Fig. 1. Entanglement generation. (a) In the intraparticle entanglement, the polarization and OAM subsystems are made to interact using a q-plate. The two-dimensional state |ψ⟩1 is initialized with the right polarization |R⟩=|0⟩, while the qudit |ψ⟩2 is prepared with a null OAM value |0⟩. The action of the unitary operator consists of increasing or decreasing the OAM value in a polarization-dependent way. (b) In the interparticle regime, two photons characterized by defined states in the hybrid space composed of polarization and OAM interfere using a beam splitter. Fixing the elements of the computational basis as |0⟩=|L,−2⟩ and |1⟩=|R,2⟩, both |ψ⟩1 and |ψ⟩2 are initialized with the qubit state |0⟩, and after postselecting on the coincidence counts a probabilistic entangling quantum gate is implemented. It is worth noting that considering separately the polarization and OAM Hilbert spaces of both photons, the proposed apparatus implements a four-qubit gate.

Fig. 2. Source HOM interference and second-order correlation function. (a) (Left) The single-photon source is a commercial device (Quandela): InGaAs quantum-dot based bright emitters are embedded in (right) electrically contacted micropillars. The source is pumped with a near-resonant (Δλ=−0.6nm) FWHM 10 ps 79 MHz-pulsed laser (red arrow). The single photons (red dots) are emitted at a wavelength of 927.8 nm and are directly coupled to an SMF. (b) Through a standard Hanbury Brown and Twiss setup, we measure the second-order autocorrelation histogram of our QD-based source as a function of the delay. We obtain a single-photon purity of g(2)(0)=(1.26±0.05)%. (c) Normalized correlation histogram, obtained via an HOM interference experiment, through which we measure a two-photon interference fringe visibility between subsequent single photons emitted by the QD source of VHOM=(93.05±0.06)%. Moreover, following Ref. 84, we obtain an indistinguishability value of Ms=(95.5±0.1)%.

Fig. 3. Experimental setup. Single-photon states at a wavelength of 927.8±0.2nm are generated using a QD source pumped with a shaped 79 MHz-pulsed laser at 927.2 nm. Then, a fiber BS splits the photons between the two arms of the setup, and after passing through a PBS, the input states have horizontal polarization and OAM eigenvalue m=0. In both paths, a series of QWP, HWP, and q-plate are used to produce OAM modes of the form reported in Eq. (2), while in one of the arms, a delay line (τ) is inserted in order to synchronize on the BS the photons emitted in different pulses of the pump beam. The intraparticle regime is investigated removing the fiber BS and performing all the experiment on a single line, involving the first input and output of the BS, as shown in the below panel. On the other hand, in the interparticle experiment, the photons are sent to the fiber BS, and the gate is implemented interfering on the second BS. After passing through the BS, the state of the photons is analyzed, coupled to SMFs and detected by APDs. The measurement setup consists in two different stages, a series of q-plate, QWP, HWP, and PBS are used to study the OAM states of the photons coupled with the polarization, while a QWP, an HWP, and a PBS compose the polarization analysis setup. In the interparticle regime, only OAM analysis is performed on the photon pairs, while in the intraparticle regime both analysis setups are used to separately investigate the polarization and OAM content of single photons, as shown in the lower panel.

Fig. 4. Intraparticle entangled state: (a) intensity and polarization patterns of the Bell states basis in the combined OAM and polarization space. As highlighted by the red box, we focused our attention on the |Φ+⟩ state. (b) Real and (c) imaginary parts of the measured density matrix for the |Φ+⟩ state reconstructed via quantum state tomography. The fidelity between the reconstructed state and the theoretical one is equal to F=0.9714±0.0007, where the standard deviations are estimated through a Monte Carlo approach assuming a Poissonian statistics.

Fig. 5. HOM interference for OAM states: measured coincidences at the output of the final BS, see Fig. 3, for different input states in the hybrid space of OAM and polarization. A perfect HOM interference can be obtained only when the photon states are indistinguishable from the point of view of the observer. By knowing the BS action on circular polarization and OAM (see the Supplementary Material), we observe a near-unitary visibility when the photons are prepared in the same eigenstate of the BS reflection operation |Φ+⟩a|Φ+⟩b (panel d), or when the initial states have opposite circular polarization and OAM value [panel (b)]; while near-zero visibility is reported in panels (a) and (c) for initial states |R,2⟩a|R,2⟩b and |Φ+⟩a|Φ−⟩b, respectively. Moreover, we also analyze the hybrid configuration in which one photon is prepared in the state |R,2⟩ and the other in the VV state |Φ+⟩. In the latter case, the expected number of coincidences is half of the one obtained for distinguishable photons [panel (e)]. The reported intensity patterns are associated to constructive and destructive interference for both initial states |R,2⟩a|L,−2⟩b and |Φ+⟩a|Φ+⟩b.

Fig. 6. Interparticle entangled state. (a) Real and (b) imaginary parts of the measured density matrix for the two-photon state in the hybrid OAM-polarization space reported in Eq. (7), these are reconstructed via quantum state tomography. The fidelity between the reconstructed state and the theoretical one is equal to F=0.935±0.002, where the standard deviation is estimated through a Monte Carlo approach assuming a Poissonian statistics.