## 3 Experimental Setup  
A schematic of our experimental apparatus is shown in Figure 3. The setup includes three parts: the optical devices for entangled state generation, the servo-locking devices, and the balanced homodyne detection (HD) system. _The optical devices part_: A single-frequency continuous-wave laser (Cobolt Lasers, Solna, Sweden) at 532 nm was used to pump a NOPO and NOPA from opposite directions. The NOPO and NOPA shared a bowtie-type ring cavity consisting of two concave mirrors (radius of curvature, 50 mm) and two flat mirrors. One of the flat mirrors was a partial transmittance mirror and worked as the output coupler, while the other had high reflectance (\(\sim\)99.9%) at all three wavelengths (pump, signal, and idler). From previous analysis, although high escape efficiency is beneficial to entanglement, the drawback is that it will boost the threshold of the NOPO. There is a compromise between escape efficiency and threshold pump power. In this case, the output coupler was coated for partial transmission (\(\sim\)10.2%) at 0.8 \(\upmu\)m and 1.5 \(\upmu\)m, and high reflectivity at 532 nm (99.6%). The two input couplers (concave mirrors) were coated for high reflectivity at 0.8 \(\upmu\)m and 1.5 \(\upmu\)m (\(>\)99.9%), and partial transmission at 532 nm (\(\sim\)44%). Here, the design of the partial transmission at 532 nm for the input couplers helps to decrease the NOPO threshold effectively by about 2.2 times. A 20-mm long periodically poled KTiOPO4 (PPKTP) crystal (Raicol Crystals Ltd, Rosh Ha'Ayin, Israel) placed between the two concave mirrors served as the second-order nonlinear medium. The crystal was especially anti-reflection coated (\(<\)0.1%) at both the signal and the idler wavelength in order to reduce intracavity losses as much as possible. The beam waist size inside the crystal was about 30 \(\upmu\)m, to allow for optimal focusing of the Gaussian pump beams. To avoid the residual reflection effects from the crystal's surfaces on the stable operation of the NOPO, a parallelogram PPKTP was used, where both end faces were cut at an angle of two degrees. Inside the cavity, the two directions of the down-converted beams propagated along exactly the same path, and thus share the same linear and nonlinear losses, phase shift, phase-matched conditions, and so on. Therefore, when the cavity length is stabilized, the forward NOPA and backward NOPO both oscillate on a single signal-idler mode pair with the same frequency. _The servo-locking devices_: To enable generation of entanglement, the ring cavity was controlled in resonance using a dither-locking servo-control system. To this end, the leaked light from one of the high reflectance flat mirror of the cavity was detected by a photodetector (PD2) and fed into the servo system. The signal-injected NOPA is phase insensitive, and no relative phase-locking is required between the signal and the pump fields [6]. In order to measure the quantum correlation between the amplitude (phase) quadratures, we needed to precisely lock the relative phase at zero or half \(\pi\) between the signal (idler) mode and the corresponding local oscillator (LO) beam in two sets of HD simultaneously. As shown in Section 2, the fluctuations of the relative phase projected noise from the anti-squeezed combined quadratures into the squeezed combined quadratures. For the phase-locking of the half \(\pi\) relative phase, the direct current (DC) signals from each homodyne detection were used as error signals, which were further processed by proportional-integral-derivative controllers (PID) (Stanford Research Systems Inc., Sunnyvale, CA, USA) and amplified to drive the piezoelectric transducers PZT1 and PZT2. For the phase locking of the zero relative phase, we employed a dither-locking approach. In the phase-locking loop, a low pass filter with a 100 kHz cut-off frequency filtered the high-frequency background noise and improved the signal-to-noise ratio.