# Characterization of a Quantum Random Number Generator Based on Vacuum Fluctuations
## 4 Results and Discussion
### Estimation of Entropy Due to Detection Unbalancing  
As shown through Equation (12), an increase in the QCNR can be made by minimizing the electronic noise, for example, by controlling the setup temperature or using a detector with a lower noise floor. Unfortunately, such methods increase costs and introduce complexity to the implementation. Alternatively, a simple increase in the laser's optical power can be made, since the shot noise variance increases linearly with the impinging power. Unfortunately, this is not always possible, as excess noise quadratically dependent on the LO power also appears as a consequence of an unbalanced detection, either due to a BS imbalance, different path losses, or different photodetector's efficiencies between each path. To experimentally verify the presence of this quadratic power dependency, the detector's characterization curve was obtained for an unbalanced condition by introducing a VOA before the BS1 and varying the LO optical power, as can be observed in Figure 4. Here, each variance was evaluated over \(15.625\times 10^{7}\) noise samples, and the electronic noise level was assumed to remain constant throughout the experimental process. As expected by the theoretical description, the variance, \(\sigma^{2}\), follows a quadratic dependency with the input power at BS2, \(P_{\text{LO}}\)[42]:  
\[\sigma^{2}=aP_{\text{LO}}^{2}+bP_{\text{LO}}+c. \tag{18}\]  
A regression with \(r^{2}=0.9853\) allows to obtain the coefficients: \(c=7.652\times 10^{-7}\,\text{V}^{2}\), \(b=2.708\times 10^{-7}\,\text{V}^{2}/\text{mW}\), \(a=3.277\times 10^{-7}\,\text{V}^{2}/\text{mW}^{2}\). This clearly highlights the quadratic dependency of the variance with the input power and illustrates the importance of achieving a well-balanced detection. As represented in Figure 4, the electronic noise level is independent of the optical power. Nonetheless, for an unbalanced detection, two distinct regions emerge. For low LO powers, the variance is dominated by these intrinsic electronic noise contributions and its increase can be mainly attributed to shot noise, which grows linearly with \(P_{\text{LO}}\). For higher impinging power, however, the excess RIN noise becomes more significant and constitutes the majority of the total noise due to the observed quadratic dependency. As described in Section 2, its impact is dependent on the balancing condition of the detection scheme. At a given LO power, this manifests itself as an additional entropy contribution, \(H_{\text{unbal}}\), dependent on the scheme unbalancing, that should be correctly quantified to avoid overestimating the available quantum entropy, thus compromising the generator output. With this objective, we assessed the entropy contributions as a function of the homodyne detection unbalance, as represented in Figure 5.  
Here, the LO power was fixed at approximately 1.33 dBm to avoid saturating the TIA output signal. Posteriorly, the path attenuation was changed through the employment of a VOA, so that we can iterate over the detection's asymmetry \(\Delta P=\frac{P_{1}}{P_{\text{total}}}\), defined as the ratio between the optical power measured at the output without the VOA, \(P_{1}\), and the total impinging power, \(P_{\text{total}}\). It should be noted that this is theoretically distinct from simply varying the BS transmittance since the total power that reaches the photodetectors decreases with the additional attenuation. The electronic and quantum noise contribution were calculated as described in Section 3 for practical asymmetries of 50.75%, 55.28%, 59.72%, 64.90%, and 68.93%, yielding respective quantum contributions of 0.0836, 0.1085, 0.6793, 1.083, and 1.298 bit from a total entropy of 5 bits. Additionally, a comparison with the theoretically expected values, obtained through Equation (11), is also presented, considering a measured RIN of -135 dB/Hz. The measured entropy, as can be seen in Figure 5a, appears to increase with the detection unbalancing.