Spatiotemporal Single-Photon Airy Bullets

Uninhibited control of the complex spatiotemporal quantum wave function of a single photon has so far remained elusive even though it can dramatically increase the encoding flexibility and thus the information capacity of a photonic quantum link. By fusing temporal waveform generation in an atomic ensemble and spatial single-photon shaping, we hereby demonstrate for the first time complete spatiotemporal control of a propagation invariant $(2+1)\text{D}$ Airy single-photon optical bullet. These correlated photons are not only self-accelerating and impervious to spreading as their classical counterparts, but can be concealed and revealed in the presence of strong classical stray light.

Figure 1

Experimental scheme for the observation of spatiotemporal Airy biphotons. (a) Optical setup. Two classical circularly polarized ($\sigma$), counterpropagating beams with frequencies ${\omega }_p$ (Airy) and ${\omega }_c$ (Gaussian) generate circularly polarized Stokes, ${\omega }_S$, and anti-Stokes, ${\omega }_{AS}$, entangled photons via spontaneous four-wave mixing. A space-to-time mapping generates nonclassical temporal Airy biphotons $\mathrm{Ai}(\tau )$. The ${\omega }_{AS}$ photons are subsequently shaped into a nondiffracting Airy wave packet to form a three-dimensional spatiotemporal photon, $\mathrm{Ai}(x)\mathrm{Ai}(y)\mathrm{Ai}(\tau )$. Single-photon counting modules (SPCM) register the photons that have been filtered by Fabry-Pérot (FP) cavities. A classical noise pattern is superimposed for correlated photon concealing. (b) Double-$\mathrm{\Lambda }$ configuration for spontaneous four-wave mixing. The 780 nm 1D-Airy pump beam, ${\omega }_p$, has a detuning of ${\mathrm{\Delta }}_p=120\mathrm{MHz}$, while the 795 nm coupling beam, ${\omega }_c$, is on resonance. The twin photons ${\omega }_S$ (780 nm) and ${\omega }_{AS}$ (795 nm) are spontaneously emitted. (c) Simulated formation of spatiotemporal quantum Airy wave packets as the photon number is increased. The temporal (left) and the spatial (right) wave packets emerge simultaneously.

Figure 2

Observation of temporal Airy biphotons. (a) Temporal Airy biphoton. Normalized counts of the observed biphotons (shaded pink), numerically predicted waveform (orange solid line) for $d=150$ and ${\mathrm{\Omega }}_c=2\pi \times 10\mathrm{MHz}$, and analytical prediction, from Eq. (1), (dash dotted green line) for a pump having a beam profile $f_p(z)=\mathrm{Ai}(z/z_0)\exp (az/z_0)$, with $z_0=18.7\mathrm{mm}$, and $a\approx 0.1$. (b) (top) Two- and (bottom) three-lobe Airy waveforms (shaded pink) and their corresponding self-correlations, $g_c^{(2)}(\tau )$ (black squares). The biphoton generation rate for the two- and three-lobe waveforms during the SFWM is $470{\mathrm{s}}^{-1}$ and $374{\mathrm{s}}^{-1}$. The error bars are calculated from the one standard deviation of the photon counts. (c) Locally probing the biphoton phase. Observed normalized counts when the Airy pump is imposed as a whole (shaded pink), with only the first lobe (shaded green), and only the second lobe (shaded blue). (d) Simulated (dash dotted green line), and analytical (orange solid line) Airy waveforms for high optical depths ($d=500$). Experimentally obtained (e) temporal waveform density $|\psi (\tau ){|}^2$ as a function of $d$ and fixed ${\mathrm{\Omega }}_c=2\pi \times 10\mathrm{MHz}$, and (f) temporal waveform as a function of the normalized coupling Rabi frequency $|{\mathrm{\Omega }}_c|/{\gamma }_{13}$ for $d=150$. The curves overlaid on the experimental data highlight the maximum (solid) and the half-maximum as determined by numerical simulations. The same color map is used throughout the Letter.

Figure 3

A three-dimensional spatiotemporal Airy photon. Isosurface of (a) the experimentally observed $|{\psi }_{xy\tau }{|}^2$ of the Airy biphotons, and (b) the corresponding theoretical wave packets. The isosurfaces range from 0.1 to 1.0. The vector lengths represent $x$, $y$, and $\tau$ scales of $100\mathrm{\mu }\mathrm{m}$, $100\mathrm{\mu }\mathrm{m}$, and 200 ns respectively. A projection of the photon probability density (c) for the temporal $\tau$ and spatial $y$ distribution along the $x$ axis, and (d) for the two-dimensional spatial $x$, $y$ distribution along the time axis $\tau$. (e) Observation of the $z$ propagation of the spatially self-bending nonclassical Airy photons along the diagonal ($x=y$). The theoretically expected parabolic trajectories (solid for the maximum and dashed for the half-maximum) are overlaid.

Figure 4

Concealing correlated nondiffracting photons in classical light noise. (a) Airy to noise ratio $\alpha /\beta$ as a function of time window $t_w$. Blue and red lines show two measurements with different classical noise powers. The inset depicts a typical biphoton waveform for a Gaussian pump. The blue arrow corresponds to image (b) for $t_w=0.2\mathrm{\mu }\mathrm{s}$ of collecting heralded photons, and (c) is the concealed photon pattern when no trigger was used. (d) Propagation of nondiffracting correlated Airy photons over classical noise. When triggering of Stokes photons is used the self-bending of Airy photons is clearly visible. (e) Propagation of uncorrelated thermal Airy photons over classical noise. Without triggering the strong noise masks the Airy light. The overlaid theoretical curves indicate the peak (solid) and the half-maximum (dashed).