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Super-radiance reveals infinite-range dipole interactions through a nanofiber

Super-radiance reveals infinite-range dipole interactions through a nanofiber ARTICLE DOI: 10.1038/s41467-017-01994-3 OPEN Super-radiance reveals infinite-range dipole interactions through a nanofiber 1 1,2 3 1 1 P. Solano ,P. Barberis-Blostein , F.K. Fatemi , L.A. Orozco & S.L. Rolston 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 sub-radiance. When the field is confined to one dimension it enables infinite-range atom–atom interactions. Here we present the first report of infinite-range interactions between macroscopically separated atomic dipoles mediated by an optical waveguide. We use cold Rb atoms in the vicinity of a single-mode optical nanofiber (ONF) that coherently exchange evanescently coupled photons through the ONF mode. In particular, we observe super-radiance of a few atoms separated by hundreds of resonant wavelengths. The same platform allows us to measure sub-radiance, 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 many-body physics. 1 2 Joint Quantum Institute, Department of Physics and NIST, University of Maryland, College Park, MD 20742, USA. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México 04510 D.F., Mexico. Army Research Laboratory, Adelphi, MD 20783, USA. Correspondence and requests for materials should be addressed to P.S. (email: solano.pablo.a@gmail.com) NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications 1 | | | 1234567890 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 new class of quantum technologies exploits the interfaces the electromagnetic field is given by ref. 1–10 between propagating photons and cold atoms . Recent _ ½ ρ ¼iH ; ρ þL½ρ: ð2Þ eff Arealizations using optical nanofibers (ONFs) platforms include optical isolators, switches, memories, and reflectors . The effective Hamiltonian H of the dipolar interaction between eff These devices guide the electromagnetic field, a feature that could atoms and the Lindblad super operator L in Eq. (2) modify two allow engineering and control a collective time evolution of atomic properties: the resonance frequency and the spontaneous macroscopically separated subsystems. States that evolve as a decay rate, respectively. They are given by whole with dynamics different to that of the independent sub- systems are called collective states. These states emerge from H ¼  hΩ σ σ ; eff ij j i ð3Þ atoms interacting via a common mode of the electromagnetic i;j field, and their generation and control can enable adittional tools 12–18 for atomic-based technologies and the study of many-body 19, 20 y y y physics . L½ρ¼  hγ 2σ ρσ  σ σ ρ  ρσ σ ; ij j j j i i i ð4Þ For an ensemble of N two-level atoms, in the single excitation i;j limit, with σ σ being the atomic lowering (raising) operator for an ðÞ γ þiΩ t excitation of the i-th atom. Ω is the rate of photons exchanged 2 α ij ji Ψ ðtÞ / e c g g  e   g α αj 1 2 j N ð1Þ between atoms and γ is the term responsible for collective ij j¼1 radiative decays, where γ is the single atom decay rate. The decay ii of an excitation in such a system, that leads to a collective state as represents the α-th collective state of the system, where γ and Ω in Eq. (1), depends on the coupling amplitudes and relative phase α α are its collective decay and frequency shift, respectively, and between the atoms given by γ . ij γ t c e is the probability of having an excitation in the When atoms are far apart in free space, their interaction is αj j¼1 atoms. When γ is larger (shorter) than the natural radiative mediated by a propagating field with an expanding wavefront, 21, 22 decay time γ , the system is super- (sub-)radiant . For free and a separation of few wavelengths is enough to make the space coupling, collective states emerge for atom–atom separa- interaction negligible. As atoms get closer together, Ω in Eq. (3) ij tions smaller than a few wavelengths . By externally exciting the diverges, reducing the coherence of a system with more than two atoms, super-radiant states are readily observed, but because sub- atoms. These constraints can be circumvented by using longer radiant states are decoupled from the electromagnetic vacuum wavelengths with larger atomic dipole moments, such as Rydberg 24 26 field, they are challenging to produce . atoms , or long-range phonon modes, implemented with trap- 27, 28 The master equation that describes the dynamics of an ped ions . However, these techniques are limited to sub- ensemble of atomic dipoles, of density matrix ρ, coupled through wavelength distances. When the field is confined to one bc 1.0 0.5 (rad) (1D) /  / 0 1D 12 12 0.0 –0.5 –1.0 Fig. 1 Position-dependent atom–atom coupling along the optical nanofiber. a Schematic of an ONF as a platform for generating single photon collective atomic states, excited from the side by a weak probe of polarization V or H. When two atoms are close together, the dipolar interaction is mostly mediated by the modes of the electromagnetic field radiating outside the nanofiber. This is a limited-range interaction that decays inversely with distance. When the atoms are widely separated, the guided mode of an ideal ONF mediates the interaction for arbitrary distances. b, c Show the atom–atom interaction rate γ (see Eq. (4)) experienced by an atom around the fiber given another atom at the position denoted by the white cross (see “Methods” section for the details of the calculation). Its amplitude is shown for a longitudinal and a transversal cut (specified by dashed black lines). Both plots share the color scale, but in b the interaction rate is normalized by the single atom total decay rate γ and in c by the decay rate into the guided mode γ . Along the z-axis, the interaction 0 1D ðradÞ among atoms through free space radiation modes decreases as γ / sinðÞ kjj Δz =kΔz (with k being the wavenumber and Δz the separation between two ð1DÞ atoms). The infinite interaction through the ONF-guided mode changes as γ / cosðÞ β Δz cosðΔϕÞ (with β being the propagation constant of the 12 0 resonant-guided mode and Δϕ the angle difference in cylindrical coordinates). The wavelength λ sets the scale in b, c 2 NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications | | | NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 ARTICLE ðradÞ dimension, it enables infinite-range interactions. This has been from modes radiating outside the ONF, γ , and from the ð1DÞ 29, 30 25 observed for atoms in an optical cavity . guided mode, γ (see Fig. 1b, c). In particular, we observe Waveguides offer an alternative by confining the mediating sub-radiant decay rates of proximal atoms interacting through the field, where the extent of the interactions is not limited by the radiated modes and super-radiant decay rates of atoms interact- cavity size and the field can propagate unaltered for a broad range ing through the guided mode over distances of hundreds of 31, 32 of frequencies , facilitating the coupling of atoms separated by resonant wavelength. many wavelengths (see Fig. 1). Dipole–dipole interactions, given by Ω , are finite for atoms along the waveguide, removing a ij practical limit for creating super-radiant states of a large number Results of atoms. Super-radiance of atoms around a waveguide has been Experimental setup. We overlap a cold atomic cloud of Rb observed , but its long-range interaction feature has not been atoms from a magneto-optical trap (MOT) with a 240 nm radius proven or explored. Such effect has been implemented with ONF. This ONF is single mode at the D2 resonant wavelength of superconducting waveguides and two artificial atoms one wave- 780 nm. After the MOT is turned off, the atoms form a cold length apart , but has not been realized for many atoms at multi- thermal gas around the ONF. They are prepared in the F = 1 wavelength distances in the optical regime. ground level by an external free propagating beam. A repumper We present the implementation of collective atomic states beam driving the F = 1 → F = 2 transition propagates through the through infinite-range interactions via a one-dimensional nano- nanofiber, leaving in the F = 2 ground state-only atoms that photonic waveguide. We use a few atoms evanescently coupled to interact with the ONF-guided mode. By detuning the repumper a single-mode ONF, observing super- and sub-radiant radiative below resonance, we address atoms near the nanofiber (whose decays of a single excitation in the system, evidence of collective levels have been shifted by van der Waals interactions) such that behavior. Atoms around the ONF interact at short and long the atomic density distribution peaks at ~30 nm away from the distances (see Fig. 1a), the latter mediated by the ONF-guided surface. A weak free space probe pulse, propagating perpendicular mode. The dipolar interaction that leads to a collective decay is to the fiber, excites atoms for 50 ns using the F = 2 → F′ = 3 separated into two contributions of the electromagnetic field: transition. After the probe turns off (extinction ratio better than –1 –2 –1 –3 –2 –4 –3 0 5 10 15 20 25 30 t / –4 0 5 10 15 20 0 t / –2 –4 0 5 10 15 20 25 30 t / Fig. 2 Measured super- and sub-radiant decay of excited atoms near the optical nanofiber. a Normalized rate of photons detected through the ONF mode (blue circles in a logarithmic scale) as a function of time in units of natural lifetime (τ = 1/γ = 26.24 ns) with 5 ns bins. The signal is taken after a probe 0 0 beam polarized along the nanofiber turns off. In this realization OD = 0.66 ± 0.05. The individual statistical error bars are not plotted but they are taken into account for the normalized residuals in b. The number of counts at t = 0 exceeds 10 . We see two distinct slopes (red and green), at short and long times. The initial slope (red) deviates toward decay rates faster than γ , a signature of super-radiance. The second slope (green) comes from the natural post-selection of purely sub-radiant states. The red dashed (green dashed) line is the best fit to a pure exponential decay of the initial (final) decay. The decay rate of the fit at short times is 1.10 ± 0.02 γ , and 0.13 ± 0.01 γ for the fit at longer times, with one-sigma error. The one-sigma fractional systematic 0 0 errors are ±0.01. The full description of the measured temporal evolution of the system involves averaging over many different decay rates through Monte Carlo methods (explained in “Methods” section). The solid black line is a simulation of 7 atoms along the ONF, with reduced χ of 1.60. b The red circles, green circles, and black diamonds are the normalized residuals of the exponential fits to the initial decay, final decay, and the theoretical model. c Shows two different decay signals from an excitation driving the atoms with light polarized along (cyan rectangles) and perpendicular (blue triangles) to the ONF for 25 ns bins. When the driving field is polarized along the ONF, we observe super- and sub-radiance, and when it is polarized perpendicular to the ONF the super-radiance increases and the sub-radiance decreases. This feature is qualitatively captured by the theoretical model NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications 3 | | | Normalized Log normalized count rate residuals 10 Log normalized count rate 10 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 Average number of atoms 1.10 1.0 0.8 1.05 0.6 0.4 1.00 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 z (mm) 0.95 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Optical density Fig. 3 Super-radiant decay as a function of atom number including separated clouds. a Relationship of the decays as a function of average number of atoms (OD) along the optical nanofiber. The normalized fast decay rates are plotted as a function of the OD (lower abscissa) and N (upper abscissa) measured through the ONF-guided mode. The blue circles correspond to the signals from a single cloud of atoms. We split the atomic cloud in two (as shown in b). The dashed light and dotted dark green diamonds, and the solid red square correspond to the right, left, and the combination of both atomic clouds, respectively. The systematic errors (not shown) are estimated to be 1% for the decay rates and smaller than 20% for the atom number. The plotted error bars represent the statistical uncertainty of the fitting to an exponential decay. The gray region represents the one-sigma confidence band of a linear fitto the data. The red dashed line is the theoretical prediction, and the red shaded region represents a confidence interval set by a fractional error of 1%. The curve goes below γ/γ = 1 because the natural decay rate is modified given the geometry of the ONF and the alignment of the atomic dipoles (Purcell effect) . b Separated atom clouds show long-range interactions. The top of the figure shows in black and white a fluorescence image of a split MOT. The white dotted line represents the ONF location. The fluorescence signal of the split MOT along the nanofiber is plotted as a function of position. The dashed light (dotted dark) green dashed lines is the intensity distribution of the right (left) atomic cloud when the other one is blocked. The solid red line is the intensity distribution when both clouds are present. The separation between the center of both clouds is 318 ± 1 μm, given by standard error of the mean of a Gaussian fit. This distance is equivalent to 408 wavelengths 1:2 × 10 in one atomic natural lifetime), we collect photons A full description of the temporal evolution of the entire data spontaneously emitted into the ONF mode to measure the decay sample requires numerical (Monte Carlo) methods, as the solid time using time-correlated single photon counting. black line in Fig. 2 shows. We use the average number of atoms Collective states can be tailored by positioning the atoms in a (N) as the only free parameter for this simulation, allowing for particular arrangement. This kind of control has been challenging variations of the background up to one sigma. The two-sigma to implement for atoms trapped close enough to the ONF (tens of deviation between simulation and data (see Fig. 2b from 7 to 15 nanometers) to ensure significant mode coupling. However, τ ) could come from otherwise a longer living sub-radiant state collective states are still observed when atoms from a MOT are that gets prematurely destroyed because atoms fall onto the ONF, free to go near the ONF. Their random positioning leads to emitting the excitation into the guided mode. The initial state probabilistic super- or sub-radiant states on each experimental preparation—the polarization of the incoming pulse that realization. Sub-radiant states have lifetimes much longer than produces the collective one-photon state—can favor super- or most other processes, favoring their observation. Super-radiance sub-radiant states, as Fig. 2c shows. In general, the free space can be measured as an enhanced decay rate at short times. Both atom–atom coupling is larger for dipoles driven along the ONF (z effects can provide quantitative experimental evidence of in the direction set in Fig. 1b), favoring sub-radiance, and the collective states. ONF-mediated coupling is larger for dipoles driven perpendicular to the ONF, favoring super-radiance. An important difference between sub- and super-radiant decay rates in ONF is that the latter increases as a function of N. We can Observation of super- and sub-radiance. Figure 2 shows a vary N from one to six by changing the MOT density, and typical signal of the atomic decay as measured through the ONF. quantify it through the OD of the ONF mode. n OD = Nγ /γ , Its time dependence can be described by two distinct exponential eff 1D 0 where n is the mode effective refractive index, and in our case decays. The slow decay (green dashed line in Fig. 2a) corresponds eff n ≈ 1.15. We measure the transmission spectrum through the to an average of sub-radiant decays due to pairs of atoms located eff ONF to extract the OD. The decay rate increases with N,as within a wavelength, i.e., free space interaction (Fig. 1b). Infinite- shown by the blue circles in Fig. 3, indicating super-radiance. The range interactions also produce sub-radiant decay rates. However, gray region represents the one-sigma confidence bands of a linear these events are obscured by the dominant signal of slower decays fit to the data showing a linear dependence of the super-radiant produced from free space interactions. In our case γ ≈ 0.13γ ,so 1D 0 decay rate for increasing N. The theoretical model implemented sub-radiance from infinite-range interactions is limited to γ − for the fit shown in Fig. 2 (solid black line) also predicts a linear γ ≈ 0.87γ . This is a factor of six faster than the observed sub- 1D 0 dependence on N of the decay rate γ at short times. The red radiant rates (green dashed line in Fig. 2a). Sub-radiance of atoms dashed line in Fig. 3a shows this prediction, corroborating the interacting in free space has been observed in a very optically theory with the experiment. dense cloud of atoms , 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 Evidence of infinite-range interactions. The average spacing presence of super-radiant initial states. between atoms is larger than a wavelength for most of the 4 NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications | | | Fluorescent signal (arb. units) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 ARTICLE realizations, meaning that infinite-range interactions are always stamped relative to a trigger signal coming from the pulse generator. We use time- correlated single photon counting to extract the decay rate of a single excitation present. However, to provide an unambiguous proof of infinite- in the system, eliminating after-pulsing events from the record. range interactions, we split the atomic cloud in two (see Fig. 3b). When atoms are around the nanofiber, they tend to adhere due to van der We see that two atomic clouds separated by more than 400 Waals forces. After a few seconds of having the ONF exposed to rubidium atoms it wavelengths present the same super-radiant collective behavior as gets coated, suppressing light propagation. To prevent this, we use 500 μW of 750 nm blue-detuned light (Coherent Ti:Sapph 899) during the MOT-on stage to create a function of the OD as a single atomic cloud. This shows that the a repulsive potential that keeps the atoms away from the ONF surface. This is relevant parameter is the total OD (or N) along the ONF mode, intense enough to heat the ONF and accelerate the atomic desorption from the regardless the separation between atoms. surface. The blue-detuned 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 Discussion producing a signal that looks like a long decay, an effect known as radiation Optically guided modes can be used to mediate atom–atom trapping. This effect can obscure sub-radiant signals. However, the small ODs involved in the experiment allow us to neglect contributions from radiation interactions, creating macroscopically delocalized collective trapping. We confirm this assumption by observing the same temporal evolution of atomic states. We use the super-radiant behavior of distant atoms the signal at constant OD for several detunings of the probe beam in a range of ±3 as evidence of infinite-range interaction, but other interesting linewidths . collective quantum properties remain to be tested. The practical The atomic lifetime can also be altered by modification of the electromagnetic limits of infinite-range interactions are an open question, since in environment of the atoms in the presence of an ONF, i.e., the Purcell effect. However, this effect is characterized separately and well understood. More principle optical fibers can be easily connected and rerouted along importantly, it does not depend on the number of atoms, in contrast with the several meters. An intriguing next step is the study of quantum super-radiant behavior. systems beyond the Markov approximation, coupling atoms at Further evidence of collective states can be found in the resonance spectrum of distance greater than what light travels in an atomic lifetime. 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 Moreover, by achieving fine control on the positioning of the degenerate levels. This effect is in principle visible in the transmission spectrum. In interacting particles, and/or using the directional coupling pro- our particular case, the frequency splitting is a small percentage of the linewidth. duced by chiral atom–light interaction , one can engineer Broadening mechanisms and other systematic errors prevent us from clearly desired states tailored to address specific applications. The observing such signal. However, a line-shape dependence on N can be inferred from the statistical analysis of the fit of the spectrum to a Lorentzian. This effect implementation of infinite-range interactions opens new possi- might enable the exploration of features of collective states in the spectral domain. bilities for quantum technologies and many-body physics. Given ONFs can provide chiral atom–light coupling . Even though this is a the application of one-dimensional waveguides in photonic-based promising feature of the platform, it requires a particular positioning of the atoms quantum technologies, we envision infinite-range interactions as and a preparation of their internal state. This first exploration of infinite-range the natural next step toward interconnecting quantum systems on 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 scales suitable for practical applications. consider crucial to our measurements. Methods Theoretical model. We follow the work of Svidzinsky and Chang to implement Experimental methods. A tapered single mode ONF, with waist of 240 ± 20 nm the theoretical simulations of the experiment. Consider the Hamiltonian of N radius and 7 mm length, is inside an ultrahigh vacuum (UHV) chamber, where it atoms interacting with an electromagnetic field in the rotating-wave approximation overlaps with a cloud of cold Rb atoms (less than half a millimeter width) created N hi 87 X X from a MOT. The MOT is loaded from a background gas produced by a Rb y iðÞ ωω t H ¼  hG σ ^ ^ a e þ h:c: ð5Þ int kj j dispenser. Acousto optic modulators (AOMs) control the amplitude and fre- j¼1 quencies of the MOT beams. After the atomic cloud loading reaches steady state, the MOT beams are extinguished. A free space propagating depump beam, reso- where σ ^ is the lowering operator for atom j; ^ a is the photon creation operator in nant with the F = 2 → F′ = 2 transition (150 μs duration) prepares all atoms in the k the mode k-th; ω and ω are the frequencies of atomic resonance and k-th mode of cloud in the F = 1 ground state. A 0.4 nW fiber-repump beam, detuned below 0 the field, respectively. This is a general expression for the Hamiltonian, which leads resonance by 15 MHz to the F = 1 → F′ = 2 transition, propagates through the ONF to the master equation in Eq. (2) after some approximations. The sum on j is done during the entire cycle. It pumps back to the F = 2 ground state only those atoms over the atoms and the sum on k goes over the electromagnetic field modes, guided close enough to the ONF to interact with the guided mode. This detuning repumps into the nanofiber and radiated outside. These modes can be found in the work of only those atoms close enough to the ONF surface to experience an energy shift P P R Le Kien et al. . The sum over the guided modes is ¼ dω, where f and due to the van der Waals interaction with the dielectric body. This produces a μ f ;p narrow density distribution of atoms of 5 nm width centered around 30 nm away p are the propagation direction and polarization in the circular basis (plus or from the surface. We wait 300 μs until the AOMs reach maximum extinction. The minus) of the guided mode, respectively, and μ stands for modes with different P P R R 1 k atomic cloud free falls and expands around the ONF for 2.5 ms creating a cold parameters (ω, f, p). The sum over the radiated modes is ¼ dω dβ; ν m;p 0 k thermal gas (~150 μK), where each atom interacts with the nanofiber mode for where m is the mode order, k is the wavenumber, β is the projection of the wave ~1.5 μs . The atomic density reduction due to the cloud expansion limits the vector along the fiber or propagation constant, and ν stands for modes with dif- P P P probing time of the cycle. The atoms are excited by pulses of a weak probe beam ferent parameters (ω, β, m, p). Then the total sum is ¼ þ . The elec- k μ ν incident perpendicularly to the nanofiber (see Fig. 1a) and linearly polarized tromagnetic field modes and their relative coupling strength have been previously along the ONF for the data set shown in Fig. 3. The pulses are resonant with the studied . The coupling frequencies G for the guided and radiated modes can be kj F = 2 → F′ = 3 transition of the D2 line and created with a double-passed Pockels cell written as: (Conoptics 350–160), with a pulse extinction ratio better than 1:2000 in one atomic sffiffiffiffiffiffiffiffiffiffiffiffi natural lifetime that remains at least an order of magnitude below the atomic decay hi ωβ′ ðμÞ i f βz þpϕ signal for more than 20 lifetimes. The on–off stage of the light pulses is controlled ðÞ j j ð6Þ G ¼ d  e r ; ϕ e μj j j 4πϵ  h with an electronic pulse generator (Stanford Research Systems DG645). The probe 0 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 rffiffiffiffiffiffiffiffiffiffiffiffi hi ðνÞ i βz þmϕ effects. Only those atoms that interact with the ONF-guided mode are in the F = 2 ðÞ j G ¼ d  e r ; ϕ e ð7Þ νj j j j 4πϵ  h ground state and will be excited by the probe beam. During the probing time, we 0 send a train of 50 ns probe pulses every 1 μs. The probe is a 7 mm 1/e diameter (μ,ν) collimated beam. After 2 ms of probing (~2000 pulses), the probe beam is turned where β′ = dβ/dω, d is the dipole moment of the j-th atom, and e are the off and the MOT beams are turned back on. During the probing time, the atomic electric field profile function (or spatial dependence of the amplitude) of the guided density remains constant. We wait 20 ms after the MOT reloads and repeat the and radiated modes (μ and ν). cycle. The average acquisition time for an experimental realization is around 5 h, Atoms interact with each other mediated by the electromagnetic field. The giving a total of about 1 × 10 probe pulses. The photons emitted into the nanofiber interaction between the atomic dipoles is proportional to the product of the and those emitted into free space are independently collected with avalanche atom–light coupling frequencies of the form G G , where k labels the mediating ki kj photodiodes (APDs, laser components COUNT-250C-FC, with less than 250 dark field mode (the repetition of the letter implies summation if there is more than one counts per second). The TTL pulses created from photons detected by APD are mode) and i and j label the i-th and j-th atom. It is possible to identify two processed with a PC time-stamp card (Becker and Hickl DPC-230) and time contributions from the coupling of atoms to the dynamics of the system, a NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications 5 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 dispersive and a dissipative one, as shown in Eq. (2). The dispersive part The electromagnetic field operator for the guided modes is ref. sffiffiffiffiffiffiffiffiffiffi contributes to the unitary evolution of the system (see Eq. (3)), and it can be ðradÞ ð1DÞ ðradÞ ð1DÞ ðþÞ hωβ′ ðμÞ iðωtf βzpϕÞ decomposed as Ω ¼ Ω þ Ω , where Ω and Ω come from the ^ ij ij ij ij ij E ¼ i dω ^ a e e : ð14Þ guided 4πε 0 0 interaction of the i-th and j-th atoms mediated by the radiated and guided modes, fp respectively. Ω is usually called the dipole–dipole coupling frequency. The ij dissipative part contributes to the decay of the system (see Eq. (4)), and it can be The formal solution of the Heisenberg equation for ^ a ðtÞ in the Markov and ðradÞ ð1DÞ ðradÞ ð1DÞ rotating-wave approximation is decomposed as γ ¼ γ þ γ , where γ and γ come from the interaction ij ij ij ij ij of the i-th and j-th atoms mediated by the radiated and guided modes, respectively. ^ a ðtÞ¼ ^ a ðÞ t þ 2π G δωðÞ  ω σ ^ ðtÞ; μ μ 0 0 j μj ð15Þ For simplicity, here we focus only on the case where atoms are regarded as two- level systems prepared in an initial state with induced atomic dipoles aligned along the ONF (z-axis). This is a reasonable approximation for atoms weakly driven by The substitution of this expression into Eq. (14) gives the guided field operator as a an external probe polarized along z. In a realistic scenario, the light scattered by the function of the dipole operators. fiber and by the multi-level internal structure of the atoms can mix the light Assuming that the guided modes are initially empty and that all the dipoles are polarization. The computation of such a system becomes cumbersome and only oriented in the z direction and at the same distance from the ONF, the intensity of contributes to correction to the dominant effect. A description given by two-level the guided field as a function of the atomic dipole operators is atoms aligned along the z-axis allows us to quantitatively capture the physical DE ðÞ ðþÞ 2 2 phenomena while keeping the mathematical description simple. For atoms placed ^ ^ E E ¼Ejj ðrÞ jj dðtÞ ; ð16Þ guided guided in the position r = (r , ϕ , z ) with reduced dipole moment d , we obtain i i i i i ′  where 2ω β ð1DÞ 0 0 ðÞ μ ðÞ μ 0 0 ð8Þ γ ¼ d d e ðÞ r e r cos ϕ  ϕ cos β z  z X i j i j i j ij z z i j 0 ðeÞ i βz þϕ ϵ  h ðÞ j j dðtÞ¼ e b ; ð17Þ X 0 2ω ðradÞ 0 ðνÞ ðνÞ γ ¼ d d dβe ðÞ r e r ´ cos m ϕ  ϕ cos β z  z ð9Þ 2 hω γ ðrÞ i j i j i j 2 0 ij z z i j 1D ϵ  h jj EðrÞ ¼ ; ð18Þ n cε A ðrÞ eff 0 eff ð1DÞ ðÞ μ considering γ ðrÞ¼ γ ðrÞ from Eq. (8) and A ðrÞ¼ n e ðrÞ to be 1D effðzÞ eff z ω β ii ð1DÞ 0 ðÞ μ ðÞ μ 0 0 Ω  d d e ðÞ r e r cos ϕ  ϕ sin β z  z ð10Þ ij i j i j i j 0 i j z z the effective mode area of the z component of the electric field . Equation (18) ϵ  h relates the total radiated power into the waveguide with the energy radiated per unit time, i.e., I(r)A (r) = ħω γ (r), where I(r) is the intensity of the radiated eff(z) 0 1D where μ parametrizes the guided modes on resonance. The dispersive component field. ðradÞ of the interaction given by the radiated modes as Ω is a complicated expression ij Equation (16) shows that the measured intensity corresponds to the one and hard to solve even numerically. We follow the work of Le Kien et al. and use produced by N classical dipoles with different phases, different positions, and ðeÞ ðradÞ 40 amplitudes given by the probability of being in the excited state b . the free space value of Ω throughout the calculation as a reasonable ij ðradÞ ð1DÞ approximation. γ = γ with γ the single atom natural decay rate. γ and γ ii 0 0 12 12 are plotted in Fig. 1b, c, respectively, for an atom fixed at r = (240 + 30) nm, 0, 0) Theoretical methods. We use Monte Carlo simulations, randomly positioning N atoms around the ONF. The position of each atom is given in cylindrical coor- (240 nm being the ONF radius and 30 nm the distance of the atom to the surface). ðradÞ ðradÞ dinates by r = (r , ϕ , z ), where r = (240 + 30) nm, ϕ ∈ [0, 2π], and z is obtained i 0 i i 0 i i When atoms are too close to each other, the radiated terms Ω and γ ij ij from a Gaussian distribution with a FWHM of 200 μm, determined by the atomic ð1DÞ ð1DÞ ðradÞ ðradÞ dominate over the guided ones (Ω and γ ), with Ω diverging and γ cloud size. The radial position of the atoms is fixed, determined by the experi- ij ij ij ij approaching the total decay rate. With a low number of atoms randomly mental procedure of repumping the atoms close to the nanofiber surface. In our distributed along the ONF, the effects of short-range interaction are small but still case, all the atoms are at a constant radial position of 30 nm away from the surface observable. of an ONF of 240 nm radius, with γ /γ ≈ 0.13. This is a good approximation given 1D 0 For simplicity, we are interested in the decay of only one excitation in a system the narrow radial distribution of the atoms (~5 nm), as explained in the experi- of two-level atoms, however, generalizations to multi-level atoms can be found in mental methods. the literature . Such system is represented by the state 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 X X the ground state except one with an induced atomic dipole. The initial phase ðgÞ ðeÞ ji Ψ ¼ b ðtÞji g g  g ji 1 þ b ðtÞ g g  e  g ji 0 1 2 N k 1 2 j N ð11Þ k j between the atoms depends on their position; assuming an excitation pulse with a k ;k j¼1 μ ν 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 ðgÞ where k is the sum over the guided (radiated) modes, b is the probability the intensity of the guided field, Eq. (16). We use these results to take the mean of μ(ν) amplitude of all the atoms being in the ground state and one excitation in the k-th the intensity of the guided field as a function of time. Typically, 100,000 realizations ðeÞ mode of the field, and b is the probability amplitude of having zero excitation in are required to converge to a level of precision higher than what it is visible in the field and an excitation in the i-th atom. Assuming that we start the cycle with Figs. 2 and 3. If the mean of the intensity guided field is normalized, there is no ðgÞ the excitation in the atoms, i.e., b ð0Þ¼ 0, we can write the Schrödinger equation dependence on the amplitude of the initial induced dipole in the weak excitation ðeÞ in the Markov approximation for the coefficients b ðtÞ in a matrix form as ref. limit. There is a correspondence between super-radiance (sub-radiance) BðtÞ¼ ΓBðtÞ ð12Þ configurations and constructive (destructive) interference of the field emitted by the dipoles into the ONF (see Eq. (17)); meaning that super-radiant ðeÞ configurations contributes more than sub-radiant configurations when taking the where B(t) is a vector with entries given by the b ðtÞ, and Γ is a non-hermitian mean over all the realizations for an electric field detected through the ONF (Eq. symmetric matrix with entries 2Γ = γ + iΩ , representing the couplings between ij ij ij (16)). the i-th and j-th atoms calculated from the optical nanofiber modes, radiated and The theoretical model prediction for different dipole moment orientations guided. The eigenvalues η of Eq. (12) give the possible decay rates of the system. relative to the ONF qualitatively agrees with the observed experimental behavior: These are the collective sates mentioned in Eq. (1). The eigenvectors form a basis The long-term sub-radiance disappears on our signal-to-background-ratio window fg ji B that allows us to write the state of the system as when exciting with vertically polarized light (see of Fig. 2c). A sensitivity analysis to N the ONF radius shows no significant changes in the predictions up to a ±10 nm X X ðgÞ η t ji Ψ ¼ b ðtÞji g g  g ji 1 þ c e ji B ji 0 1 2 N k α α ð13Þ variation. k ;k α¼1 μ ν Data availability. The data that support the findings of this study are available where the coefficients c are given by the initial state. In contrast with Eq. (1), here α from the authors on reasonable request. we have also included the states with one excitation in the field. Following this approach, the many-body problem, of calculating the decay of Received: 24 July 2017 Accepted: 30 October 2017. one excitation distributed among N interacting atoms, becomes an eigenvalue 2 2N problem in a Hilbert space of dimension N instead of 2 . This speeds the Published online: xx xxx 2017 calculations, allowing us to compute the decay rate of the system with Monte Carlo simulations for a large N in random positions. 6 NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications | | | NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 ARTICLE References 31. Shahmoon, E., Grišins, P., Stimming, H. P., Mazets, I. & Kurizki, G. Highly 1. Thompson, J. D. et al. Coupling a single trapped atom to a nanoscale optical nonlocal optical nonlinearities in atoms trapped near a waveguide. Optica 3, cavity. Science 340, 1202–1205 (2013). 725–733 (2016). 2. Sayrin, C. et al. Nanophotonic optical isolator controlled by the internal state of 32. Chang, D. E., Jiang, L., Gorshkov, A. V. & Kimble, H. J. Cavity QED with cold atoms. Phys. Rev. X 5, 041036 (2015). atomic mirrors. New J. Phys. 14, 063003 (2012). 3. Tiecke, T. G. et al. Nanophotonic quantum phase switch with a single atom. 33. van Loo, A. 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A 72, 063815 adaptation, distribution and reproduction in any medium or format, as long as you give (2005). appropriate credit to the original author(s) and the source, provide a link to the Creative 26. Bendkowsky, V. et al. Observation of ultralong-range Rydberg molecules. Commons license, and indicate if changes were made. The images or other third party Nature 458, 1005–1008 (2009). material in this article are included in the article’s Creative Commons license, unless 27. Richerme, P. et al. Non-local propagation of correlations in quantum systems indicated otherwise in a credit line to the material. If material is not included in the with long-range interactions. Nature 511, 198–201 (2014). article’s Creative Commons license and your intended use is not permitted by statutory 28. Bohnet, J. G. et al. Quantum spin dynamics and entanglement generation with hundreds of trapped ions. Science 352, 1297–1301 (2016). regulation or exceeds the permitted use, you will need to obtain permission directly from 29. Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase the copyright holder. To view a copy of this license, visit http://creativecommons.org/ transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 licenses/by/4.0/. (2010). 30. Bohnet, J. G. et al. A steady-state superradiant laser with less than one © The Author(s) 2017 intracavity photon. Nature 484,78–81 (2012). NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications 7 | | | http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nature Communications Springer Journals

Super-radiance reveals infinite-range dipole interactions through a nanofiber

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ARTICLE DOI: 10.1038/s41467-017-01994-3 OPEN Super-radiance reveals infinite-range dipole interactions through a nanofiber 1 1,2 3 1 1 P. Solano ,P. Barberis-Blostein , F.K. Fatemi , L.A. Orozco & S.L. Rolston 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 sub-radiance. When the field is confined to one dimension it enables infinite-range atom–atom interactions. Here we present the first report of infinite-range interactions between macroscopically separated atomic dipoles mediated by an optical waveguide. We use cold Rb atoms in the vicinity of a single-mode optical nanofiber (ONF) that coherently exchange evanescently coupled photons through the ONF mode. In particular, we observe super-radiance of a few atoms separated by hundreds of resonant wavelengths. The same platform allows us to measure sub-radiance, 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 many-body physics. 1 2 Joint Quantum Institute, Department of Physics and NIST, University of Maryland, College Park, MD 20742, USA. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México 04510 D.F., Mexico. Army Research Laboratory, Adelphi, MD 20783, USA. Correspondence and requests for materials should be addressed to P.S. (email: solano.pablo.a@gmail.com) NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications 1 | | | 1234567890 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 new class of quantum technologies exploits the interfaces the electromagnetic field is given by ref. 1–10 between propagating photons and cold atoms . Recent _ ½ ρ ¼iH ; ρ þL½ρ: ð2Þ eff Arealizations using optical nanofibers (ONFs) platforms include optical isolators, switches, memories, and reflectors . The effective Hamiltonian H of the dipolar interaction between eff These devices guide the electromagnetic field, a feature that could atoms and the Lindblad super operator L in Eq. (2) modify two allow engineering and control a collective time evolution of atomic properties: the resonance frequency and the spontaneous macroscopically separated subsystems. States that evolve as a decay rate, respectively. They are given by whole with dynamics different to that of the independent sub- systems are called collective states. These states emerge from H ¼  hΩ σ σ ; eff ij j i ð3Þ atoms interacting via a common mode of the electromagnetic i;j field, and their generation and control can enable adittional tools 12–18 for atomic-based technologies and the study of many-body 19, 20 y y y physics . L½ρ¼  hγ 2σ ρσ  σ σ ρ  ρσ σ ; ij j j j i i i ð4Þ For an ensemble of N two-level atoms, in the single excitation i;j limit, with σ σ being the atomic lowering (raising) operator for an ðÞ γ þiΩ t excitation of the i-th atom. Ω is the rate of photons exchanged 2 α ij ji Ψ ðtÞ / e c g g  e   g α αj 1 2 j N ð1Þ between atoms and γ is the term responsible for collective ij j¼1 radiative decays, where γ is the single atom decay rate. The decay ii of an excitation in such a system, that leads to a collective state as represents the α-th collective state of the system, where γ and Ω in Eq. (1), depends on the coupling amplitudes and relative phase α α are its collective decay and frequency shift, respectively, and between the atoms given by γ . ij γ t c e is the probability of having an excitation in the When atoms are far apart in free space, their interaction is αj j¼1 atoms. When γ is larger (shorter) than the natural radiative mediated by a propagating field with an expanding wavefront, 21, 22 decay time γ , the system is super- (sub-)radiant . For free and a separation of few wavelengths is enough to make the space coupling, collective states emerge for atom–atom separa- interaction negligible. As atoms get closer together, Ω in Eq. (3) ij tions smaller than a few wavelengths . By externally exciting the diverges, reducing the coherence of a system with more than two atoms, super-radiant states are readily observed, but because sub- atoms. These constraints can be circumvented by using longer radiant states are decoupled from the electromagnetic vacuum wavelengths with larger atomic dipole moments, such as Rydberg 24 26 field, they are challenging to produce . atoms , or long-range phonon modes, implemented with trap- 27, 28 The master equation that describes the dynamics of an ped ions . However, these techniques are limited to sub- ensemble of atomic dipoles, of density matrix ρ, coupled through wavelength distances. When the field is confined to one bc 1.0 0.5 (rad) (1D) /  / 0 1D 12 12 0.0 –0.5 –1.0 Fig. 1 Position-dependent atom–atom coupling along the optical nanofiber. a Schematic of an ONF as a platform for generating single photon collective atomic states, excited from the side by a weak probe of polarization V or H. When two atoms are close together, the dipolar interaction is mostly mediated by the modes of the electromagnetic field radiating outside the nanofiber. This is a limited-range interaction that decays inversely with distance. When the atoms are widely separated, the guided mode of an ideal ONF mediates the interaction for arbitrary distances. b, c Show the atom–atom interaction rate γ (see Eq. (4)) experienced by an atom around the fiber given another atom at the position denoted by the white cross (see “Methods” section for the details of the calculation). Its amplitude is shown for a longitudinal and a transversal cut (specified by dashed black lines). Both plots share the color scale, but in b the interaction rate is normalized by the single atom total decay rate γ and in c by the decay rate into the guided mode γ . Along the z-axis, the interaction 0 1D ðradÞ among atoms through free space radiation modes decreases as γ / sinðÞ kjj Δz =kΔz (with k being the wavenumber and Δz the separation between two ð1DÞ atoms). The infinite interaction through the ONF-guided mode changes as γ / cosðÞ β Δz cosðΔϕÞ (with β being the propagation constant of the 12 0 resonant-guided mode and Δϕ the angle difference in cylindrical coordinates). The wavelength λ sets the scale in b, c 2 NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications | | | NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 ARTICLE ðradÞ dimension, it enables infinite-range interactions. This has been from modes radiating outside the ONF, γ , and from the ð1DÞ 29, 30 25 observed for atoms in an optical cavity . guided mode, γ (see Fig. 1b, c). In particular, we observe Waveguides offer an alternative by confining the mediating sub-radiant decay rates of proximal atoms interacting through the field, where the extent of the interactions is not limited by the radiated modes and super-radiant decay rates of atoms interact- cavity size and the field can propagate unaltered for a broad range ing through the guided mode over distances of hundreds of 31, 32 of frequencies , facilitating the coupling of atoms separated by resonant wavelength. many wavelengths (see Fig. 1). Dipole–dipole interactions, given by Ω , are finite for atoms along the waveguide, removing a ij practical limit for creating super-radiant states of a large number Results of atoms. Super-radiance of atoms around a waveguide has been Experimental setup. We overlap a cold atomic cloud of Rb observed , but its long-range interaction feature has not been atoms from a magneto-optical trap (MOT) with a 240 nm radius proven or explored. Such effect has been implemented with ONF. This ONF is single mode at the D2 resonant wavelength of superconducting waveguides and two artificial atoms one wave- 780 nm. After the MOT is turned off, the atoms form a cold length apart , but has not been realized for many atoms at multi- thermal gas around the ONF. They are prepared in the F = 1 wavelength distances in the optical regime. ground level by an external free propagating beam. A repumper We present the implementation of collective atomic states beam driving the F = 1 → F = 2 transition propagates through the through infinite-range interactions via a one-dimensional nano- nanofiber, leaving in the F = 2 ground state-only atoms that photonic waveguide. We use a few atoms evanescently coupled to interact with the ONF-guided mode. By detuning the repumper a single-mode ONF, observing super- and sub-radiant radiative below resonance, we address atoms near the nanofiber (whose decays of a single excitation in the system, evidence of collective levels have been shifted by van der Waals interactions) such that behavior. Atoms around the ONF interact at short and long the atomic density distribution peaks at ~30 nm away from the distances (see Fig. 1a), the latter mediated by the ONF-guided surface. A weak free space probe pulse, propagating perpendicular mode. The dipolar interaction that leads to a collective decay is to the fiber, excites atoms for 50 ns using the F = 2 → F′ = 3 separated into two contributions of the electromagnetic field: transition. After the probe turns off (extinction ratio better than –1 –2 –1 –3 –2 –4 –3 0 5 10 15 20 25 30 t / –4 0 5 10 15 20 0 t / –2 –4 0 5 10 15 20 25 30 t / Fig. 2 Measured super- and sub-radiant decay of excited atoms near the optical nanofiber. a Normalized rate of photons detected through the ONF mode (blue circles in a logarithmic scale) as a function of time in units of natural lifetime (τ = 1/γ = 26.24 ns) with 5 ns bins. The signal is taken after a probe 0 0 beam polarized along the nanofiber turns off. In this realization OD = 0.66 ± 0.05. The individual statistical error bars are not plotted but they are taken into account for the normalized residuals in b. The number of counts at t = 0 exceeds 10 . We see two distinct slopes (red and green), at short and long times. The initial slope (red) deviates toward decay rates faster than γ , a signature of super-radiance. The second slope (green) comes from the natural post-selection of purely sub-radiant states. The red dashed (green dashed) line is the best fit to a pure exponential decay of the initial (final) decay. The decay rate of the fit at short times is 1.10 ± 0.02 γ , and 0.13 ± 0.01 γ for the fit at longer times, with one-sigma error. The one-sigma fractional systematic 0 0 errors are ±0.01. The full description of the measured temporal evolution of the system involves averaging over many different decay rates through Monte Carlo methods (explained in “Methods” section). The solid black line is a simulation of 7 atoms along the ONF, with reduced χ of 1.60. b The red circles, green circles, and black diamonds are the normalized residuals of the exponential fits to the initial decay, final decay, and the theoretical model. c Shows two different decay signals from an excitation driving the atoms with light polarized along (cyan rectangles) and perpendicular (blue triangles) to the ONF for 25 ns bins. When the driving field is polarized along the ONF, we observe super- and sub-radiance, and when it is polarized perpendicular to the ONF the super-radiance increases and the sub-radiance decreases. This feature is qualitatively captured by the theoretical model NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications 3 | | | Normalized Log normalized count rate residuals 10 Log normalized count rate 10 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 Average number of atoms 1.10 1.0 0.8 1.05 0.6 0.4 1.00 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 z (mm) 0.95 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Optical density Fig. 3 Super-radiant decay as a function of atom number including separated clouds. a Relationship of the decays as a function of average number of atoms (OD) along the optical nanofiber. The normalized fast decay rates are plotted as a function of the OD (lower abscissa) and N (upper abscissa) measured through the ONF-guided mode. The blue circles correspond to the signals from a single cloud of atoms. We split the atomic cloud in two (as shown in b). The dashed light and dotted dark green diamonds, and the solid red square correspond to the right, left, and the combination of both atomic clouds, respectively. The systematic errors (not shown) are estimated to be 1% for the decay rates and smaller than 20% for the atom number. The plotted error bars represent the statistical uncertainty of the fitting to an exponential decay. The gray region represents the one-sigma confidence band of a linear fitto the data. The red dashed line is the theoretical prediction, and the red shaded region represents a confidence interval set by a fractional error of 1%. The curve goes below γ/γ = 1 because the natural decay rate is modified given the geometry of the ONF and the alignment of the atomic dipoles (Purcell effect) . b Separated atom clouds show long-range interactions. The top of the figure shows in black and white a fluorescence image of a split MOT. The white dotted line represents the ONF location. The fluorescence signal of the split MOT along the nanofiber is plotted as a function of position. The dashed light (dotted dark) green dashed lines is the intensity distribution of the right (left) atomic cloud when the other one is blocked. The solid red line is the intensity distribution when both clouds are present. The separation between the center of both clouds is 318 ± 1 μm, given by standard error of the mean of a Gaussian fit. This distance is equivalent to 408 wavelengths 1:2 × 10 in one atomic natural lifetime), we collect photons A full description of the temporal evolution of the entire data spontaneously emitted into the ONF mode to measure the decay sample requires numerical (Monte Carlo) methods, as the solid time using time-correlated single photon counting. black line in Fig. 2 shows. We use the average number of atoms Collective states can be tailored by positioning the atoms in a (N) as the only free parameter for this simulation, allowing for particular arrangement. This kind of control has been challenging variations of the background up to one sigma. The two-sigma to implement for atoms trapped close enough to the ONF (tens of deviation between simulation and data (see Fig. 2b from 7 to 15 nanometers) to ensure significant mode coupling. However, τ ) could come from otherwise a longer living sub-radiant state collective states are still observed when atoms from a MOT are that gets prematurely destroyed because atoms fall onto the ONF, free to go near the ONF. Their random positioning leads to emitting the excitation into the guided mode. The initial state probabilistic super- or sub-radiant states on each experimental preparation—the polarization of the incoming pulse that realization. Sub-radiant states have lifetimes much longer than produces the collective one-photon state—can favor super- or most other processes, favoring their observation. Super-radiance sub-radiant states, as Fig. 2c shows. In general, the free space can be measured as an enhanced decay rate at short times. Both atom–atom coupling is larger for dipoles driven along the ONF (z effects can provide quantitative experimental evidence of in the direction set in Fig. 1b), favoring sub-radiance, and the collective states. ONF-mediated coupling is larger for dipoles driven perpendicular to the ONF, favoring super-radiance. An important difference between sub- and super-radiant decay rates in ONF is that the latter increases as a function of N. We can Observation of super- and sub-radiance. Figure 2 shows a vary N from one to six by changing the MOT density, and typical signal of the atomic decay as measured through the ONF. quantify it through the OD of the ONF mode. n OD = Nγ /γ , Its time dependence can be described by two distinct exponential eff 1D 0 where n is the mode effective refractive index, and in our case decays. The slow decay (green dashed line in Fig. 2a) corresponds eff n ≈ 1.15. We measure the transmission spectrum through the to an average of sub-radiant decays due to pairs of atoms located eff ONF to extract the OD. The decay rate increases with N,as within a wavelength, i.e., free space interaction (Fig. 1b). Infinite- shown by the blue circles in Fig. 3, indicating super-radiance. The range interactions also produce sub-radiant decay rates. However, gray region represents the one-sigma confidence bands of a linear these events are obscured by the dominant signal of slower decays fit to the data showing a linear dependence of the super-radiant produced from free space interactions. In our case γ ≈ 0.13γ ,so 1D 0 decay rate for increasing N. The theoretical model implemented sub-radiance from infinite-range interactions is limited to γ − for the fit shown in Fig. 2 (solid black line) also predicts a linear γ ≈ 0.87γ . This is a factor of six faster than the observed sub- 1D 0 dependence on N of the decay rate γ at short times. The red radiant rates (green dashed line in Fig. 2a). Sub-radiance of atoms dashed line in Fig. 3a shows this prediction, corroborating the interacting in free space has been observed in a very optically theory with the experiment. dense cloud of atoms , 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 Evidence of infinite-range interactions. The average spacing presence of super-radiant initial states. between atoms is larger than a wavelength for most of the 4 NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications | | | Fluorescent signal (arb. units) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 ARTICLE realizations, meaning that infinite-range interactions are always stamped relative to a trigger signal coming from the pulse generator. We use time- correlated single photon counting to extract the decay rate of a single excitation present. However, to provide an unambiguous proof of infinite- in the system, eliminating after-pulsing events from the record. range interactions, we split the atomic cloud in two (see Fig. 3b). When atoms are around the nanofiber, they tend to adhere due to van der We see that two atomic clouds separated by more than 400 Waals forces. After a few seconds of having the ONF exposed to rubidium atoms it wavelengths present the same super-radiant collective behavior as gets coated, suppressing light propagation. To prevent this, we use 500 μW of 750 nm blue-detuned light (Coherent Ti:Sapph 899) during the MOT-on stage to create a function of the OD as a single atomic cloud. This shows that the a repulsive potential that keeps the atoms away from the ONF surface. This is relevant parameter is the total OD (or N) along the ONF mode, intense enough to heat the ONF and accelerate the atomic desorption from the regardless the separation between atoms. surface. The blue-detuned 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 Discussion producing a signal that looks like a long decay, an effect known as radiation Optically guided modes can be used to mediate atom–atom trapping. This effect can obscure sub-radiant signals. However, the small ODs involved in the experiment allow us to neglect contributions from radiation interactions, creating macroscopically delocalized collective trapping. We confirm this assumption by observing the same temporal evolution of atomic states. We use the super-radiant behavior of distant atoms the signal at constant OD for several detunings of the probe beam in a range of ±3 as evidence of infinite-range interaction, but other interesting linewidths . collective quantum properties remain to be tested. The practical The atomic lifetime can also be altered by modification of the electromagnetic limits of infinite-range interactions are an open question, since in environment of the atoms in the presence of an ONF, i.e., the Purcell effect. However, this effect is characterized separately and well understood. More principle optical fibers can be easily connected and rerouted along importantly, it does not depend on the number of atoms, in contrast with the several meters. An intriguing next step is the study of quantum super-radiant behavior. systems beyond the Markov approximation, coupling atoms at Further evidence of collective states can be found in the resonance spectrum of distance greater than what light travels in an atomic lifetime. 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 Moreover, by achieving fine control on the positioning of the degenerate levels. This effect is in principle visible in the transmission spectrum. In interacting particles, and/or using the directional coupling pro- our particular case, the frequency splitting is a small percentage of the linewidth. duced by chiral atom–light interaction , one can engineer Broadening mechanisms and other systematic errors prevent us from clearly desired states tailored to address specific applications. The observing such signal. However, a line-shape dependence on N can be inferred from the statistical analysis of the fit of the spectrum to a Lorentzian. This effect implementation of infinite-range interactions opens new possi- might enable the exploration of features of collective states in the spectral domain. bilities for quantum technologies and many-body physics. Given ONFs can provide chiral atom–light coupling . Even though this is a the application of one-dimensional waveguides in photonic-based promising feature of the platform, it requires a particular positioning of the atoms quantum technologies, we envision infinite-range interactions as and a preparation of their internal state. This first exploration of infinite-range the natural next step toward interconnecting quantum systems on 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 scales suitable for practical applications. consider crucial to our measurements. Methods Theoretical model. We follow the work of Svidzinsky and Chang to implement Experimental methods. A tapered single mode ONF, with waist of 240 ± 20 nm the theoretical simulations of the experiment. Consider the Hamiltonian of N radius and 7 mm length, is inside an ultrahigh vacuum (UHV) chamber, where it atoms interacting with an electromagnetic field in the rotating-wave approximation overlaps with a cloud of cold Rb atoms (less than half a millimeter width) created N hi 87 X X from a MOT. The MOT is loaded from a background gas produced by a Rb y iðÞ ωω t H ¼  hG σ ^ ^ a e þ h:c: ð5Þ int kj j dispenser. Acousto optic modulators (AOMs) control the amplitude and fre- j¼1 quencies of the MOT beams. After the atomic cloud loading reaches steady state, the MOT beams are extinguished. A free space propagating depump beam, reso- where σ ^ is the lowering operator for atom j; ^ a is the photon creation operator in nant with the F = 2 → F′ = 2 transition (150 μs duration) prepares all atoms in the k the mode k-th; ω and ω are the frequencies of atomic resonance and k-th mode of cloud in the F = 1 ground state. A 0.4 nW fiber-repump beam, detuned below 0 the field, respectively. This is a general expression for the Hamiltonian, which leads resonance by 15 MHz to the F = 1 → F′ = 2 transition, propagates through the ONF to the master equation in Eq. (2) after some approximations. The sum on j is done during the entire cycle. It pumps back to the F = 2 ground state only those atoms over the atoms and the sum on k goes over the electromagnetic field modes, guided close enough to the ONF to interact with the guided mode. This detuning repumps into the nanofiber and radiated outside. These modes can be found in the work of only those atoms close enough to the ONF surface to experience an energy shift P P R Le Kien et al. . The sum over the guided modes is ¼ dω, where f and due to the van der Waals interaction with the dielectric body. This produces a μ f ;p narrow density distribution of atoms of 5 nm width centered around 30 nm away p are the propagation direction and polarization in the circular basis (plus or from the surface. We wait 300 μs until the AOMs reach maximum extinction. The minus) of the guided mode, respectively, and μ stands for modes with different P P R R 1 k atomic cloud free falls and expands around the ONF for 2.5 ms creating a cold parameters (ω, f, p). The sum over the radiated modes is ¼ dω dβ; ν m;p 0 k thermal gas (~150 μK), where each atom interacts with the nanofiber mode for where m is the mode order, k is the wavenumber, β is the projection of the wave ~1.5 μs . The atomic density reduction due to the cloud expansion limits the vector along the fiber or propagation constant, and ν stands for modes with dif- P P P probing time of the cycle. The atoms are excited by pulses of a weak probe beam ferent parameters (ω, β, m, p). Then the total sum is ¼ þ . The elec- k μ ν incident perpendicularly to the nanofiber (see Fig. 1a) and linearly polarized tromagnetic field modes and their relative coupling strength have been previously along the ONF for the data set shown in Fig. 3. The pulses are resonant with the studied . The coupling frequencies G for the guided and radiated modes can be kj F = 2 → F′ = 3 transition of the D2 line and created with a double-passed Pockels cell written as: (Conoptics 350–160), with a pulse extinction ratio better than 1:2000 in one atomic sffiffiffiffiffiffiffiffiffiffiffiffi natural lifetime that remains at least an order of magnitude below the atomic decay hi ωβ′ ðμÞ i f βz þpϕ signal for more than 20 lifetimes. The on–off stage of the light pulses is controlled ðÞ j j ð6Þ G ¼ d  e r ; ϕ e μj j j 4πϵ  h with an electronic pulse generator (Stanford Research Systems DG645). The probe 0 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 rffiffiffiffiffiffiffiffiffiffiffiffi hi ðνÞ i βz þmϕ effects. Only those atoms that interact with the ONF-guided mode are in the F = 2 ðÞ j G ¼ d  e r ; ϕ e ð7Þ νj j j j 4πϵ  h ground state and will be excited by the probe beam. During the probing time, we 0 send a train of 50 ns probe pulses every 1 μs. The probe is a 7 mm 1/e diameter (μ,ν) collimated beam. After 2 ms of probing (~2000 pulses), the probe beam is turned where β′ = dβ/dω, d is the dipole moment of the j-th atom, and e are the off and the MOT beams are turned back on. During the probing time, the atomic electric field profile function (or spatial dependence of the amplitude) of the guided density remains constant. We wait 20 ms after the MOT reloads and repeat the and radiated modes (μ and ν). cycle. The average acquisition time for an experimental realization is around 5 h, Atoms interact with each other mediated by the electromagnetic field. The giving a total of about 1 × 10 probe pulses. The photons emitted into the nanofiber interaction between the atomic dipoles is proportional to the product of the and those emitted into free space are independently collected with avalanche atom–light coupling frequencies of the form G G , where k labels the mediating ki kj photodiodes (APDs, laser components COUNT-250C-FC, with less than 250 dark field mode (the repetition of the letter implies summation if there is more than one counts per second). The TTL pulses created from photons detected by APD are mode) and i and j label the i-th and j-th atom. It is possible to identify two processed with a PC time-stamp card (Becker and Hickl DPC-230) and time contributions from the coupling of atoms to the dynamics of the system, a NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications 5 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01994-3 dispersive and a dissipative one, as shown in Eq. (2). The dispersive part The electromagnetic field operator for the guided modes is ref. sffiffiffiffiffiffiffiffiffiffi contributes to the unitary evolution of the system (see Eq. (3)), and it can be ðradÞ ð1DÞ ðradÞ ð1DÞ ðþÞ hωβ′ ðμÞ iðωtf βzpϕÞ decomposed as Ω ¼ Ω þ Ω , where Ω and Ω come from the ^ ij ij ij ij ij E ¼ i dω ^ a e e : ð14Þ guided 4πε 0 0 interaction of the i-th and j-th atoms mediated by the radiated and guided modes, fp respectively. Ω is usually called the dipole–dipole coupling frequency. The ij dissipative part contributes to the decay of the system (see Eq. (4)), and it can be The formal solution of the Heisenberg equation for ^ a ðtÞ in the Markov and ðradÞ ð1DÞ ðradÞ ð1DÞ rotating-wave approximation is decomposed as γ ¼ γ þ γ , where γ and γ come from the interaction ij ij ij ij ij of the i-th and j-th atoms mediated by the radiated and guided modes, respectively. ^ a ðtÞ¼ ^ a ðÞ t þ 2π G δωðÞ  ω σ ^ ðtÞ; μ μ 0 0 j μj ð15Þ For simplicity, here we focus only on the case where atoms are regarded as two- level systems prepared in an initial state with induced atomic dipoles aligned along the ONF (z-axis). This is a reasonable approximation for atoms weakly driven by The substitution of this expression into Eq. (14) gives the guided field operator as a an external probe polarized along z. In a realistic scenario, the light scattered by the function of the dipole operators. fiber and by the multi-level internal structure of the atoms can mix the light Assuming that the guided modes are initially empty and that all the dipoles are polarization. The computation of such a system becomes cumbersome and only oriented in the z direction and at the same distance from the ONF, the intensity of contributes to correction to the dominant effect. A description given by two-level the guided field as a function of the atomic dipole operators is atoms aligned along the z-axis allows us to quantitatively capture the physical DE ðÞ ðþÞ 2 2 phenomena while keeping the mathematical description simple. For atoms placed ^ ^ E E ¼Ejj ðrÞ jj dðtÞ ; ð16Þ guided guided in the position r = (r , ϕ , z ) with reduced dipole moment d , we obtain i i i i i ′  where 2ω β ð1DÞ 0 0 ðÞ μ ðÞ μ 0 0 ð8Þ γ ¼ d d e ðÞ r e r cos ϕ  ϕ cos β z  z X i j i j i j ij z z i j 0 ðeÞ i βz þϕ ϵ  h ðÞ j j dðtÞ¼ e b ; ð17Þ X 0 2ω ðradÞ 0 ðνÞ ðνÞ γ ¼ d d dβe ðÞ r e r ´ cos m ϕ  ϕ cos β z  z ð9Þ 2 hω γ ðrÞ i j i j i j 2 0 ij z z i j 1D ϵ  h jj EðrÞ ¼ ; ð18Þ n cε A ðrÞ eff 0 eff ð1DÞ ðÞ μ considering γ ðrÞ¼ γ ðrÞ from Eq. (8) and A ðrÞ¼ n e ðrÞ to be 1D effðzÞ eff z ω β ii ð1DÞ 0 ðÞ μ ðÞ μ 0 0 Ω  d d e ðÞ r e r cos ϕ  ϕ sin β z  z ð10Þ ij i j i j i j 0 i j z z the effective mode area of the z component of the electric field . Equation (18) ϵ  h relates the total radiated power into the waveguide with the energy radiated per unit time, i.e., I(r)A (r) = ħω γ (r), where I(r) is the intensity of the radiated eff(z) 0 1D where μ parametrizes the guided modes on resonance. The dispersive component field. ðradÞ of the interaction given by the radiated modes as Ω is a complicated expression ij Equation (16) shows that the measured intensity corresponds to the one and hard to solve even numerically. We follow the work of Le Kien et al. and use produced by N classical dipoles with different phases, different positions, and ðeÞ ðradÞ 40 amplitudes given by the probability of being in the excited state b . the free space value of Ω throughout the calculation as a reasonable ij ðradÞ ð1DÞ approximation. γ = γ with γ the single atom natural decay rate. γ and γ ii 0 0 12 12 are plotted in Fig. 1b, c, respectively, for an atom fixed at r = (240 + 30) nm, 0, 0) Theoretical methods. We use Monte Carlo simulations, randomly positioning N atoms around the ONF. The position of each atom is given in cylindrical coor- (240 nm being the ONF radius and 30 nm the distance of the atom to the surface). ðradÞ ðradÞ dinates by r = (r , ϕ , z ), where r = (240 + 30) nm, ϕ ∈ [0, 2π], and z is obtained i 0 i i 0 i i When atoms are too close to each other, the radiated terms Ω and γ ij ij from a Gaussian distribution with a FWHM of 200 μm, determined by the atomic ð1DÞ ð1DÞ ðradÞ ðradÞ dominate over the guided ones (Ω and γ ), with Ω diverging and γ cloud size. The radial position of the atoms is fixed, determined by the experi- ij ij ij ij approaching the total decay rate. With a low number of atoms randomly mental procedure of repumping the atoms close to the nanofiber surface. In our distributed along the ONF, the effects of short-range interaction are small but still case, all the atoms are at a constant radial position of 30 nm away from the surface observable. of an ONF of 240 nm radius, with γ /γ ≈ 0.13. This is a good approximation given 1D 0 For simplicity, we are interested in the decay of only one excitation in a system the narrow radial distribution of the atoms (~5 nm), as explained in the experi- of two-level atoms, however, generalizations to multi-level atoms can be found in mental methods. the literature . Such system is represented by the state 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 X X the ground state except one with an induced atomic dipole. The initial phase ðgÞ ðeÞ ji Ψ ¼ b ðtÞji g g  g ji 1 þ b ðtÞ g g  e  g ji 0 1 2 N k 1 2 j N ð11Þ k j between the atoms depends on their position; assuming an excitation pulse with a k ;k j¼1 μ ν 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 ðgÞ where k is the sum over the guided (radiated) modes, b is the probability the intensity of the guided field, Eq. (16). We use these results to take the mean of μ(ν) amplitude of all the atoms being in the ground state and one excitation in the k-th the intensity of the guided field as a function of time. Typically, 100,000 realizations ðeÞ mode of the field, and b is the probability amplitude of having zero excitation in are required to converge to a level of precision higher than what it is visible in the field and an excitation in the i-th atom. Assuming that we start the cycle with Figs. 2 and 3. If the mean of the intensity guided field is normalized, there is no ðgÞ the excitation in the atoms, i.e., b ð0Þ¼ 0, we can write the Schrödinger equation dependence on the amplitude of the initial induced dipole in the weak excitation ðeÞ in the Markov approximation for the coefficients b ðtÞ in a matrix form as ref. limit. There is a correspondence between super-radiance (sub-radiance) BðtÞ¼ ΓBðtÞ ð12Þ configurations and constructive (destructive) interference of the field emitted by the dipoles into the ONF (see Eq. (17)); meaning that super-radiant ðeÞ configurations contributes more than sub-radiant configurations when taking the where B(t) is a vector with entries given by the b ðtÞ, and Γ is a non-hermitian mean over all the realizations for an electric field detected through the ONF (Eq. symmetric matrix with entries 2Γ = γ + iΩ , representing the couplings between ij ij ij (16)). the i-th and j-th atoms calculated from the optical nanofiber modes, radiated and The theoretical model prediction for different dipole moment orientations guided. The eigenvalues η of Eq. (12) give the possible decay rates of the system. relative to the ONF qualitatively agrees with the observed experimental behavior: These are the collective sates mentioned in Eq. (1). The eigenvectors form a basis The long-term sub-radiance disappears on our signal-to-background-ratio window fg ji B that allows us to write the state of the system as when exciting with vertically polarized light (see of Fig. 2c). A sensitivity analysis to N the ONF radius shows no significant changes in the predictions up to a ±10 nm X X ðgÞ η t ji Ψ ¼ b ðtÞji g g  g ji 1 þ c e ji B ji 0 1 2 N k α α ð13Þ variation. k ;k α¼1 μ ν Data availability. The data that support the findings of this study are available where the coefficients c are given by the initial state. In contrast with Eq. (1), here α from the authors on reasonable request. we have also included the states with one excitation in the field. Following this approach, the many-body problem, of calculating the decay of Received: 24 July 2017 Accepted: 30 October 2017. one excitation distributed among N interacting atoms, becomes an eigenvalue 2 2N problem in a Hilbert space of dimension N instead of 2 . 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Science 352, 1297–1301 (2016). regulation or exceeds the permitted use, you will need to obtain permission directly from 29. Baumann, K., Guerlin, C., Brennecke, F. & Esslinger, T. Dicke quantum phase the copyright holder. To view a copy of this license, visit http://creativecommons.org/ transition with a superfluid gas in an optical cavity. Nature 464, 1301–1306 licenses/by/4.0/. (2010). 30. Bohnet, J. G. et al. A steady-state superradiant laser with less than one © The Author(s) 2017 intracavity photon. Nature 484,78–81 (2012). NATURE COMMUNICATIONS 8: 1857 DOI: 10.1038/s41467-017-01994-3 www.nature.com/naturecommunications 7 | | |

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Published: Nov 30, 2017

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