Fig. 1. (a) Experimental setup for the generation and distribution of OAM multiplexed CV quantum entanglement and steering in a lossy or noisy channel. Pr, probe beam; Conj, conjugate beam; LO−l,P and LOl,C, local oscillators of Pr and Conj fields; AM, amplitude modulator; PM, phase modulator; GL, Glan-laser polarizer; GT, Glan–Thompson polarizer; PBS, polarization beam splitter; HWP, half-wave plate; VPP, vortex phase plate; M, mirror; BS, 50:50 beam splitter; BHD1, BHD2, balanced homodyne detectors; SA, spectrum analyzer. (b) Double-Λ energy-level structure for the FWM process in a cesium vapor cell. Δ, one-photon detuning.

Fig. 2. (a) and (b) Beam patterns of the OAM multiplexed CV entanglement for l=1 and l=2 in a lossy channel and corresponding transmitted patterns through a tilted lens. (c) Dependence of PPT values of the OAM multiplexed CV entanglement on transmission efficiency η for l=0, l=1, and l=2 in lossy channels. The PPT value is 0.46±0.01 at η=1. Curves and data points show theoretical predictions and experimental results, respectively. Error bars of experimental data represent one standard deviation and are obtained based on the statistics of the measured data.

Fig. 3. Dependence of PPT values of the OAM multiplexed CV entanglement on transmission efficiency η for l=0, l=1, and l=2 in noisy channels. Three different amounts of excess noise δ=0.15 (black), δ=0.5 (red), and δ=1 (blue) are compared. The light blue plane shows the boundary for sudden death of entanglement where the PPT value equals 1. The three vertical dashed lines indicate corresponding transmission efficiencies where entanglement starts to disappear. Curves and data points show theoretical predictions and experimental results, respectively. Error bars of experimental data represent one standard deviation and are obtained based on the statistics of the measured data.

Fig. 4. Quantum steerabilities of OAM multiplexed CV entangled state distributed in a (a) lossy or (b) noisy channel. The excess noise shown in (b) is δ=0.15. Solid and dashed curves show theoretical predictions of GA→B and GB→A, respectively. Data points show experimental results. Error bars of experimental data represent one standard deviation and are obtained based on the statistics of the measured data.

Fig. 5. Detailed experimental schematic for distributing OAM multiplexed CV entanglement in a noisy channel. The lossy channel is realized by blocking the auxiliary beam. D-shaped mirrors (DMs) are utilized to combine or separate light beams with small distances. HWP, half-wave plate; PBS, polarization beam splitter; EOM, electro-optic modulator; VPP, vortex phase plate; GL, Glan-laser polarizer; GT, Glan–Thompson polarizer; Pr, probe beam; Conj, conjugate beam; AM, amplitude modulator; PM, phase modulator; M, mirror; DM, D-shaped mirror; PZT, piezoelectric ceramics; BS, 50:50 beam splitter; BHD, balanced homodyne detector; SA, spectrum analyzer.

Fig. 6. Measured quantum correlation noises for initially generated OAM multiplexed CV entangled states carrying topological charges (a) l=0, (b) l=1, and (c) l=2, respectively. Brown curve at 0 dB shows the SNL. The other six curves show the noise variances of Δ2X^−l,C, Δ2Y^−l,C, Δ2X^l,P, Δ2Y^l,P, as well as the noise variances of their joint amplitude or phase quadrature Δ2(X^l,P−X^−l,C) and Δ2(Y^l,P+Y^−l,C). These six curves are all normalized to the same SNL. All the measurements are performed at 1.2 MHz. The electronic noise of the BHDs and the background noise from leaked pump fields are subtracted from the SNL and signals, respectively.

Fig. 7. Images of OAM modes of the Pr beam and Conj beam generated from the FWM process. (a) l=0; (b) l=1; (c) l=2. From left to right: Conj beam, leaked pump beam, and Pr beam.