Observation of vacuum-induced collective quantum beats
Hyok Sang Han, Ahreum Lee, Kanupriya Sinha, Fredrik K. Fatemi, Steven L. Rolston
OObservation of vacuum-induced collective quantum beats
Hyok Sang Han, Ahreum Lee, Kanupriya Sinha, ∗ Fredrik K. Fatemi,
3, 4 and S. L. Rolston
1, 4, † Joint Quantum Institute, University of Maryland and the NationalInstitute of Standards and Technology, College Park, Maryland 20742, USA Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA U.S. Army Research Laboratory, Adelphi, Maryland 20783, USA Quantum Technology Center, University of Maryland, College Park, MD 20742, USA
We demonstrate collectively enhanced vacuum-induced quantum beat dynamics from a three-levelV-type atomic system. Exciting a dilute atomic gas of magneto-optically trapped Rb atoms witha weak drive resonant on one of the transitions, we observe the forward-scattered field after a suddenshut-off of the laser. The subsequent radiative dynamics, measured for various optical depths of theatomic cloud, exhibits superradiant decay rates, as well as collectively enhanced quantum beats.Our work is also the first experimental illustration of quantum beats arising from atoms initiallyprepared in a single excited level as a result of the vacuum-induced coupling between excited levels.
Introduction. —Quantum beats are a well-studied phe-nomenon that describes the interference between spon-taneously emitted radiation from two or more excitedlevels, resulting in a periodic modulation of the radiatedfield intensity [1]. This has been a valuable spectroscopictool to measure the energy difference between excitedlevels across many experimental platforms such as atoms[2, 3], molecules [4], semiconductors [5], and quantumdots [6, 7].Although quantum beats have been extensively stud-ied, here we demonstrate two new aspects: (i) quantumbeats without an initial superposition of excited levels,and (ii) enhanced beat amplitudes due to collective emis-sion of light [8, 9]. In a typical quantum beat experi-ment, an excitation pulse with sufficient bandwidth tospan the energy spacing between multiple excited atomiclevels is used to create an initial coherent superposition.The beat signal amplitude is proportional to the coher-ence between the excited levels, and in the absence ofan initial superposition, one might expect no quantumbeats. This notion was challenged in [10, 11], predictingthat the vacuum electromagnetic (EM) field can createthe required coherence between the excited atomic lev-els. However, experimental observation of such vacuum-induced quantum beats is challenging due to the com-peting requirements on the level structure: The excitedlevels separation needs to be large compared to the nat-ural linewidth to enable the initialization of only one ofthe levels, which, in turn, reduces the strength of thevacuum-induced coupling.We experimentally address this using the well-separated Rb P / F = 3 and 4 hyperfine levelsas our excited levels and using a long enough (200 ns)excitation pulse such that any coherence due to theturn-on edge decays away, leaving the atomic popula-tion in a single excited level. Detecting the forward-scattered mode (see Fig. 1 (a)) allows us to observe theradiation from a timed-Dicke state [12–14]. We theo-retically illustrate that for such a collective state, thequantum beat dynamics can be cooperatively enhanced by the constructive interference between the transitionprocesses in different atoms. The collective amplificationof the forward-scattered beat signal allows us to observevacuum-induced quantum beats and serves as an experi-mental proof of collective effects in quantum beats. Suchcollective enhancement may also be used to amplify smallsignals that are otherwise unobservable. Model. —Let us consider a system of three-level V-type Rb atoms, with the ground level | i = (cid:12)(cid:12) S / , F = 3 (cid:11) and the two excited levels | i = (cid:12)(cid:12) P / , F = 4 (cid:11) and | i = (cid:12)(cid:12) P / , F = 3 (cid:11) (see Fig. 1 (b)). The frequency dif-ference between the excited levels is ω = 2 π ·
121 MHz,and the optical transition wavelength between the groundand the excited levels is λ = 780 nm. We observe the for-ward scattering, where the phase factor of the field frompropagation within the atomic cloud is exactly compen-sated by the phases of the atomic dipoles initially in-duced by the drive [12]. The damping rates of atomiclevels originating from second-order coupling between | j i and | l i is Γ jl = −→ d j ·−→ d l ω j πε (cid:126) c , where −→ d j and ω j are thetransition dipole moments and the transition frequencybetween | j i and | i , respectively. Note that Γ rep-resents second-order coupling between the excited statesvia vacuum-induced decay and absorption [11], while Γ and Γ describe the normal decay of the excited states.Assuming that all the transition dipole moments are realand parallel to each other Γ ≈ √ Γ Γ . In our sys-tem Γ = 2 π · . P / level and Γ = Γ , as | i decays to | i onlyfractionally with the branching ratio 5/9 [15].The atoms are initialized in a symmetric state with ashared single excitation in | i . After a sudden turn-off ofthe drive field, the atomic ensemble starts to decay dueto its interaction with the vacuum field modes, whichcouple the excited levels to reveal quantum beating. An-alytically solving collective atomic and field dynamics inthe experimental regime where the excited atomic levelsare well-separated from each other (Γ ( N ) jl (cid:28) ω ), we findthe intensity of light emitted from the ensemble as (see a r X i v : . [ qu a n t - ph ] F e b Supplemental Material) I ( t ) I = e − Γ ( N )22 t + I b e − Γ ( N )avg t sin ( ω t + φ ) , (1)where we have defined the total collective decay rate asΓ ( N ) jl ≡ (1 + N f )Γ jl , with f corresponding to the angularemission factor in to the forward scattered modes and N corresponding to the effective number of atoms emittingcollectively [16]. We have assumed here that the atomsemit collectively in the forward direction as a result of thephase coherence due to the timed-Dicke state, while theemission in the remainder of the modes is independent.Γ ( N )avg ≡ (cid:16) Γ ( N )22 + Γ ( N )33 (cid:17) / I b = Γ ( N )33 ω ≈
59 Γ ( N )22 ω , (2)and the beat phase is defined as φ = arctan Γ ( N )22 ω ! . (3)The first term of Eq. (1) represents the collective decayfrom | i , with a cooperatively enhanced amplitude anddecay rate relative to a single atom. The second termaccounts for the small but non-negligible beat which de-cays away with an enhanced average rate Γ ( N )avg . Thisresult shows that vacuum-induced quantum beats in theabsence of an initial superposition of excited atomic levelscan exhibit collective effects, generalizing the single atomquantum trajectory prediction in [11]. From Eq. (2) weobserve that the collective nature of the quantum beatoriginates from the virtual coupling between the excitedlevels as indicated by the cross-damping term Γ . Experiment. —Fig. 1 (a) shows the schematic of the ex-periment. A cold atomic cloud of ∼ Rb atomsis produced by a magneto-optical trap (MOT) withGaussian-shaped atomic density distribution having a1 /e diameter of ∼ ρλ (cid:28)
1, where ρ is the spatial atomic den-sity, meaning that the separation between atoms is muchlarger than the photon wavelength. An excitation beamwith 1 /e diameter of 1.6 mm is overlapped with thecloud whose transmitted light is collected by a single-mode (SM) fiber 0.6-meter away in the forward direction.For the observation of the spontaneous emission, theMOT lasers are turned off for 200 µ s during which atomsinitialized in | i are illuminated by a train of excitationpulses that resonantly drive the | i↔| i transition. Thepeak intensity of the excitation beam is ∼ × timessmaller than the saturation intensity I s = 3.9 mW/cm of the transition [15], delivering less than one photon perpulse on average, ensuring that the system is well withinthe single-excitation regime. Each excitation pulse is FIG. 1. (a)
Experimental setup. A linearly polarized excita-tion beam containing a train of pulses illuminates a cold Rbatomic cloud produced by the MOT. The photons scatteredby the cloud in the forward direction are coupled into thesingle-mode (SM) fiber, counted by the avalanche photodiode(APD) and histogrammed to obtain the atomic radiative de-cay profile. (b)
Relevant energy levels of Rb atom. Theexcitation beam (780 nm) resonantly drives the | i↔| i tran-sition. Γ and Γ are the decay rates of the excited levels | i and | i , respectively, to the ground level | i . turned on (off) for 200 ns (800 ns) with >
30 dB extinctionand a 3.5-ns fall-time controlled by two fibered Mach-Zehnder intensity modulators (EOSPACE AZ-0K5-10-PFA-PFA-780) in series. We derive the optical Blochequations for the atoms in the presence of the drive andsolve those numerically to obtain estimates for the pop-ulation in level | i (see Supplemental Material). In thesteady state at the end of the excitation pulse, the atomicensemble is mostly in the ground state, with a small pop-ulation of ∼ − in | i . The population in | i , and thecoherence between level | i and | i is negligibly small.Modeling the 3.5 ns laser turn off edge as a cosine-fourthfunction, we find that it generates negligible amplitude in | i due to the small Rabi frequency and short evolutiontime.After the driving field is switched off, spontaneouslyemitted photons coupled to the SM fiber are countedby an avalanche photodiode (APD) and histogrammedwith 0.5-ns resolution. By detecting only those photonscoupled to the SM fiber, we effectively filter out incoher-ent fluorescence, owing to the small collection solid angle( ≈ × − sr). The atomic velocity v ≈
120 nm/ µ scorresponding to the Doppler temperature T D ≈ µ Kgives negligible motion compared to the optical wave-length (780 nm) within the time scale of the emissionprocess (1 / Γ ( N )22 ≤
26 ns). After the repetition of 200pulses within 200- µ s, the MOT lasers are turned back -300 -200 -100 0 10000.51 0 10 20 30 4000.51 FIG. 2. (a)
Examples of the histogrammed photon counts for various optical depths (OD), representing the forward-modeintensity, normalized to that of the excitation pulse. As the excitation pulse is abruptly (within ≈ I proportional to the OD. (b) The decay profiles after the flashpeak are zoomed-in for analysis. The intensity of each curve is further normalized to the exponential decay amplitude I ( ≈ flash peak size shown in (a)). The error bars represent the shot-noise limit of the photon counts. The overlaid solid lines fitthe data using Eq. (1) whose results are displayed in Fig. 3 in detail. The black dashed line represents the single-atom decaycurve I ( t ) = e − Γ t (Γ = 2 π · . on to recover and maintain the atomic cloud for 1.8 msbefore a new measurement cycle begins, repeating thewhole sequence every 2 ms. For typical histogrammeddata, we run the sequence continuously for 30 minutes,comprising 2 × excitation pulses.Examples of histogrammed photon counts are shownin Fig. 2 (a) where I ( t ) represents the intensity of theforward-scattered light normalized to the steady-stateintensity of the excitation pulse. The atomic samplesare almost transparent at the sharp switch-on edge ofthe excitation pulse due to its broad spectral compo-nents, but the transmission soon decays to a steady-statevalue T s , which we use to calculate the optical depth(OD = − ln T s ). We vary the OD of the MOT cloudbetween 0.5 and 4.5 by adjusting the injection currentrunning through the rubidium dispensers (SAES Get-ters RB/NF/7/25) between 3.5 A and 6.5 A to increaseatomic background pressure. The steady state transmis-sion T s results from the destructive interference betweenthe driving field and the field coherently radiated (with π -phase shift) in the forward direction by the atomicdipoles. When the driving field is switched off, only theatomic radiation field remains in the forward direction,resulting in a sudden intensity jump (“flash”), which hasbeen intensively investigated in recent studies [17–19].The flash peak intensity, which is proportional to theOD, represents the intensity I of the overall decay as inEq. (2).The decay profiles after the flash peak are magnified inFig. 2 (b) for detailed analysis. Each curve is normalized to the exponential decay amplitude I (see Eq. (1)), sothe enhanced decay rates and the relative beat intensi-ties for different OD can be easily compared. For com-parision, the single-atom decay curve I ( t ) = e − Γ t withno collective enhancement is also shown (black dashedline). We first note that a higher OD results in an en-hanced decay rate demonstrating the collective nature ofthe emission process. The quantum beat signal is appar-ent as a sinusoidal modulation of the exponential decay.This illustrates the occurrence of quantum beats in theabsence of an initial superposition between the excitedlevels. To verify the frequency of the observed beat sig-nal, we first remove the exponential decay profile from thedata and then fast-Fourier transform (FFT) the residual.The FFT results (see inset) confirm that the observedbeat frequency is ω as expected.The solid lines in Fig. 2 (b) fit the modulated decaycurves using Eq. (1) with I b , Γ ( N )22 , and φ as fitting pa-rameters, and the fit results for the full range of OD be-tween 0.5 and 4.5 are presented in Fig. 3. In the inset, thelinear dependence of the enhancement factor Γ ( N )22 / Γ onOD displays the collective nature of the emission process,in agreement with the superradiant behavior [13, 16, 20–23]. The blue solid line fitting the data provides a lin-ear relation Γ ( N )22 / Γ = 1 . · OD + 1 . I b is plotted as a functionof Γ ( N )22 / Γ in Fig. 3 (a). The blue shaded region rep-resents the one-sigma confidence band of the linear fitto the data, displaying the amplification of the quantum FIG. 3. (a)
The relative beat intensity I b is plotted as a function of Γ ( N )22 / Γ for various OD. The plotted error bars representthe one-sigma confidence interval of the fitting to the modulated decay curves. The shaded region displays the one-sigmaconfidence band of a linear fit to the data. The red solid line plots Eq. (2). The inset shows a linear dependence of Γ ( N )22 / Γ onthe OD. (b) The beat phase φ subtracted by the common offset φ is presented. The shaded region represents the one-sigmaconfidence band of the fitting of Eq. (4) to the data. beat due to the increasing number of cooperative atoms.The red solid line plotting Eq. (2) is in good agreementwith the data, confirming the validity of our model.The measured beat phase φ is displayed in Fig. 3 (b)fit to φ = arctan η · Γ ( N )22 Γ ! + φ . (4)The fitted value of φ = 0 .
17 is presumably due to thetransient intensity of the driving field during the switch-off time. From the fit, η = 1 . × − is almostthree times larger than its expected value of Γ /ω =5 . × − (see Eq. (3)). We note that non-equilibriumdynamics during the switch-off time can produce an addi-tional OD-dependent phase delay, potentially resulting ina larger η value than expected, which is not captured byour current model. Such an additional phase can be usedto characterize the non-equilibrium dynamics of emissionduring the transient time, the study of which is left tofuture work. Discussion. —We have demonstrated collective quan-tum beats in a spontaneous emission process withoutan initial superposition of the excited levels in a three-level atomic system. The collective nature of the for-ward emission results in an enhanced coupling betweenthe excited levels, manifested in cooperatively amplifiedquantum beats We observe that the enhancement fac-tor Γ ( N )22 / Γ for the collective decay rate increases withthe atomic OD. The relative beat intensity also scaleswith Γ ( N )22 / Γ , in excellent agreement with our theoreti-cal prediction. It signifies a combination of two differentquantum interference phenomena featuring interplay be-tween multi-level atomic structure and multi-atom col- lective effects which which has been the focus of manytheoretical studies [25–27].The collective enhancement of quantum beats can be avaluable tool in precision spectroscopy by enhancing beatamplitudes in systems with small signals. It can also beutilized as a source of strongly correlated photons. Forexample, previous works have illustrated that a systemof three-level V type atoms in an interferometric setup,as in the case of a “quantum beat laser” [28, 29], can ex-hibit strong correlations in the two-frequency emission[30, 31]. It has been suggested as a means of gener-ating or amplifying entanglement in the radiated fieldmodes [32, 33]. These proposed schemes rely on the co-herence between the excited atomic levels, therefore re-quiring a strong classical drive to induce such coherences.Vacuum-induced collective quantum beats can circum-vent the need for a classical drive, thereby avoiding addi-tional noise, while facilitating a collective signal enhance-ment.Our study of collective quantum effects can be read-ily combined with waveguide optics to study interactionsbetween distant atomic ensembles [34–40]. Recent stud-ies have shown that such delocalized collective states canexhibit surprisingly rich non-Markovian dynamics [41–47]. A challenge in observing such exotic dynamics isthat the quantum optical correlation between the multi-ple emitters is highly sensitive to the position of individ-ual atoms, requiring sub-wavelength precision. Replacingthe optical frequency by the beat RF frequency could al-low one to bypass the strict requirements on controllingthe atomic positions. An experimental investigation ofcollective effects in non-Markovian regimes with multi-level atomic ensembles coupled to optical nanofibers iswithin the scope of our future works [40]. Acknowledgments .—We thank Hyun Gyung Lee andHuan Q. Bui for technical support and fruitful discus-sions. We are also grateful to Pablo Solano and JonathanHoffman for helpful comments. This research is sup-ported by the Army Research Laboratory’s MarylandARL Quantum Partnership W911NF-17-S-0003 and theJoint Quantum Institute (70NANB16H168). ∗ [email protected] † [email protected][1] E. T. 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Supplemental Material
MODEL
We consider a collection of N three-level V-type atoms located at the same position. We label the ground state as | i and the two excited states as | i and | i , and the transition frequency from level j to i as ω ij . A weak drive fieldwhich is resonantly tuned to ω prepares the atomic system in a timed-Dicke state. As the drive field is turned off,we detect the photons emitted from the cloud in the forward direction. In the experiment, the atomic cloud has afinite size, but for theoretical simplicity we can assume it to be point-like ensemble interacting each other through thevacuum field modes. This is because we are measuring the forward scattering, where any phases of emitted photonsdue to the atomic position distribution is exactly compensated by the phases initially imprinted on the atoms bythe drive field [12]. Additionally, the transitions | i ↔ | i and | i ↔ | i interact with the field effectively with thesame phase considering that the atomic cloud size is much smaller compared to 2 πc/ω . We note that while theforward-scattered field is collectively enhanced, the decay rate of the atoms arising from interaction with the rest ofthe modes is not cooperative [20].The atomic Hamiltonian H A and the vacuum field Hamiltonian H F are H A = N X m =1 X j =2 , (cid:126) ω j ˆ σ + m,j ˆ σ − m,j ,H F = X k (cid:126) ω k ˆ a † k ˆ a k , (S5)where ˆ σ ± m,j is the raising/lowering operator acting on m th atom and j th level, ˆ a † k and ˆ a k are the field cre-ation/annihilation operators of the corresponding frequency mode ω k , and N refers to the effective number of atomsacting cooperatively in the forward direction.First, we prepare the atomic system by a weak drive field. The atom-drive field interaction Hamiltonian is H AD = − N X m =1 X j =2 , (cid:126) Ω mj (cid:0) ˆ σ + m,j e − iω D t + ˆ σ − m,j e iω D t (cid:1) . (S6)Here, ω D is the drive frequency and Ω mj ≡ −→ d mj · −→ (cid:15) D E D is the Rabi frequency of j th level, where −→ d mj is the dipolemoment of | j i ↔ | i transition of m th atom, −→ (cid:15) D is the polarization unit vector of the drive field, and E D is theelectric field of the drive field. Given that the atomic ensemble is driven with the common field in our experiment, wewill assume that the atomic dipoles are aligned with the drive and each other. We can thus omit the atomic labels towrite Ω j .The interaction Hamiltonian describing the atom-vacuum field interaction, under the rotating wave approximation,is given as H AV = − N X m =1 X j =2 , X k (cid:126) g m,j ( ω k ) (cid:16) ˆ σ + m,j ˆ a k + ˆ σ − m,j ˆ a † k (cid:17) . (S7)Here, the atom-field coupling strength g m,j ( ω k ) ≡ −→ d mj · −→ (cid:15) k q ω k (cid:126) ε V , where −→ (cid:15) k is the polarization unit vector of thefield mode, ε is the vacuum permittivity, and V is the field mode volume. As justified previously, the atomic dipolesare aligned to each other and we write g j ( ω k ). Also, note that the sum over k only refers to the forward-scatteredmodes. The spontaneous emission arising from the rest of the modes is to be considered separately later. DRIVEN DYNAMICS
We consider here the driven dynamics of a atoms. Moving to the rotating frame with respect to the drive frequency,and tracing out the vacuum field modes, we can write the following Born-Markov master equation for the atomicdensity matrix: d ˆ ρ A dt = − i (cid:126) h b H A + b H AD , ˆ ρ A i − N X m,n =1 X i,j =2 , Γ ( D ) ij,mn (cid:2) ˆ ρ A b σ + m,i b σ − n,j + b σ + m,i b σ − n,j ˆ ρ A − b σ − n,j ˆ ρ A b σ + m,i (cid:3) , (S8) -6 -5 -4 -3 -2 -1 0 1 200.20.40.60.811.2 FIG. S1. (a)
The drive field intensity (red circles) at turn-off edge characterized as the truncated cos (cid:0) π t − t τ (cid:1) function (redsolid line) bridging the on and off state of the intensity. Here, t = − τ = 3 . ≈ t = 0 as shown in Fig. 2 (b), to further remove the residual drive intensity and thetransient effect from our measurement. where b H A = − P Nm =1 P j =2 , (cid:126) ∆ j b σ + m,j b σ − m,j is the free atomic Hamiltonian and b H AD = − P Nm =1 P j =2 , (cid:126) Ω mj (cid:0)b σ + m,j + b σ − m,j (cid:1) is the atom-drive interaction Hamiltonian in the rotating frame, with∆ j ≡ ω j − ω D . The driven damping rates are defined as Γ ( D ) ij,mn ≡ −→ d mi ·−→ d nj ω D πε (cid:126) c , with the indices i, j referringto the atomic levels, and m, n to different atoms.Using the above master equation, one can obtain the following optical Bloch equations for the case of a single atom: ∂ t ρ = i Ω ( ρ − ρ ) − Γ ( D )33 ρ − Γ ( D )23 ρ − Γ ( D )23 ρ (S9a) ∂ t ρ = i Ω ( ρ − ρ ) − Γ ( D )22 ρ − Γ ( D )23 ρ − Γ ( D )23 ρ (S9b) ∂ t ρ = − i Ω ( ρ − ρ ) − i Ω ( ρ − ρ ) + Γ ( D )33 ρ + Γ ( D )22 ρ + Γ ( D )23 ( ρ + ρ ) (S9c) ∂ t ρ = − i Ω ρ − i Ω ( ρ − ρ ) − Γ ( D )33 − i ∆ ! ρ − Γ ( D )23 ρ (S9d) ∂ t ρ = i Ω ρ + i Ω ( ρ − ρ ) − Γ ( D )33 i ∆ ! ρ − Γ ( D )23 ρ (S9e) ∂ t ρ = − i Ω ρ − i Ω ( ρ − ρ ) − Γ ( D )22 − i ∆ ! ρ − Γ ( D )23 ρ (S9f) ∂ t ρ = i Ω ρ + i Ω ( ρ − ρ ) − Γ ( D )22 i ∆ ! ρ − Γ ( D )23 ρ (S9g) ∂ t ρ = − i Ω ρ + i Ω ρ − Γ ( D )22 + Γ ( D )33 − iω ! ρ − Γ ( D )23 ρ + ρ ) (S9h) ∂ t ρ = i Ω ρ − i Ω ρ − Γ ( D )22 + Γ ( D )33 iω ! ρ − Γ ( D )23 ρ + ρ ) , (S9i)where we have defined the single atom driven damping rate as Γ ( D ) ij ≡ −→ d i ·−→ d j ω D πε (cid:126) c .Numerically solving Eq. (S9a)–Eq. (S9i) along with the normalization condition ρ + ρ + ρ = 1 gives us the steadystate density matrix ρ S for the atom. Substituting our experimental parameters, we get the populations: ρ S, ≈ ρ S, ≈ − , and ρ S, ≈
1. The absolute value of the coherences are: | ρ S, | ≈ | ρ S, | ≈ − , and | ρ S, | ≈ N ≈ −
10, assuming the collective driven damping rate to be Γ ( D ) ij ( N ) ≈ (1 + N f )Γ ( D ) ij with phenomenological value f=1 and the collective Rabi frequency to be Ω j ≈ √ N Ω j . Thus we can conclude thatthe atomic ensemble is well within the single excitation regime in | i .The 3.5 ns time window of laser extinction has broad spectral component and may excite extra population to | i and | i . We numerically simulate the optical Bloch equation for this time window to find the density matrix after thelaser turn-off. We model the laser turn-off shape as cos (see Fig. S1) and vary the Rabi frequency accordingly. Notethat this is a calculation for estimate purposes and may not convey the full dynamics in the laser extinction period.Within the numerical precision limit which is set by the evolution time step (10 − ns) multiplied by Γ ij ≈ .
01 GHz,we obtain the following density matrix values after the turn-off: ρ ≈ ρ ≈ ρ ≈ ρ ≈ ρ ≈ − , and ρ ≈ − − − . Thus the laser turn-off edge doesn’t produce any significant excitation in | i . QUANTUM BEAT DYNAMICS
As the drive field is turned off, the system evolves with the atom-vacuum field interaction Hamiltonian. Moving tothe interaction representation with respect to H A + H F , we get the interaction Hamiltonian in the interaction picture:˜ H AV = − N X m =1 X j =2 , X k (cid:126) g j ( ω k ) (cid:16) ˆ σ + m,j ˆ a k e i ( ω j − ω k ) t + ˆ σ − m,j ˆ a † k e − i ( ω j − ω k ) t (cid:17) , (S10)Initially the system shares one excitation in | i symmetrically, and the EM field is in the vaccum state such that | Ψ(0) i = 1 √ N N X m =1 ˆ σ + m, | · · · i |{ }i . (S11)As the system evolves due to the atom-vacuum field interaction, it remains in the single-excitation manifold of totalatom + field Hilbert space, as one can see from the interaction Hamiltonian (Eq. (S10)): | Ψ( t ) i = N X m =1 X j =2 , c m,j ( t )ˆ σ + m,j + X k c k ( t )ˆ a † k | · · · i |{ }i . (S12)Now we solve the Schrödinger equation to find the time evolution of the atom + field system under the atom-fieldinteraction using Eqs.(S12) and (S10) to obtain ∂ t c m,j ( t ) = i X k g j ( ω k ) e i ( ω j − ω k ) t c ω k ( t ) , (S13a) ∂ t c ω k ( t ) = i N X m =1 X j =2 , g j ( ω k ) e − i ( ω j − ω k ) t c m,j ( t ) . (S13b)Formally integrating Eq. (S13)(b) and plugging it in Eq. (S13)(a), we have ∂ t c m,j ( t ) = − X k g j ( ω k ) e i ( ω j − ω k ) t Z t d τ N X n =1 X l =2 , g l ( ω k ) e − i ( ω l − ω k ) τ c n,l ( τ ) . (S14)We observe that c m, ( t )’s ( c m, ( t )’s) have the same initial conditions and the same evolution equation, thus we canjustifiably define c ( t ) ≡ c m, ( t ) ( c ( t ) ≡ c m, ( t )).Assuming a flat spectral density of the field and making the Born-Markov approximation we get ∂ t c ( t ) = − Γ ( N )22 c ( t ) − Γ ( N )23 e iω t c ( t ) , (S15a) ∂ t c ( t ) = − Γ ( N )33 c ( t ) − Γ ( N )32 e − iω t c ( t ) , (S15b)0where we have defined Γ ( N ) jl ≡ Γ jl + N f Γ jl , with Γ jl = −→ d j ·−→ d l ω l πε (cid:126) c as the generalized decay rate into the quasi-isotropic modes and N f Γ jl as the collective decay rate in the forward direction [16, 20]. The factor f representsthe geometrical factor coming from restricting the emission to the forward scattered modes. We emphasize here thatthe emission into all the modes (not specifically the forward direction) denoted by Γ jl is added phenomenologicallyand is not collective. Considering that the atomic dipole moments induced by the drive field are oriented along thepolarization of the driving field, we can obtain Γ = √ Γ Γ , which can be extended to Γ ( N )23 = q Γ ( N )22 Γ ( N )33 .To solve the coupled differential equations, we take the Laplace transform of Eq. (S15)(a) and (b): s ˜ c ( s ) = c (0) − Γ ( N )22 c ( s ) − Γ ( N )23 c ( s − iω ) , (S16a) s ˜ c ( s ) = c (0) − Γ ( N )33 c ( s ) − Γ ( N )32 c ( s + iω ) , (S16b)where we have defined ˜ c j ( s ) ≡ R ∞ c j ( t ) e − st d( t ) as the Laplace transform of c j ( t ). Substituting the initial conditions,we obtain the Laplace coefficients as˜ c ( s ) = 1 √ N s + Γ ( N )33 − iω s + (Γ ( N )avg − iω ) s − iω
23 Γ ( N )22 , (S17a)˜ c ( s ) = − Γ ( N )32 √ N s + (Γ ( N )avg + iω ) s + iω
23 Γ ( N )33 . (S17b)And the poles of the denominators are, respectively, s (2) ± = − Γ ( N )avg iω ± iδ , (S18a) s (3) ± = − Γ ( N )avg − iω ± iδ , (S18b)where we have defined Γ ( N )avg = Γ ( N )33 +Γ ( N )22 , Γ d = Γ ( N )33 − Γ ( N )22 , and δ = r ω − (cid:16) Γ ( N )avg (cid:17) + 2 iω Γ ( N )d . The real partof the above roots corresponds to the collective decay rate of each of the excited states, while the imaginary partcorresponds to the frequencies. The fact that δ is generally a complex number unless Γ = Γ means that we willhave modification to both the decay rate and the frequency. To see this more clearly, we can expand δ up to secondorder in Γ ( N ) jl /ω , considering we are working in a spectroscopically well-separated regime (Γ ( N ) jl (cid:28) ω ); δ ≈ ω − Γ ( N )23 ω ! + i Γ ( N ) d Γ ( N )23 ω ! , (S19)the above poles become s (2)+ = − Γ ( N )33 ( N )d Γ ( N )22 ω ! + iω − Γ ( N )23 ω ! , (S20a) s (2) − = − Γ ( N )22 − Γ ( N )d Γ ( N )33 ω ! + iω Γ ( N )23 ω ! , (S20b) s (3)+ = − Γ ( N )33 ( N )d Γ ( N )22 ω ! − iω Γ ( N )23 ω ! (S20c) s (3) − = − Γ ( N )22 − Γ ( N )d Γ ( N )33 ω ! − iω − Γ ( N )23 ω ! . (S20d)1The atomic state coefficients in time domain are c ( t ) = 12 √ N δ e − Γ ( N )avg t/ e iω t/ h ( − i Γ ( N ) d − ω + δ ) e iδt/ + ( i Γ ( N ) d + ω + δ ) e − iδt/ i , (S21a) c ( t ) = i Γ ( N )32 √ N δ e − Γ ( N )avg t/ e − iω t/ h e iδt/ − e − iδt/ i . (S21b)Again, expanding δ under the condition Γ ( N ) jl (cid:28) ω , we get c ( t ) = 1 √ N e − Γ ( N )22 t/ − Γ ( N )23 ω ! δ ∗ δ e − Γ ( N )33 t/ e iω t , (S22a) c ( t ) = − i Γ ( N )32 √ N δ h e − Γ ( N )22 t/ e − iω t − e − Γ ( N )33 t/ i . (S22b)Note that the collection of N atoms behaves like one “super-atom” which decays with a rate that is N -times thatof an individual atom in the forward direction. We note that the system is not only superradiant with respect tothe transition involving the initially excited level, but also with respect to other transitions as well as a result ofthe vacuum-induced coupling between the levels. Most population in | i decays with the decay rate Γ ( N )22 , and smallamount of it decays with Γ ( N )33 and has corresponding level shift ω . In | i are the equal amount of componentsdecaying with Γ ( N )22 (and level shifted − ω ) and Γ ( N )33 . The small but nonzero contribution of | i makes beating offrequency about ω . FIELD INTENSITY
The light intensity at position x and time t (assuming the atom is at position x = 0 and it starts to evolve at time t = 0) is I ( x, t ) = (cid:15) c h Ψ( t ) | ˆ E † ( x, t ) ˆ E ( x, t ) | Ψ( t ) i , (S23)where the electric field operator is ˆ E ( x, t ) = Z ∞−∞ d k E k ˆ a k e ikx e − iω k t . (S24)Plugging in the electric field operator and the single-excitation ansatz (Eq. (S12)), we obtain the intensity up to aconstant factor: I ( x, t ) ’ N (cid:12)(cid:12)(cid:12)(cid:12) e − iω τ c ( τ ) + Γ Γ c ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) Θ( τ ) , (S25)where τ = t − | x/v | .Substituting Eqs. (S21)(a) and (b) in the above and approximating δ in the regime Γ ( N ) jl (cid:28) ω , we get I ( τ ) I = e − Γ ( N )22 τ + Γ ( N )33 ω ! e − Γ ( N )33 τ + Γ ( N )33 ω e − Γ ( N )avg τ sin( ω τ + φ ) , (S26)where I is a normalization factor which increases as the number of atom increases. Neglecting the small second termin the right hand side, we get the relative beat intensity normalized to the main decay amplitude:beat amp. = Γ ( N )33 ω , (S27)2and the beat phase φ : φ = arctan Γ ( N )22 ω ! . (S28)We see that even if there was no population in level 3 in the beginning, the vacuum field builds up a coherence betweenlevel 2 and level 3 to make a quantum beat. This is in line with the quantum trajectory calculation of single atomcase [11], where the individual decay rates are replaced with collective decay rates. We can verify that the collectiveeffect manifests in the beat size and the beat phase. DATA ANALYSIS IN FIG. 2 (B)
The modulated decay profiles of the flash after the peak are magnified in Fig. 2 (b). The purpose of the figure is tovisually compare the decay rate and the relative beat intensity I b , so we normalize each curve with the exponentialdecay amplitude such that the normalized intensity starts to decay from ≈ t = 0. In practice, we fit the I ( t )shown in Fig. 2 (a) after t = 0 using Eq. (1) to get I for each curve, to get I ( t ) /I curves as in Fig. 2 (b). Note that,more precisely, it is the fitting curve that decays from I ( t ) /I ≈
1, not the experimental data. In fact, the plotteddata tend to be lower than the fitting curves near t = 0, due to the effect of the transient behavior around the flashpeak.The inset displays the FFT of the beat signal shown in the main figure. We first subtract from I ( t ) /I data theexponential decay profile the first term of the fitting function Eq. (1) as well as the dc offset. The residual, which isa sinusoidal oscillation with an exponentially decaying envelop, is the beat signal represented by the second term ofEq. (1). The FFT of the beat signal has the lower background at ω = 0 due to the pre-removal of the exponential decayand the offset. The linewidth of each spectrum is limited by the finite lifetime of the beat signal, which correspondsto Γ ( N )avg)avg