Quantum Information Transfer from Spin to Orbital Angular Momentum of Photons

The optical “spin-orbit” coupling occurring in a suitably patterned nonuniform birefringent plate known as a “$q$ plate” allows entangling the polarization of a single photon with its orbital angular momentum (OAM). This process, in turn, can be exploited for building a bidirectional “spin-OAM interface,” capable of transposing the quantum information from the spin to the OAM degree of freedom of photons and vice versa. Here, we experimentally demonstrate this process by single-photon quantum tomographic analysis. Moreover, we show that two-photon quantum correlations such as those resulting from coalescence interference can be successfully transferred into the OAM degree of freedom.

Figure 1

A Ti:sapphire mode-locked laser converted by second harmonic generation (SHG) into a beam with wavelength ${\lambda }_p=397.5\mathrm{nm}$. This field pumps a nonlinear crystal of $\beta$-barium borate (BBO) which emits a single-mode biphoton state with $H$ and $V$ polarizations and $\lambda =795\mathrm{nm}$, filtered by the interference filter (IF) with $\Delta \lambda =6\mathrm{nm}$ and then coupled to a single-mode fiber [19]. The gray dot-dashed box has been optionally inserted to prepare a single-photon state triggered by detector $D_T$. Birefringent quartz crystals ($Q$) having different thicknesses were used to introduce a controlled temporal delay between the two photons. After setting the input polarization by means of a suitably oriented quarter wave plate, the photons were sent through the $q$ plate (QP) and the output OAM states were analyzed with the help of a hologram (Hol) and a polarization analysis set (PA). In OAM-to-spin conversion experiments, Hol and QP were interchanged. To measure (or prepare) OAM states in the basis $l=\pm 2$, a double-fork hologram has been used [inset (A)], so that the OAM state of the first diffracted modes is shifted by $\Delta l=\pm 2$, while the undiffracted zero-order beam has $\Delta l=0$. The photons on the first diffracted modes are then coupled to single-mode fibers which select output states with $l=0$ and convey them to the detectors $D_A$ and $D_B$. Hence, the detection of a photon in $D_A$ ($D_B$) corresponds to a photon incident on the hologram with OAM $l=+2$ ($l=-2$). The first-order diffraction efficiency of the hologram was $\sim 10\%$. The measurement (or preparation) of OAM in superposition states has been realized by adopting the other holograms shown in the inset. [The hologram (B) refers to $|d_{+}{⟩}_l$, (C) to $|d_R{⟩}_l$, (D) to $|d_{-}{⟩}_l$. $|d_L{⟩}_l$ was also analyzed by hologram (C) after reversing its orientation.]

Figure 2

(a),(b) Experimental density matrices for the single-photon entangled state. The computational values $\{0,1\}$ are associated to the $\{|R⟩,|L⟩\}$ polarization states and to $\{|+2⟩,|-2⟩\}$ for the orbital angular momentum $l$ for the first and the second qubit, respectively. The incoming state on the QP is (a) $|H{⟩}_{\pi }|0{⟩}_l$ and (b) $|V{⟩}_{\pi }|0{⟩}_l$. The average experimental concurrence is $C=(0.95\pm 0.02)$. (c),(d) Experimental Poincaré sphere both for the OAM (c) and $\pi$ (d) degrees of freedom obtained after the $\pi \to l$ and the $l\to \pi$ transferrer, respectively. Experimentally we carried out single-qubit tomography to determine the Stokes parameters for the $\pi$ and the analogous parameters for the $l$ degrees of freedom. The mean fidelity values are (c) $F=(98\pm 1)\%$ and (d) $F=(97\pm 1)\%$.

Figure 3

(I),(II) Schematic representation of the two transferrer devices based on the $q$ plate (QP). (III) (a) Double transfer $\pi \to l\to \pi$ and (b) experimental real part of the matrix $\chi$ represented in the Pauli operator basis, the imaginary elements are $\approx 0$.

Figure 4

(a) Coincidence counts between $[D_A,D_B]$ versus the temporal delay $t_d$, for the state $|{\Psi }_{\mathrm{out}}⟩$. The continuous line shows the best fit of the experimental data. (b) Coincidence counts $[D_A,D_{A^{\prime }}]$ versus $t_d$. (c) Coincidence counts $[D_A,D_B]$ versus $t_d$ [$V=(0.91\pm 0.01)$] for the state $|{\Phi }_{\mathrm{out}}⟩$. (d) Coincidence counts $[D_A,D_{A^{\prime }}]$ in different OAM bases.