Fig. 1. Schematic of the underlying principle involved in generating the CV quadripartite spatial-mode Gaussian cluster state. (a) Pump laser drives the multimode OPA to produce entanglement of two spatial modes, HG01 and HG10, within one beam. By performing the transformation U, the multimode entanglement is transformed into a cluster state containing four spatial orthogonal modes, HG01, HG10, HG45°, and HG135°. (b) Square representation in the graph-state picture. Each cluster node, corresponding to a spatial mode, is represented by a circle. Neighboring nodes are connected by lines and represent the bipartite entanglement between two spatial modes.

Fig. 2. Experimental layout for the generation and characterization of the spatial-mode cluster state. Entanglement is generated using a multimode OPA and measured by BHD detection with spatial tailored local oscillators. LG10 pump reconstruction is shown in the right panel. NOPA: nondegenerate optical parametric amplifier; LO: local oscillator; DBS: dichroic beam splitter; PBS: polarization beam splitter; HWP: half-wavelength plate; QWP: quarter-wave plate; BHD: balanced homodyne detection; DP: Dove prism; SA: spectrum analyzer; FQ-PM: four-quadrant phase mask; MC: HG11−LG10 mode converter.

Fig. 3. Measured quantum correlations of the cluster state in the mode basis (a) HG01, (b) HG10, (c) HG45°, and (d) HG135° using the fundamental mode pump. Trace (2) is the shot noise limit (SNL), and traces (1) and (3) are the correlation variance and anti-correlation variance, respectively, normalized to SNL. Measurement settings: resolution bandwidth 300 kHz, video bandwidth 390 Hz.

Fig. 4. Enhanced entanglement for the (a) HG01, (b) HG10, (c) HG45°, and (d) HG135° modes with LG10 pumping. Trace (2) is SNL; traces (1) and (3) are the correlation variances normalized to SNL. Measurement settings: resolution bandwidth 300 kHz, video bandwidth 390 Hz.

Fig. 5. Experimental measurement of inseparability for the (a) HG01, (b) HG10, (c) HG45°, and (d) HG135° modes with LG10 pumping. Values below unity indicate entanglement.