## 2 Delay-Free Phase Noise QRNG Scheme  
### Principle of Scheme  
The schematic setup of our scheme is shown in Figure 1. Two laser diodes with very close center wavelength are used as random source. The intensity of lasers are carefully tuned by variable optical attenuator (VOA) to the exact same level, then optical signals of two lasers are injected into a 50:50 beam splitter (BS). Electric fields at the input ports (P1, P2) of the BS could be written as:  
\[E_{1}(t) = \mathbf{E}_{1}(t)\exp\left[i\left(\omega_{1}t+\varphi_{1}(t) \right)\right] \tag{1}\] \[E_{2}(t) = \mathbf{E}_{2}(t)\exp\left[i\left(\omega_{2}t+\varphi_{2}(t) \right)\right] \tag{2}\]Since the amplitude of laser is tunable and could be very stable during the experiment, the amplitude part of field could be regarded as a constant in our scheme, which means \(\mathbf{E}_{i}(t)=\mathbf{E}_{i},i=1,2\). Two signals are interfered at the beam splitter, thus field at the output ports (P3, P4) of the BS could be written as:  
\[E_{3}(t) = \sqrt{T}E_{1}(t)+\sqrt{1-T}E_{2}(t) \tag{3}\] \[E_{4}(t) = \sqrt{1-T}E_{1}(t)-\sqrt{T}E_{2}(t) \tag{4}\]  
where \(T\) refers to the transmittance, thus \(R=1-T\) refers to the reflectivity of the beam splitter. Signals of port P3 and P4 take \(t_{1},t_{2}\) time before arriving at the detectors (D1, D2) respectively. Hence, the intensity of electric signal at detector D1 (which comes from port P3) is:  
\[ \begin{array}{lll}I_{D1}(t+t_{1})&=&E_{3}(t+t_{1})E_{3}^{*}(t+t_{1})\\ &=&TE_{1}^{2}+(1-T)E_{2}^{2}\\ &&+2\sqrt{T(1-T)}E_{1}E_{2}\exp\left[i\left(\omega_{1}(t+t_{1})+\varphi_{1}(t+t _{1})-\omega_{2}(t+t_{1})-\varphi_{2}(t+t_{1})\right)\right]\end{array}  \tag{5}\]  
while the electric signal at detector D2 (which comes from port P4) has a similar expression:  
\[ \begin{array}{lll}I_{D2}(t+t_{2})&=&(1-T)E_{1}^{2}+TE_{2}^{2}\\ &&-2\sqrt{T(1-T)}E_{1}E_{2}\exp\left[i\left(\omega_{1}(t+t_{2})+\varphi_{1}(t+ t_{2})-\omega_{2}(t+t_{2})-\varphi_{2}(t+t_{2})\right)\right]\end{array}  \tag{6}\]  
Apparently \(I_{D1}(t)\) includes both DC term and AC term:  
\[I_{D1}(t)=[I_{D1}(t)]_{DC}+[I_{D1}(t)]_{AC} \tag{7}\]  
According to (5), the DC term is \(TE_{1}^{2}+(1-T)E_{2}^{2}\), and AC term is the rest. Therefore, if we use AC coupling photo-detectors in our scheme, we can only keep the AC terms in the following analysis and processing. After being detected by photo-detectors, the optical signal turns into an electric signal and the intensity \(I_{D1}(t)\) is converted into optoelectric current \(i_{D1}(t)\) proportionally:  
\[ \begin{array}{lll}[i_{D1}(t)]_{AC}&\propto&[I_{D1}(t)]_{AC}\\ &=&+2\sqrt{T(1-T)}E_{1}E_{2}\exp\left[i\left((\omega_{1}-\omega_{2})(t+t_{1}) +(\varphi_{1}(t+t_{1})-\varphi_{2}(t+t_{1}))\right)\right]\end{array}  \tag{8}\]  
\[ \begin{array}{lll}[i_{D2}(t)]_{AC}&\propto&[I_{D2}(t)]_{AC}\\ &=&-2\sqrt{T(1-T)}E_{1}E_{2}\exp\left[i\left((\omega_{1}-\omega_{2})(t+t_{2}) +(\varphi_{1}(t+t_{2})-\varphi_{2}(t+t_{2}))\right)\right]\end{array}  \tag{9}\]  
Finally, the voltage signals \(V_{D1}(t),V_{D2}(t)\) are combined in a mixer by frequency mixing (only the real part is considered here), and the new signal is denoted as \(V_{mix}\)  
Figure 1: The schematic setup of phase noise QRNG scheme with two lasers. Laser Diode: Distribution Feedback Laser Diode with tunable center wavelength (as random source); VOA: Variable Optical Attenuator; BS: 50:50 Beam Splitter; PD1, PD2: Photodiode detectors; MIX: Frequency Mixer; FIL: Band-pass Filter; ADC: Analog-to-Digital Converter; EXT: Randomness Extractor.