2.1.

Single-Photon Source

The single-photon source is a quantum-dot-based emitter embedded in an electrically controlled cavity on a commercially available Quandela e-Delight-LA photonic chip. A single self-assembled InGaAs QD is surrounded by a two Bragg reflectors made of ${\mathrm{GaAs}/\mathrm{Al}}_{0.95}{\mathrm{Ga}}_{0.05}\mathrm{As}$ $\lambda /4$ layers with 36 (16) pairs for the bottom (top) and positioned in the center of a micropillar.⁶³ The micropillar is connected to a larger circular structure that is electrically contacted, enabling the tuning of the emission frequency of the QD via the Stark effect. The sample is kept at 4 K in a low-vibration closed-cycle He cryostat Attocube - Attodry800. The QD source is pumped with a 79 MHz-pulsed laser shaped with a QShaper (Quandela) 4f pulse shaper to select a specific wavelength and achieve a bandwidth of $\sim 100\mathrm{pm}$. The optical excitation of the QD is achieved in an LA phonon-assisted configuration with a laser at 927.2 nm blue-detuned from the transition,⁸³ which enables single-photon generation by exciton emission at $(927.8\pm 0.2)\mathrm{nm}$ [Fig. 2(a)]. The emitted photons are directly coupled in single-mode fiber (SMF) and spectrally separated from the residual pumping laser with bandpass filters. At the output of the e-Delight-LA system, we measure a single-photon count rate of $R_{\det }=4\mathrm{MHz}$. The fibered brightness of the single-photon source depends mainly on the coupling efficiency into the SMF, the spectral separation transmission of the single-photon stream from the pump laser—whose effects we estimate in an overall efficiency of ${\eta }_{\text{setup}}\sim 52\%$—and the detector efficiency, estimated to be around ${\eta }_{\det }\sim 38\%$. Using these figures, we estimate a first lens brightness of $B=\frac{R_{\det }}{R_{\mathrm{exc}}{\eta }_{\det }{\eta }_{\text{setup}}}\sim 26\%$, where $R_{\mathrm{exc}}$ is the pump frequency. The overall quality of the single-photon generation can be characterized by measuring the multi-photon emission and indistinguishability. Using a standard Hanbury Brown and Twiss setup, we measured a second-order autocorrelation of $g^{(2)}(0)=(1.26\pm 0.05)\%$. Such a figure is computed by normalizing the zero-time delay coincidences to the side peaks coincidences between two consecutive near-resonant excitations [Fig. 2(b)]. We also measured the indistinguishability between photons successively emitted by the QD, through an HOM interference experiment.⁵¹ Two consecutively emitted photons are split by a beam splitter (BS) and coupled in SMFs, whose length is chosen to delay one of them by $\approx 12.5\mathrm{ns}$ to ensure temporal overlap on a second BS. At its outputs, photons are collected in avalanche photodiode detectors (APDs) to record photon coincidence counts. Therefore, we evaluate a two-photon interference visibility derived from the correlation histogram [Fig. 2(c)] as $V_{\mathrm{HOM}}=1-2\frac{C_0}{{⟨C⟩}_{t\to \infty }}$, where $C_0$ is the counts when the two photons are synchronized and ${⟨C⟩}_{t\to \infty }$ is the average peak counts for relative temporal delays larger than one repetition rate of the laser. We measure an interference visibility $V_{\mathrm{HOM}}=(93.05\pm 0.06)\%$, which can be corrected to account for unwanted multi-photon components,⁸⁴ resulting in a photon indistinguishability equal to $M_s=(95.5\pm 0.1)\%$.

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 ($\mathrm{\Delta }\lambda =-0.6\mathrm{nm}$) 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 $g^{(2)}(0)=(1.26\pm 0.05)\%$. (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 $V_{\mathrm{HOM}}=(93.05\pm 0.06)\%$. Moreover, following Ref. 84, we obtain an indistinguishability value of $M_s=(95.5\pm 0.1)\%$.

2.2.

Experimental Implementation of OAM-Based Platform

We experimentally implemented a flexible platform for the study of single and multi-photon properties capable of implementing a probabilistic entangling quantum gate. A visual scheme of the setup is shown in Fig. 3.

Fig. 3

Experimental setup. Single-photon states at a wavelength of $927.8\pm 0.2\mathrm{nm}$ 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 ($\tau$) 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.

For this purpose, the stream of single photons generated by the QD is preliminarily split through a fiber BS, and OAM encoding is performed separately on the two outputs. In particular, the input state, $|H,0⟩$, having horizontal polarization and null OAM value, is selected through single mode fibers and polarizing beam splitters (PBSs). In the engineering stage, by placing a set of wave plates together with a q-plate on each arm, we are able to independently generate two distinct OAM-encoded single-photon states. In particular, a q-plate is a thin film of birefringent material (in our case, nematic liquid crystals) characterized by a nonuniform distribution of the optic axis across the plane transverse to the light propagation direction. The angle between the optic axis and the horizontal axis $x$ of the device follows the relation $\alpha (\varphi )={\alpha }_0+q\varphi$, where $\varphi$ is the azimuthal angle in the transverse plane, ${\alpha }_0$ is the optic axis orientation for $\varphi =0$, and $q$ is the topological charge, i.e., the winding number of the optic axis for $\varphi \in [\mathrm{0,2}\pi ]$. Owing to the inhomogeneity of its optic axis distribution and to the resulting Pancharatnam–Berry geometric phases, the q-plate develops on the OAM degree of freedom of single photons an action that depends on their polarization, according to the following expression:^59,60

Therefore, an optical setup consisting of a quarter-wave plate (QWP), a half-wave plate (HWP), and a q-plate with $q=1$, acting on the input state $|H,0⟩$, is able to engineer arbitrary superpositions of $|L,-2⟩$ and $|R,2⟩$ as given by

Subsequently, the two arms are synchronized by introducing a fixed delay in fiber and a tunable delay in air and then sent to a bulk BS used to probabilistically generate an entangled state between the two photons in the hybrid space of OAM and SAM by a postselection on the measured events.

Finally, in both single- and two-photon experiment, the state reconstruction is performed by using q-plates and polarization tomography setups comprising a QWP and an HWP followed by a PBS. In fact, the OAM tomography setup is implemented by adding a q-plate in front of the polarization tomography setup to convert the correlations present in the OAM degree of freedom on the polarization space [Eq. (1)]. The efficacy of this approach was demonstrated in Ref. 85 by exploiting pure-phase holograms displayed on photographic plates. In particular, in the intraparticle regime, the fiber BS is removed, and the entangled state is generated along the lower arm of the interferometer. Both the polarization and OAM analysis of such states is performed along one BS output by inserting the polarization tomography setups followed by the OAM tomography one. Instead, the analysis of the interparticle entangled state is performed by placing only the OAM tomography setups on each BS output. After the projection, the photons are collected in SMFs and detected using APDs. This scheme is used both to study the entanglement content of the states through Bell inequalities violation and to perform quantum state tomographies.

3.1.

Vector Vortex Beam: Intrasystem Entanglement

The first investigation regards the generation of VV beams encoded into the single-photon states generated by the QD source. The VV beams are superpositions of two different OAM beams associated with orthogonal circular polarizations [see, for example, Eq. (2)]. Here, the two systems individuated by the OAM eigenstates $\{|-2⟩,|2⟩\}$ and the polarization states $\{|R⟩,|L⟩\}$ can be exploited for encoding two qubits. In this way, it is possible to define a complete basis of maximally entangled states between these two degrees of freedom. The set of Bell-like states is reported in Fig. 4, in which the nonuniform polarization distribution in the transverse plane is highlighted.

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 $|{\mathrm{\Phi }}^{+}⟩$ state. (b) Real and (c) imaginary parts of the measured density matrix for the $|{\mathrm{\Phi }}^{+}⟩$ state reconstructed via quantum state tomography. The fidelity between the reconstructed state and the theoretical one is equal to $\mathcal{F}=0.9714\pm 0.0007$, where the standard deviations are estimated through a Monte Carlo approach assuming a Poissonian statistics.

In our setup, to increase the generation rate, the signal is sent only in one of the two arms of the interferometer by removing the first fiber BS (see Fig. 3). The VV beams are prepared by making horizontally polarized photons passing subsequently through a QWP, an HWP, and a q-plate with $q=1$. In this way, the state produced by the device is described by $\theta =\pi /2$ in Eq. (2) and a value of $\psi$ which depends on the ${\alpha }_0$ of the q-plate optic axis. This additional phase term is compensated by a further HWP (not shown in Fig. 3) in order to have $\psi =0$. The final entangled state between OAM and polarization will be

Although such an entanglement structure is not associated with nonlocal properties (since it is encoded in a single carrier), these correlations can be detected using Bell-like inequalities. We refer to such type of quantum correlations as intraparticle entanglement.

The adoption of a nearly deterministic single-photon source allows us to perform the intraparticle analysis without the need of heralding measurements or postselection. The latter are unavoidable procedures for generating single-photon states with high purity via probabilistic sources. This reduces drastically the losses allowing to reach a rate of $\simeq 99\mathrm{kHz}$ of VV states generation (see the Supplementary Material for further details). The quality of the state and of its entanglement structure has been certified by the measurement stage setup shown and described in Sec. 2. In particular, we performed a quantum state tomography by analyzing the OAM and the polarization independently via cascaded measurement stages as in the lower panel of Fig. 3. The resulting density matrix is reported in Fig. 4, and the relative fidelity, computed by subtracting for dark counts, is $\mathcal{F}=0.9714\pm 0.0007$. Moreover, we also certified the intraparticle entanglement by evaluating a CHSH-like inequality. Collecting data for 20 s, we obtained a raw violation of $S^{(\mathrm{r}\mathrm{a}\mathrm{w})}=2.736\pm 0.008$, which exceeds the separable bound by 92 standard deviations, while the value obtained by subtracting the background signal is $S=2.792\pm 0.008$, which exceeds the classical bound by 99 standard deviations. The results are summarized in Table 1.

Table 1

Experimental results. The results are obtained both for the intraparticle and interparticle regimes. The measurement acquisition time T, the generation rate, the values for the Bell parameter (S), and the fidelity are reported. In particular, the violation S(raw) is computed using raw data, while the parameter S is obtained by subtracting the background signal or the accidental coincidence. The fidelity value is computed by comparing the reconstructed density matrix with the triplet Bell state.

3.2.

Certification of Photon States Generation

In quantum information processes, an important computational resource relies on the capability of manipulating multiple photons and making them interact. Therefore, in this section, we assess the capacity of codifying specific OAM states on photons generated by subsequent pulsed pumping of the QD. This is performed by evaluating the visibility of HOM interference in a BS, which is equivalent to a pairwise overlap estimation in a SWAP test.⁸⁶ There are some previous examples of HOM experiments with single-photon states carrying OAM,^39,87 but our tests are among the first to be applied to vector beams generated by a deterministic single-photon source.

Let us first briefly review the effect of an unbiased BS on the field annihilation and creation operators, $\widehat{a},{\widehat{a}}^{†}$ and $\widehat{b},{\widehat{b}}^{†}$. The relation between input modes $\{a,b\}$ and output modes $\{c,d\}$ can be expressed as (Fig. 2)

By considering two photons at the two inputs of the BS, the signature of the interference is a change in the probability to detect photons in different outputs (see the Supplementary Material). In particular, two photons are indistinguishable if their states, associated with each degree of freedom, are the same from the point of view of the observer. To approach this condition in our setup, a delay line is used to synchronize the photons in the temporal domain. This is mandatory because the two single photons are emitted by the QD at different times. However, when the photons are characterized by an OAM value different from zero and superposed polarizations, it is necessary to take into account the effect of reflections. Indeed, in a physical BS, the semireflective mirror flips the elicity of both OAM and polarization. In other words, after one reflection, we have $\{|R⟩,|L⟩\}\to \{|L⟩,|R⟩\}$ and $|\pm 2⟩\to |\mp 2⟩$, while horizontal and vertical polarizations are eigenstates of this operation with eigenvalues of opposite signs. Then, we have that the creation operators are changed as follows:

Since the indistinguishability of photons generated by the source has already been checked in Sec. 2.1, here we are interested in computing the overlap between VV states encoded in different photons. As for the previous analysis, the OAM and polarization degrees of freedom are controlled through a series of QWP, HWP, and q-plate placed in each arm of the interferometer. This allows us to prepare the desired state for each photon.

Considering the BS action in Eq. (5), we expect no interference when the two photons are prepared as ${|R,2⟩}_a{|R,2⟩}_b$, since the reflected beam and the transmitted one in the outputs $c$ and $d$ will display orthogonal states. Conversely, the HOM effect occurs when the initial state is ${|R,2⟩}_a{|L,-2⟩}_b$. The correlation histograms, obtained via an HOM interference experiment, for both input states ${|R,2⟩}_a{|R,2⟩}_b$ and ${|R,2⟩}_a{|L,-2⟩}_b$ are reported in Figs. 5(a) and 5(b). The visibility of such HOM experiments quantifies the variation, from the maximum to minimum overlapping between the wave functions, of the probability of detecting photons in different outputs. The obtained visibilities are $V_{|R,2⟩,|R,2⟩}=-4\%\pm 1\%$ and $V_{|R,2⟩,|L,-2⟩}=90.1\%\pm 0.3\%$, respectively. We repeat the same interference scheme with VV states, such as $|{\mathrm{\Phi }}^{+}⟩$ and $|{\mathrm{\Phi }}^{-}⟩$ (see Fig. 4). For these classes of states, we note that they are symmetric with respect to the BS operation. This means that the reflected photon and the transmitted one always display the same state if they are indistinguishable at the input faces of the BS (see the Supplementary Material). The resulting HOM correlations for the input state ${|{\mathrm{\Phi }}^{+}⟩}_a{|{\mathrm{\Phi }}^{-}⟩}_b$ are reported in Fig. 5(c) and the achieved visibility is equal to $V_{|{\mathrm{\Phi }}^{+}⟩,|{\mathrm{\Phi }}^{-}⟩}=0.70\%\pm 0.10\%$, as expected. On the contrary, when the two photons are prepared in the same VV states such as ${|{\mathrm{\Phi }}^{+}⟩}_a{|{\mathrm{\Phi }}^{+}⟩}_b$, the theoretical HOM visibility is 1. We measured $V_{|{\mathrm{\Phi }}^{+}⟩,|{\mathrm{\Phi }}^{+}⟩}=88.2\%\pm 0.3\%$, as reported in Fig. 5(d).

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 $|{\mathrm{\Phi }}^{+}{⟩}_a|{\mathrm{\Phi }}^{+}{⟩}_b$ (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 $|{\mathrm{\Phi }}^{+}{⟩}_a|{\mathrm{\Phi }}^{-}{⟩}_b$, 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 $|{\mathrm{\Phi }}^{+}⟩$. 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 $|{\mathrm{\Phi }}^{+}{⟩}_a|{\mathrm{\Phi }}^{+}{⟩}_b$.

A further peculiar configuration is when the interfering input states are neither equal nor orthogonal, for which we expect a $V=\frac{1}{2}$. This is the case of two photons prepared in the input ports as $|{\mathrm{\Phi }}^{+}{⟩}_a|R,2{⟩}_b$. The measured visibility is $V_{|{\mathrm{\Phi }}^{+}⟩,|R,2⟩}=44.5\%\pm 0.6\%$ [Fig. 5(e)].

3.3.

Two-Photon Quantum Gate: Intersystem Entanglement

The configuration described in the previous section can be exploited to implement a multi-qubit probabilistic quantum gate able to generate an entangled state in the hybrid space composed by OAM and polarization. In particular, by postselecting on the two-photon coincidence events resulting from the preparation ${|R,2⟩}_a{|R,2⟩}_b$, and noticing that one of the outputs is affected by a further reflection that introduces a phase $\pi$ between horizontal and vertical polarization, and inverts the OAM value, the following state is generated:

Therefore, this state exhibits quantum correlations that could be detected by performing a Bell-like test that is used as an entanglement witness. In particular, we evaluated a CHSH-like inequality performing the projective measurements placing the OAM measurement stage, shown in Fig. 3, on both BS outputs. Collecting data for 400 s and with a coincidence rate of 146 Hz, we obtained a raw violation of $S^{(\text{raw})}=2.516\pm 0.006$, which exceeds the classical bound by 86 standard deviations. The value obtained by subtracting the accidental coincidences, mainly due to background noise and quantified measuring coincidences between signals in the region between two consecutive peaks (with a relative delay of $\sim 6\mathrm{ns}$), is $S=2.779\pm 0.006$, which exceeds the separable bound by 130 standard deviations.

Moreover, we also performed a quantum state tomography of the state using the same experimental configuration. The measurement apparatus selects on the 4D subspace $\{|R,2{⟩}_1,|L,-2{⟩}_1\}\otimes \{|R,2{⟩}_2,|L,-2{⟩}_2\}$ in which the interparticle entangled state lives. In fact, it contains the significantly nonzero elements of the whole density matrix of the states. Other terms outside such a subspace come from the imperfect conversion of the OAM by the q-plates. The probability of such components in the state can be kept very low, and thus can be neglected, by tuning the $\delta$ parameters of the q-plates toward the $\pi$ value. Subtracting for accidental coincidences, the estimated fidelity changes from ${\mathcal{F}}^{(\text{raw})}=0.869\pm 0.002$ to $\mathcal{F}=0.935\pm 0.002$. The retrieved density matrix is shown in Fig. 6. The results are summarized in Table 1. It is worth noting that the decrease in the coincidence rate is mainly due to the coupling efficiency into SMFs in the detection stage of about 45% (see the Supplementary Material for further details). This lower value depends on both the limited conversion efficiency of the QPs and on the higher divergence to which beams endowed with OAM are subjected. Other loss effects that need to be included are the q-plate transmission of about 75% acting on each photon. Therefore, looking toward gates with more than two photons, where the weight of the losses increases exponentially in the number of photons, the rate could be improved by compensating for losses due to the divergence and adopting experimental devices with limited losses.

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 $\mathcal{F}=0.935\pm 0.002$, where the standard deviation is estimated through a Monte Carlo approach assuming a Poissonian statistics.