Abstract
Atoms interact with each other through the electromagnetic field, creating collective states that can radiate faster or slower than a single atom, i.e., super and subradiance. When the field is confined to one dimension it enables infiniterange atom–atom interactions. Here we present the first report of infiniterange interactions between macroscopically separated atomic dipoles mediated by an optical waveguide. We use cold ^{87}Rb atoms in the vicinity of a singlemode optical nanofiber (ONF) that coherently exchange evanescently coupled photons through the ONF mode. In particular, we observe superradiance of a few atoms separated by hundreds of resonant wavelengths. The same platform allows us to measure subradiance, a rarely observed effect, presenting a unique tool for quantum optics. This result constitutes a proof of principle for collective behavior of macroscopically delocalized atomic states, a crucial element for new proposals in quantum information and manybody physics.
Introduction
A new class of quantum technologies exploits the interfaces between propagating photons and cold atoms^{1,2,3,4,5,6,7,8,9,10}. Recent realizations using optical nanofibers (ONFs) platforms include optical isolators, switches, memories, and reflectors^{11}. These devices guide the electromagnetic field, a feature that could allow engineering and control a collective time evolution of macroscopically separated subsystems. States that evolve as a whole with dynamics different to that of the independent subsystems are called collective states. These states emerge from atoms interacting via a common mode of the electromagnetic field, and their generation and control can enable adittional tools for atomicbased technologies^{12,13,14,15,16,17,18} and the study of manybody physics^{19, 20}.
For an ensemble of N twolevel atoms, in the single excitation limit,
represents the αth collective state of the system, where γ _{ α } and Ω_{ α } are its collective decay and frequency shift, respectively, and \(\mathop {\sum}\nolimits_{j = 1}^N \left {c_{\alpha j}} \right^2e^{  \gamma _\alpha t}\) is the probability of having an excitation in the atoms. When γ _{ α } is larger (shorter) than the natural radiative decay time γ _{0}, the system is super (sub)radiant^{21, 22}. For free space coupling, collective states emerge for atom–atom separations smaller than a few wavelengths^{23}. By externally exciting the atoms, superradiant states are readily observed, but because subradiant states are decoupled from the electromagnetic vacuum field, they are challenging to produce^{24}.
The master equation that describes the dynamics of an ensemble of atomic dipoles, of density matrix ρ, coupled through the electromagnetic field is given by ref. ^{25}
The effective Hamiltonian H _{eff} of the dipolar interaction between atoms and the Lindblad super operator \({\cal L}\) in Eq. (2) modify two atomic properties: the resonance frequency and the spontaneous decay rate, respectively. They are given by
with σ _{ i } \(\left( {\sigma _i^\dagger } \right)\) being the atomic lowering (raising) operator for an excitation of the ith atom. Ω_{ ij } is the rate of photons exchanged between atoms and γ _{ ij } is the term responsible for collective radiative decays, where γ _{ ii } is the single atom decay rate. The decay of an excitation in such a system, that leads to a collective state as in Eq. (1), depends on the coupling amplitudes and relative phase between the atoms given by γ _{ ij }.
When atoms are far apart in free space, their interaction is mediated by a propagating field with an expanding wavefront, and a separation of few wavelengths is enough to make the interaction negligible. As atoms get closer together, Ω_{ ij } in Eq. (3) diverges, reducing the coherence of a system with more than two atoms. These constraints can be circumvented by using longer wavelengths with larger atomic dipole moments, such as Rydberg atoms^{26}, or longrange phonon modes, implemented with trapped ions^{27, 28}. However, these techniques are limited to subwavelength distances. When the field is confined to one dimension, it enables infiniterange interactions. This has been observed for atoms in an optical cavity^{29, 30}.
Waveguides offer an alternative by confining the mediating field, where the extent of the interactions is not limited by the cavity size and the field can propagate unaltered for a broad range of frequencies^{31, 32}, facilitating the coupling of atoms separated by many wavelengths (see Fig. 1). Dipole–dipole interactions, given by Ω_{ ij }, are finite for atoms along the waveguide, removing a practical limit for creating superradiant states of a large number of atoms. Superradiance of atoms around a waveguide has been observed^{7}, but its longrange interaction feature has not been proven or explored. Such effect has been implemented with superconducting waveguides and two artificial atoms one wavelength apart^{33}, but has not been realized for many atoms at multiwavelength distances in the optical regime.
We present the implementation of collective atomic states through infiniterange interactions via a onedimensional nanophotonic waveguide. We use a few atoms evanescently coupled to a singlemode ONF, observing super and subradiant radiative decays of a single excitation in the system, evidence of collective behavior. Atoms around the ONF interact at short and long distances (see Fig. 1a), the latter mediated by the ONFguided mode. The dipolar interaction that leads to a collective decay is separated into two contributions of the electromagnetic field: from modes radiating outside the ONF, \(\gamma _{{\mathrm{12}}}^{{\mathrm{(rad)}}}\), and from the guided mode, \(\gamma _{{\mathrm{12}}}^{{\mathrm{(1D)}}}\) ^{25} (see Fig. 1b, c). In particular, we observe subradiant decay rates of proximal atoms interacting through the radiated modes and superradiant decay rates of atoms interacting through the guided mode over distances of hundreds of resonant wavelength.
Results
Experimental setup
We overlap a cold atomic cloud of ^{87}Rb atoms from a magnetooptical trap (MOT) with a 240 nm radius ONF. This ONF is single mode at the D2 resonant wavelength of 780 nm. After the MOT is turned off, the atoms form a cold thermal gas around the ONF. They are prepared in the F = 1 ground level by an external free propagating beam. A repumper beam driving the F = 1 → F = 2 transition propagates through the nanofiber, leaving in the F = 2 ground stateonly atoms that interact with the ONFguided mode. By detuning the repumper below resonance, we address atoms near the nanofiber (whose levels have been shifted by van der Waals interactions) such that the atomic density distribution peaks at ~30 nm away from the surface. A weak free space probe pulse, propagating perpendicular to the fiber, excites atoms for 50 ns using the F = 2 → F′ = 3 transition. After the probe turns off (extinction ratio better than 1:2 × 10^{3} in one atomic natural lifetime), we collect photons spontaneously emitted into the ONF mode to measure the decay time using timecorrelated single photon counting.
Collective states can be tailored by positioning the atoms in a particular arrangement. This kind of control has been challenging to implement for atoms trapped close enough to the ONF (tens of nanometers) to ensure significant mode coupling. However, collective states are still observed when atoms from a MOT are free to go near the ONF. Their random positioning leads to probabilistic super or subradiant states on each experimental realization. Subradiant states have lifetimes much longer than most other processes, favoring their observation. Superradiance can be measured as an enhanced decay rate at short times. Both effects can provide quantitative experimental evidence of collective states.
Observation of super and subradiance
Figure 2 shows a typical signal of the atomic decay as measured through the ONF. Its time dependence can be described by two distinct exponential decays. The slow decay (green dashed line in Fig. 2a) corresponds to an average of subradiant decays due to pairs of atoms located within a wavelength, i.e., free space interaction (Fig. 1b). Infiniterange interactions also produce subradiant decay rates. However, these events are obscured by the dominant signal of slower decays produced from free space interactions. In our case γ _{1D} ≈ 0.13γ _{0}, so subradiance from infiniterange interactions is limited to γ _{0} − γ _{1D} ≈ 0.87γ _{0}. This is a factor of six faster than the observed subradiant rates (green dashed line in Fig. 2a). Subradiance of atoms interacting in free space has been observed in a very optically dense cloud of atoms^{24}, but we can observe it even for optical densities (OD) as small as 0.3. The fast decay rate (red dashed line in Fig. 2a) is larger than the natural decay rate, showing the presence of superradiant initial states.
A full description of the temporal evolution of the entire data sample requires numerical (Monte Carlo) methods, as the solid black line in Fig. 2 shows. We use the average number of atoms (N) as the only free parameter for this simulation, allowing for variations of the background up to one sigma. The twosigma deviation between simulation and data (see Fig. 2b from 7 to 15 τ _{0}) could come from otherwise a longer living subradiant state that gets prematurely destroyed because atoms fall onto the ONF, emitting the excitation into the guided mode. The initial state preparation—the polarization of the incoming pulse that produces the collective onephoton state—can favor super or subradiant states, as Fig. 2c shows. In general, the free space atom–atom coupling is larger for dipoles driven along the ONF (z in the direction set in Fig. 1b), favoring subradiance, and the ONFmediated coupling is larger for dipoles driven perpendicular to the ONF, favoring superradiance.
An important difference between sub and superradiant decay rates in ONF is that the latter increases as a function of N. We can vary N from one to six by changing the MOT density, and quantify it through the OD of the ONF mode. n _{eff}OD = Nγ _{1D}/γ _{0}, where n _{eff} is the mode effective refractive index, and in our case n _{eff} ≈ 1.15. We measure the transmission spectrum through the ONF to extract the OD. The decay rate increases with N, as shown by the blue circles in Fig. 3, indicating superradiance. The gray region represents the onesigma confidence bands of a linear fit to the data showing a linear dependence of the superradiant decay rate for increasing N. The theoretical model implemented for the fit shown in Fig. 2 (solid black line) also predicts a linear dependence on N of the decay rate γ at short times. The red dashed line in Fig. 3a shows this prediction, corroborating the theory with the experiment.
Evidence of infiniterange interactions
The average spacing between atoms is larger than a wavelength for most of the realizations, meaning that infiniterange interactions are always present. However, to provide an unambiguous proof of infiniterange interactions, we split the atomic cloud in two (see Fig. 3b). We see that two atomic clouds separated by more than 400 wavelengths present the same superradiant collective behavior as a function of the OD as a single atomic cloud. This shows that the relevant parameter is the total OD (or N) along the ONF mode, regardless the separation between atoms.
Discussion
Optically guided modes can be used to mediate atom–atom interactions, creating macroscopically delocalized collective atomic states. We use the superradiant behavior of distant atoms as evidence of infiniterange interaction, but other interesting collective quantum properties remain to be tested. The practical limits of infiniterange interactions are an open question, since in principle optical fibers can be easily connected and rerouted along several meters. An intriguing next step is the study of quantum systems beyond the Markov approximation, coupling atoms at distance greater than what light travels in an atomic lifetime. Moreover, by achieving fine control on the positioning of the interacting particles, and/or using the directional coupling produced by chiral atom–light interaction^{10}, one can engineer desired states tailored to address specific applications. The implementation of infiniterange interactions opens new possibilities for quantum technologies and manybody physics. Given the application of onedimensional waveguides in photonicbased quantum technologies, we envision infiniterange interactions as the natural next step toward interconnecting quantum systems on scales suitable for practical applications.
Methods
Experimental methods
A tapered single mode ONF, with waist of 240 ± 20 nm radius and 7 mm length, is inside an ultrahigh vacuum (UHV) chamber, where it overlaps with a cloud of cold ^{87}Rb atoms (less than half a millimeter width) created from a MOT. The MOT is loaded from a background gas produced by a ^{87}Rb dispenser. Acousto optic modulators (AOMs) control the amplitude and frequencies of the MOT beams. After the atomic cloud loading reaches steady state, the MOT beams are extinguished. A free space propagating depump beam, resonant with the F = 2 → F′ = 2 transition (150 μs duration) prepares all atoms in the cloud in the F = 1 ground state. A 0.4 nW fiberrepump beam, detuned below resonance by 15 MHz to the F = 1 → F′ = 2 transition, propagates through the ONF during the entire cycle. It pumps back to the F = 2 ground state only those atoms close enough to the ONF to interact with the guided mode. This detuning repumps only those atoms close enough to the ONF surface to experience an energy shift due to the van der Waals interaction with the dielectric body. This produces a narrow density distribution of atoms of 5 nm width centered around 30 nm away from the surface. We wait 300 μs until the AOMs reach maximum extinction. The atomic cloud free falls and expands around the ONF for 2.5 ms creating a cold thermal gas (~150 μK), where each atom interacts with the nanofiber mode for ~1.5 μs^{34}. The atomic density reduction due to the cloud expansion limits the probing time of the cycle. The atoms are excited by pulses of a weak probe beam incident perpendicularly to the nanofiber (see Fig. 1a) and linearly polarized along the ONF for the data set shown in Fig. 3. The pulses are resonant with the F = 2 → F′ = 3 transition of the D2 line and created with a doublepassed Pockels cell (Conoptics 350–160), with a pulse extinction ratio better than 1:2000 in one atomic natural lifetime that remains at least an order of magnitude below the atomic decay signal for more than 20 lifetimes. The on–off stage of the light pulses is controlled with an electronic pulse generator (Stanford Research Systems DG645). The probe power is kept low, i.e., saturation parameter s < 0.1, to ensure a single photon excitation while staying in the limit of low excitation and avoiding photon pileup effects. Only those atoms that interact with the ONFguided mode are in the F = 2 ground state and will be excited by the probe beam. During the probing time, we send a train of 50 ns probe pulses every 1 μs. The probe is a 7 mm 1/e ^{2} diameter collimated beam. After 2 ms of probing (~2000 pulses), the probe beam is turned off and the MOT beams are turned back on. During the probing time, the atomic density remains constant. We wait 20 ms after the MOT reloads and repeat the cycle. The average acquisition time for an experimental realization is around 5 h, giving a total of about 1 × 10^{9} probe pulses. The photons emitted into the nanofiber and those emitted into free space are independently collected with avalanche photodiodes (APDs, laser components COUNT250CFC, with less than 250 dark counts per second). The TTL pulses created from photons detected by APD are processed with a PC timestamp card (Becker and Hickl DPC230) and time stamped relative to a trigger signal coming from the pulse generator. We use timecorrelated single photon counting^{35} to extract the decay rate of a single excitation in the system, eliminating afterpulsing events from the record.
When atoms are around the nanofiber, they tend to adhere due to van der Waals forces. After a few seconds of having the ONF exposed to rubidium atoms it gets coated, suppressing light propagation. To prevent this, we use 500 μW of 750 nm bluedetuned light (Coherent Ti:Sapph 899) during the MOTon stage to create a repulsive potential that keeps the atoms away from the ONF surface. This is intense enough to heat the ONF and accelerate the atomic desorption from the surface. The bluedetuned beam is turned off at the same time as the MOT beams, so the probed atoms are free to get close to the nanofiber.
Photons from the probe beam can be scattered multiple times by the atoms producing a signal that looks like a long decay, an effect known as radiation trapping. This effect can obscure subradiant signals. However, the small ODs involved in the experiment allow us to neglect contributions from radiation trapping. We confirm this assumption by observing the same temporal evolution of the signal at constant OD for several detunings of the probe beam in a range of ±3 linewidths^{24}.
The atomic lifetime can also be altered by modification of the electromagnetic environment of the atoms in the presence of an ONF, i.e., the Purcell effect. However, this effect is characterized separately^{36} and well understood. More importantly, it does not depend on the number of atoms, in contrast with the superradiant behavior.
Further evidence of collective states can be found in the resonance spectrum of the system (see Eqs. (2) and (3)). The dispersive part of the interaction modifies the resonance frequencies of the system, due to avoiding crossing of otherwise degenerate levels. This effect is in principle visible in the transmission spectrum. In our particular case, the frequency splitting is a small percentage of the linewidth. Broadening mechanisms and other systematic errors prevent us from clearly observing such signal. However, a lineshape dependence on N can be inferred from the statistical analysis of the fit of the spectrum to a Lorentzian. This effect might enable the exploration of features of collective states in the spectral domain.
ONFs can provide chiral atom–light coupling^{10}. Even though this is a promising feature of the platform, it requires a particular positioning of the atoms and a preparation of their internal state. This first exploration of infiniterange interactions involves detecting only on one end of the ONF and azimuthally averaging the atomic position, preventing studies of chiral effects that we do not consider crucial to our measurements.
Theoretical model
We follow the work of Svidzinsky and Chang^{37} to implement the theoretical simulations of the experiment. Consider the Hamiltonian of N atoms interacting with an electromagnetic field in the rotatingwave approximation
where \(\hat \sigma _j\) is the lowering operator for atom j; \(\hat a_{\mathbf{k}}^\dagger\) is the photon creation operator in the mode kth; ω _{0} and ω are the frequencies of atomic resonance and kth mode of the field, respectively. This is a general expression for the Hamiltonian, which leads to the master equation in Eq. (2) after some approximations. The sum on j is done over the atoms and the sum on k goes over the electromagnetic field modes, guided into the nanofiber and radiated outside. These modes can be found in the work of Le Kien et al.^{25}. The sum over the guided modes is \(\mathop {\sum}\nolimits_\mu = \mathop {\sum}\nolimits_{f,p} {\int}_0^\infty \rm{d}\omega\), where f and p are the propagation direction and polarization in the circular basis (plus or minus) of the guided mode, respectively, and μ stands for modes with different parameters (ω, f, p). The sum over the radiated modes is \(\mathop {\sum}\nolimits_\nu = \mathop {\sum}\nolimits_{m,p} {\int}_0^\infty \rm{d}\omega {\int}_{  k}^k \rm{d}\beta\); where m is the mode order, k is the wavenumber, β is the projection of the wave vector along the fiber or propagation constant, and ν stands for modes with different parameters (ω, β, m, p). Then the total sum is \(\mathop {\sum}\nolimits_k = \mathop {\sum}\nolimits_\mu + \mathop {\sum}\nolimits_\nu\). The electromagnetic field modes and their relative coupling strength have been previously studied^{25}. The coupling frequencies G _{ kj } for the guided and radiated modes can be written as:
where β′ = dβ/dω, d _{ j } is the dipole moment of the jth atom, and e ^{(μ,ν)} are the electric field profile function (or spatial dependence of the amplitude) of the guided and radiated modes (μ and ν).
Atoms interact with each other mediated by the electromagnetic field. The interaction between the atomic dipoles is proportional to the product of the atom–light coupling frequencies of the form G _{ ki } G _{ kj }, where k labels the mediating field mode (the repetition of the letter implies summation if there is more than one mode) and i and j label the ith and jth atom. It is possible to identify two contributions from the coupling of atoms to the dynamics of the system, a dispersive and a dissipative one, as shown in Eq. (2). The dispersive part contributes to the unitary evolution of the system (see Eq. (3)), and it can be decomposed as \({\mathrm{\Omega }}_{ij} = {\mathrm{\Omega }}_{ij}^{({\mathrm{rad}})} + {\mathrm{\Omega }}_{ij}^{({\mathrm{1D}})}\), where \({\mathrm{\Omega }}_{ij}^{({\mathrm{rad}})}\) and \({\mathrm{\Omega }}_{ij}^{({\mathrm{1D}})}\) come from the interaction of the ith and jth atoms mediated by the radiated and guided modes, respectively. Ω_{ ij } is usually called the dipole–dipole coupling frequency. The dissipative part contributes to the decay of the system (see Eq. (4)), and it can be decomposed as \(\gamma _{ij} = \gamma _{ij}^{({\mathrm{rad}})} + \gamma _{ij}^{({\mathrm{1D}})}\), where \(\gamma _{ij}^{({\mathrm{rad}})}\) and \(\gamma _{ij}^{({\mathrm{1D}})}\) come from the interaction of the ith and jth atoms mediated by the radiated and guided modes, respectively. For simplicity, here we focus only on the case where atoms are regarded as twolevel systems prepared in an initial state with induced atomic dipoles aligned along the ONF (zaxis). This is a reasonable approximation for atoms weakly driven by an external probe polarized along z. In a realistic scenario, the light scattered by the fiber and by the multilevel internal structure of the atoms can mix the light polarization. The computation of such a system becomes cumbersome and only contributes to correction to the dominant effect. A description given by twolevel atoms aligned along the zaxis allows us to quantitatively capture the physical phenomena while keeping the mathematical description simple. For atoms placed in the position r _{ i } = (r _{ i }, ϕ _{ i }, z _{ i }) with reduced dipole moment d _{ i }, we obtain
where μ _{0} parametrizes the guided modes on resonance. The dispersive component of the interaction given by the radiated modes as \({\mathrm{\Omega }}_{ij}^{({\mathrm{rad}})}\) is a complicated expression and hard to solve even numerically. We follow the work of Le Kien et al.^{38} and use the free space value of \({\mathrm{\Omega }}_{ij}^{({\mathrm{rad}})}\) throughout the calculation as a reasonable approximation. γ _{ ii } = γ _{0} with γ _{0} the single atom natural decay rate. \(\gamma _{12}^{({\mathrm{rad}})}\) and \(\gamma _{12}^{({\mathrm{1D}})}\) are plotted in Fig. 1b, c, respectively, for an atom fixed at r _{1} = (240 + 30) nm, 0, 0) (240 nm being the ONF radius and 30 nm the distance of the atom to the surface). When atoms are too close to each other, the radiated terms \({\mathrm{\Omega }}_{ij}^{({\mathrm{rad}})}\) and \(\gamma _{ij}^{({\mathrm{rad}})}\) dominate over the guided ones (\({\mathrm{\Omega }}_{ij}^{({\mathrm{1D}})}\) and \(\gamma _{ij}^{({\mathrm{1D}})}\)), with \({\mathrm{\Omega }}_{ij}^{({\mathrm{rad}})}\) diverging and \(\gamma _{ij}^{({\mathrm{rad}})}\) approaching the total decay rate. With a low number of atoms randomly distributed along the ONF, the effects of shortrange interaction are small but still observable.
For simplicity, we are interested in the decay of only one excitation in a system of twolevel atoms, however, generalizations to multilevel atoms can be found in the literature^{39}. Such system is represented by the state
where k _{ μ ( ν ) } is the sum over the guided (radiated) modes, \(b_{\mathbf{k}}^{(g)}\) is the probability amplitude of all the atoms being in the ground state and one excitation in the kth mode of the field, and \(b_j^{(e)}\) is the probability amplitude of having zero excitation in the field and an excitation in the ith atom. Assuming that we start the cycle with the excitation in the atoms, i.e., \(b_{\mathbf{k}}^{(g)}(0) = 0\), we can write the Schrödinger equation in the Markov approximation for the coefficients \(b_i^{(e)}(t)\) in a matrix form as ref. ^{37}
where B(t) is a vector with entries given by the \(b_i^{(e)}(t)\), and Γ is a nonhermitian symmetric matrix with entries 2Γ_{ ij } = γ _{ ij } + iΩ_{ ij }, representing the couplings between the ith and jth atoms calculated from the optical nanofiber modes, radiated and guided. The eigenvalues η _{ α } of Eq. (12) give the possible decay rates of the system. These are the collective sates mentioned in Eq. (1). The eigenvectors form a basis \(\left\{ {\left {B_\alpha } \right\rangle } \right\}\) that allows us to write the state of the system as
where the coefficients c _{ α } are given by the initial state. In contrast with Eq. (1), here we have also included the states with one excitation in the field.
Following this approach, the manybody problem, of calculating the decay of one excitation distributed among N interacting atoms, becomes an eigenvalue problem in a Hilbert space of dimension N ^{2} instead of 2^{2N}. This speeds the calculations, allowing us to compute the decay rate of the system with Monte Carlo simulations for a large N in random positions.
The electromagnetic field operator for the guided modes is ref. ^{25}
The formal solution of the Heisenberg equation for \(\hat a_\mu (t)\) in the Markov and rotatingwave approximation is
The substitution of this expression into Eq. (14) gives the guided field operator as a function of the dipole operators.
Assuming that the guided modes are initially empty and that all the dipoles are oriented in the z direction and at the same distance from the ONF, the intensity of the guided field as a function of the atomic dipole operators is
where
considering \(\gamma _{{\mathrm{1D}}}(r) = \gamma _{ii}^{{\mathrm{(1D)}}}(r)\) from Eq. (8) and \(A_{{\mathrm{eff}}(z)}(r) = \left {n_{{\mathrm{eff}}}e_z^{\left( {\mu _0} \right)}(r)} \right^{  2}\) to be the effective mode area of the z component of the electric field^{25}. Equation (18) relates the total radiated power into the waveguide with the energy radiated per unit time, i.e., I(r)A _{eff(z)}(r) = ħω _{0} γ _{1D}(r), where I(r) is the intensity of the radiated field.
Equation (16) shows that the measured intensity corresponds to the one produced by N classical dipoles with different phases, different positions, and amplitudes given by the probability of being in the excited state \(b_j^{(e)}\) ^{40}.
Theoretical methods
We use Monte Carlo simulations, randomly positioning N atoms around the ONF. The position of each atom is given in cylindrical coordinates by r _{ i } = (r _{0}, ϕ _{ i }, z _{ i }), where r _{0} = (240 + 30) nm, ϕ _{ i } ∈ [0, 2π], and z _{ i } is obtained from a Gaussian distribution with a FWHM of 200 μm, determined by the atomic cloud size. The radial position of the atoms is fixed, determined by the experimental procedure of repumping the atoms close to the nanofiber surface. In our case, all the atoms are at a constant radial position of 30 nm away from the surface of an ONF of 240 nm radius, with γ _{1D}/γ _{0} ≈ 0.13. This is a good approximation given the narrow radial distribution of the atoms (~5 nm), as explained in the experimental methods.
The initial state will depend on the amplitude and phase of the excitation beam. We assume that the initial state corresponds to a superposition of all the atoms in the ground state except one with an induced atomic dipole. The initial phase between the atoms depends on their position; assuming an excitation pulse with a wave vector perpendicular to the fiber, each atom initial phase can be calculated from its coordinates. For each random realization, we solve Eq. (12) and calculate the intensity of the guided field, Eq. (16). We use these results to take the mean of the intensity of the guided field as a function of time. Typically, 100,000 realizations are required to converge to a level of precision higher than what it is visible in Figs. 2 and 3. If the mean of the intensity guided field is normalized, there is no dependence on the amplitude of the initial induced dipole in the weak excitation limit.
There is a correspondence between superradiance (subradiance) configurations and constructive (destructive) interference of the field emitted by the dipoles into the ONF (see Eq. (17)); meaning that superradiant configurations contributes more than subradiant configurations when taking the mean over all the realizations for an electric field detected through the ONF (Eq. (16)).
The theoretical model prediction for different dipole moment orientations relative to the ONF^{25} qualitatively agrees with the observed experimental behavior: The longterm subradiance disappears on our signaltobackgroundratio window when exciting with vertically polarized light (see of Fig. 2c). A sensitivity analysis to the ONF radius shows no significant changes in the predictions up to a ±10 nm variation.
Data availability
The data that support the findings of this study are available from the authors on reasonable request.
References
 1.
Thompson, J. D. et al. Coupling a single trapped atom to a nanoscale optical cavity. Science 340, 1202–1205 (2013).
 2.
Sayrin, C. et al. Nanophotonic optical isolator controlled by the internal state of cold atoms. Phys. Rev. X 5, 041036 (2015).
 3.
Tiecke, T. G. et al. Nanophotonic quantum phase switch with a single atom. Nature 508, 241–244 (2014).
 4.
Shomroni, I. et al. Alloptical routing of single photons by a oneatom switch controlled by a single photon. Science 345, 903–906 (2014).
 5.
Volz, J., Scheucher, M., Junge, C. & Rauschenbeutel, A. Nonlinear π phase shift for single fibreguided photons interacting with a single resonatorenhanced atom. Nat. Photonics 8, 965–970 (2014).
 6.
Hood, J. D. et al. Atomatom interactions around the band edge of a photonic crystal waveguide. Proc. Natl Acad. Sci. 113, 10507–10512 (2016).
 7.
Goban, A. et al. Superradiance for atoms trapped along a photonic crystal waveguide. Phys. Rev. Lett. 115, 063601 (2015).
 8.
Gouraud, B., Maxein, D., Nicolas, A., Morin, O. & Laurat, J. Demonstration of a memory for tightly guided light in an optical nanofiber. Phys. Rev. Lett. 114, 180503 (2015).
 9.
Sayrin, C., Clausen, C., Albrecht, B., Schneeweiss, P. & Rauschenbeutel, A. Storage of fiberguided light in a nanofibertrapped ensemble of cold atoms. Optica 2, 353–356 (2015).
 10.
Lodahl, P. et al. Chiral quantum optics. Nature 541, 473–480 (2017).
 11.
Solano, P. et al. in Advances In Atomic, Molecular, and Optical Physics Vol. 66 (eds Arimondo, E., Lin, C. C. & Yelin, S. F.) 439–505 (Academic Press, New York, 2017).
 12.
Kimble, H. J. The quantum internet. Nature 453, 1023–1030 (2008).
 13.
Scully, M. O. Single photon subradiance: quantum control of spontaneous emission and ultrafast readout. Phys. Rev. Lett. 115, 243602 (2015).
 14.
AsenjoGarcia, A., MorenoCardoner, M., Albrecht, A., Kimble, H. J. & Chang, D. E. Exponential improvement in photon storage fidelities using subradiance and selective radiance in atomic arrays. Phys. Rev. X 7, 031024 (2017).
 15.
Ruostekoski, J. & Javanainen, J. Emergence of correlated optics in onedimensional waveguides for classical and quantum atomic gases. Phys. Rev. Lett. 117, 143602 (2016).
 16.
Ruostekoski, J. & Javanainen, J. Arrays of strongly coupled atoms in a onedimensional waveguide. Phys. Rev. A 96, 033857 (2017).
 17.
Bettles, R. J., Gardiner, S. A. & Adams, C. S. Cooperative eigenmodes and scattering in onedimensional atomic arrays. Phys. Rev. A 94, 043844 (2016).
 18.
Le Kien, F. & Hakuta, K. Cooperative enhancement of channeling of emission from atoms into a nanofiber. Phys. Rev. A 77, 013801 (2008).
 19.
Hung, C.L., GonzálezTudela, A., Cirac, J. I. & Kimble, H. J. Quantum spin dynamics with pairwisetunable, longrange interactions. Proc. Natl Acad. Sci. 113, E4946–E4955 (2016).
 20.
Douglas, J. S. et al. Quantum manybody models with cold atoms coupled to photonic crystals. Nat. Photonics 9, 326–331 (2015).
 21.
Dicke, R. H. Coherence in spontaneous radiation processes. Phys. Rev. 93, 99 (1954).
 22.
Scully, M. O. & Svidzinsky, A. A. The super of superradiance. Science 325, 1510–1511 (2009).
 23.
DeVoe, R. G. & Brewer, R. G. Observation of superradiant and subradiant spontaneous emission of two trapped ions. Phys. Rev. Lett. 76, 2049 (1996).
 24.
Guerin, W., Araújo, M. O. & Kaiser, R. Subradiance in a large cloud of cold atoms. Phys. Rev. Lett. 116, 083601 (2016).
 25.
Le Kien, F., Gupta, S. D., Nayak, K. P. & Hakuta, K. Nanofibermediated radiative transfer between two distant atoms. Phys. Rev. A 72, 063815 (2005).
 26.
Bendkowsky, V. et al. Observation of ultralongrange Rydberg molecules. Nature 458, 1005–1008 (2009).
 27.
Richerme, P. et al. Nonlocal propagation of correlations in quantum systems with longrange interactions. Nature 511, 198–201 (2014).
 28.
Bohnet, J. G. et al. Quantum spin dynamics and entanglement generation with hundreds of trapped ions. Science 352, 1297–1301 (2016).
 29.
Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 (2010).
 30.
Bohnet, J. G. et al. A steadystate superradiant laser with less than one intracavity photon. Nature 484, 78–81 (2012).
 31.
Shahmoon, E., Grišins, P., Stimming, H. P., Mazets, I. & Kurizki, G. Highly nonlocal optical nonlinearities in atoms trapped near a waveguide. Optica 3, 725–733 (2016).
 32.
Chang, D. E., Jiang, L., Gorshkov, A. V. & Kimble, H. J. Cavity QED with atomic mirrors. New J. Phys. 14, 063003 (2012).
 33.
van Loo, A. F. et al. Photonmediated interactions between distant artificial atoms. Science 342, 1494–1496 (2013).
 34.
Grover, J. A., Solano, P., Orozco, L. A. & Rolston, S. L. Photoncorrelation measurements of atomiccloud temperature using an optical nanofiber. Phys. Rev. A 92, 013850 (2015).
 35.
O’Connor, D. & Phillips, D. TimeCorrelated Single Photon Counting (Academic Press, London, 1984).
 36.
Solano, P. et al. Alignmentdependent decay rate of an atomic dipole near an optical nanofiber. Preprint at http://arxiv.org/abs/1704.08741 (2017)
 37.
Svidzinsky, A. & Chang, J.T. Cooperative spontaneous emission as a manybody eigenvalue problem. Phys. Rev. A 77, 043833 (2008).
 38.
Le Kien, F. & Rauschenbeutel, A. Nanofibermediated chiral radiative coupling between two atoms. Phys. Rev. A 95, 023838 (2017).
 39.
Hebenstreit, M., Kraus, B., Ostermann, L. & Ritsch, H. Subradiance via entanglement in atoms with several independent decay channels. Phys. Rev. Lett. 118, 143602 (2017).
 40.
Araújo, M. O., Guerin, W. & Kaiser, R. Decay dynamics in the coupleddipole model. J. Modern Opt. http://dx.doi.org/10.1080/09500340.2017.1380856 (2017).
Acknowledgements
We are grateful to A. AsenjoGarcia, H. J. Carmichael, D.E. Chang, J. P. Clemens, M. FossFeig, M. Hafezi, B.D. Patterson, W.D. Phillips, and P.R. Rice for the useful discussions. We give special thanks to P. Zoller who besides discussing the topic of the paper helped us improve the manuscript. This research is supported by the National Science Foundation of the United States (NSF) (PHY1307416); NSF Physics Frontier Center at the Joint Quantum Institute (PHY1430094); the USDOC, NIST, Joint Quantum Institute (70NANB16H168); and the Office of the Secretary of Defense of the United States, Quantum Science and Engineering Program.
Author information
Affiliations
Contributions
P.S., F.K.F., L.A.O. and S.L.R. conceived the project. P.S. realized the measurements. P.B.B. and P.S. developed the theoretical model. All authors discussed the results, contributed to the data analysis, and worked together on the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Solano, P., BarberisBlostein, P., Fatemi, F.K. et al. Superradiance reveals infiniterange dipole interactions through a nanofiber. Nat Commun 8, 1857 (2017). https://doi.org/10.1038/s41467017019943
Received:
Accepted:
Published:
Further reading

Coherent amplification and inversion less lasing of surface plasmon polaritons in a negative index metamaterial with a resonant atomic medium
Scientific Reports (2021)

Waveguide bandgap engineering with an array of superconducting qubits
npj Quantum Materials (2021)

A subradiant optical mirror formed by a single structured atomic layer
Nature (2020)

Large array of Schrödinger cat states facilitated by an optical waveguide
Nature Communications (2020)

Measurement and simulation of atomic motion in nanoscale optical trapping potentials
Applied Physics B (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.