Unraveling disorder-induced optical dephasing in an atomic ensemble
Yizun He, Qingnan Cai, Lingjing Ji, Zhening Fang, Yuzhuo Wang, Liyang Qiu, Lei Zhou, Saijun Wu, Stefano Grava, Darrick E. Chang
UUnraveling disorder-induced optical dephasing in an atomic ensemble
Yizun He , ∗ Qingnan Cai , Lingjing Ji , Zhening Fang , Yuzhuo Wang , Liyang Qiu , Lei Zhou , † and Saijun Wu ‡ Department of Physics, State Key Laboratory of Surface Physics and Key Laboratory of Micro andNano Photonic Structures (Ministry of Education), Fudan University, Shanghai 200433, China.
Stefano Grava , ,(cid:63) and Darrick E. Chang , § ICFO-Institut de Ciencies Fotoniques, The Barcelona Instituteof Science and Technology, 08860 Castelldefels, Barcelona, Spain. ICREA-Instituci´o Catalana de Recerca i Estudis Avan¸cats, 08015 Barcelona, Spain. (Dated: January 27, 2021)Quantum light-matter interfaces, based upon ensembles of cold atoms or other quantum emitters,are a vital platform for diverse quantum technologies and the exploration of fundamental quantumphenomena. Most of our understanding and modeling of such systems are based upon macroscopictheories, wherein the atoms are treated as a smooth, quantum polarizable medium. Although itis known that such approaches ignore a number of microscopic details, such as the granularity ofatoms, dipole-dipole interactions and multiple scattering of light, the consequences of such effects inpractical settings are usually mixed with background macroscopic effects and difficult to quantify. Inthis work we demonstrate a time-domain method to measure microscopically-driven optical effects ina background-free fashion, by transiently suppressing the macroscopic dynamics. With the method,we reveal a microscopic dipolar dephasing mechanism that generally limits the lifetime of the opticalspin-wave order in a random gas. Theoretically, we show the dephasing effect emerges from thestrong resonant dipole interaction between close-by atomic pairs.
I. INTRODUCTION
Quantum light-matter interfaces [1], based upon en-sembles of cold atoms or other quantum emitters, arebeing actively pursued as a platform for various quantumtechnologies, including quantum memories for light [2, 3],nonlinear optical devices operating at the single-photonlevel [4–6], and quantum sensors [7, 8]. Our theoreti-cal understanding of atom-light interactions in such sys-tems is largely based upon the Maxwell-Bloch equa-tions (MBE) [1, 9–12], where the atoms are treated asa smooth, polarizable medium that interacts with opti-cal fields. This macroscopic picture has been challengedby theoretical considerations based on refined treatmentsof microscopic details, such as atomic granularity, dipole-dipole interactions, and multiple scattering. Within thelinear optics regime, highly nontrivial effects are pre-dicted such as modifications of refractive indices andscattering rates [13–18], coherent back-scattering [19–21],and 3D Anderson localization of light [22–24]. It is im-portant to understand these and similar microscopically-driven anomalous optical effects, both within and beyondthe linear optics regime, so as to refine our knowledge ofquantum light-matter interactions and practically to im-prove ensemble-based quantum optical technologies.To experimentally quantify such microscopic effects,the measurements need to be carefully designed to isolateany effects being well-described by the standard MBE. ∗ (cid:63) : Equal contribution † [email protected] ‡ [email protected] § [email protected] Apart from a meticulous choice of observables [24], ver-ification of subtle microscopic effects may rely on sub-stantial pre-knowledge of the samples such as the atomicdensity distribution [25], or via a side-by-side compar-ison between state-of-art atomic and numerical experi-ments [15–17]. The numerical comparison approach islikely limited to moderate sample sizes L and atom num-bers N computationally, even if the modeling is classi-cal [26].In this work, we demonstrate a novel background-freemeasurement of microscopically-driven anomalous opti-cal effects, made possible by actively suppressing themacroscopic dynamics. With this technique, we unveila previously unidentified dephasing mechanism of dipolespin waves on an optical transition, due to microscopicfluctuations of resonant dipole interactions in an other-wise ideal ensemble of laser-cooled atoms. Our measure-ment scheme relies on being able to shift the wavevec-tor k of collective spin wave excitations in the atomicmedium [27, 28]. In particular, while spin waves withwavevectors | k | = ω/c matched to the dispersion relationof light are naturally generated by laser pulses, they natu-rally also experience non-trivial spatio-temporal dynam-ics such as propagation or superradiant emission alongthe direction k . In contrast, we apply a recently de-veloped, robust geometric phase imprinting technique tocreate highly mismatched spin waves | k (cid:48) | (cid:54) = ω/c [29],which are essentially free of collective evolution underMBE, but can be mapped to light on demand in a re-versible manner for optical measurements. By studyingthe time-dependent decay of the mismatched spin waves,we unveil a dephasing effect that should generally existin any ensemble-based light-matter interface, and whichcan be significant even at moderate density ρ < | k | . a r X i v : . [ phy s i c s . a t o m - ph ] J a n Theoretically, we show the dephasing mechanism arisesfrom the combination of random atomic positions and thenear-field interactions between optically excited atoms.Our results should have implications for quantum tech-nologies based on atom-light interfaces, for example, im-posing bounds on atomic densities in order to achievedesired fidelities. Furthermore, while we specifically in-vestigate dephasing here, we believe that our powerfulmethod to suppress macroscopic dynamics will generallyfacilitate investigations of diverse other anomalous opti-cal effects [22–25]. Our work also represents a first steptoward precisely measuring interacting spin dynamics inthe optical domain in a way similar to microwave NMRresearch [30–33], but involving the much stronger electricdipole interaction, and for radiation at a much shorterwavelength.The remainder of the paper is structured as follows.In Sec. II we review the existing formal theory of therelaxation dynamics of optical spin waves and previewour main theoretical results. In Sec. III, we present ex-perimental measurements of the local-density-dependentdipolar dephasing of phase-mismatched spin waves, andprovide a theoretical model of the dephasing effect. Weconclude this work with an expanded discussion on boththe measurement method and the microscopic dipolardephasing effect in the optical domain.
II. OPTICAL SPIN WAVE RELAXATION
We begin by introducing a well-known formalism forlight-matter interactions, which in principle accountsfully for the granularity of atoms and multiple scatter-ing. For concreteness, we consider an ensemble of N two-level atoms with ground and excited states | g j (cid:105) , | e j (cid:105) that couple to light via an electric dipole transition (seeAppendix for discussion of more realistic level struc-tures). The atoms are randomly distributed at location r j , j = 1 , · · · , N according to a smooth density profile ρ ( r ) with a spatial width σ . The atoms can interactvia the emission and re-absorption of photons, whose re-sultant dynamics can be derived from the Hamiltonian H eff = (cid:80) i,j ˆ V i,j DD with [34]ˆ V i,j DD = ω eg ε c d ∗ eg · G ( r ij , ω eg ) · d eg σ + i σ − j . (1)Here σ − j = | g j (cid:105)(cid:104) e j | and σ + j = ( σ − j ) † are the singleatom spin-lowering and raising operators respectively,and d eg = d eg ˆ x is the transition dipole matrix element,assumed to have linear polarization. The free-space elec-tromagnetic Green’s tensor G ( r ij , ω eg ) with r ij ≡ r i − r j physically describes how light propagates from a pointsource at r j to another point r i , thus encoding thephoton-mediated interaction. Important to later consid-erations, G contains both a far-field, radiating compo-nent G far ( r ij ) ∼ /r ij and a G near ( r ij ) ∼ /r ij near-fieldcomponent. The Hamiltonian of Eq. (1) is non-Hermitian, and in-cludes both coherent interactions and collective emission.In the case of an isolated, excited atom, this Hamil-tonian predicts a spontaneous emission rate of Γ e = ω eg | d eg | / π (cid:126) ε c . Generally, a complete quantum de-scription of dissipation would require a master equation,but the non-Hermitian Hamiltonian is sufficient in thesingle-excitation regime of interest here [35].Physically, any dissipation of atomic excitations mustbe in the form of emitted photons. The specific propertiesof the emitted light are encoded in the quantum electricfield operator [35, 36]ˆ E s ( r ) = ω eg ε c N (cid:88) i G ( r − r i , ω eg ) · d eg σ − i . (2)We now discuss how this formalism can be ap-plied to investigate the emission and dynamical prop-erties of single-excitation spin waves, which take theform | ψ k (cid:105) = S + ( k ) | g , g , · · · , g N (cid:105) with S + ( k ) =(1 / √ N ) (cid:80) j e i k · r j σ + j [27, 28]. Such states (or coherentstate superpositions of these quasi-bosonic excitations)are naturally generated by incoming optical fields. Ofparticular interest here will be the survival ratio betweensuch a state prepared initially, | ψ (0) (cid:105) = | ψ k (cid:105) , and theevolved state under Eq. (1) at later times, O k ( t ) = |(cid:104) ψ k | ψ ( t ) (cid:105)| . (3)One expects that the dominant contribution to the ini-tial decay of O k ( t ) will be due to collective emission.Specifically, in typical situations, the spin wave is ex-cited by a traveling optical pulse, with the wavevector | k | = k , k = ω eg /c matching the dispersion relationof light. In that case, the spin wave emits in analogousfashion as a phased array antenna – the emission of lightby different atoms in the forward k direction, within anarrow superradiant emission solid angle Ω s ∼ ( k σ ) − (see Appendix C 1 for a rigorous definition), adds con-structively, while interference of the randomly positionedatoms in other directions washes out when averagedover microscopic configurations. This enhancement canbe seen by directly calculating the single-photon spa-tial wave function, ε k ( r ) = (cid:104) g , g , · · · , g N | E s ( r ) | ψ k (cid:105) viaEq. (2) as illustrated in Fig. 1(c). Correspondingly, theinitial decay rate of spin-wave population is given byΓ k = − (cid:104) ψ k | H eff | ψ k (cid:105) ), and can readily be evaluatedto be Γ k = Γ e + OD(ˆ k )Γ e / { r j } [27, 37, 38]. The first and sec-ond terms reflect the random emission into most of 4 π (which occurs at the single-atom rate), and the collec-tively enhanced emission in the forward direction, respec-tively. Here, OD(ˆ k ) = (cid:82) OD( r ⊥ ) d r ⊥ / (cid:82) OD( r ⊥ ) d r ⊥ is the resonant optical depth of the atomic gas OD( r ⊥ )averaged over the r ⊥ plane perpendicular to the k -direction [37, 38].In Fig. 1(e) we plot simulated dynamics of the sur-vival ratio O k ( t ) and the superradiant intensity I k ( t ) Ω 𝑠 መ𝑆 + (𝐤) መ𝑆 + (𝐤′) (a) (b) (c)(d) 𝑂 𝐤 ′ ( 𝑡 ) e x p ( Γ 𝑒 𝑡 ) Γ 𝑒 𝑡 (f)(e) 𝐼 𝐤 (MBE) 𝐼 𝒌 (CDM) 𝑂 𝐤 (CDM) 𝑂 𝐤 (MBE) 𝑂 𝐤′ (CDM) |Re(𝜺 𝐤 (𝐫))| 𝑂 𝐤 ( 𝑡 ) e x p ( Γ 𝑒 𝑡 ) 𝐼 𝐤 ( 𝑡 ) e x p ( Γ 𝑒 𝑡 ) 𝑂 𝐤′ (MBE) 𝑂 𝐤′ (CDM,(ii))𝑂 𝐤′ (CDM,(iii)) FIG. 1. Decay of singly excited optical spin waves in a classical random gas due to resonant dipole interactions. Fig. (a,b)illustrate the spin-wave order initiated in a Gaussian distributed random 2-level gas for | k | = ω eg /c and | k (cid:48) | = 2 . ω eg /c ,respectively. The corresponding electric fields | Re( ε k ( r )) | , calculated over a two-dimensional cut at the sample center, aresimulated with the coupled dipole model (CDM) and plotted in (c),(d). In (e),(f), we plot the time evolution of the spin wavesurvival ratio for the phase-matched and mismatched cases, respectively. Here, the survival ratios are normalized by O k ( t ) e Γ e t and O k (cid:48) ( t ) e Γ e t , to compensate for any trivial decay that can be attributed to the single-atom, independent spontaneous emissionrate. We have calculated the survival ratio using both CDM (dashed red) and the Maxwell-Bloch equations (MBE, dashedgray). In (e), we also plot the simulated emission intensity I k ( t ) within the superradiant solid angle Ω s , which is normalized to I k ( t ) = 1 at t = 0. In (f) we plot the survival ratio as simulated by CDM, but with close-by pairs (with | r ij | < λ eg / π ) in thesample removed (dashed blue curve, CDM (ii)). Finally as comparison, with the dotted green curve (CDM (iii)) we illustratethe effect of removing the same number of atoms, but in a completely random fashion. The simulations above contain N = 532atoms with a peak density at the center of the Gaussian distribution being ρ λ eg ≈
5, detailed in Appendix C 2. both of which are subject to spatio-temporal evolutionof the spin wave as it propagates through the ensemble.Here I k ( t ) = (cid:82) Ω s | ε k ( r , t ) | d Ω is the emission integratedover the full superradiant angle Ω s , which is more exper-imentally accessible than O k ( t ). One sees that in con-trast to an expected nearly exponential decay of O k ( t ),the decay of intensity I k ( t ) deviates from being exponen-tial at long times, which is due to an angular redistri-bution of the forward emission associated with the col-lective spin-wave dynamics (similar effects were found inRef. [39]). We obtain these quantities by directly evolvingthe spin wave under Eq. (1) via the Schrodinger equation,which is efficient as the single-excitation Hilbert spaceis of size ∼ N . More specifically, the amplitudes β j for | ψ ( t ) (cid:105) = 1 / √ N (cid:80) j β j ( t ) σ + j | g , g , · · · , g N (cid:105) obey thesame set of N equations as a classical resonantly cou-pled dipole model (CDM, see Appendix C) [37]. Impor-tantly, although we have reached all these conclusions us-ing Eq. (1) where granularity is accounted for, they canalso be derived within the conventional MBE for smoothand classical fields (cid:104) ˆ E s (cid:105) and (cid:104) ˆ P (cid:105) = (cid:104) (cid:80) j d eg σ − j δ ( r − r j ) (cid:105) while treating atoms as a continuous medium [15, 39] (seeAppendix B). Indeed, the deviation from MBE due to theextra microscopic physics contained in the CDM simula-tions seems quite subtle in Fig. 1(e).Separately, we can consider a similar scenario, butstarting from a spin wave with wavevector | k (cid:48) | > ω/c strongly mismatched from radiation (Figs. 1(b)(d)(f)). Because of the phase mismatch, there is no directionalong which emission will constructively interfere, afteraveraging over microscopic configurations of a randomgas, and similar calculations based on energy emissionpredict an initial decay rate of Γ k (cid:48) = Γ e [40], i.e. , anexponential decay with a precisely known rate accordingto spontaneous emission from single, isolated atoms.The main discovery in this work is that the above con-clusion is incomplete . Instead, we show experimentallyand argue theoretically that the initial decay of opticalspin waves generally behaves as O k ( t ) ≈ e − Γ k t , withΓ k = Γ k + γ (cid:48) deviating from the standard MBE pre-diction by a local density-dependent rate γ (cid:48) ∝ ρλ eg Γ e ( λ eg = 2 π/k ). This additional term arises due to de-phasing, and only appears by accounting for the effectof granularity in the dynamics of the dipole-dipole in-teractions, in particular, between close pairs of atoms.We show γ (cid:48) can be significant even for a dilute gas with ρ < k . While we present a complete theoretical picturein Sec. IV, an initial glimpse into the effect is provided bycomparing CDM and MBE simulations of the spin-wavedynamics. In particular, while the survival ratio O k ( t )is seen to be trivial within the MBE (dashed gray curveof Fig. 1(f)), the CDM predictions (dashed red curve)strongly deviate and decay faster. The important role ofclose-by pairs (having a distance less than λ eg / π , about3% percent of all atoms in the simulation) is illustratedby removing such pairs from the ensemble, which results −1 −2 −3 −4 n o r m a l i z e d i n t e n s i t y 𝐤 𝑝 − 2𝐤 𝑐 = |𝐤 𝑝 |𝐤 𝑝 − 4𝐤 𝑐 ≠ |𝐤 𝑝 | 𝐤 𝑠′ 𝐤 𝑝 𝐤 𝑝 𝐤 𝑠 detection probe 𝑥 𝑧 lens 𝐤 𝑐 −𝐤 𝑐 𝐤 𝑝 𝐤 𝑠 (a) |𝑒〉 |𝑎〉 control (795nm) Ω 𝑐 |𝑔〉 probe(780nm) Ω 𝑝 (𝐄 𝑝 )5𝑃 (b) ̂ Rb (c) ………… control "A""B" 𝜇m Δ𝑡 Δ𝑡 𝜏 𝑝 𝜏 𝑐 𝑒 𝑖𝐤 𝑐 ⋅𝐫 𝑒 −𝑖𝐤 𝑐 ⋅𝐫 𝑒 𝑖𝐤 𝑐 ⋅𝐫 𝑒 −𝑖𝐤 𝑐 ⋅𝐫 𝜏 𝑐 𝜏 𝑐 𝜏 𝑐 𝑒 𝑖𝐤 𝑝 ⋅𝐫 generation 𝑡 |Ω |0 𝑇 i 𝑒 𝑖𝐤 𝑐 ⋅𝐫 𝑒 −𝑖𝐤 𝑐 ⋅𝐫 𝜏 𝑐 𝜏 𝑐 I II III (e) redirection shift out recall (d) 𝑥 𝑦𝑧𝑥 𝑦𝑧
FIG. 2. Measurement scheme of optical spin wave decay. (a)Schematic of the setup. Shapes of atomic samples labeled as“A” and “B” are illustrated with absorption images. Also il-lustrated are the directions and wavevectors associated withthe exciting probe beam, the control pulses, and the directionfor detection of emission. (b) Atomic level diagram and lasercoupling scheme. The probe beam couples to the | g (cid:105) - | e (cid:105) tran-sition with the levels indicated for Rb, while fast controlpulses couple to an auxiliary | g (cid:105) - | a (cid:105) transition. (c, d): Angu-lar distribution of the light emission for the phase matched S + ( k s = k p − k c ), and mismatched S + ( k (cid:48) s = k p − k c )spin-wave, as predicted by CDM simulations. (e): Tim-ing diagram for shifting a S + ( k p ) spin-wave excitation to k s = k p − k c (interval I), k (cid:48) s = k p − k c (interval II) andback to k s = k p − k c (interval III). in a survival ratio (dashed blue curve) that goes back tothe MBE results at short times. In contrast, removingthe same percentage of atoms randomly (dashed green)results in almost no difference, compared to CDM sim-ulations of the original ensemble. Together, these simu-lations clearly show the dramatic effect that “freezing”macroscopic dynamics can have, in order to observe mi-croscopic optical phenomena. III. MEASUREMENTS AND ANALYSIS
We follow the control and measurement protocol inRef. [29] to investigate the decay dynamics of phasematched and mis-matched spin-wave order in a laser-cooled gas. The optical spin waves are defined on the5 S / , F = 2 to 5 P / , F (cid:48) = 3 hyperfine transition of Rb, with the Zeeman sub-levels labeled as | g (cid:105) and | e (cid:105) , respectively (Fig. 2(b)). A short probe pulse with wavevector k p and duration τ p is applied to resonantlyexcite the | g (cid:105) − | e (cid:105) electric dipole transition, with a weakRabi frequency Ω p with pulse area θ p = Ω p τ p (cid:28) k associated with the resulting spin wave, and thus thephase-matching condition, by cyclically driving the aux-iliary | g (cid:105) − | a (cid:105) D1 transition ( | a (cid:105) labels the 5 P / , F (cid:48) sub-levels). In particular, we successively drive populationinversions from | g (cid:105) → | a (cid:105) and back | a (cid:105) → | g (cid:105) with apair of pulses on the D1 transition, with the first andsecond pulses having wavevectors ± k c and ∓ k c , respec-tively (Fig. 2(a)). Although all atoms initially in | g (cid:105) windup back in the same state, the difference in local phases ofthe pulses seen by each atom leads each atom to pick up anon-trivial, spatially dependent geometric phase. It canbe readily shown [38] that this phase patterning exactlyleads to a wavevector shift k p → k p ∓ k c of the spin-waveexcitation S + ( k p ). We finely align the control direction k c to ensure that the new direction | k s | = ω eg /c , with k s = k p − k c , is also phase-matched, and thus the spinwave preferentially emits in the k s direction (Fig. 2(c)).This has the advantage that the spin-wave populationcan be read out by the detection of superradiant emis-sion, but without any background [42–46] caused by theprobe pulse that now propagates in a different direction.After generating the phase matched spin wave S + ( k s ),we investigate the dynamics of phase-mismatched spinwaves by immediately applying a second pair of controlpulses to shift to the new wavevector k (cid:48) s = k s − k c , where | k (cid:48) s | = 2 . ω eg /c . After waiting for an interrogation time T i for the S + ( k (cid:48) s ) spin wave to accumulate dynamics (es-pecially from the near-field dipole-dipole interactions ofinterest), a backward shift k → k + 2 k c is applied to con-vert the spin wave back to k s , onto the light cone, to re-call the superradiance (Figs. 2(c)(d)(e)). In general, thestrength of the superradiant emission immediately fol-lowing the interrogation time and recall, I k s ( T i ), decaysas a function of increasing T i . Recording this strength forvarious t = T i directly reveals the survival ratio O k (cid:48) s ( t ) ofthe phase-mismatched spin wave and allows us to inferthe dephasing rate γ (cid:48) .The measurements start with an optically trappedsample of Rb atoms prepared by laser cooling, moder-ate evaporation and then an adiabatic compression of thesample [29]. The final atomic sample at a temperature of T ∼ µ K is released from the trap for spin-wave gen-eration, control, and superradiance measurements. Weproduce atomic samples with different combinations ofdipole trapping potentials, resulting in nearly sphericalsamples of type “A” and elongated samples of type “B”,as illustrated in Fig. 2a. Both samples share a similarsize along the x and y directions with a Gaussian radius σ ∼ µ m. The Gaussian radius along z for the type“B” sample is elongated to l z ≈ µ m. Benefiting froma stronger optical confinement, the type “A” sample canreach a peak density of ρ ≈ × / cm with the adia-batic compression (with σ slightly below 4 µ m). On the 𝑘𝑘 𝑘 S u p e rr a d i a n t s i g n a l 𝐼 𝐤 𝑠 [ c o u n t s ] 𝑡 [ns] (b)(a) 𝑘OD 𝐤 𝑠 = 2.7𝑘OD 𝐤 𝑠 = 12 OD 𝐤 𝑠 d e c a y r a t e / Γ 𝑒 𝑠 (c) 𝑠 FIG. 3. Superradiant dynamics for phase-matched S + ( k s )spin-wave excitation. Time-dependent fluorescence counts arehistogrammed into curves in (a) and (b) for type “A” and “B”samples, respectively. Blue curves are predictions by CDMwith no freely adjustable parameters. For comparison, thegray dashed lines indicate the spontaneous emission rate e − Γ e t of an isolated atom. The expected nearly exponential decaydynamics of the spin wave survival ratio O k s ( t ), also obtainedby CDM, are shown with red curves. The measured initialdecay rate of the superradiant intensity is plotted vs. OD(ˆ k s )in (c). The blue and red lines indicate the initial decay ofthe superradiant intensity I k and survival ratio O k by CDMsimulations, respectively. other hand, the type “B” sample can reach a higher opti-cal depth along k s , which is close to the z direction, withOD(ˆ k s ) ∼
12 with merely N ∼ × atoms. A. Decay of phase-matched spin waves
We first investigate dynamics of superradiant emissionassociated with S + ( k s ) spin-wave excitations. For thispurpose, after the S + ( k p ) spin-wave generation (and af-ter a ∆ t ≈ . k → k − k c shift, and record the resulting super-radiant emission along k s afterward. Typical results of I k s ( t ) are given in Fig. 3. The measurement results arecompared with CDM simulations (Appendix C 3, MBE simulations produce essentially identical initial decay dy-namics, as in Fig. 1(e)). As detailed in the Appendix D 2,here OD(ˆ k s ) is estimated from absorption imaging mea-surements (Appendix D 2,D 3). It is known that the I k s superradiance signal deviates from the decay of the spinwave itself as a result of small-angle diffraction that re-shapes the superradiance profile [38, 39]. The reshapingeffect generally leads to a more rapid initial decay of I k s ,which is followed by a non-exponential tail. The excellentagreement between experimental curves and CDM sim-ulations, with no freely adjustable parameters, suggeststhe accuracy of our measurements.From numerical simulations we find the initial decayrate for the superradiant intensity I k ( t ) should approx-imately follow ˜Γ k = (1 + κ OD(ˆ k ))Γ e with κ ≈ . k s ). As illustrated in Fig. 3(c),the initial decay rates retrieved from measurements oftype “A” and “B” samples follow closely the prediction(1 + 0 . k s )), without freely adjustable parameters.These results also confirm, albeit indirectly, the MBE-predicted Γ k s = (1 + OD(ˆ k s ) / e scaling of phase-matched spin-wave initial decay (Eq. (3)). However, thetiny difference between the CDM and MBE predictions,as illustrated in Fig. 1(c), is difficult to distinguish ex-perimentally. B. Decay of phase-mismatched spin waves
To unravel the microscopic dephasing dynamics pre-dicted by CDM, we now proceed with the full spin-wavecontrol sequence (Fig. 2(e)) to investigate the phase-mismatched S + ( k (cid:48) s ) excitation. Typical superradiancesignals during such measurements, with interrogationtimes T i = 0 . , . , . I k s ( t ) has two peaks. The first peak corre-sponds to the interval I in Fig. 2(e), and arises immedi-ately following the generation of the spin wave S + ( k p ) bythe probe pulse, and the re-direction by the first pair ofcontrol pulses to orient this spin wave S + ( k s ) along thephase-matched (and detected) k s direction. The signalthen effectively vanishes once the second pair of controlpulses is applied, to shift to a phase-mismatched excita-tion S + ( k (cid:48) s ), where it remains for a time ∼ T i until itis recalled back to S + ( k s ) to produce the second peak(interval III). Not surprisingly, once recalled back to aphase-matched state, the superradiant intensity I k s de-cays at an superradiant rate that is enhanced by largeOD(ˆ k s ) (similar to the data of Fig. 3). More impor-tant for us, however, is the decay of the peak intensity ∼ I k s ( T i ) versus interrogation time T i , which directly re-veals how the phase-mismatched spin wave decays duringthe interval T i , i.e. , the decay of the survival ratio O k (cid:48) s 𝑡 [ns] S u p e rr a d i a n t s i g n a l 𝐼 𝐤 𝑠 [ c o u n t s ] (b)(a) 𝑘OD 𝐤 𝑠 = 2.3𝑘OD 𝐤 𝑠 = 6.7 FIG. 4. Extrapolating the mismatched spin wave survivalratio O k (cid:48) s ( t ) from decay of the recalled superradiant emission.In Fig. (a), we plot the measured superradiant intensity I k s ( t )versus time t , for a typical type “A” sample, and for differentinterrogation times T i = 0 . , . , . e − Γ e t of a single, isolated atom. (b) Same as (a), but for atypical type “B” sample. (Eq. (3)) for the mismatched spin wave. For the con-ditions plotted, this I (cid:48) k s ( T i ) decay rate is hardly distin-guishable from the single-atom spontaneous emission rateΓ e (dashed gray line) and aside from the atomic densityis insensitive to the sample shape. This independenceto shape is expected, due to the absence of determinis-tic propagation and associated macroscopic phenomena,given that the spin wave is highly mismatched from ra-diation, | k (cid:48) s | = 2 . ω eg /c .To look for deviation of the decay of O k (cid:48) s ( t ) from thesingle-atom spontaneous emission rate, we repeat themeasurements of Figs. 4(a)(b) at various sample densi-ties, and fit the initial decay rate of the peak intensity ∼ I k s ( T i ) vs. T i to an exponential. The background-freemeasurement method is key, as one can specifically searchfor any deviation from the precisely known single-atomemission rate Γ e . We collect the associated I k s ( t ) sig-nals from ∼ repeated measurements over ∼
20 hours.The data are then grouped according to the peak den-sity ρ estimated from absorption imaging analysis (Ap-pendix D). We take great care to suppress systematicerrors arising from drifts of sample conditions and im-perfections of spin-wave control [38]. Steps implementedto suppress errors include inline characterization of eachsample (Appendix D 2), implementing rapid T i scans forrelative measurements, careful sampling of T i to avoid D Γ 𝐤 𝑠 ′ / Γ 𝑒 𝜂 MBE
FIG. 5. The initial decay rates of the spin wave survival ra-tio, plotted vs estimated dimensionless peak density parame-ter η = ρ λ eg . The error bars reflects the full statistical andsystematic uncertainties. The solid line gives the predictionfrom the dipolar dephasing theory of Eq. (7) with ¯ η = η / √ k (cid:48) s = Γ e is instead the prediction based on MBE,which ignores microscopic effects associated with atomic gran-ularity. tons by atoms decaying from 5 P / (Appendix D 4). Theestimated decay rate Γ k (cid:48) s with peak density ρ and the as-sociated dimensionless density parameter η = ρ λ eg areplotted in Fig. 5. A density-dependent dephasing rate γ (cid:48) ≈ . η Γ e of the survival ratio O k (cid:48) s ( t ) can be ex-tracted from the data. This deviation of Γ k (cid:48) s from theMBE-predicted rate Γ k (cid:48) s = Γ e is the main experimentalresult of this work. We now explain the physical originof this density-dependent decay. C. Pair-wise dipolar dephasing induced spin-wavedecay
Here, we introduce a theoretical model that quanti-tatively reproduces the observed density-dependent de-phasing rate as in Fig. 5. We begin by considering asimpler problem, involving just a pair of two-level atomsseparated by a distance r (cid:46) k − . In that case, the dipole-dipole interaction Hamiltonian given by Eq. (1) is dom-inated by the ∼ /r near-field component. To be morespecific, we can explicitly separate out the Green’s func-tion terms that are proportional to ∼ /r , obtaining G near ( r, θ ) = 14 πk r (cid:0) θ − (cid:1) (4)where θ is the angle between the dipole polarization (lin-early polarized along ˆ x ) and the distance between twoatoms r . The near-field contribution is real, and thus thecorresponding interaction is purely coherent and Hermi-tian. In the single-excitation manifold, this interactionis diagonalized by symmetric and anti-symmetric wavefunctions, |±(cid:105) = ( | eg (cid:105) ± | ge (cid:105) ) / √
2, which experience op-posite frequency shifts ω ± ( r ) = ± e (3 cos θ − / k r relative to the bare atomic transition frequency.We now turn to the evolution of a spin wave in amany-atom system, due to dipole-dipole interactions. Weconsider the decay of spin-wave survival ratio O k ( δt ) = |(cid:104) ψ k | e − iH eff δt | ψ k (cid:105)| within a short interval δt (cid:28) / Γ e .We begin by dividing H eff of Eq. (1) into Hermitianand anti-Hermitian parts, H r = ( H eff + H † eff ) / H a = ( H eff − H † eff ) /
2, which describe coherent and dissi-pative interactions, respectively. Examining the coherentinteractions first, the 1 /r scaling of the near field im-plies that close-by neighbors will interact with each othermore strongly than with all other atoms combined [47].Thus, we can approximately diagonalize H r by isolatingclose-by neighbors and simply diagonalizing these pairsexactly as we have described for the two-atom problem,while treating all other interactions between atoms asa (negligible) perturbation. In other words, an approx-imate complete basis of single-excitation eigenstates isgiven by | e i (cid:105) with energy ω ≈ | e i g j (cid:105) ± | g i e j (cid:105) ) / √ ω ± ( r ij ) for all atoms i, j that form close-bypairs. The short-time dynamics can then be evaluated bydecomposing the initial spin wave in this basis. Further-more, we assume that the spin wave has negligible over-lap with the anti-symmetric pairs, which is justified giventhe relatively long wavelength of the spin wave in com-parison with the inter-atomic separation of close neigh-bors. Then, the time-dependent survival ratio becomes O k ( δt ) ≈ | (cid:82) P ( ω + ) e − iω + δt | , where P ( ω + ) is the prob-ability distribution of finding close-by pairs with energy ω + . Utilizing the probability distribution f (2) ( r ) to findthe nearest neighbor of a random gas of uniform density ρ at relative position r , f (2) ( r ) = ρe − π r ρ , (5)it can be shown that the high-frequency tails of P ( ω + )behave as P ( ω + ) = ξη Γ e πω . (6)Here η = ρλ eg is the local density parameter, and ξ isa numerical factor that depends on details of the dipo-lar interaction. Beyond the 2-level model, in Table I ofAppendix A 3 we list the value of ξ for various models in-cluding those taking into account hyperfine interactions.In particular, ξ = 0 . / π for the F = 2 − F (cid:48) = 3 tran-sition of Rb.Importantly, the scaling P ( ω + ) ∝ /ω is due to the1 /r near-field interaction in a random gas [48]. Thisscaling guarantees that the survival ratio experiences an initial exponential decay due to near-field interactions of O k ( δt ) as e − γ (cid:48) δt , with γ (cid:48) = ξη Γ e . (7) Returning to the spontaneous emission arising from theanti-Hermitian term H a , mathematically, at short timesits effect on evolution commutes with that of H r ( e.g. ,by considering a Suzuki-Trotter expansion [49]). As theinitial ensemble-averaged spontaneous emission rate of aphase-mismatched spin wave is simply that of a singleatom, Γ e [40], we can conclude that the total initial de-cay of the survival ratio is given by Γ k (cid:48) s = Γ e (1 + ξη ). Tomatch to experiment, we must also account for the factthat the atomic ensemble has a Gaussian rather thanuniform density distribution. Defining η = ρ λ eg as thepeak density, and η = η / √ k (cid:48) s = Γ e (1 + ξ ¯ η ) for the Gaussian distribution.Using the above stated value of ξ , in Fig. 5 we see ex-cellent agreement of this model with the experimentallymeasured values of survival ratio, when plotted as a func-tion of peak density parameter η = ρ λ eg .While we have analytically derived the short-time de-cay of a phase-mismatched spin wave, one might ask overwhat time scale does this decay deviate from being expo-nential. Investigating this question, we numerically findthat for similar parameters as the present experimentwith ρ (cid:28) k and moderate N , the exponential decay isrobust for relatively long times t (cid:38) / Γ e (see CDM sim-ulations in Figs. 1(e)(f) and Appendix C) during whichthe experimental measurements of γ (cid:48) are performed. Theexponential decay suggests that the spin states orthogo-nal to the initial | ψ k (cid:105) , being gradually populated by thespin relaxation process, hardly re-populate | ψ k (cid:105) duringthis initial time t .Beyond this initial decay, we expect long time spin-wave evolution and the deviation from exponential dy-namics to be an interesting and complex problem for fu-ture work. The deviation from exponential likely arises acombination of effects involving the low-frequency spec-trum of the distribution P ( ω ± ), and collective effects( e.g. , radiation trapping) that arise from the far-field in-teractions contained in H r and H a . For example, we ex-pect that any population that becomes radiation trappedis likely to be divided equally among all possible spinwaves, presenting a lower limit on how small the popula-tion in | ψ k (cid:105) can be at long times. Finally, it is worthnoting that in the high-density limit η (cid:29)
1, whereeach atom sees many other neighbors within a distance r (cid:46) k − , “renormalized” strongly interacting close pairscan be subsequently renormalized many times over withother neighbors. This intuitive picture can in fact beused to quantify the behavior of the low-frequency partof P ( ω ± ) [47], and thus perhaps can provide additionalinsight into the long-time dynamics. IV. DISCUSSIONSA. Implications of the dipolar spin-wave dephasing
The strong near-field resonant dipole interaction be-tween closely spaced pairs of atoms should be a univer-sal property of dense atomic ensembles [13, 18, 50–52].Its dephasing effect on spin-wave order, as analyzed inSec. III C, is of direct and widespread relevance to theperformance of quantum light-matter interfaces, wheredephasing is typically seen as being detrimental. Thissuggests that there could be upper bounds on the maxi-mum atomic densities tolerable in order to reach a givenfidelity. Furthermore, as total atom number (or opti-cal depth) is also an important resource [1], this in turnwould set a limit on the minimum system size, whichhas implications in efforts to make compact quantum de-vices based upon ensembles. Interestingly, while manydephasing mechanisms are simply “technical”, this oneis seemingly fundamental to randomly positioned atomicgases. To overcome this limitation, one might resort toatomic arrays [53] where the fluctuations of near-fieldinteractions are controlled, as has been demonstrated re-cently [54].
B. Unraveling microscopic optical spin dynamics
We recall that phase-matched collective radiation is re-sponsible for nearly all macroscopic coherent optical phe-nomena, from coherent scattering in linear optics to effi-cient multi-wave mixing in nonlinear optics, and is well-described by the Maxwell-Bloch Equations [55]. Methodsto elucidate interactions beyond MBE have been devel-oped in the field of nonlinear optics [56–59], typically byspectroscopically isolating the optical response of mul-tiple quantum excitations [56, 60, 61]. In contrast, ourmethod relies on temporarily breaking the phase match-ing condition to nullify the collective radiation and theassociated far-field interactions. The method is able toisolate the spin wave dynamics driven by short-range in-teractions in an otherwise ideal gas of cold atoms, forbackground-free measurements with a precision unlim-ited by collective radiation damping. Although in thiswork we have focused on dynamics of weakly excited spinwaves in the linear optics regime, the method is readilyapplicable to nonlinear spin waves where the effects ofquantum interactions have been predicted to be particu-larly rich [62–68].To best clarify the importance of suppressing superra-diant dynamics in the optical domain for resolving micro-scopic effects beyond MBE, we can compare with the mi-crowave domain, where collective radiation is much lessimportant. In particular, it was found more than 70 yearsago that the T time of magnetic resonances is limited bythe magnetic spin-spin interactions [30, 69, 70], an effectthat shares the same essential physics of near-field dipo-lar interaction as in this work. In Ref. [69], Van Vleck noted that for the dipolar dephasing problem in the op-tical domain, “it may not be safe to neglect, as we do,the influence of Doppler and radiation broadening”. To-day, while Doppler broadening can easily be suppressedby using cold atomic gases (also see Appendix C 4), thepresence of radiation broadening background becomes amajor obstacle. In particular, for a macroscopic samplewith size L (cid:29) λ eg , the collective broadening associatedwith Γ k ∼ OD( k )Γ e is larger than the typical micro-scopic rate γ (cid:48) ∼ η Γ e by a factor of L/λ eg . The fact thatΓ k and wave dynamics are all sensitive to the sampledensity distribution, only makes it more difficult to re-solve the microscopic rate of interest directly. For thisreason, we believe our technique will be generally use-ful to resolving microscopic spin dynamics in cold atoms.Conversely, we expect that the ability to access spin-wavestates in motionless atoms with suppressed macroscopicdynamics will generate the discovery of interesting effectsbeyond those in NMR, due to the more prominent role offar-field dipolar interactions and radiation in the opticaldomain [22–25, 71, 72].We note that superradiant emission of optical dipolescan also be suppressed by engineering the electromag-netic environment, for example, by periodically dress-ing a slow-light medium to suppress the light propaga-tion [73–75]. Similar to the phase-mismatched spin wavesin this work, we expect the propagation-free optical ex-citations in those experiments to be sensitive to micro-scopic interactions. However, the method of temporalphase-matching control in this work does not require ad-ditional dressing fields, and may thus help to unravelgeneric microscopic dynamics of resonantly interactingoptical dipoles. C. Outlook
In this work we have demonstrated a general methodto probe microscopic dynamics in cold atomic ensemblesin a background-free fashion, by managing the phase-matching condition in the time domain so as to tran-siently suppress macroscopic collective dynamics asso-ciated with optical spin-wave excitations. With thismethod, we have measured a density-dependent dipo-lar dephasing rate of optical spin waves. The previouslyoverlooked dephasing effect is fundamental to a randomlypositioned atomic gas. We have developed a quantitativetheory to explain the dephasing effect.Straightforward extension of this observation to atomicgases near quantum degeneracy would unravel interest-ing effects related to quantum statistics and correlationsin atomic positions [25]. Furthermore, we envision am-ple opportunities to uncover and investigate many-bodydipolar interaction effects in cold atomic ensembles be-yond MBE, once collective, macroscopic dynamics aresuppressed. After more than 70 years since a similar ob-servation was made in the microwave domain [30, 69, 70],we hope that the observation of optical dipolar spin-wavedephasing in this work will contribute to novel develop-ments of many-body physics in quantum optics, similarto its counterpart in NMR.
ACKNOWLEDGMENTS
We are grateful to Prof. J. V. Porto and Prof.I. Bloch for helpful discussions. We acknowl-edge support from National Key Research Pro-gram of China under Grant No. 2016YFA0302000,No. 2017YFA0304204, and No. 2017YFA0303504, fromNSFC under Grant No. 12074083, No. 11574053,and No. 11734007, from Natural Science Founda-tion of Shanghai (NO. 20JC1414601), from the Eu-ropean Union’s Horizon 2020 research and innovationprogramme, under European Research Council grantagreement No. 639643 (FOQAL), FET Open grantagreement No. 899275 (DAALI), and Quantum Flag-ship project 820445 (QIA); MINECO Severo Ochoaprogram CEX-2019-000910-S, AEI Europa ExcelenciaProgram EUR2020-112155 (ENHANCE), CERCA Pro-gramme/Generalitat de Catalunya, Fundacio PrivadaCellex, Fundacio Mir-Puig; Plan Nacional Grant ALIQS,funded by MCIU, AEI, and FEDER; and Secretariad’Universitats i Recerca del Departament d’Empresa iConeixement de la Generalitat de Catalunya, co-fundedby the European Union Regional Development Fundwithin the ERDF Operational Program of Catalunya(project QuantumCat, Ref. 001-P-001644).
Appendix A: Derivation of the effective frequencydistribution
Here, we provide a more detailed derivation of theprobability distribution of effective resonance frequencies P ( ω + ) of close-by atomic pairs, presented in Eq. (6) ofthe main text, which leads to the density-dependent spin-wave dephasing rate. We start with the simplest case oftwo-level atoms, before discussing the case of multilevelatoms.
1. The effective frequency distribution of stronglyinteracting pairs
As mentioned in the main text, we consider the near-field interaction as being the dominant process that con-tributes to dephasing, and furthermore approximatelydiagonalize it into the following complete basis in thesingle-excitation manifold: | e i (cid:105) with energy ω ≈ | e i g j (cid:105) + | g i e j (cid:105) ) / √ ω ± ( r ij ) for all atoms i, j that form close-by pairs. As also stated in the maintext, we can trivially evaluate the dynamics induced bythe near-field interactions by projecting the initial spin wave into this basis, and assuming that overlap with anti-symmetric pair states is negligible.More precisely, we start from the probability distribu-tion of nearest neighbors in a random gas of density ρ [76], f (2) ( r ) = ρe − π r ρ (A1)which gives the probability of finding the closest neighborat a position r (such that (cid:82) d r f (2) ( r ) = 1), given oneatom at the origin. Within the approximations statedabove and further assigning to each close-by pair a fre-quency ω + ( r ) in the symmetric branch, the spin wavesurvival ratio is formally given by O k ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d r f (2) ( r ) e − iω + ( r ) t (cid:12)(cid:12)(cid:12)(cid:12) . (A2)Here, as we are primarily interested in the short-distanceand high-frequency contribution of particularly nearbypairs, we need not impose any specific distance cut-off ( e.g. , r < k − ) in the integral.To proceed further, it is convenient to introduce thechange of variables ω + = ω + ( r ) = ω + ( r, θ ), to convertthe integrand into the Fourier transform of the frequencyprobability distribution P ( ω + ) – that of strongly inter-acting and symmetrically excited pairs in the ensemble.As ω + ( r, θ ) changes sign at | cos θ | = 1 / √
3, the fre-quency distribution needs to be piecewise defined, P ( ω + ) = Γ e π ηω × (cid:82) / √ d (cos θ ) h ( θ ) e − Γ e π h ( θ ) ηω + ω + > − (cid:82) / √ d (cos θ ) h ( θ ) e − Γ e π h ( θ ) ηω + ω + < h ( θ ) = 3 cos θ − P ( ω + ) is not symmetric, it can beshown that the high-frequency tails are symmetric andbehave asymptotically like P ( ω + ) ±∞ ∼ ξη Γ e πω (A4)Here, we have defined η = ρλ eg as the local density pa-rameter, and ξ = 1 / (6 π √
3) is a numerical factor that de-pends on the details of the atomic structure, here specif-ically evaluated within the 2-level approximation.
2. Gaussian distribution
The calculation of the previous section assumes aninfinite and homogeneous atomic cloud. In an experi-ment, an ensemble generally follows a position depen-dent distribution ρ ( r ), which can be generally takeninto account substituting a mean density ρ → ρ = (cid:82) ρ ( r ) d r / (cid:82) ρ ( r ) d r in Eq. (5). For a Gaussian distri-bution the mean density is related to the peak density as ρ = ρ / √ − − − − − m g = m e =
13 11523 815 1525 35 2515 815 23115 35 FIG. 6. Multilevel atomic structure, corresponding to the F g = 2 → F e = 3 transition of the D Rb probed inthe experiment. The states are labeled by their Zeeman quan-tum number m g ( e ) , while in orange we indicate the strengthof the allowed transitions, as characterized by the squaredClebsh-Gordan coefficients | C ge | . r ˆ x ˆ x ˆ σ − ˆ σ − (a) r θ ˆ x ˆ x ˆ σ − ˆ σ − ˆ π ˆ σ + (b) FIG. 7. (a) Resonant dipole-dipole interactions in the molec-ular basis. When the molecular axis, defined by the distancebetween the atoms r , aligns with the quantization axis ˆ x , theinteraction preserves the total angular momentum projectionalong ˆ x . Thus if one atom emits a σ − photon, the second atomcan only absorb it on a σ − transition. (b) In an arbitrary ori-entation of the molecular axis respect to the quantization axis(for example ˆ x · ˆ r = cos θ ) this is no longer true. In particulara photon emitted by one atom can generically drive all thepossible transitions of the second atom, depending on θ .
3. Hyperfine atoms
Our calculation thus far relies on the approximationof an atom as a two-level system. However, real atomshave a complex multilevel structure. This is specificallyillustrated in Fig. 6 for our case of interest, involvingthe maximum angular momentum ground ( F g = 2) andexcited ( F e = 3) state manifolds of the D2 transitionof Rb. Here, the set of ground (excited) states { g ( e ) } consist of all possible Zeeman levels | m g ( e ) | ≤ F g ( e ) alongsome given quantization axis. As before the goal will beto find the exact eigenstates of the two-atom problem,and use the symmetric states as an approximate basis todiagonalize the spin-wave of a many-atom system. Wewill show that this leads simply to a modification of the dephasing coefficient ξ .In the presence of multiple ground and excited states,the dipole-dipole interaction (Eq. (1) for two-level atoms)can be readily generalized to [77]ˆ V ijeg,e (cid:48) g (cid:48) = ω eg d F e F g ε c e ∗ g (cid:48) e (cid:48) · G ( r ij , ω eg ) · e ge C g (cid:48) e (cid:48) C ge σ ie (cid:48) g (cid:48) σ jge . (A5)As before, it describes photon emission and re-absorptionbetween two atoms at a distance r ij and the la-bels g, e, g (cid:48) , e (cid:48) refer to arbitrary Zeeman levels. Thestrengths of these dipole transitions depends on a re-duced dipole matrix element d F e F g that is independentof the Zeeman levels, Clebsch-Gordan coefficients C ge = (cid:104) F g , m g | F e , m e ; 1 , m g − m e (cid:105) , and the overlap of the emit-ted/collected photon polarization with the spherical basisof choice, given by e ge = e m g − m e = − (ˆ z + i ˆ y ) / √ , m g − m e = 1ˆ x, m g − m e = 0(ˆ z − i ˆ y ) / √ , m g − m e = − π transitions ( m g − m e = 0) with the beam polarization ˆ x . The totalspontaneous emission rate of any one of the excitedstates (equal for all states) can be related to these quan-tities by Γ e = (cid:80) g | C ge | ω eg d FeFg π (cid:126) ε c . In particular, by con-sidering the “closed transition” with | C ge | = 1 we haveΓ e = ω eg d FeFg π (cid:126) ε c .As before, we will consider the specific case of twoatoms sufficiently close to each other ( r (cid:28) k − ) thatthe interaction of Eq. (A6) is dominated by the coherentnear-field component of the Green’s function (comparewith Eq. (4) for two-level atoms).The form of Eq. (A5) greatly simplifies when the quan-tization axis ˆ x aligns with the natural “molecular” axis,defined as being the vector r connecting the two atoms,as represented in Fig. 7a. In this case, the interactionis only non-zero when excited atom emits on a transi-tion ( σ − for example) that is equal to the transition ofthe second, ground-state atom as it absorbs the photon,thus preserving the projection of the total angular mo-mentum along the quantization axis ˆ x . In an arbitraryconfiguration, however, as in Fig. 7b where the molecularand quantization axes do not agree, this is no longer true,and the ground state atom can be excited along any tran-sition once absorbing the photon. As a consequence, al-ready the two-body problem appears to be complex sinceeigenstates of the Hamiltonian (A5) necessarily involvenon-trivial superpositions of multiple Zeeman states. Wewill therefore approach the problem numerically.First, we diagonalize (A5) within the single-excitationmanifold for a pair of atoms at fixed distance r , obtain-ing n = 2 × (2 F e + 1) × (2 F g + 1) non-trivial eigenstates, { ψ i } i =1 ,...n , and eigenvalues, { ω i } i =1 ,...n . The eigenval-ues can only depend on the distance between the atoms r (not on the angle θ ) and, just considering the near field1interaction, will have the general form ω i ( r ) = C i Γ e k r (A7)where C i is generically a non trivial combination of CGcoefficients.The angular part will appear in the projection over theeigenstates as we now discuss. As before we assume thatan ˆ x polarized short pulse excites the atomic pair in asymmetric state given by | x + (cid:105) = S † | g , g (cid:105) S † = 1 √ (cid:0) σ e ,g + σ e ,g (cid:1) (A8)such that m e = m g , m e = m g . Each atom is assumedto initially be in a randomly chosen Zeeman sublevel g , ,and to obtain observables we will average over all thepossible sublevel configurations.While the excited state looks simple because of ourconvenient choice of the polarization basis, generally itwill not be an eigenstate of the dipole-dipole interac-tion Hamiltonian, but will have some overlap with them h i ( θ ) = |(cid:104) x + | ψ i (cid:105)| , such that all the modes would natu-rally contribute to the dephasing in the time evolution.Then, making the same assumptions as before for theevolution of a many-atom spin wave, the survival ratio ofEq. A2 for two-level atoms naturally generalizes to O k ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d r f (2) ( r ) (cid:88) i h i ( θ ) e − iω i ( r ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (A9)Further defining the integrals H i = (cid:82) π d cos θ h i ( θ ) andperforming multiple change of variables ω + = ω i ( r ) ac-cording to the sign of the eigenvalue it is again possible todefine the frequency distribution of strongly interacting,symmetrically excited pairs of atoms: P ( ω + ) = Γ e π ηω × (cid:80) C i > H i C i e − C i Γ e π ηω + ω + > − (cid:80) C i < H i C i e − C i Γ e π ηω + ω + < P ( ω + ) ±∞ ∼ ξη Γ e πω (A11)where now the dephasing coefficient ξ introduced in themain text, which depends on the details of the atomicstructure generalizes to ξ = 16 π (cid:88) C i > H i C i . (A12)This quantity is evaluated numerically in Table I for var-ious different hyperfine transitions. This thus generalizesthe result for 2-level atoms ξ = π √ that we found inEq. (A12). transition 2-level 0 → → → → → πξ / √ .
77 0 .
69 0 .
66 0 . ξ (multiplied by 6 π in the Table) forthe density-dependent dephasing rate γ (cid:48) = ξη Γ e , as calculatedfor different atomic structures. These include two-level atoms,and transitions F g → F e involving ground and excited statemanifolds with angular momenta F g and F e , respectively. Appendix B: MBE simulations
The Maxwell-Bloch equations (MBE) describe coher-ent coupling between continuous optical media (describedby optical Bloch equations) and optical fields (describedby Maxwell’s equations), and is one of most common ap-proaches to describe light-atom interactions [55]. In thisAppendix we derive the MBE from the 2-level spin model(Eqs. (1)(2)) in the classical, weak excitation limit. Wethen outline the numerical methods for solving the MBEand to simulate spin wave dynamics as in Figs. 1(e)(f) ofthe main text.
1. The model
It can readily be verified that Eq. (2) is the solution tothe Maxwell wave equation ∇ × ∇ × ˆ E s − ω eg c ˆ E s = ω eg ε c ˆ P ( r ) , (B1)where the polarization density takes the form ˆ P ( r ) = (cid:80) j d eg σ − j δ ( r − r j ) for two-level atoms. As we are inter-ested in linear dynamics, it suffices to consider the singlyexcited atomic state | ψ ( t ) (cid:105) = 1 √ N (cid:88) j β j σ + j | g , · · · , g N (cid:105) , (B2)and define a single-photon wave function ε = (cid:104) g , · · · , g N | ˆ E s | ψ ( t ) (cid:105) . Then, Eq. (B1) becomes ∇ × ∇ × ε − ω eg c ε = ω eg ε c p ( r , t ) , (B3)with p ( r , t ) = 1 / √ N (cid:80) j d eg β j δ ( r − r j ). In the Maxwell-Bloch equations, it is assumed that the granularity ofatoms can be ignored, and the atomic medium treatedas a smooth density distribution ρ ( r ) = (cid:104) (cid:80) j δ ( r − r j ) (cid:105) .The fields and polarizations can then be smoothed aswell, ε and p . In that case, the field (Maxwell) equationbecomes ∇ × ∇ × ε − ω eg c ε = ω eg ε c p , (B4)while the Bloch equation describing the evolution of thepolarization in response to the field is i ˙ p = − i Γ e p + i Γ e ε ε ( r ) · χ ( r ) , (B5)2where the linear susceptibility is χ ( r ) = ρ ( r ) α /ε with α = 2 i | d eg | / (cid:126) Γ e .
2. Simulating the experiment
Our goal now is to solve the Maxwell-Bloch equa-tions (B4) and (B5), starting from a phase-matchedspin wave as the initial excitation p ( r , t = 0) =( d eg / √ N ) ρ ( r ) e i k s · r . For a smooth density distribution, ρ ( r ) = ρ e − ( x + y ) / σ − z / l z with k s σ, k s l z (cid:29)
1, wecan assume that the field and polarization have slowly-varying envelope ˜ ε and ˜ p , related to the original quan-tities by ε = ˜ ε e i k s · r and p = ˜ p e i k s · r . In that case, thewave equation significantly simplifies,2 ik s ∂ Z ˜ ε = −∇ ⊥ ˜ ε − ω eg ε c ˜ p . (B6)For notational convenience, we have defined a Z -direction to align with that of the spin wave direction k s , while ∇ ⊥ denotes the divergence operator in the XY -plane. The boundary condition for the field is given by˜ ε → Z → −∞ . To simulate the coupled equa-tions (B5) and (B6), we discretize a simulation space with L X = L Y = 10 σ, L Z = 10 l z into N X × N Y × N Z
3D grids.The grid size is chosen to support the slowly varying ˜ ε ,˜ p only, and do not need to be very fine. Practically, wefind N X = N Y = N Z = 400 allows for convergence forboth type “A” and “B” samples in this work. We directlysolve Eqs. (B5)(B6) in the time domain and use this toconstruct the superradiant field emission ˜ ε ( x, y, z, t ). InFig. 1(e) and 1(f), the MBE simulations are performedwith the same parameters as those for CDM simulations,which are detailed in Appendix C 2. Appendix C: Coupled Dipole Models
As our goal is to study microscopic effects beyond theMBE, here we briefly introduce the coupled dipole mod-els (CDM) that we use to take into account atomic gran-ularity. We present CDM for two-level atoms, as well asfor isotropic atoms ( F g = 0 → F e = 1) and also a “hy-brid” model, which we use to approximately take intoaccount the hyperfine structure of Rb atoms on the F g = 2 → F e = 3 transition. Equipped with the CDM,we also analyze the effects of atomic motion on the decayof spin-wave order.
1. CDM for 2-level model
As discussed in the main text, we consider N two-levelatoms with positions r j , with resonant dipolar interac-tion specified by Eq. (1). To study spin-wave dynamicswithin the linear excitation regime, it suffices to considerthe singly excited wave function | ψ ( t ) (cid:105) as in Eq. (B2). The Schr¨odinger equation for the amplitudes β j ( t ) isreadily found to be [37]:˙ β j = − Γ e β j + i λ eg Γ e (cid:88) l (cid:54) = j G xx ( r jl , ω eg ) β l . (C1)To simulate the experiment with the coupled dipoleequations, we set Γ e = 1 / . λ eg = 780 nmto correspond to Rb. The positions r j are randomlyand independently sampled according to a Gaussian dis-tribution. To initialize a spin wave excitation withwavevector k , we accordingly set the initial amplitudesas β j ( t = 0) = e i k · r j . With { β j ( t ) } evolving accordingto Eq. (C1), the spin-wave surviving ratio by Eq. (3) isevaluated as O k ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N (cid:88) j β j ( t ) e − i k · r j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (C2)The instantaneous single-photon wave function ε ( r , t ) = (cid:104) g , ..., g N | ˆ E s ( r ) | ψ ( t ) (cid:105) is given by ε ( r , t ) = ω eg ε c N (cid:88) j G ( r − r j , ω eg ) · d eg β j ( t ) . (C3)Having introduced the CDM, we now explain howthese simulations can be used to justify the field collec-tion setup of our experiment. In particular, we can re-peat the CDM calculations over many microscopic spatialconfigurations { r j } of atoms, to obtain an accurate meanfield solution ε ( r , t ) = (cid:104) ε ( r , t ) (cid:105) . Note that this averagingonly retains the part of the field that has a coherent phaserelationship with the spin wave, while eliminating thefield that has a phase that randomly depends on the mi-croscopic configuration. For a phase-matched spin wavewith | k | = ω eg /c , we define the superradiance solid angleΩ s as the solid angle around k over which a substantialfraction ( e.g. ε ( r , t ) emission is di-rected, and find that it is practically set by Ω s = ( k σ ) − for the Gaussian distribution ρ ( r ) ∼ e − r / σ . Exper-imentally, the emission from large samples within thissmall solid angle Ω s and for N (cid:29) I k ( t ) ∝ (cid:82) Ω s | ε ( r , t ) | d Ω as the super-radiant intensity. We perform an average over manymicroscopic spatial configurations { r j } to obtain (cid:104) I k ( t ) (cid:105) and (cid:104) O k ( t ) (cid:105) as the final simulated observables to comparewith the experimental data.Finally, while we have fixed the initial amplitudes β j ( t = 0) by hand in the method described above, theequations can readily be modified to explicitly account3for a weak probe pulse to initially excite the atomic am-plitudes β j . For the dilute sample with ρ (cid:28) k and withthe nanosecond probe pulse in this work, we find that thedifference in the simulated results is negligible.
2. CDM for isotropic model
We now introduce the isotropic model of light-atominteraction, characterized by F g = 0 → F e = 1. Eachatom thus has a unique ground state | g, m g = 0 (cid:105) andthree excited states | e, m e = 0 , ± (cid:105) . The spin model byEq. (1) is modified asˆ V i,je ; e (cid:48) = ω eg ε c d ∗ eg · G ( r ij , ω eg ) · d eg ˆ σ ieg ˆ σ jge (cid:48) , (C4)and the effective Hamiltonian H eff = (cid:80) i,j,e,e (cid:48) ˆ V i,je ; e (cid:48) ismodified accordingly. Here e , e (cid:48) label the three de-generate excited states ( | m = ± , (cid:105) ). Similar tothe 2-level model, we expand the singly excited state | ψ ( t ) (cid:105) = (cid:80) j,e β j,e σ + j,e | g , ..., g N (cid:105) , to obtain the coupledipole model [37] as˙ β je = − Γ e β je + 32 iλ eg Γ e (cid:88) l (cid:54) = j (cid:88) e (cid:48) e ∗ e · G ( r jl , ω eg ) · e e (cid:48) β le (cid:48) . (C5)Comparing with the 2-level model in the main text, theisotropic model of Eqs. (C4)(C5) more closely mimics theinteraction of real atoms, by allowing resonant exchangeof vector photons of all polarization. The Eqs. (C2)(C3)are straightforwardly generalized to evaluate the emissionprofile and the spin-wave survival ratio, and this isotropicmodel is applied to the simulations in Fig. 1 of the maintext. In the simulation of Fig. 1, the atomic sample has N = 532 motionless atoms in a Gaussian density distri-bution ρ ( r ) ∝ e − ( x + y ) / σ − z / l z with σ = 1 . λ eg and l z = 3 . λ eg . In each configuration of the CDM sim-ulation, the N atoms are again randomly and indepen-dently sampled in space according to the density distri-bution ρ ( r ). In particular, the electric fields | Re( ε k ( r )) | in Fig. 1(c) and 1(d) are simulated for a single micro-scopic spatial configuration, while the results by CDMin Fig. 1(e) and 1(f) are averaged over 1000 microscopicconfigurations.
3. CDM for a hybrid model based on hyperfinetransitions
To rigorously treat dipole-dipole interactions for manyatoms in the presence of hyperfine structure is numer-ically unfeasible, even if restricting to just a single ex-citation. This is because such interactions generally donot conserve the total projection of angular momentumonto a given axis, as discussed in Appendix A 3 (also seeFig. 7). Thus, over time, dynamics will generally allowa large fraction of the single-excitation Hilbert subspace r θ ˆ x A ˆ x B ˆ σ + ˆ π ˆ σ − FIG. 8. Schematic illustration of resonant dipole-dipole inter-actions within the “hybrid model” of a F g = 2 − F e = 3 hy-perfine transition. The two atoms A,B are initiated with ran-domly chosen ground states, here illustrated to be m g = 0 , A here, after certain evolution time) is only al-lowed to occur if the transition is back toward the initialground state (solid arrows), while those effectively inducingRaman transitions (dashed) are forbidden by setting themto have zero interaction strength. Within this model, thestrengths of the various allowed transitions still vary accord-ing to the physical Clebsch-Gordan coefficients, here illus-trated by the different thickness of the arrows indicating thepossible transitions of atom B. to become occupied. However, this subspace is expo-nentially large (being dominated by the ∼ (2 F g + 1) N − possible state combinations of the ground state atoms).One approximation scheme to avoid this exponentialcomplexity was provided in Ref. [26], which we adopthere. In particular, we assume that the initial state be-fore excitation is given by a product state of Zeemanground states, | ψ ( t = 0) (cid:105) = | g , ..., g N (cid:105) , with g j specify-ing a specific Zeeman sub-level of atom j . These states g j will be randomly sampled and the results averaged,to obtain observables. Crucially, it is further assumedthat dipole-dipole interactions can never induce transi-tions to different states g (cid:48) j , i.e. the ground state g j iseffectively the unique ground state of atom j . The effec-tive level structure and interactions between two atomsunder this approximation are illustrated in Fig. 8. Thenumber of ground and excited states to keep track of peratom is then identical to the case of isotropic atoms. Themain difference compared to that case, however, is thatthe Clebsch-Gordan coefficients of the hyperfine inter-action Hamiltonian Eq. (A5) are still kept, and appearin the subsequent equations of motion. Thus, the time-dependent singly excited state (within the above approxi-mations) is | ψ ( t ) (cid:105) = (cid:80) j,a β j,a ( t ) σ je = g − a,g | g , ..., g N (cid:105) with a = 0 , ± β ja = − Γ e β ja + 32 iλ eg Γ e × (cid:88) l (cid:54) = j (cid:88) gj − ej = agl − el = a (cid:48) e ∗ g j e j · G ( r jl , ω eg ) · e g l e l β la (cid:48) C g j e j C g l e l . (C6)To simulate the experiments with Eq. (C6), for a givenconfiguration, we randomly assign to each atom one of4the Zeeman sub-levels with g j = ± , ± ,
0, to accountfor our unpolarized Rb atomic gas in the F = 2 hyper-fine state. After solving for dynamics and observablesfor this configuration, we repeat and perform an averageover both ground state samplings { g j } and spatial { r j } configurations to calculate the observables ε ( r , t ), O k ( t ), I k ( t ) of interest.The hybrid model accounts for interactions mediatedby coherent light scattering. Therefore, we expect themodel to quite accurately predict the coherent evolutionof phase-matched spin waves where the coherent scatter-ing dominates the dynamics, particularly in large sam-ples. Indeed, we find that Eq. (C6) can well reproducethe experimental observations using in situ measured ρ ( r )parameters. One example is shown in Fig. 3(b) in themain text. Here the estimated Gaussian radii are σ x , σ y , σ z along the x , y , z axis are 4.7 µ m, 4.7 µ m, 40 µ m,respectively. The measured atom number is N ≈ σ x = ισ x , ˜ σ y = ισ y , ˜ σ z = ισ z , and ˜ N = ι N with ι = 0 .
36 and ˜ N = 6000, with the optical depthOD(ˆ k s ) unchanged. We have verified that the increaseddensity by 1 /ι does not significantly affect the superra-diant dynamics [37], by re-running the simulating with ι between 0 .
12 and 0 .
36. Similar simulations are also com-pared to the Fig. 3(a)(c) measurements, with excellentagreement.More generally, since the hybrid model overlooks the m g -changing photon exchange channels, we expect themodel to underestimate the dephasing/decoherence ofspin-wave order. Here the insufficient modeling is bench-marked by comparing the simulation with the exact pair-analysis in Appendix A 3. In particular, comparing withthe ξ = 0 . / π factor from the full model, the hybridmodel here predicts a ξ factor which is ∼
25% smaller,and thus a slower density-dependent initial decay. Be-yond the initial decay analysis, it is difficult to compu-tationally evaluate how this inaccuracy evolves. Never-theless, we have carried out CDM simulations at typicalatomic sample densities in this work, over the interroga-tion time T i of interest (Fig. 2, Fig. 4), to verify that sim-ilar to the isotropic model (Appendix C 2,Fig. 1(f)), thehybrid model predicts an initial spin-wave decay whichis nearly ideally exponential.
4. Impact of atomic motion
All the discussions so far in this work assume that thedipole spin waves are excited in a motionless gas of atoms.Here, we estimate the errors associated with this assump-tion.We first analyze the impact of atomic motion to thenear-field interaction associated with Eq. (4). For a ther-mal ensemble, the relative motion of an atomic pair leadsto a position change of δr = v T δt during the spin-waveevolution time δt . Furthermore, considering the eigenfre-quencies of an interacting pair at close distances, ω ± ( r ) = ± e (3 cos θ − / k r , one sees that they create a vander Waals force that accelerates the relative motion, lead-ing to a velocity change δv ∼ v r × | ω + δt | /k r and an as-sociated relative displacement δr ∼ v r δt × | ω + δt | / k r .Here v r = (cid:126) k /m ∼ D v T ≈
50 mm/s is the thermal veloc-ity of our T ∼ µ K Rb sample. The validity of the P ( ω + ) ∼ /ω scaling analysis of the frequency distribu-tion of nearby pairs, calculated in Appendix A, requires apositional change δr , (cid:28) r , which sets an upper boundto | ω + | during the observation time δt . This bound isgiven by | ω + | (cid:28) min(Γ e (Γ e k v r δt ) − / , Γ e ( k v T δt ) − ).For example, for δt = T i = 30 ns, we find | ω + | (cid:28) × Γ e is required. It is also worth pointing out that P ( ω + ) for ω + beyond this upper bound mostly affect the dynamicsof O k ( t ) within | ω + | t ∼
1. For such a short initial time,we have O k ( t ) ≈ τ D = 1 /k v T [42], which is at the microsecond level inthis experiment, and is expected to affect negligibly thespin-wave decay [29].Finally, to verify that dipole-dipole interactions dur-ing repeated spin-wave measurements (Appendix D 1) donot modify the pair distribution (Eq. (6)), we numer-ically simulate the relative motion of the pairs duringthe repeated probe and control sequences. The simu-lation splits the steps of evaluating the hybrid model(Eq. (C6)), calculating the classical forces on atom j as f j = (cid:104)−∇ j H eff (cid:105) , and updating the external atomic mo-tion. Here a driving term (cid:126) / (cid:80) j Ω c ( r j , t ) | a j (cid:105)(cid:104) g j | + h.c. and dipolar interaction mediated by exchange of | g (cid:105) − | a (cid:105) excitations are added to H eff [38]. Within the numeri-cal model, we find no noticeable change of the pair dis-tribution at short distance from the initial random pairdistribution during typical measurement times. Appendix D: Experimental details
This Appendix provides additional details of the mea-surement procedure and data analysis in this work.
1. Measurement of superradiance I k s ( t ) The atomic sample is prepared every ∼ y − z plane,and a dimple trap with a focused 840 nm beam alongthe x direction (Fig. 9) [29]. The probe pulse to drivethe spin wave is chosen to have duration τ p = 5 ns,long enough to suppress the excitation of the off-resonant F = 2 − F (cid:48) = 1 , θ p ∼ . S + ( k s ) emission, detected by a multi-mode fibercoupled single photon counter, to be less than one dur-ing each measurement. By repeating the measurements N exp ∼ − times to histogram the time-dependentcounts, the redirected emission I k s ( t ) can be recon-structed, such as those in Figs. 3(a)(b)( N exp = 20000and N exp = 40000) and Figs. 4(a)(b) ( N exp = 60000and N exp = 90000) in the main text. Practically, with N e ∼ − rounds of the sample preparation cycles,we let each atomic sample to support up to N rep = 72spin-wave measurements so that N exp = η g × N e × N rep to enhance the measurement speed. Here the η g factor isthe fraction of the N rep measurements to be grouped to-gether based on specific atomic sample conditions such asOD and ρ . The grouping is thus based on the estimatedevolution of atomic density distribution ρ ( r ) during the N rep measurements, which is carefully characterized asdescribed in the next section. The period T rep = 690 nsbetween each measurement is long enough to allow effi-cient optical pumping, and to ensure independent N rep measurements.
2. Optical depth and atomic density measurements
To investigate the dependence of the spin-wave de-cay dynamics on the optical depth OD and density ρ , it is crucial to accurately estimate the atomic den-sity distribution ρ ( r ) for all the N exp measurements de-scribed above, over hours of repeated taking of data,and for various combinations of spin wave generation,control, and measurement sequences (Fig. 2e). We ad-dress this task using a combination of sample shapepre-characterization and inline atom number monitor-ing measurements (Fig. 9a) [38]. Taking the sym-metry of the optical trapping potential, the methodassumes a Gaussian density distribution ρ ( r , t j ) ∼ Nσ x σ y σ z e − x σ x − y σ y − z σ z . Here N ( t j ) , σ x ( t j ) , σ y ( t j ) , σ z ( t j )are to be determined at each spin-wave measurementtime t j = t + ( j − × T rep . The time t (cid:46) µ s isthe interval between the dipole trap release and the firstspin-wave measurement.First, we use in situ and time-of-flight (TOF) absorp-tion images to pre-characterize the atomic density dis-tribution ρ ( r ) for the first instance of the j = 1 , ..., N rep spin-wave measurements. Specifically, we time the imag-ing pulse at t probe = t + t tof , typically with t tof from 0up to 500 µ s, for absorption imaging along the x direc-tion with 20 µ s exposure time (Fig 9(b)). By fitting theabsorption images with Gaussian distributions, the TOFimage data allow us to extract temperature of the gas T ,y and T ,z and to estimate the absolute width σ y and σ z ofthe in situ sample. We further take an auxiliary absorp-tion image along z to obtain the ratio σ x /σ y (Fig. 9(a)),which is kept close to unity by maintaining a relativelyweak dimple trap confinement. To minimize image aber- 𝜌𝜌 [ c m − ] (b) 𝑡𝑡 [ 𝜇𝜇 s] 𝑥𝑥𝑧𝑧 AuxiliaryBeam z CCDAuxiliaryCCD (a)
Imaging Beam x m m m m FIG. 9. Absorption imaging characterization of atomic sam-ple during the N rep spin wave measurements. (a): The ab-sorption imaging setup. The main imaging setup along the x direction has an aberration free numerical aperture NA ≈ . ∼ µ m spatial resolution. Example absorption images oftype “A” and “B” samples immediately after the dipole traprelease are given. The auxiliary imaging setup along z withNA ≈ . σ x /σ y by observing the far field diffraction patterns. (b) Evolutionof estimated ρ ( t ) (solid lines) for atomic sample undergoingfree flight (blue) and subjected to repeated optical controland measurements (red). The top array of images are for freeTOF atomic sample with t tof = 0 , µ s, with 20 µ s expo-sure time. For the 2nd image, the estimated peak density at60 µ s is marked with a blue circle. The bottom array of imagesare taken immediately after 0 , , , , , , , , , , µ s exposure time. Theestimated peak densities at the beginning of imaging exposuresare marked with red circles. All the densities are estimatedrelative to the first data point denoted by the solid circle,which is measured without spin-wave control. ration errors, we use out-of-focus diffractive images toestimate the ratio. Finally, by assuming T ,x = T ,y ,we can estimate the parameters σ x,y,z ( t ), as well as theassociated OD(ˆ k s ) and ρ for the first spin-wave mea-surement.Next, to retrieve σ x,y,z ( t j ) with 1 < j ≤ N rep , we re-peat the atomic sample pre-characterization for selected t j > t . Specifically, we choose to measure σ x,y,z ( t j ) atthe j = 6 , , , , , , , , ,
72 instances. We6then fit these σ ( t j )-data with σ ( t ) = σ ( t ) + k B T m ( t − t ) + k B T K m ( t − t ) , with T and K as parameters, forall the x , y and z directions. The last term in the fitfunction accounts for heating by the optical control andrepumping pulses. The fit helps to estimate the time-dependent density distribution parameters σ x,y,z ( t j ) forall t j of interest. The selective absorption image mea-surements also verify that the atom number N is ap-proximately conserved during the N rep = 72 measure-ments over the ∼ µ s measurement time. The timedependent ρ ( t j ) (Fig. 9(b)) and OD( t j ) can then be es-timated.Finally, to account for shot-to-shot fluctuations ofatom number N during the hours long data-taking pro-cess, we record absorption images for each sample afterthe spin-wave measurements, and estimate the fluctua-tion and drifts with the absorption signal magnitudes.
3. Analysis of the I k s ( t ) signals High quality signals I k s ( t ) with N exp ∼ arecollected to study superradiant dynamics as those inFigs. 3(a)(b). The measurements are compared withthe CDM model detailed in Appendix C 3. The simu-lation inputs are simply the ρ ( r ) parameters from theabsorption imaging analysis (Appendix D 2), with whichwe obtain excellent agreement between theory and ex-periments. To retrieve the initial decay rates for the su-perradiant emission ˜Γ k s , we fit the I k s ( t ) curves between0 < t <
10 ns with an exponential. The fit uncertaintiesare reflected in the error bars of the ˜Γ k s vs OD plots.The measurements of the dephasing rate γ (cid:48) as in Fig. 5require careful selection of the interrogation times T i , asspecified in Appendix D 4, so as to avoid a hyperfine in-terference effect due to the imperfect spin-wave controls.The much larger number of different experiments pro-hibit us from assigning large N exp to each one. We there-fore restrict N exp to be at the 10 level. To retrieve the I k s ( T i ) amplitudes for samples with specific OD(ˆ k s ) and ρ , we integrate I k s ( T i + t ) for 0 < t <
50 ns and subtractthe background (by estimating photon counts at large t )to obtain the recalled superradiant photon number n ( T i ).We then fit n ( T i ) vs T i to obtain Γ k (cid:48) s at specific ρ . Weeffectively account for photon shot noise, by numericallygenerating a Poissonian distribution around n ( T i ) as de-rived data sets, and repeat the fits many times. The fituncertainties are reflected in the error bars of the Γ k (cid:48) s vs ρ plots.
4. Management of imperfect spin-wave control
The use of nanosecond pulses on the D D 𝑡 [ns] 𝑡 [ns] 𝑡 [ns] C o u n t s 𝑇 i [ns] R e c a ll e d 𝐼 𝐤 [ a r b . u n i t s ] 𝑇 i [ns] 𝑇 i [ns] FIG. 10. The sampling strategy of interrogation times T i toobtain the spin wave decay rate Γ k (cid:48) s from measurements of thesuperradiant signal I k s ( T i ), in presence of an oscillating re-call efficiency due to a hyperfine interference effect [38]. Top:example data sets of I k s ( T i ) for the measurements of Γ k (cid:48) s inFig. 5. Bottom: simulated oscillatory recalled I k s amplitudesdue to imperfect control associated with 5 P / hyperfine split-ting. The red and blue markers set the T i for the correspond-ing red and blue I k s ( T i ) measurement curves on the top. The control is not perfect, and the imperfections havebeen systematically studied in Ref. [38]. The spin-waveshifts k → k ± k c in this work share the same efficiencyof ∼
75% as those in Ref. [29, 38].Starting from the S + ( k p ) excitation, the multiple im-perfect control pulses on the D1 transition generallylead to multiple spin waves in the atomic ensemble with k = k p ± n k c at integer n . Some of the spin-wave statesmay be reached from different control pathways. Theexample of relevance here is that leading to the recallefficiency oscillation as a function of interrogation time T i at a frequency of ∆ D , hfse / π = 814 . / T i (marked by thered and blue markers in the bottom plots) within each N rep run to extract the decay rate. The first set is for T i = 4 πm / ∆ D , hfse + t off with m being integers, andthe second set is for T i = 4 πm / ∆ D , hfse + t off with m being half integers. We adjust t off to ensure the two setswith coherently enhanced and reduced recall efficiencies,respectively. We then take the mean value of the respec-tively fitted Γ k (cid:48) s as the final Γ k (cid:48) s value for Fig. 5.Other than the recall interference effect, the existenceof multiple spin waves in the atomic ensemble is harmlessto the measurement of the spin wave under control. Thisis because the dipole spin waves are all within the linearoptics regime in this work and their evolutions are in-dependent. However, over hours long experimental mea-surement time, drifts and fluctuations of the imperfect D T i measurements to be withinadjacent j = 1 , ..., N rep spin-wave measurements whichare separated by less than 1 µ s intervals. The rapid pa-7rameter scan ensures nearly identical interaction betweenlight and the atomic sample for consistent retrieval of therelative signal magnitudes.Finally, we note that part of the imperfect D P / hy-perfine level during the multiple | g (cid:105) → | a (cid:105) → | g (cid:105) opticalcontrol cycles [38]. During the interrogation time T i , partof the atoms trapped in the 5 P / states gradually decayinto the 5 S / , F = 2 ground states through D P / atomsdecaying to the ground states increases with T i to morestrongly attenuate I k s ( T i ), this effect could have madethe mismatched spin-wave lifetime appear shorter in ourmeasurements. We estimate the re-absorption effect, bynumerically solving Eq. (C6) and in each time step prob-abilistically adding ground-state atoms according to the ρ ( r ) distribution. The time-dependent “flux” of addedatoms obeys N add = 58 ∆ N (1 − e − Γ D1 t ). The 5 / N (cid:46) . N is thetrapped 5 P / population immediately after the shift-outoperation, estimated by a full simulation of the multi-ple pulse control dynamics [38]. The estimation showsthat the re-absorption effect does lead to increased mea-surement values of the dephasing rate γ (cid:48) , particularlyfor the type “B” sample with large OD. For the type“A” sample and for the measurements shown in Fig. 5,this systematic should have introduced less than 20% er-rors, well within our overall experimental uncertainty. Infuture work, this time-dependent photon re-absorptioneffect may be suppressed by better optical pumping toreduce the population trapping [38], and by controllingthe spin waves using an auxiliary | e (cid:105) − | a (cid:105) transition notto share the same ground state with the weakly exciteddipole spin wave. [1] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Quantuminterface between light and atomic ensembles , Rev. Mod.Phys. , 1041 (2010).[2] P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, andJ. Laurat, Highly-efficient quantum memory for polar-ization qubits in a spatially-multiplexed cold atomic en-semble , Nat. Commun. , 363 (2018).[3] Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du,H. Yan, and S. L. Zhu, Efficient quantum memory forsingle-photon polarization qubits , Nat. Photonics , 346(2019).[4] H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, andS. Hofferberth, Single-photon transistor mediated byinterstate rydberg interactions , Phys. Rev. Lett. ,053601 (2014).[5] D. Tiarks, S. Baur, K. Schneider, S. D¨urr, and G. Rempe,
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