Spontaneous photon-pair generation at the nanoscale

Optical nanoantennas have shown a great capacity for efficient extraction of photons from the near to the far-field, enabling directional emission from nanoscale single-photon sources. However, their potential for the generation and extraction of multi-photon quantum states remains unexplored. Here we demonstrate experimentally the nanoscale generation of two-photon quantum states at telecommunication wavelengths based on spontaneous parametric down-conversion in an optical nanoantenna. The antenna is a crystalline AlGaAs nanocylinder, possessing Mie-type resonances at both the pump and the bi-photon wavelengths and when excited by a pump beam generates photonpairs with a rate of 35 Hz. Normalized to the pump energy stored by the nanoantenna, this rate corresponds to 1.4 GHz/Wm, being one order of magnitude higher than conventional on-chip or bulk photon-pair sources. Our experiments open the way for multiplexing several antennas for coherent generation of multi-photon quantum states with complex spatial-mode entanglement and applications in free-space quantum communications and sensing.


Introduction
Correlated photon-pairs are essential building blocks for photon entanglement (1,2), which underpins many quantum applications, including secure networks, enhanced measurement and lithography, and quantum information processing (3). One of the most versatile techniques for the generation of correlated photons is the process of spontaneous parametric down-conversion (SPDC) (4). The latter SPDC allows for an arbitrary choice of energy and momentum correlations between the generated photons, robust operation at room temperature, as well as for spatial and temporal coherence between simultaneously pumped multiple SPDC sources.
Alternative approaches based on atom-like single photon emitters, such as solid-state fluorescent atomic defects (5), quantum dots (6,7), and 2D host materials (8,9), have reached a high degree of frequency indistinguishability, purity and brightness (6,7). However, this comes with the expense of operation at cryogenic temperatures and lack of spatial coherence between multiple quantum emitters. These features might limit possible applications and reduce the potential for device scalability. Furthermore, the small size of the atomic sources often requires complex schemes aimed at coupling to optical nanoantennas and improving the photon extraction efficiency (8).
The miniaturization of SPDC quantum-light sources to micro and nanoscale dimensions is a continuing quest, as it enables denser integration of functional quantum devices. Traditionally, bulky cm-sized crystals were utilized for SPDC, entailing the difficulty of aligning multiple optical elements after the SPDC crystal, while offering relatively low photon-pair rates (4).
As a first step of miniaturization, SPDC was realized in low-index-contrast waveguides, which allowed confining light down to several square micrometers transversely to the propagation direction, significantly enhancing the conversion efficiency (10). However, this approach still requires centimetres of propagation length, which makes the on-chip integration with other elements challenging (11). The introduction of high-index contrast waveguides and ring resonators allowed for shrinking the sizes necessary for SPDC to millimetres (12), and to tens of micrometers (13). However, further miniaturisation down to the nanoscale requires conceptually different approaches.
For a long time, plasmonic nanoantennas have been considered as a favorable platform for enhancing single-photon emission (14,15) and nonlinear interactions (16)(17)(18)(19)(20). However, the limited volume of the plasmonic modes, the losses and the centrosymmetric nature of plasmonic materials, result in a relatively low second order nonlinear conversion efficiency. Dielectric nanoantennas have thus emerged as a an alternative nanoscale nonlinear platform (21)(22)(23)(24).
The strong enhancement of the nonlinear processes observed in them is largely due to the absence of material absorption and the excitation of Mie-type bulk resonances (25). The highest conversion efficiency to date has been achieved employing III-V semiconductor nanostructures, such as AlGaAs which is a non-centrosymmetric material with high quadratic nonlinear susceptibility. In particular, second-harmonic generation efficiencies up to 10 −4 have been recently demonstrated (26)(27)(28)(29)(30), six orders of magnitude higher than in plasmonics.
Despite the rapid recent progress, the generation of quantum light with nonlinear nanoantennas has not been reported to date. Such nanoscale multi-photon quantum sources would offer an unexplored avenue for applications of highly indistinguishable and spatially reconfigurable quantum states, through the spatial multiplexing of several coupled nanoantennas. Until now, the big question on whether a single nanoscale antenna can generate measurable pairs of photons with non-classical polarization and energy correlations remains open.
Here, we demonstrate experimentally the generation of spontaneous photon pairs from a single AlGaAs disk nanoantenna exhibiting Mie-type resonances at both pump and bi-photon wavelengths. As such, the generation of photon pairs is a result of the correlations between these two sub-wavelength magnetic dipole modes. The observed photon-pair generation rate is 35 Hz, which, per unit volume, is higher than other SPDC light sources. Our SPDC source offers room temperature operation and the possibility of both engineering the radiation pattern and obtaining coherent interference between multiplexed sources, thanks to its inherent capacity to shape the subwavelength electromagnetic mode fields.

Results and Discussions
A schematic of our nanoantenna photon-pair source is shown in Fig. 1a wavelengths. The simulated linear scattering efficiency is defined as the scattering cross section C sca normalized by the cross area of the nanocylinder πr 2 : Q sca = C sca /πr 2 . It is shown in Fig. 1c along with the two leading multipolar contributions of the scattering. In the infrared region of the spectrum, where the signal and idler photon pairs are generated, the nanocylinder exhibits a magnetic dipolar resonance, which is the lowest order Mie-mode, featuring a Q-factor of nine (Fig. 1c). For the spectral region of the pump 760 − 790 nm, we have another strong resonance with a Q factor of 52, represented by a peak in the scattering efficiency spectrum ( Fig. 1c). This is dominated by the electric dipole moment of the antenna, although it also contains higher-order multipolar contributions (not shown). The strong internal fields at the Mie-type resonances allow for strong enhancement of the nonlinear frequency mixing processes and also imposes a spectral selection for the frequencies of the generated photons.
The SPDC process in the nanocylinder can result in the emission of photon pairs with nontrivial correlations, associated with different angular and polarization components. In order to experimentally determine the optimal conditions for photon-pair generation and ultimately for optimum SPDC efficiency, usually one uses the technique of quantum state tomography (31).
However, due to weakness of nonlinear process, the bi-photon rate tends to be low, thereby resulting in long time acquisition of the photon counting statistics, as well as lack of correlation precision. Therefore, optimizing the experimental parameters directly through SPDC measurements is impractical and we need an alternative solution.
To solve this issue, we resort to the quantum-classical correspondence between SPDC and its reversed process, namely sum-frequency generation (SFG), where the generated sum-frequency and pump waves propagate in opposite directions to the SPDC pump, signal and idler (32,33).
Such quantum-classical correspondence is applicable to any quadratic nonlinear structures and allows the classical estimation of the SPDC generation bi-photon rates through the relation Here, Φ p is the SPDC pump flux, λ p , λ s and λ i are the pump, signal and idler wavelengths, and ∆λ is the nonlinear resonance bandwidth at the signal/idler wavelengths. The efficiency which is comparable to the SHG efficiency obtained in earlier measurements (26)(27)(28). As shown in the BFP images, the SFG radiation patterns strongly depend on signal and idler polarization combinations, however the general observation is that the SFG signal is emitted under angle, off-axis to the nanocylinder. This is due to the symmetry of the nonlinear tensor, as previously reported for SHG in Refs. (28,34). Detection of coincidences between photons generated through SPDC in the nanocylinder is illustrated in Fig. 1a. In the experiment we use a CW pump laser, with a power of 2 mW at the wavelength of 785 nm. The generated photon pairs are expected to have a large spectral bandwidth of about 150 nm, due to the broad magnetic dipole resonance in the IR spectral range, as shown in Fig. 1c. This bandwidth is quite broad with respect to conventional SPDC sources, which have typical sub-nm or few-nm bandwidth. This broad bandwidth offers a range of advantages, including a short temporal width for timing-critical measurements, such as for temporal entanglement (35), or for SPDC spectroscopy (36). It also dictates a sub-100 fs temporal width of the generated photons, which is much shorter than the coincidence window τ c (see Methods).
The measured coincidences for an H-polarized CW pump are presented in Figure 3a (for details see Sec. 8 of Supplementary Information). Importantly, this rate is significantly higher than the reference measurements of the AlOx/GaAs substrate without the nanocylinder

Materials and Methods
Sample Fabrication.
We fabricate crystalline AlGaAs monolithic nanoantenna, since this material platform provides strong second order nonlinear susceptibility. The fabrication steps follow the procedure devel- ijk with i = j = k. We predict the SPDC output and correlations based on the quantum-classical analogy between the SPDC and the SFG processes (32). For the SFG process, assuming an undepleted pump approximation, we follow two steps. Firstly, we simulate the linear scattering at the fundamental wavelengths λ s and λ i . The bulk nonlinear polarization induced inside the particle is then employed as a source for the electromagnetic simulation to obtain the generated SF field. Based on the calculated SF field and the field at the signal and idler wavelengths λ s and λ i , we further obtain the SPDC output correlations based on the quantum-classical analogy, as shown in Eq. (1).

SFG measurements.
Pulsed signal and idler beams are derived from a broadband femtosecond laser with a repetition rate of 80 MHz (Toptica, FibrePro). The two pulses are generated by spectrally slicing the 100 fs long pulses (bandwidth of 80 nm) into two paths at central wavelengths of 1520 and 1560 nm.
After appropriate polarization conversion via the use of half and quarter wave plates, the two pulses are recombined with a 50:50 beam splitter and focussed onto the nanoantenna at normal incidence by a 0.7 NA objective (as shown in Fig. S1 of the Supplementary Information). The reflected SFG radiation was collected in reflection through the same objective, separated from the signal and idler by a dichroic mirror. A short-pass filter at 800 nm is subsequently used for removal of the photoluminescence from the substrate, while a long-pass filter at 600 nm is used to remove the third harmonic emission component from the nanocylinder. The SFG emission was then acquired with a spectrometer or with a cooled camera in the real space. An additional confocal lens focusing at the objective back focal plane is used for imaging the emission pattern in the Fourier space.

SPDC measurements.
The statistics of photons generated in SPDC can be characterised by measuring the second-order correlation g (2) = R c /(R 1 R 2 τ c ) using a beam splitter and two single-photon detectors at both outputs of this beam splitter (13). Here R c is the rate of coincidences between the detectors, corresponding to SPDC, while R acc = R 1 R 2 τ c is the accidental coincidence rate that includes the count rates on each detector R 1 and R 2 . The coincidence time window is τ c . We expect maximum g (2) at the zero time delay for the simultaneous arrival of the two photons (43).
In our experiments, we used a CW pump beam at a central wavelength of 785 nm and a linewidth of < 10 MHz, horizontally polarized, to pump the nanoantenna at normal incidence via a 0.7 NA objective (as shown in Fig. S4 of the Supplementary Information). The choice of the CW laser is justified by the fact that the SPDC photon-pair generation rate scales linearly with the average pump power. Furthermore, the CW operation allows us to eliminate the timecorrelated noise if a pulsed pump was used, since a CW source has a flat temporal profile and a coherence time > 100 ns, which is larger than the measured coincidence time range of 40 ns.
The reflected SPDC signal and idler photon pairs were collected in reflection through the same objective, separated by the pump with a dichroic mirror, and further filtered in free-space from the residual pump with the use of three long-pass filters at 1100 nm. The photon pairs were then separated by a 50:50 beam splitter into two paths and coincidences were measured with two gated InGaAs avalanche photo-diodes (IDQ230) and a time-tagging module (ID801-TDC).
The detectors are coupled with multi-mode fibres and operate at −90 • C with an efficiency of 10% and dark counts of 5 Hz. The counting scheme consisted of a coincidence window of 300 bins with a bin width of τ c = 162 ps.