Superradiance from Lattice-Confined Atoms inside Hollow Core Fibre

Unravelling superradiance, also known as superfluorescence, relies on an ensemble of phase-matched dipole oscillators and the suppression of inhomogeneous broadening. Here we report on a novel superradiance platform that combines an optical lattice free from the ac Stark shift and a hollow-core photonic crystal fibre, enabling an extended atom-light interaction over $2 \mathrm{mm}$ free from the Doppler effect. This system allows controlling the atom spatial distribution and spectral homogeneity whilst efficiently coupling the radiation field to an optical fibre. The experimentally-observed and theoretically-corroborated temporal, spectral and spatial dynamic behaviours of the superradiance, e.g., superradiance ringing and density-dependent frequency shift, demonstrate a unique interplay between the trapped atoms and the fibre-guided field with multiple transverse modes. Our theory indicates the resulting temporal evolution of the guided light shows a minimal beam radius of $3.1 \mathrm{\mu m}$ that is three times smaller than that of the lowest-loss fibre mode.

atom spatial distribution and spectral homogeneity whilst efficiently coupling the radiation field to an optical fibre. The experimentally-observed and theoretically-corroborated temporal, spectral and spatial dynamic behaviours of the superradiance, e.g., superradiance ringing and density-dependent frequency shift, demonstrate a unique interplay between the trapped atoms and the fibre-guided field with multiple transverse modes. Our theory indicates that the resulting temporal evolution of the guided light shows a minimal beam radius of 3.1 µm which is three times smaller than that of the lowest-loss fibre mode.
Superradiance (SR) has been the subject of active research since the pioneering work of Dicke in 1954 1 . Such a collective effect arises from an ensemble of two-level emitters spontaneously locking their dipole oscillations in phase 2 , giving rise to an enhanced decay rate SR multiple times faster than the spontaneous emission rate 0 of individual emitters and a strong peak intensity of radiation in quadratic proportion to the excited-state population. The other SR signature manifests as a collective frequency shift and broadening, which are underpinned by fascinating physical phenomena such as the energy-level shift due to virtual photon exchange (collective Lamb shift) 3 or van der Waals dephasing 2 . In addition to its undisputed importance in fundamental science, SR also has substantial applications in various fields. For example, lasing based on SR has been proved to be a potentially outstanding frequency reference because of its strong immunity to 2 environmental perturbations 4,5 . In quantum information, SR is utilized in realizing quantum memories 6,7 and single photon sources 8 . Also, SR has been recently used for cooling mechanisms by means of cooperative emission 9,10 .
Despite the observations of SR in various gaseous and solid-state samples [11][12][13][14][15][16][17][18][19] , capturing the full picture of SR properties within a single experimental platform still remains an ongoing target. The chief challenge in unravelling the SR dynamics is to find the best trade-off between maximising the number of phase-locked emitters coupled to the common radiation field and preserving the inter-emitter phase correlation. The enhancement of SR emission rate scales as the former while the latter is a prerequisite for the SR generation. The conventional SR schemes adopt an ensemble of emitters densely packed within a sub-radiation-wavelength volume, i.e., Dicke regime [11][12][13][14] , to achieve the SR. Recently, a large number of atoms confined inside optical cavities [4][5][6][7][8][9][10] or coupled to waveguides 19 have been a successful alternative for SR generation. This is based on the fact that the fraction of photon energy emitted into the SR mode (1 + ) ⁄ approaches unity for the atomic number ≫ 1 even when the single-atom cooperativity parameter = SR 0 ⁄ , which is a geometric parameter characterizing the emissive coupling of an atom to the cavity mode, is much less than unity 8 .
Research at the interdisciplinary frontier between the fields of cold atoms and photonic materials reports promising results in controlling ultracold atoms with photonic waveguides 20 , which holds the potential of offering a strong cooperative enhancement while keeping the low atomic density. Indeed, ultracold atoms are especially attractive materials since they possess the distinct feature of being precisely controllable by optical lattice 21 . In particular, when ultracold atoms are confined in an optical lattice tuned to the magic wavelength, the optical excitation of atomic transition is not only free from the Doppler shift but also free from the lattice-field-induced ac Stark shift 22 , offering an ideal platform for investigating SR based on well-isolated atomic systems. In addition, low-loss photonic waveguides allow a strongly enhanced one-dimensional (1D) atom-light interaction, as they allow the interaction length much longer than the Rayleigh range R = 0 2 0 ⁄ at the wavelength 0 while still keeping a small light-beam waist radius 0 .
Successful combination of ultracold atoms and photonic waveguide has been exemplified by the works with nanofibers 23,24 , photonic waveguides 19 , and hollow core fibers 25 . The long-range atomatom interaction is accomplished via guided photons of the so-called alligator photonic crystal 3 waveguide 19 , and the resulting single-atom cooperativity reaches = 0.34. Ultracold atoms are successfully loaded inside a hollow-core photonic crystal fibre (HCPCF) and trapped in a 1D magic-wavelength optical lattice 25 , revealing 10-kHz-wide linewidth of the transition without being affected by wall-atom collisions.
In the present work, we report on the observation of SR from ultracold atoms confined in the magic optical lattice inside the HCPCF 26 . We couple an HCPCF-transmitted SR to a singlemode fibre to measure the signal by employing a highly sensitive heterodyne technique, which allows investigating temporal, spectral and spatial dynamic behaviours of SR with a single shot measurement. We observe a ringing and accelerated decay that is up to 55 times faster than the spontaneous emission of single atoms. By analysing the inphase/quadrature components of the signal, we measure a density-dependent frequency shift. Finally, we discuss the multi-mode dynamics of SR in the HCPCF, where a superposition of specific guided modes contributes to the photon-mediated SR, complemented with theoretical predictions. The dense cigar-shaped atomic cloud restricts the guided SR field mainly within a diameter three times smaller than that of the fundamental fibre mode. For an in-fibre atom number of = 2.5 × 10 5 with the singleatom cooperativity = 4.1 × 10 −5 available for the HCPCF, the -atom cooperativity parameter 8 can be as high as 10, leading to a near-unity coupling efficiency to the guided mode of the hollow core fibre. Figure 1a shows the schematic of our experiment. A 32-mm-long hypocycloid-core-shaped Kagome-lattice HCPCF 26 is horizontally placed in a vacuum chamber.

Experimental scheme
The inner diameter of the hollow core fibre reaches 40 μm. The optical fibre guides light via the inhibited-coupling mechanism 27 and supports multiple transverse modes whose transverse profiles are very close to those of a dielectric capillary 28 . The Gaussian-like fundamental LP01 core mode has a 1 2 ⁄ waist radius 0 = 11.8 μm (1⁄ radius is 8.34 μm). 88 Sr atoms are laser cooled and trapped in a two-stage magneto-optical trap (MOT) 29 at ≈ −1 mm (the origin ofcoordinates is set at the entrance end of the HCPCF) and loaded into an optical lattice which passes through the HCPCF (see Methods). The counter-propagating linearly-polarized lattice lasers match the LP01 mode and operate at the wavelength L = 813 nm. A bias magnetic field B 0 = (0.14 mT) ̂ is applied to define the quantization axis. We focus on the 0 1 − 4 1 3 ( = 0) intercombination transition of 88 Sr at the frequency 0 = 2 × 435 THz with the natural linewidth 0 = 2 × 7.5 kHz and the dipole moment = 0.087 0 (elementary charge and Bohr radius 0 ). The lattice laser polarization is tuned to cancel out the ac Stark shift difference between 0 1 and 1 3 ( = 0) states caused by the optical lattice 21,25 .
The loaded atoms are further transported to the position ≈ 2 mm inside the hollow-core fibre by a moving lattice with its depth of 300 K 25 . By turning off one of the two lattice lasers, the atomic gas, which is radially guided by a travelling-wave lattice laser, spreads axially to a full width at half maximum (FWHM) . Then, the lattice laser is switched on again to recapture the atoms. We consider two cloud widths, i.e., unexpanded ( = 0.87 mm) and expanded ( = 2.1 mm) clouds (see Fig. 1b . We observe that the SR field propagates in the same direction as the pump light, which is consistent with the directed spontaneous emission of the timed Dicke state 3,31 . The radiation power SR ( ) ∝ ∬ | SR ( , )| 2 2 is typically on the order of nW, requiring the low noise detection. To address this issue, we utilize the balanced-heterodyne technique 32 (see Fig. 1c). 5 The SR output is coupled to a single-mode fibre (SMF) and mixed with a local oscillator LO ( , ) =̂L O ( ) cos LO by a 50:50 fibre coupler (see Methods). The power and the frequency of local oscillator are on the order of 1 mW and LO = 0 − Ω 0 with Ω 0 = 2 × 50 MHz respectively. The relation among 0 , p , and LO is summarized in Fig. 1d. Furthermore, the SMF acts as a spatial mode filter to mainly extract the LP01 component from the SR output Thanks to the heterodyne detection, we achieve the measurement dynamic range over 5 orders of magnitude. Such sensitive detection scheme will allow accessing orders of magnitude weaker subradiant process, which usually falls behind the SR process 35,36 , by extending the measurement time sufficiently longer than 0 −1 . Figure 2b shows the normalized burst width bw / 0 as a function of the number of atoms , indicating the characteristic feature of the SR or cooperative emission that is proportional to . A linear fit to the data determines the single-atom cooperativity bw = bw 0 = 4.9 × 10 −4 .

Frequency shift
The signal RF ( ) carries also frequency information. Indeed, the amplitude of  Fig. 3c, may indicate a frequency chirp in the emitted signal. Further investigation on the chirping effect will be given elsewhere. In the following, we analyse Δ SR in the latter method as illustrated in Fig. 3c, assuming Δ SR to be constant.
We investigate the conditions that may affect the frequency shift Δ SR of the SR field. Figure 3d shows the dependence of Δ SR on the atomic density for the resonant pumping Δ p = 0. It is seen that Δ SR is negative and |Δ SR | grows up linearly with . A linear fit finds the coefficient for the frequency shift to be Δ SR /~− 1 × 10 9 Hz cm 3 , which is consistent with the values measured in the absorption spectrum 25,37 , suggesting the emission and absorption spectra are subject to the same density shift. We also measure Δ SR as a function of the pump-field detuning Δ p with the same amplitude and duration of the pump pulse as that of the -pulse with Δ p = 0. As shown in Fig. 3e, for a given density , Δ SR stays nearly constant over a wide range of Δ p , which indicates that the shift Δ SR arises entirely from the density shift and is independent of the pumping frequency p . For = 2 × 10 5 atoms confined in the unexpanded lattice, the mean site-occupancy is about 10 2 and the interatomic separation is 300 nm, leading to a collective Lamb shift plus a Lorentz-Lorenz shift 38 of about 0 . The frequency shift observed in 7 experiment well exceeds any of them. Indeed, the resonant dipole-dipole interactions (RDDIs) between the atoms in the same lattice site primarily contribute to the observed density-dependent frequency shift. The linear dependence between Δ SR and is well reproduced by our numerical simulation (see below and Supplementary Note 5).

Efficiency of superradiance
We further consider the efficiency of SR that weighs the energy transfer from the pump field to the SR light. In the spontaneous emission, the emitted photons are randomly oriented in free space and thereby hardly coupled to the fibre modes. In contrast, the SR shows a well-defined direction depending upon the sample's geometry. The multitransverse-mode propagation in the HCPCF may support an in-fibre beam which matches the atomic cloud aligned along the fibre axis. As a result, the light power output from the fibre provides the attenuated pumping pulse by the absorption of atoms and the SR field emitted from the atoms. We investigate the efficiency of SR by measuring the total radiation power SR ( ) for the applied pump power p ( ).
The measurement procedure is illustrated in Fig. 1a. The light output from the HCPCF is split into two paths by a beam splitter. In one arm, a photomultiplier tube (PMT) measures the total light intensity to determine the coupling efficiency of the SR to multi-spatial modes while in the other arm, the beam enters the SMF. The light output from the SMF is in a superposition state of the fibre LP01, LP02, and LP03 modes, where the LP01 component dominates the weight.
This light beam is detected by an avalanche photodiode (APD) so as to measure the coupling efficiency 0 , which is mostly determined by the LP01 mode, i.e., the efficiency of transferring the pumping power to a specific mode (see Methods). The photon counting mode is applied for both measurements, where the neutral-density (ND) filters are inserted in front of the detectors to attenuate the signal down to the photon-counting level.  Figure 4b shows the dependence of on the atomic number for the unexpended (black solid squares) and expanded (red solid circles) atomic clouds. It is seen that in both cases goes up monotonically and approaches unity asymptotically as is increased. The absorbed energy in the pumping process can be converted into either the SR radiation or the spontaneously emitted radiation. The former, whose power scales as 2 SR with the average SR emission rate SR of single atom, is collectively coupled to the HCPCF with an efficiency . In contrast, the latter is Similarly, we obtain the coupling efficiency 0 of the superposition state of the fibre LP01, LP02, and LP03 modes, as shown in Fig. 4b by open square symbols. It is found that 0 is maximized around m ≈ 10 5 and then decreases as is further increased, suggesting that fibre higher-order modes take up more SR energy for > m . The SR rate SR may be separated into two parts, which respectively contribute to the fundamental LP01-and high-order-mode SR fields, i.e., SR = SR (f) + SR (h) , and 0 = ( SR (f) SR ⁄ ) . Since the atoms emit the guided light within the cross-sectional area of the atomic cloud, whose radius a is much smaller than the LP01-mode beam radius 0 , the higher-order fibre modes (such as LP02 and LP03) with their central-peak radii smaller than 0 get enhanced prior to LP01 and hence SR (f) < SR . For a larger , less power is transferred to LP01 and SR (f) becomes smaller, resulting in a reduced efficiency 0 . Our numerical simulation qualitatively confirms the similar dependence of and 0 on (see below and Supplementary Note 5). We find that is about ten times smaller than (bw) that is derived based on the first SR burst. This may be attributed to the fact that the first SR burst does not behave exponentially.

9
To understand the subtle mechanism behind the above experimental observations, we need to simulate the collective atom-light interplay under the experimental detection conditions. A similar in-fibre SR model has been studied in a recent theoretical work 39 , where the physical system is simplified and only the collective decay of excited atoms, rather than the photon emission, is focused on. Here, we model the  The SR events are represented by a Gaussian-like pattern with a beam radius smaller than 0 .
This profile results from a linear combination of the LP01, LP02 and LP03 modes. Hence, the SR in multimode HCPCF can sustain a strong atom-light interplay, which differs from the situation of the atoms coupled to a single-mode fibre or a cavity 4,5 . In addition, around each node SR ( , ) ≈ 0 of SR, the transverse profile becomes different from the Gaussian-like distribution, indicating the strong influence from the non-central-peak fibre modes in Group II. Only when away from the nodes, the modes in Group I get enhanced prior to the modes in Group II.
Moreover, from the numerical simulation one may derive the SR decay rate bw corresponding to the first SR burst. Our theoretical result predicts a linear dependence of bw on the number of atoms (see Fig. 5c), bw 0 ⁄ ≈ 5.4 × 10 −4 , consistent with the experimental measurement in Fig. 2b. 11 Our numerical model also reproduces other experimental results as illustrated in Fig. 6.
Performing the Fourier transform on the numerically-simulated amplitude SR ( , ), one obtains the SR spectrum ( ) ∝ |∫ [ ∬ SR ( , ) 2 ] ( − 0 ) ∞ p | 2 (see Fig. 6a), from which the frequency shift ∆ SR is read out by deducting the envelope-oscillation frequency introduced by the ringing behaviour. The dependence of ∆ SR on the average on-site atomic density is plotted in Fig. 6b, which agrees with the experimental results. We also theoretically reproduce the counterparts of Fig. 4a and 4b in Fig. 6c and 6d. The atom-light interaction length can be extended via expanding the atomic cloud inside the fibre by shutting off either of lattice lasers to remove the periodic lattice potential. Then, the atomic cloud experiences a Gaussian-like optical potential in the − plane and an axial expansion with the velocity satisfying the Maxwell-Boltzmann distribution. After a short duration, the lattice laser, which has been turned off, is switched on again. The atomic cloud with an expanded axial width is recaptured by the lattice potential. During the expanding process, the atom loss is negligible. Figure 1b shows the distribution of the atomic cloud with and without 80-ms-long expansion. We then adiabatically reduced the lattice power to 20 mW, which is one tenth of the initial power, to moderate the density shift and residual light shift for observing SR.  Fig. 3c). Assuming the frequency shift Δ SR is time independent, the plot can be formulated as ( ) ∆ SR + 0 , where ( ) = √ I 2 ( ) + Q 2 ( ) denotes the distance from the point I ( ) + Q ( ) to the origin of the coordinate and 0 is the initial phase, because the phase components are demodulated with the frequency of 0 (see Fig. 1d and Fig. 3a). If we rotate the frame around the origin of the coordinate at a frequency of ′ and with a phase offset of ′, the plot may be re-expressed as ( ) (Δ SR − ′) +( 0 − ′) . By assigning ′ = ∆ SR and ′ = 0 , the point moves only on the real axis. To find ∆ SR from two phase components, we prepare an evaluation function that integrates the square of the imaginary part with respect to time and adopt ′ that minimizes the evaluation function related to Δ SR . The blue line in Fig. 3c shows the plot in the rotating frame that suppresses the imaginary part and ∆ SR is determined to be ∆ SR ≈ −2 × 93(10) kHz. Here the uncertainty is estimated by the frequency where the value of the evaluation function is doubled.

Data availability
All data and computer code supporting the findings of this study are available from the corresponding author on reasonable request.
in the radial plane and the axial direction, respectively. Here ( , ) + ( , )̂ corresponds to the spatial position of the ( , )-th atom. ̂ is the unit vector in the direction of the -axis. 32 For the atoms in the same lattice site, the interatomic distance may be shorter than 0 , leading to the virtual-photon-mediated resonant dipole-dipole interaction 2 . In contrast, the interaction between two atoms in different lattice sites is weak and negligible because of

Supplementary Note 4. Multiple transverse modes
The HCPCF used in experiment supports multimode propagation. For the convenience here we apply the scalar wave approximation to the linearly polarized mode classification, LP , . and are the indices corresponding to the azimuthal and radial field variation, respectively. Thus, the HE11 mode is labelled as LP01. The fibre guides via Inhibited Coupling mechanism 3 . Consequently, its modes are leaky and suffer of confinement loss. Confinement loss coefficient quantifies the fraction of power lost by the mode due to the leakage in the fibre cladding per unit of length.
Supplementary Figure 1  To keep our computation load to a reasonable level for solving our equations of motion, we limit the number of transverse modes to a maximum of 9, i.e., to the ones shown in Supplementary Fig. 1. It is noteworthy that when choosing the most dominant modes in the SR dynamics, both the loss of the fibre modes and their coupling strength with the atomic cloud have been taken into account: (i) Indeed, for the fibre modes with the comparable confinement losses, the modes, whose intensity distributions show a maximum at the centre of the fibre cross section (i.e., LP 0, ), have the stronger coupling strength with the atomic cloud than those whose intensity distributions have a zero at the centre of the fibre core (i.e., LP , with > 0); and (ii) The confinement loss increases with high orders and . Consequently, the atoms hardly interact with the modes, whose intensity distributions peak at the central point of the fibre core but have the peak diameters smaller than 2 a or do not have central peaks at all.

Supplementary Note 5. Numerical simulation
The time evolution of the whole system can be numerically simulated based on Supplementary eq. S10a, S10b and S21 via the fourth-order Runge-Kutta technique. ] in the axial direction (see Supplementary Fig. 2a and   2b). For the -th lattice site, the average on-site atomic density is where we have defined the atomic density = a ( a . The resulting ( ) is depicted in Fig. 6a of the main text, where the spectrum peak is red shifted from the atomic resonance 0 by ~2 × 152 kHz.
However, this frequency shift also includes the offset frequency ~2 × 54 kHz of the envelope varying. Consequently, the SR (carrier) frequency shift Δ SR is equal to −2 × 98 kHz, which reasonably agrees with the experimental results in Fig. 3d of the main text. The numericallyderived dependence of Δ SR on the atomic density is shown in Fig. 6b of the main text.
From the simulation results, one can further compute the superradiance efficiency. The total power of the pump field is given by p ( ) ∝ ∬|E p (r, )| 2 r while the total power of the SR field output from the fibre is SR ( ) ∝ ∬ |E SR (r, )| 2 r. Figure 6c of the main text shows the time-dependent p ( ) and SR ( ) for the unexpanded atomic cloud ( = 0.87 mm) with = 9.4 × 10 4 . Within the -pulse period, p ( ) is higher than SR ( ) and the difference ab = ∫ ( p ( ) − SR ( )) p 0 corresponds to the energy absorbed by the atomic cloud. In contrast, em = ∫ SR ( ) ∞ p denotes the SR-field energy emitted by the excited atoms. We should point out that SR may start before p . The SR efficiency defined in the main text is then given by 39 = em ab ⁄ . Figure 6d of the main text plots the dependence of on the atomic number for the unexpanded cloud (i.e., = 0.87 mm), corresponding to Fig. 4b of the main text. We see that goes up as is increased and is saturated eventually. Curve fitting leads to = ( ) (1 + ) ⁄ with the coupling coefficient = 0.87 and the single-atom cooperativity parameter = 3.6 × 10 −5 . We find that presented in Fig. 6d of the main text is lower than that of Fig. 4b in the main text. This is mainly attributed to the difference between the calculated transverse fibre eigenmodes =0,…,8 (r) and those propagating in the real fibre. In addition, the insufficient number of the fibre modes, whose intensity profiles do not peak at the centre of the fibre core, joining in the atom-light interaction may also reduce the numerically-simulated efficiency .
We also calculate the fundamental-mode radiation efficiency 0 , where the estimated coupling efficiencies for different HC-PCF eigenmodes to the single-mode fibre are: 0.60 for | 0 (r)| 2 , 0.33 for | 5 (r)| 2 , 0.05 for | 5 (r)| 2 , and 0.00 for others. Figure 6d of the main text also depicts the numerical results of 0 corresponding to Fig. 4b of the main text.