Abstract
We investigate collective emission from coherently driven ultracold ^{88}Sr atoms. We perform two sets of experiments using a strong and weak transition that are insensitive and sensitive, respectively, to atomic motion at 1 μK. We observe highly directional forward emission with a peak intensity that is enhanced, for the strong transition, by >10^{3} compared with that in the transverse direction. This is accompanied by substantial broadening of spectral lines. For the weak transition, the forward enhancement is substantially reduced due to motion. Meanwhile, a densitydependent frequency shift of the weak transition (∼10% of the natural linewidth) is observed. In contrast, this shift is suppressed to <1% of the natural linewidth for the strong transition. Along the transverse direction, we observe strong polarization dependences of the fluorescence intensity and line broadening for both transitions. The measurements are reproduced with a theoretical model treating the atoms as coherent, interacting radiating dipoles.
Introduction
Understanding interactions between light and matter in a dense atomic medium is a longstanding problem in physical science^{1,2} since the seminal work of Dicke^{3}. In addition to their fundamental importance in optical physics, such interactions play a central role in enabling a range of new quantum technologies including optical lattice atomic clocks^{4} and quantum networks^{5}.
The key ingredient in a dense sample is dipole–dipole interactions that arise from the exchange of virtual photons with dispersive and radiative contributions, and their relative magnitude varies between the nearfield and farfield regimes. The dispersive (real) part is responsible for collective level shifts and the radiative (imaginary) part gives rise to line broadening and collective superradiant emission^{6,7,8}. Intense theoretical efforts have been undertaken over many years, to treat the complex interplay between the dispersive and radiative dynamics^{9,10,11,12,13,14,15,16,17,18}. However, experimental demonstrations that provide a complete picture to clarify these interactions have been elusive.
Collective level shifts and line broadening arising from the real and imaginary parts of dipole–dipole interactions have recently been observed in both atomic^{19,20,21,22,23} and condensed matter^{24} systems. The modification of radiative decay dynamics at low excitation levels has also been observed using short probe pulses^{25,26,27,28}, and interaction effects were manifested in coherent backscattering^{29,30}. Although simple models of incoherent radiation transport have often been used to describe light propagation through opaque media^{31,32} and radiation trapping in laser cooling of dense atomic samples^{33}, coherent effects arising from atom–atom interactions, which are necessary to capture correlated manybody quantum behaviour induced by dipolar exchange, are beginning to play a central role. For example, the dipole–dipole interaction is responsible for the observed dipolar blockade and collective excitations in Rydberg atoms^{34,35,36,37,38,39,40,41}; it may also place a limit to the accuracy of an optical lattice clock and will require nontrivial lattice geometries to overcome the resulting frequency shift^{42}. Previous theoretical efforts have already shown that physical conditions such as finite sample size, sample geometry and the simultaneous presence of dispersive and radiative parts can play crucial roles in atomic emission^{10,11,12,13,43,44,45}.
In this work we use millions of Sr atoms in optically thick ensembles, taking advantage of the unique level structure of Sr to address motional effects, to study these radiative and dispersive parts simultaneously. We demonstrate that a single, selfconsistent, microscopic theory model can provide a unifying picture for the majority of our observations. These understandings can help underpin emerging applications based on manybody quantum science, such as latticebased optical atomic clocks^{4,46,47}, quantum nonlinear optics^{39}, quantum simulations^{48} and atomic ensemblebased quantum memories^{49}.
Results
Experimental setup
Bosonic alkalineearth atoms with zero nuclear spin have simple atomic structure compared with the more complex hyperfine structure present in typical alkali metal atoms that complicates the modelling and interpretation of the experimental observations. For example, ^{88}Sr atoms have both a strong ^{1}S_{0}−^{1}P_{1} blue transition (λ=461 nm) and a spinforbidden weak ^{1}S_{0}−^{3}P_{1} red transition (λ=689 nm), with a strict fourlevel geometry (Fig. 1a). When the atoms are cooled to a temperature of ∼1 μK, Doppler broadening at 461 nm is ∼55 kHz, which is almost three orders of magnitude smaller than the blue transition natural linewidth, Γ=32 MHz. To an excellent approximation, atomic motion is negligible for atomic coherence prepared by the 461nm light. To the contrary, the red transition with a natural linewidth Γ=7.5 kHz is strongly affected by atomic motion. By comparing the behaviours of the same atomic ensemble observed at these two different wavelengths (Fig. 1b), we can thus collect clear signatures of motional effects on coherent scattering and dipolar coupling^{50,51}.
We use the experimental scheme shown in Fig. 1a, to perform a comprehensive set of measurements of fluorescence intensity emitted by a dense sample of ^{88}Sr atoms. The sample is released from the trap and then illuminated with a weak probe laser. We vary the atomic density, cloud geometry, observation direction and polarization state of the laser field, and we report the system characteristics using three key parameters as follows: the peak scattered intensity, the linewidth broadening and the line centre shift. For example, along the forward and transverse directions we observe different values of intensity and linewidth broadening, as well as their dependence on light polarization (see Fig. 1c). We also observe motional effects on the red transition in contrast to the same measurements on the blue transition.
In the experiment, up to 20 million ^{88}Sr atoms are cooled to ∼1 μK in a twostage magnetooptical trap, the first based on the blue transition and the second on the red transition. The atomic cloud is then released from the magnetooptical trap and allowed to expand for a variable time of flight (TOF), which allows us to control its optical depth and density. They are subsequently illuminated for 50(100) μs with a largesize probe beam resonant with the blue (red) transition (Fig. 2a). The resulting scattered light is measured with two detectors far away from the cloud (see Fig. 1a). One detector is along the forward direction (detector D_{F}) and the other along the transverse direction (detector D_{T}, offset by ∼10°). For a short TOF, the atomic cloud is anisotropic and has an approximately Gaussian distribution with an aspect ratio of R_{x}:R_{y}:R_{z}=2:2:1, where R_{{x,y,z}} are the rootmeansquared radii. We define OD as the on resonance optical depth of the cloud, , where R_{⊥} depends on the direction of observation with R_{⊥,T}=R_{x}=R_{y} and R_{⊥,F}=(R_{z}R_{y})^{1/2} for the transverse and forward directions respectively, N is the atom number and k is the laser wavevector for the atomic transition (see Supplementary Note 1).
Forward observations
The coherent effect manifests itself most clearly in the forward direction (Fig. 2). To separate the forward fluorescence from the probe beam, we focus the probe with a lens (L_{1}) after it has passed through the atomic cloud and then block it with a beam stopping blade, which can be translated perpendicular to the probe beam (Fig. 2a inset). The same lens (L_{1}) also collimates the atomic fluorescence so that it can be imaged onto D_{F}. The position of the beam stopper can be used to vary the angular range of collected fluorescence, characterized by the angle (θ) between and the edge of the beam stopper (see Methods). Using the maximum atom number available in the experiment, the measured intensity I_{x,0}(θ) is normalized to that collected at θ_{max}=7.5 mRad. Both the blue (square) and red (triangle) transition results are displayed in Fig. 2a. For the blue transition, we observe a 1,000fold enhancement of the normalized intensity for θ<0.5 mRad. The enhancement is also present for the red transition, but it is reduced by nearly two orders of magnitude at small θ due to the motional effect. On the other hand, the wider angular area of enhancement is attributed to the longer wavelength of the red transition. The forward intensity strongly depends on the atom number. In Fig. 2b, we present measurements of the forward intensity I_{x} versus the transverse intensity I_{z} at a fixed θ=2 mRad for different atom numbers. The intensities are normalized to those obtained at the peak atom number as used in Fig. 2a. To the firstorder approximation, the transverse fluorescence intensity scales linearly with the atom number. Hence, the forward intensity of both the blue and red transitions scales approximately with the atom number squared.
In the forward direction, we have also investigated the linewidth broadening of the blue transition as a function of the atomic OD. By scanning the probe frequency across resonance, we extract the fluorescence linewidth, which is found to be determined primarily by the OD of the cloud (open squares in Fig. 2c). For the range of 0<OD<20, the lineshape is Lorentzian (see insets); however, the observed lineshape starts to flatten at the centre for OD>20. We have also varied the atom number by a factor of four, and to an excellent approximation the linewidth data are observed to collapse to the same curve when plotted as a function of OD (open triangles).
Transverse observations
For independent emitters, the forward fluorescence should have no dependence on the probe beam polarization; however, the transverse fluorescence (along ) should be highly sensitive to the probe polarization and it is even classically forbidden if the probe is polarized. However, multiple scattering processes with dipolar interactions can completely modify this picture by redistributing the atomic population in the three excited magnetic states and thus scrambling the polarization of the emitted fluorescence. Consequently, even for a polarized probe there should be a finite emission along (see Fig. 1c), with an intensity that increases with increasing OD. Our experimental investigation of the fluorescence properties along the transverse direction is summarized in Fig. 3. Under the same OD along , the polarized probe beam (square) gives rise to a much more broadened lineshape for the blue transition than the polarized probe beam does (triangle), as shown in Fig. 3a. Meanwhile, the peak intensity ratio of I_{ypol}/I_{zpol} decreases significantly with an increasing OD, indicating the rapidly rising fluorescence with a polarized probe when OD increases (Fig. 3b). For the red transition, the existence of Doppler broadening requires the lineshape data to be fitted to a Voigt profile. With the Doppler linewidth Δ_{D} fixed from the thermal velocity measured in free expansion, the Voigt profile determines the line centre and the Lorentzian linewidth with the Gaussian linewidth determined by the temperature. Figure 3c displays the Lorentzian linewidth obtained with a polarized red probe showing a strong increase of the linewidth with OD.
Spectral broadening and shift
To a good approximation, the dependence of the linewidth on OD along the forward and transverse directions (for the classically allowed polarization in the single scattering limit) is similar. However, owing to the anisotropic aspect ratio of the cloud, for the same TOF, the OD is lower along than along . This is responsible for the smaller broadenings measured along than along . The classically forbidden polarization direction, on the other hand, exhibits a different scaling with OD, which is understandable given that the emission in this case comes only from multiple scattering events with dipolar interactions. The transverse linewidth broadening for the red transition is similar to that of the blue, and it does not depend sensitively on motional effects. This behaviour is in stark contrast to another important observation: the shift of the transition centre frequency. Figure 4 contrasts the linecentre frequency shift observed for ^{1}S_{0}−^{1}P_{1} (square) and ^{1}S_{0}−^{3}P_{1} (triangle, with original data reported in ref. 52 and see Supplementary Fig. 1). The blue transition frequency shift is consistent with zero at the level of 0.004Γ using an atomic density of 10^{12} cm^{−3}. However, the measured density shift for the red transition (normalized to the transition linewidth) is more than one order of magnitude larger. This densityrelated frequency shift significantly exceeds the predicted value based on general Smatrix calculations of swave collisions^{52} (2.18 × 10^{−10} Hz cm^{3} if the unitary limit is used).
Theory model
Before we turn to a microscopic model to obtain a full and consistent understanding of all these related experimental observations, we note that semiclassical models^{53} treating the atomic cloud as a continuous medium of an appropriate refractive index can give an intuitive explanation of the linewidth broadening in the forward direction. Classically, an incoming electric field is attenuated as it propagates through the medium according to the Beer–Lambert law and the forward fluorescence intensity is determined by the same mechanism. This simple semiclassical model recovers the linear dependence of the forward width for small OD and predicts a nonlinear dependence of the linewidth for large OD and a flattening of the line centre. However, we find that this semiclassical approach cannot provide explanations for most aspects of the experimental observations.
The full microscopic model builds on a set of coherently coupled dipoles. Here, each fourlevel atom is treated as a discrete radiating dipole located at a frozen position, but coupled with retarded dipole radiation, and it is driven with a weak incident laser beam. The atomic ensemble follows the Gaussian distribution observed in the experiment with the appropriate aspect ratio. By solving the master equation in the steady state, we find that the coherence, , of atom j, located at r_{j} is modified by other atoms as^{18,54,55,56,57,58,59,60}:
Here, , corresponds to the three excited states of ^{1}P_{1} or ^{3}P_{1}, with α∈{x, y, z} representing the Cartesian states. In addition, is the reduced density matrix of the atoms and δ_{γ,γ′} is the Kronecker Delta. The driving laser’s linear polarization state ξ is along or , with wavevector k along , Rabi frequency Ω^{ξ} and detuned by Δ^{α} from the transition. The function G_{α,α′}(r) accounts for the retarded pairwise dipolar interactions and is given by^{18,48,56} . The fluorescence intensity , detected at position r_{s}, can be determined^{17,18} as a function of ,
with μ is the atomic transition dipole moment and .
Discussion
To understand the forward enhancement we first consider noninteracting atoms under the zeroth order approximation. The atomic coherence is driven only by the probe field that imprints its phase and polarization onto the atoms: , where Δ=Δ^{α}. The corresponding intensity, has a Lorentzian profile. It also exhibits an N^{2} scaling and an enhanced forward emission lobe, with an angular width given by the ratio between the transition wavelength and the transverse size of the sample . The forward lobe reflects the constructive interference of the coherently emitted radiation stimulated by the laser. Outside the coherent lobe the constructive interference is quickly reduced due to the random position of atoms^{28,59,61}. The longer wavelength of the red transition corresponds to a wider angular width of the forward lobe for the red fluorescence.
Simple considerations can also give rise to a qualitative understanding of atomic motionrelated effects on forward enhancement. Again for the red transition, the Doppler effect introduces random phases accumulated by . Here, v is the thermal velocity. The dephasing reduces coherent photon emission and gives rise to a net suppression of the forward emission intensity. The suppression becomes stronger with Δ_{D}/Γ, with the Doppler width. Such a suppression is clearly observed for the red transition.
To address the linewidth broadening we now consider atoms coupled by dipolar interactions, which tend to emit collectively in an optically dense cloud. The collective emission manifests itself with a broader fluorescence linewidth. Moving to the firstorder approximation, we note that the atomic coherence acquires contributions not only from the probe beam but also from the surrounding atoms, with . Here, and For a relatively dilute cloud with average interparticle distance , the farfield interactions dominate; thus, higherorder terms beyond 1/r can be neglected. Dipolar interactions modify the fluorescence lineshape, with consequences of both a frequency shift that depends on the cloud peak density n_{0} and a line broadening that is proportional to OD: and , with and . Thus, the firstorder approximation provides an intuitive picture about the role of dipolar effects on the lineshape.
However, in a cloud with an increasingly large OD, dipolar interactions are stronger and multiple scattering processes become relevant. The firstorder perturbative analysis then breaks down^{62,63,64}. The full solution of equation (1) based on the coherent coupled dipole model becomes necessary to account for multiple scattering processes (see Methods). The first signatures arise from the forward fluorescence intensity, where its naive N^{2} scaling is reduced with an increasing atom number as a consequence of multiple scattering processes. This effect is observed in both red and blue calculations, and it is expected to be more pronounced on the red transition due to its longer wavelength. However, atomic motion leads to a lower effective OD, which tends to suppress multiple scattering processes and thus helps to partially recover the collective enhancement. The solid lines in Fig. 2a,b represent such quantitative theory calculations for both transitions, which agree with the experiment.
Meanwhile, for the linewidth broadening observed in the forward direction, it becomes evident that the scaling of the linewidth versus OD turns nonlinear at large values of OD. The experimental data falls within the shaded area in Fig. 2c, which represents the full solution with a 20% uncertainty in the experimental atom number. Multiple scattering processes are also key to the explanation of the measured fluorescence along the transverse direction, especially for the classically forbidden polarization . Indeed, for both intensity and linewidth broadening observed in the transverse direction, under either or probe polarization, the full model (shown as shaded areas in both Fig. 3a,b) reproduces well the experimental results on ^{1}S_{0}−^{1}P_{1}. Taking into account motional dephasing (see Supplementary Note 2), the transverse broadening for ^{1}S_{0}−^{3}P_{1} is also well reproduced as shown in Fig. 3c.
So far, we have shown the observed effects on linewidth and fluorescence intensity are uniquely determined by OD. However, following the arguments discussed above, the frequency shift arising from the dipolar coupling is expected to scale with atomic density, , which includes both the collective Lamb shift and the Lorentz–Lorenz shift^{50,65}. For our experimental density, this effect is ≲10^{−3}, which is consistent with the observed frequency shift for the blue transition (Fig. 4). (It is noteworthy that the role of multiple scattering processes is to further suppress this frequency shift mechanism^{50}.) In contrast, for the red transition the measured density shift (normalized to Γ) is significantly larger than what is predicted from the current treatment of interacting dipoles; it is also much bigger than the unitarity limit of swave scattering. Qualitatively, we expect that as the atoms move and approach each other, the longlived groundexcited state coherence in the red transition can be significantly modified by the collisional process and open higher partial wave channels. We can thus expect a larger collisional phase shift. This process can be further complicated by atomic recoil, light forces and Doppler dephasing^{66}.
We have shown that a coherent dipole model describes light scattering in a dense atomic medium with collective effects and multiple scatterings. The model captures the quantitative features of the experimental observations. Motional effects, as manifested in dephasing, can be captured in the model as well. Our results provide useful guides for further developments of optical atomic clocks and other applications involving dense atomic ensembles.
Methods
Coherent dipole model
Here we present the derivation of equation (1). The fundamental assumption is to treat the atoms as frozen during the interrogation. This is an excellent approximation if ħΓ is much faster than other energy scales in the problem. The latter condition is always satisfied in the case of the blue transition. For the J=0 to J=1 configuration exhibited by ^{88}Sr, we can label the J=0 ground state as and the excited J=1 states using a Cartesian basis , , . Here, the 0, ±1〉 states are the standard angular momentum ones. In the Cartesian basis, the vector transition operator for the j atom located at r_{j} can be written as . Here . On this basis, the master equation governing the evolution of the reduced density matrix of the N atom ensemble, , in the presence of an incident laser beam with linear polarization ξ can be written as^{18}:
where is the Rabi frequecy of the incident field, polarized along ξ and detuned by Δ^{α} from the atomic transition . The parameters and are the components of the elastic and inelastic dipolar interactions between a pair of atoms at position r_{j} and r_{m}, respectively, and are given by
where , y_{n}(x), j_{n}(x) are the spherical Bessel functions of the second and first kind, respectively. Here, also α, β=x, y or z represent Cartesian components. The symbol δ_{γ,γ′} is the Kronecker Delta. In the lowintensity limit, we can project the density matrix into a state space including the ground state and states with only one excitation^{57,58,59} such as . In this reduced space, the relevant equations of motion simplify to
where , and .
As we are interested in the situation of a weak probe limit, , we expand the density matrix in successive orders of Ω^{ξ}/Γ, , and keep the firstorder terms. At this level of approximation, ρ_{G,G}=1, ρ_{jα,mβ}=0 and only the optical coherences evolve in time accordingly to the following set of linear equations:
Here, G_{α,β}(r)=(f_{α,β}(r)+ig_{α,β}(r))/Γ. The steadystate solution can be obtained by setting and then solving the subsequent 3N linear equations.
Measure the enhancement of forward fluorescence
To measure the scattered light in the forward direction, we use the setup shown in the inset of Fig. 2a, to tightly focus and block the probe beam, while still collecting most of the atomic fluorescence on the CCD (chargecoupled device) camera. We focus the probe beam, after it interacts with the atoms, to a small spot with 15 μm rootmeansquared radius and block it using a beam stopping blade. We then translate the beam stopper perpendicular to the probe beam by a distance Δx from our reference point of x=0, which we define as the position of the beam stopper where we see the greatest fluorescence without saturating the CCD camera with the probe beam. As only the forward direction is particularly sensitive to positional changes, we convert this change in position to a change in angle simply using , where the first lens with a 15cm focal length collimates the fluorescence. In numerical calculations, the CCD camera is simulated as a ring area centred around the forward direction and the average intensity collected over the ring is determined. The external radius is set to be large enough to reach the angular region outside the interference cone and the inner angular radius θ_{sim}, simulating the blocking of the signal by the beam stopper, is varied accordingly to the experiment. To account for the difference between σ_{sim} and the experiment cloud size, θ_{sim} is rescaled so that we satisfy the experimental observation that at θ_{max} the enhancement factor drops to 1.
Additional information
How to cite this article: Bromley, S. L. et al. Collective atomic scattering and motional effects in a dense coherent medium. Nat. Commun. 7:11039 doi: 10.1038/ncomms11039 (2016).
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Acknowledgements
We are grateful to Paul Julienne, Chris Greene, John Cooper and Murray Holland for their important insights and stimulating discussions. This research is supported by NIST, NSF Physics Frontier Center at JILA, AFOSR, AFOSRMURI, ARO, DARPA QuASAR, NSF Center for Ultracold Atoms at HarvardMIT, ITAMP and ANR14CE260032.
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S.L.B., M.B., X.Z., T.B., T.L.N. and J.Y. contributed to the executions of the experiments. B.Z., J.S., R.K., S.F.Y., M.D.L. and A.M.R. contributed to the development of the theory model. All authors discussed the results, contributed to the data analysis and worked together on the manuscript.
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Supplementary Figure 1, Supplementary Notes 12 and Supplementary References. (PDF 212 kb)
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Bromley, S., Zhu, B., Bishof, M. et al. Collective atomic scattering and motional effects in a dense coherent medium. Nat Commun 7, 11039 (2016). https://doi.org/10.1038/ncomms11039
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DOI: https://doi.org/10.1038/ncomms11039
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