð0Þ ¼ 19ð2Þ, weget b0ð0Þ ¼ 120ð10Þ. A direct measurement gives a slightlysmaller value, b0ð0Þ ¼ 95ð5Þ (see Supplemental Material[35]). The expression of τv, given by Eq. (5), simplifies toτvð0Þ ¼ 2½b0ð0ÞΓ�−1 at resonance (b ≫ 1, θ ¼ 0) and toτvð�∞Þ ¼ 4½b0ð0ÞΓ�−1 far from resonance (b ¼ 0 andθ ¼ 0). The solid blue curve in Fig. 2 is a plot of τvðδÞfor kv=Γ ¼ 3.4. τv has a weak dependence on δ and v; itdepends mainly on b0ð0Þ, which can be much larger thanthe optical thickness bvð0Þ seen by a resonant probe atnonzero temperature. This strongly reduces the lifetime ofthe forward scattered field with respect to the atomiclifetime, Γ−1. Equation (5) has a rather simple physicalinterpretation: the second term represents the geometricalproperties of the propagation inside the medium (change inamplitude and phase shift) while the term 2=Γb0ð0Þrepresents the collective behavior of all excited dipoles.It does not depend on the atomic velocity, but only on theatomic density integrated along the laser direction, becausethere is no Doppler effect for photons scattered in theforward direction. Similarly, it does not depend on thedetuning because all dipoles decay with the same rate Γindependently of the detuning.The FID experiment is performed using an AOM as a

light switching device [see Fig 1(a)]. The experimental data

points, represented by blue open circles in Fig. 2, are inreasonable agreement with the theoretical prediction. Theevaluation of dIt=dtðt ¼ 0þÞ is performed on a shorttemporal window (∼200 ns) after switching off the incidentprobe. While the flash signal has a good signal to noiseratio [see Fig. 1(b)], the resulting dIt=dtðt ¼ 0þÞ valuesfrom this analysis are noisier. This leads to the largestatistical errors for τv. The slight positive systematic error,also associated to the determination of dIt=dtðt ¼ 0þÞ,comes from the finite response time of our experimentalscheme, of the order of 40 ns ≈ ð500ΓÞ−1. To check thelatter statement, we use Eqs. (1) and (2) to numericallycompute ItðtÞ. E0ðωÞ in Eq. (1) is determined from themeasured time evolution of the incident intensity. We thenapply, on the numerical signal, the same procedure usedexperimentally to extract τv, resulting in an excellentagreement with the experimental data (see Fig. 2).Instead of a FID experiment, we now consider an abrupt

jump of the phase of the incident field by π [see Fig. 1(c)],at constant incident intensity. The initial decay time τvbecomes (see Supplemental Material [35])

τvðδÞ ¼4

Γb0ð0Þ1 − expð−b=2Þ cos θ2 − expð−b=2Þ cos θ : ð6Þ

We plot this expression as the red dashed line in Fig. 2. Ifthe π phase jump occurs at t ¼ 0, according to Eq. (4),we have Etðt ¼ 0þÞ ¼ −E0ðt ¼ 0−Þ þ Esðt ¼ 0−Þ. Toobserve the largest possible amplitude of the transientfield, we choose the probe frequency detuning such thatthe interference between E0ðt ¼ 0−Þ and Esðt ¼ 0−Þ isdestructive. This condition is necessarily fulfilled whenthe incident field is at resonance. If bvð0Þ ≫ 1,jEsðt ¼ 0−Þj≃ jE0j, so we expect a coherent flash witha peak intensity, It ¼ 4I0. The destructive interferencecondition may also happen at a nonzero detuning if thephase rotation experienced by Es is large enough, forexample if bvð0Þ ≫ 1. In our experiment, this situationoccurs at jδj ¼ 11.3Γ (i.e., superflash regime [27]). In thiscontext, jEsðt ¼ 0−Þj≃ 1.8jE0j; thus, the flash has a peakintensity, It ≃ ð1þ 1.8Þ2I0 ≃ 7.8I0. This value is slightlybelow the maximum value 9I0 allowed by energy con-servation, achievable at larger optical thickness.The phase jump is performed using an EOM placed on

the probe laser path [see Fig. 1(a)]. The EOM is driven by ahigh voltage controller and has a slew rate ∼2.3π rad μs−1.The two experimental values (red squares), correspondingto δ ¼ 0 and jδj ¼ 11.3Γ, are shown in Fig. 2. They aresystematically higher than the theoretical prediction for anabrupt phase shift change because of the response time ofthe EOM driver. Similarly to the FID experiment, we usethe experimentally measured EOM driver output to numeri-cally compute the ItðtÞ signal. The resulting values of thedecay time (red dash-dotted line in Fig. 2) agree with theexperimental ones.

FIG. 2 (color online). Initial decay time of the coherent flashversus the probe detuning at kv=Γ ¼ 3.4. The zero temperatureresonant optical thickness is b0ð0Þ ¼ 120. The blue open circlesand plain curve are, respectively, the experimental data points andthe theoretical curve for an abrupt switch off of the probe. Thetwo horizontal black dashed lines give the theoretical predictionsat resonance and at large detuning. The red squares and dashedcurve are the experimental data points and theoretical predictionfor an abrupt phase jump of π. The blue dotted line and the reddash-dotted line are numerical predictions taking into account thefinite response time of the experimental scheme (see text for moredetails).

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We now analyze the cooperative emission when a squareperiodic π phase jump is applied. We observe a pulse trainwith a repetition time TR [see an example in Fig. 1(b)]limited by the relaxation time of the system. The co-operative emission in the forward direction dramaticallydecreases the repetition time below the atomic excited statelifetime.Bringing the probe on resonance, we plot in Fig. 3(a)

(red dots and solid curve) the intensity contrast Ic of thepulse train. We define Ic ¼ maxfItg − hIti as the differ-ence between the maximum intensity maxfItg and themean intensity, hIti ¼ 1=TR

RTR

ItðtÞdt. We observe anexcellent agreement between the experiment and thetheoretical prediction of Eqs. (1) and (2). At longrepetition time, i.e., TR ≫ Γ−1, the system reaches itssteady state before every phase jump. Hence, we measureIc ≃ 4I0 − hIti≃ 4I0. We note that hIti≃ 0 [see the blueopen circles and dashed curve in Fig. 3(a)] and most ofthe incident power is scattered out by single atomfluorescence events. In the τv ≲ TR ≲ Γ−1 intermediateregime, Ic oscillates and can reach a larger value.Moreover, the mean intensity hIti rapidly increases toits maximal value, I0. Here, the incident power is almost

perfectly transferred to the pulse train. This interestingresult can be understood considering cooperativity inforward scattering. Indeed, its characteristic relaxationtime scales like ½b0ð0ÞΓ�−1. Therefore, for b0ð0Þ ≫ 1,coherent processes relax much faster than single atomfluorescence events. The latter are quenched, leading tothe good figure of merit at a repetition time shorter thanΓ−1. In other words, the emission from the atoms isgoverned by cooperativity. For TR < τv, the repetition rateis faster than any time scale of the atomic ensemble. Eventhough the probe power is fully transmitted, the contrastIc tends to zero.At detuning jδj ¼ 11.3Γ, for long repetition time

(TR ≫ Γ−1), the pulses have a higher contrast, Ic ≃ ð1þ1.8Þ2I0 − hIti≃ 7.1I0 [see Fig. 3(b)]. The large value of themean intensity, namely, hIti≃ 0.7I0, is due to the smalloptical thickness, bvðδÞ ¼ 0.4. Hence, most of the trans-mitted power is in a continuous transmission mode andnot in the pulse train. At the intermediate repetition time(τv ≲ TR ≲ Γ−1), the pulse contrast and the figure of meritare not as good as in the resonant case.To conclude, we generate pulse trains of short repetition

time using cooperative forward emission in an opticallythick scattering medium. We can almost completely trans-fer the incident power into the high intensity contrast pulsetrain, quenching the single atom fluorescence. This meansthat, in free space, the cooperativity effect can dominateemission from a dilute atomic gas. The decay time of thepulses also weakly depends on the temperature of the gasand on the probe detuning. An interesting extension of thisstudy could be to look for quantum signatures in thecooperative emission.Finally, we employ the narrow intercombination line of

strontium as a proof of principle, where the time scales areof the order of microseconds. For future practical applica-tions, such as a high contrast pulse generator, shorterrepetition times in the picosecond or subpicosecond regimeshould be attainable. For this purpose, one has to usescattering media with higher optical thickness and/orshorter transition lifetime. The fact that cooperativity isrobust over thermal dephasing means that we can also use ahot vapor of rubidium [bvð0Þ ≈ 600 at 110 °C [36]]. We canalso use condensed matter systems, e.g., a samarium dopedfiber [bð0Þ ≈ 100 [37]], which allows us to bring thistechnique into the 1.55 μm telecommunication band.

The authors thank M. Pramod, F. Leroux, and K. Pandeyfor technical support and fruitful discussions. C. C. K.thanks the CQT and ESPCI institutions for funding histrip to Paris. This work was supported by CQT/MoEfunding, Grant No. R-710-002-016-271. R. P. acknowl-edges the support of LABEX WIFI (Laboratory ofExcellence ANR-10-LABX-24) within the FrenchProgram “Investments for the Future” under referenceANR-10-IDEX-0001-02 PSL*.

(a)

(b)

FIG. 3 (color online). Figures of merit of the generated pulsetrain (a) at resonance and (b) at jδj ¼ 11.3Γ. The red solid andblue dashed curves are the theoretical predictions for the intensitycontrast Ic=I0 and transfer efficiency hIti=I0, respectively. Thered dots and blue open circles are the corresponding experimen-tally measured values.

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*Present address: National Institute of Metrology, Beijing,China.

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Supplemental Material:Cooperative emission of a pulse train in an optically thick scattering medium

C. C. Kwong,1 T. Yang,2 D. Delande,3 R. Pierrat,4 and D. Wilkowski1, 2, 5

1School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore, Singapore.2Centre for Quantum Technologies, National University of Singapore, 117543 Singapore, Singapore

3Laboratoire Kastler Brossel, UPMC, CNRS, ENS,College de France; 4 Place Jussieu, F-75005 Paris, France

4ESPCI ParisTech, PSL Research University, CNRS,Institut Langevin, 1 rue Jussieu, F-75005, Paris, France

5MajuLab, CNRS-University of Nice-NUS-NTU International Joint Research Unit UMI 3654, Singapore

Optical thickness measurement

We employ three different methods to measure the op-tical thickness. First, we compute the theoretical trans-mission spectrum for various bv(δ) values and use theseprofiles to fit the experimentally obtained transmissiondata. This leads to an optical thickness bv(0) = 19. Sec-ond, we perform a shadow imaging experiment on the1S0 → 1P1 broad transition (λb = 461 nm, linewidthΓb = 2π × 32 MHz), where Doppler broadening is negli-gible. A collimated probe beam with a waist larger thanthe atomic cloud is sent onto the cloud, and the trans-mission signal It/I0 is measured using an electron multi-plying CCD camera (Andor iXon Ultra 897 ). Typically,the probe frequency is set at a detuning, δb = 53 MHz,to reduce the systematic error in the transmission mea-surement due to large optical thickness. The opticalthickness B is computed from the transmission signal,B = − log(It/I0), and is related to b0(0) of the intercom-bination line by b0(0) = B

(1 + 4δ2

b/Γ2b

)λ2/λ2

b . In our ex-periment, we measure a peak value of b0(0) = 95(5) usingthis method and a corresponding value of bv(0) = 15(1)using bv(0) = b0(0)g(kv/Γ). Third, we carry out shadowimaging experiment directly on the intercombination linetransition. We vary the detuning in a range of 100 kHzaround the resonance. The value of bv(0) is deduced us-ing

Et(ω) = E0(ω)ein(ω)ωL

c , (S1)

and

α(ω) = −3πΓc3

ω3

1√2πv

∫ +∞

−∞dv

e−v2/2v2

δ − kv + iΓ/2, (S2)

which are Eqs. (1) and (2) in the main text. We havebv(0) = 19(2), a value slightly larger than the one ob-tained by the second method.

Initial decay time τv

We take t = 0 as the time when the abrupt changeoccurs for the incident field E0. To calculate the initial

decay time of the cooperative forward transmitted field,we first note that we can rewrite Eq. (5) in the main textas

τv(δ) =

∣∣∣∣∣∣1− |Et(t =∞)|2/|Et(t = 0+)|2

2 Re{

dEt/dt(t=0+)Et(t=0+)

}∣∣∣∣∣∣ , (S3)

where Et(t = 0+) = E0(t = 0+) + Es(t = 0−), andEt(t =∞) is the steady state transmitted field after theabrupt change in the incident field. For the case of abruptextinction, Et(t = 0+) = Es(t = 0−) and Et(t =∞) = 0.For abrupt ignition, Et(t = 0+) = E0 and |Et(t =∞)| =|E0|e−b/2. Here, b = bv(δ) and θ = θv(δ), the same asdefined in Eq. (3) of the main text:

bv(δ) =2ω

cIm[n(ω)]L, θv(δ) =

ω

cRe[n(ω)−1]L. (S4)

For abrupt phase jump by ϕ, we have Et(t = 0+) =E0eiϕ +Es(t = 0−), ignoring the small propagation timeL/c in the medium, and |Et(t = ∞)| = |E0|e−b/2. Theforward scattered field during the steady state regime,in both cases of abrupt extinction and phase jump, isEs(t = 0−) = E0e−b/2+iθ − E0.

In the denominator of Eq. (S3), we need to com-pute the time derivative of the transmitted field at t =0+. It can be computed by considering the derivatived[Es(t)e

iωt]/dt. The forward scattered field in the time

domain, Es(t), is related to the incident field in the fre-quency domain, E0(ω′), by the following well-behavedintegral:

Es(t) =

∫e−iω

′t

[eiω′ρα(ω′)L

2c − 1

]E0(ω′)dω′. (S5)

The integration ranges of the integrals in this Supple-mental Material, when not specified, are from −∞ to∞.E0(ω′) is given for the cases of abrupt ignition, abruptextinction and abrupt phase jump of ϕ by:

E0(ω′) =iξE0

2πPV

1

ω′ − ω+ηE0

2δ(ω′ − ω). (S6)

where ω is the frequency of the probe. The Fourier vari-able corresponding to t is denoted as ω′. ξ and η are −1

S2

and 1 respectively for abrupt extinction of the probe, andeiϕ − 1 and 1 + eiϕ respectively for abrupt phase jumpof the probe field. In the case of abrupt probe ignition,both ξ and η are equal to 1. We substitute Eq. (S6)in Eq. (S5), noting that the integral involving the Diracdelta function goes to zero, to obtain

d

dt

[Es(t)e

iωt]

=ξE0

2π

∞∑p=1

1

p!

∫ (iω′ρα(ω′)L

2c

)pe−i(ω

′−ω)tdω′. (S7)

We work in the regime where δ,Γ, kv � ω0. For p = 1,the integral can be evaluated to be

ξE0

2π

∫iω′ρα(ω′)L

2ce−i(ω

′−ω)tdω′

=ξE0

2πi

b0(0)

2

Γ

2

1√2πv

∫∫dv dω′

e−i(ω′−ω)te−v

2/2v2

ω′ − ω0 − kv + iΓ/2

= −ξE0b0(0)Γ

4eiδte−Γt/2 1√

2πv

∫dv e−ikvte−v

2/2v2

= −ξE0b0(0)Γ

4eiδte−Γt/2e−k

2v2t2/2, (S8)

for t > 0. In Eq. (S8), b0(0) = 6πρc2L/ω20 . The p > 1

terms, in general, are difficult to evaluate for the generaltime dependence. Nevertheless, at t = 0+, they vanish.We take the example of the term p = 2, where essentiallywe have to deal with the following triple integral.∫∫∫

e−i(ω−ω′)te−v

2/(2v2)e−v′2/(2v2) dω′dv dv′

(ω′ − ω0 − kv + iΓ/2)(ω′ − ω0 − kv′ + iΓ/2).

(S9)We rewrite for v 6= v′,

e−i(ω−ω′)t

(ω′ − ω0 − kv + iΓ/2)(ω′ − ω0 − kv′ + iΓ/2)

=1

k(v − v′)e−i(ω−ω

′)t

ω′ − ω0 − kv + iΓ/2

− 1

k(v − v′)e−i(ω−ω

′)t

ω′ − ω0 − kv′ + iΓ/2. (S10)

The integration over ω′ of the above expression can becarried out easily, which results in 0 at t = 0+. Whenv = v′, we have an integral over a multiple pole of order2, which also goes to 0 at t = 0+. Therefore, the termwith p = 2 is zero at t = 0+. Similar argument can beextended to all orders p > 1, showing that all p > 1 termsvanish at t = 0+. Finally, we have

d

dt

[Es(t)e

iωt]

(t = 0+) = −ξE0b0(0)Γ

4. (S11)

We then use the fact that Es(t = 0+) = Et(t = 0+) −E0(t = 0+) to obtain

dEtdt

(t = 0+) = −ξE0b0(0)Γ

4− iωEt(t = 0+). (S12)

Using the above expression, we deduce the initial decaytime for the case of abrupt probe extinction,

τv(δ) =2

Γb0(0)

1 + exp(−b)− 2 exp(−b/2) cos(θ)

1− exp(−b/2) cos(θ),

(S13)which is Eq. (5) of the main text. For the case of abruptphase change, we find the initial decay time to be

τv(δ) =4

Γb0(0)

1− exp(−b/2) cos θ

2− exp(−b/2) cos θ. (S14)

This is Eq. (6) of the main text. The initial decay timeof the flash in the case of abrupt ignition is found to be

τv(δ) =2

Γb0(0)

[1− e−b

]. (S15)

We observe again the appearance of the factor 2/Γb0(0)which arises from the cooperativity among the atomicdipoles.

In the case of an abrupt phase jump, we can furtherchoose in the experiment, for ϕ to be equal to the phase ofEs(t = 0−) relative to E0(t = 0−). This choice ensuresa constructive interference after the phase jump. Thedecay time can be simplified to

τv(δ) =4

Γb0(0)

|Es(t = 0+)|E0 + |Es(t = 0+)|

. (S16)