# Characterization of a Quantum Random Number Generator Based on Vacuum Fluctuations
## 2 Theoretical Description
Nonetheless, this function can be mainly characterized by the latter contribution due to the much lower bandwidths typically presented by the amplifier. Consequently, in this analysis, an equal ideal response, \(h_{p}(t)\), was considered for both photodiodes, such that \(h_{p}(t)=\delta(t)\). Under these approximations, the impulse response function can be approximated in the frequency domain by a Butterworth filter:  
\[\left|H(w)\right|^{2}=\frac{1}{1+\big{(}\frac{w}{2\pi\Delta f}\big{)}^{2n}}, \tag{4}\]where \(n\) is the filter-order, and \(\Delta f\) is the detector's bandwidth [36]. In its turn, the strong coherent laser signal can be described as:  
\[\hat{a}_{\text{LO}}(t)=\hat{\mathtt{a}}(t)\exp(i(w_{LO}t+\theta(t))), \tag{5}\]  
where \(\hat{\mathtt{a}}(t)\) is the time-dependent photon flux, \(w_{LO}\) the frequency and \(\theta(t)\) the initial phase of the laser [33]. Here, the time-dependence of the initial phase implies the existence of phase noise. This, however, does not result in an additional noise contribution as long as no phase shift, \(\Delta\phi\), is present between the output beams of the beamsplitter. In this analysis, identical optical path lengths are considered for both output arms, such that \(\Delta\phi=0\), and its contribution can be disregarded. Considering a null mean electronic current, \(\langle\hat{i}_{e}(t)\rangle\), and \(\hat{\mathtt{a}}(t)=\sqrt{F+\Delta f_{lo}(t)}\), where \(F\) is the time-independent mean photon flux and \(\Delta f_{lo}(t)\) denotes the intensity fluctuations, the expected output voltage is given by:  
\[\langle\hat{V}_{H}(t)\rangle=qG_{\text{TIA}}\gamma F\otimes h(t), \tag{6}\]  
where:  
\[\gamma=\Big{(}\frac{1}{2}+\Delta\Big{)}\eta_{3}\eta_{\text{VOA3}}-\Big{(} \frac{1}{2}-\Delta\Big{)}\eta_{4}\eta_{\text{VOA4}}. \tag{7}\]  
The voltage variance, \(\sigma_{\text{H}}^{2}(t)\), can be obtained from the autocovariance function, \(K(\tau)\), evaluated at \(\tau=0\). For a ergodic stationary stochastic process, and assuming that:  
\[ \begin{cases}\langle i_{e}(t_{1})\hat{i}_{e}(t_{2})\rangle=2\frac{k_{\text{B} }T}{K}\delta(t_{1}-t_{2})\\ \langle\Delta f_{lo}(t_{1})\Delta f_{lo}(t_{2})\rangle=\text{RIN}F^{2}\delta(t _{1}-t_{2}),\end{cases}  \tag{8}\]  
where \(k_{\text{B}}\) is the Boltzmann constant, \(T\) the temperature, \(R\) the load resistance of the TIA, and RIN represents the average relative intensity noise (RIN) over the detector's bandwidth [36], the autocovariance function can be calculated through Equations (3) and (6) as:  
\[ \begin{split} K(\tau)=\Big{[}2G_{\text{TIA}}^{2}\frac{k_{\text{B }}T}{R}&+q^{2}G_{\text{TIA}}^{2}\beta F\\ &+q^{2}G_{\text{TIA}}^{2}\text{RIN}\gamma^{2}F^{2}\Big{]}\int_{- \infty}^{+\infty}d\tau^{\prime}\,h(t-\tau^{\prime})h(t-\tau^{\prime}+\tau), \end{split}  \tag{9}\]  
where:  
\[\beta=\Big{(}\frac{1}{2}+\Delta\Big{)}\eta_{3}\eta_{\text{VOA3}}+\Big{(} \frac{1}{2}-\Delta\Big{)}\eta_{4}\eta_{\text{VOA4}}. \tag{10}\]  
This cross-correlation can be greatly simplified through the Wiener-Khinchin theorem. Abiding by the previous definitions, we obtain the analytical expression:  
\[\sigma_{\text{H}}^{2}(t=0)=\frac{2\pi}{3}G_{\text{TIA}}^{2}\Delta f\Big{[}2 \frac{k_{\text{B}}T}{R}+q^{2}\beta F+q^{2}\text{RIN}\gamma^{2}F^{2}\Big{]}+ \frac{\delta_{q}^{2}}{12}, \tag{11}\]  
for a Butterworth's filter of order \(n=3\), considering \(t=\tau=0\). Here, we considered the quantization error of an analog to digital converter (ADC) with a bin width of \(\delta_{q}\), which yields a quantization error variance of \(\frac{\delta_{q}^{2}}{12}\)[37]. Finally, it is possible to rewrite the Equations (6) and (11) in function of the wavelength-dependent responsivity, \(\mathcal{R}(\lambda)\), and the optical power of the LO, \(P_{LO}\). Assuming that \(\eta_{3}=\eta_{4}\):  
\[ \begin{cases}\langle\hat{V}_{H}(t)\rangle=G_{\text{TIA}}\gamma^{\prime}\\ \sigma_{\text{H}}^{2}(t=0)=\frac{2\pi}{3}G_{\text{TIA}}^{2}\Delta f\Big{[}2 \frac{k_{\text{B}}T}{K}+q\beta^{\prime}+q\text{RIN}\gamma^{\prime 2}\Big{]}+ \frac{\delta_{q}^{2}}{12},\end{cases}