Coherent interaction of atoms with a beam of light confined in a light cage

Controlling coherent interaction between optical fields and quantum systems in scalable, integrated platforms is essential for quantum technologies. Miniaturised, warm alkali-vapour cells integrated with on-chip photonic devices represent an attractive system, in particular for delay or storage of a single-photon quantum state. Hollow-core fibres or planar waveguides are widely used to confine light over long distances enhancing light-matter interaction in atomic-vapour cells. However, they suffer from inefficient filling times, enhanced dephasing for atoms near the surfaces, and limited light-matter overlap. We report here on the observation of modified electromagnetically induced transparency for a non-diffractive beam of light in an on-chip, laterally-accessible hollow-core light cage. Atomic layer deposition of an alumina nanofilm onto the light-cage structure was utilised to precisely tune the high-transmission spectral region of the light-cage mode to the operation wavelength of the atomic transition, while additionally protecting the polymer against the corrosive alkali vapour. The experiments show strong, coherent light-matter coupling over lengths substantially exceeding the Rayleigh range. Additionally, the stable non-degrading performance and extreme versatility of the light cage provide an excellent basis for a manifold of quantum-storage and quantum-nonlinear applications, highlighting it as a compelling candidate for all-on-chip, integrable, low-cost, vapour-based photon delay.

mode within specific spectral domains leading to a sequence of high transmission bands (Article Figure 1b). Note that due to the absence of total internal reflection the formed leaky modes radially dissipate energy while propagating 4 , with most of their fields concentrated in the hollow core (Article Figure 1g). According to simulations, the LC concept reaches up to 99.99 % of power in the core (Article Figure 1e). The interaction length between light and core medium is determined by the length of the LC, which is 4.5 mm in the presented case. This provides 70 times longer interaction at high intensity in comparison with the Rayleigh length of a focused beam with identical beam waist. One notable feature of LC is the side-wise openness allowing the surrounding low-pressure vapour to enter the core and efficiently interact with the light field. A representative comparison of filling times between a capillary-type hollow-core structure and the LC is shown in Article Figure 1h. More than two orders of magnitude faster filling makes the LC concept a beneficial platform, in particular for vapour-based applications.

Sample fabrication.
We fabricated LCs on a silicon substrate by 3D nanoprinting with a two-photon polymerisation lithography system (Nanoscribe GT, details in Methods section). The waveguide design parameters (i.e. Ds and Λ, top inset in Article Figure 1a) are chosen according to finite element simulations, followed by experimental prototyping to simultaneously realise strong mechanical stability, efficient light guidance and an as-large-as-possible openness. Combining the alkali resistance, obtained by film coating, with the intrinsic properties of the LCsuch as strong light confinement, low optical loss, long-term stability, and the virtually instantaneous filling of roomtemperature alkali vapourone obtains a particularly attractive novel class of integrated quantum devices for experiments focusing on strong coherent light-matter interaction.

Light cage: Durability
The LC with alumina coating proves stable against the chemically highly-reactive Cs atmosphere.
The transmission spectrum in Article Figure 1b was measured ten months after the LC was placed into the cell. Light fields have been guided in the LC with intensities of the coupling laser field up to the order of Ic = 20 kW cm -2 (at Pc = 30 mW). More than 99 % of the field in the centre mode of the LC are in a distance to the confining strands of about 6 μm. Thus, only marginal interaction of the intense fields with the LC material is expected. A more thorough examination of the optical damage threshold of the LC material and structure will be required for nonlinear optical experiments which strive for high intensities in order to achieve near unity efficiencies.

II.
Gas diffusion The diffusion of atoms, for instance Rb, in a glass capillary does not follow classical, or modified, Knudsen models for transport 5 Figure 1h). Filling of the LC takes three orders of magnitude less time than centimetres-long capillary-type geometries where the filling length is given by the length of the waveguide. Due to the sidewise accessible core. Between experiments, even faster stabilisation times of less than a day were observed, which is probably due to the geometry of the LC where atoms flow through from multiple directions and not only in a linear manner as in the other structures.
The atom density N/V inside the LC was inferred from reference transmission spectra that were taken without a coupling field. At an identical oven temperature (Toven = 80 °C) the atom density inside the LC, (N/V)LC = (6.9 ± 0.3)·10 10 cm -3 , is less than a factor of two smaller than 3 mm above the LC in free space: (N/V)f.sp= (1.02 ± 0.02)·10 11 cm -3 . This is, however, a remarkably small reduction as compared to experiments in hollow-core fibres, where additional means such as light induced atomic desorption (LIAD) are essential to provide an acceptable vapour density 7 .
Compared to other hollow-core waveguide structures, the LC uniquely provides a reduced presence of adherable surfaces, drastically improves the filling efficiency with low-pressure vapour, and minimises the atom-wall collisions, which otherwise induce decoherence 8 . In particular, no additional desorption mechanism is necessary to reach sufficiently high atom densities, which prevents quantum channels with signals at the single-photon level to be filled with numerous highpower laser fields and facilitates filtering.

III. Setup: Efficiencies and losses
The setup transmission loss with the light cage was measured to be ηsetup,LC = η1,LC + η2 + η3,LC = −34.9 dB. The overall light-cage loss η1,LC = −30.1 dB was measured from just before lens L1 to behind an iris, placed right after lens L2, for a beam that is non-resonant with the caesium (Cs) atoms. See Supplementary Figure  In the case of the free-space experiment we find ηsetup,fs = η1,fs + η2 + η3,fs = − 4.95 dB that are composed off η1,fs = ηrefl = −1.3 dB, η2 = -2.1 dB being identical for the LC and the free-space experiment, and η3,fs = −1.55 dB. The difference η3,LC > η3,fs leads to the conclusion that the propagation of the probe field in the recent light cage affects the spatial shape of the mode. In the normalised spectra (see Supplementary Material Section IV Analysis) the EIT transmission goes as high as 99 % on resonance, for the highest powers, at any of the measured oven temperatures. This implies that η1 would remain unaltered in the high coupling-power regime.

Data processing
Normalisation of the data has been performed to compensate for the power changes when scanning the probe diode laser. Three data sets were taken for each scan over 20 GHz: (1)  Here, χ is the electric susceptibility of the atomic vapor in the linear regime, that can be calculated in presence of EIT as in Ref. 11 as

Integrated on-axis intensity and weighted fractional delay
The EIT windows for the present experimental conditions allow for the incoming pulses to be When considering the improved two-layer LC, the transmission efficiency of the LC is increased to τ = 42.9 %. Meanwhile, in the free-space scenario, it is reduced to only 5.6 %. In the comparatively longer vapour cell, the free-space propagating beam covers a longer distance with a larger diameter or lower intensitysince the Rayleigh length remains constant. In this case, a narrower EIT window is generated than in the LC since low intensities produce small Rabi frequencies. This eventually pushes the figure of merit to FLC = 0.89 (and Ff.sp = 0.04, respectively) while three orders of magnitude less intensity will be required. In a comparable work with a rectangular hollow-core ARROW waveguide with core dimensions 4.75 μm × 12 μm × 4 mm in Rb vapour 13 , the extended fractional delay can be extracted as F 13 ≈ 0.2 for delay times tD ≤ 16 ns. The transmission of τ = 0.44 causes strong spectral filtering of the pulse and, thus, a significantly broadened pulse width from 20 ns to 32 ns.  Figure S4). In order to define the core area (and thus the core radius R) as the area lying within a circle passing through the centres of the strands, we have chosen a circular grid (inset of Fig. S4(b)). Please note that this geometry is very similar to the hexagonal light cages used in the experiment due to the large ratio of wavelength and light cage radius and, thus, no significant differences to the experimentally studied structure are to be expected. The vertical dashed blue line in Fig. S4 shows the situation of N = 12, corresponding to the number of strands used in the experiments. R = Λ / (2·sin[π/N]). As expected, the effective mode index neff (here relative to unity) decreases for smaller core diameters due to the stronger interaction of mode and strand array ( Fig. S4(a)). The corresponding fraction-of-power simulation ( Fig. S4(b)) clearly demonstrates the capabilities of the light cage for further reduction of the geometric footprint, showing values greater than 99% for all core radii considered here. Please note that the definition of the core area in the circular light cage geometry becomes increasingly questionable when the number of strands falls below five, which causes us to stop the simulations at N = 6. The presented results have been achieved for single-ring light cage structures and an even improved performance particularly at small core extents can be anticipated in case the number of rings is increased. Moreover, a variation of the pitch Λ presumably towards smaller values can further increase the fraction of power inside the core, thus overall leaving room for future performance improvement. It should also be noted that upon improvement of the LC design 10 , attenuation rates of 1.5 dB m -1 from Ref. 8 will be within reach. In such a low loss regime, the high probe and coupling fields would be virtually constant over the whole length of the LC. This is especially interesting for nonlinear quantum effects in gaseous or liquid 16,17 optical media, where close-to-unity spatial overlap of, e.g., atoms, the single-photon probe, and, simultaneously, the strong coupling fields, are crucial 18 . Preservation of polarisation in the LC, as another necessary property for nonlinear quantum optics, can be achieved by introducing birefringence through a small asymmetry of the LC geometry.