# Characterization of a Quantum Random Number Generator Based on Vacuum Fluctuations
## 2 Theoretical Description  
In Figure 1, a schematic model of a homodyne detection scheme is illustrated. Here, the local oscillator, represented by the destruction operator \(\hat{a}_{\text{LO}}(t)\), is directed towards a lossless beamsplitter (BS), where one of the input ports is kept blocked so that only the vacuum state, \(\hat{\sigma}_{s}(t)\), is present at the second input port. Figure 1: Representation of the homodyne detector (HD) model considered. Here, each variable optical attenuator (VOA) is obtained from a lossless BS. Similarly, each photodetector can be modeled by introducing a lossless BS before a theoretical perfect photodetector. Considering a certain complex reflection coefficient, \(\mathcal{R}\), we can define the BS imbalance, \(\Delta\), as \(|\mathcal{R}|^{2}=(\frac{1}{2}+\Delta)\)[33]. Under these assumptions, the output signals, \(\hat{a}_{3,4}(t)\), can be described as given by the input-output relations for a lossless BS [34]:  
\[\hat{a}_{3}(t) =i\sqrt{\frac{1}{2}+\Delta}\;\hat{a}_{\text{LO}}(t)+\sqrt{\frac{1 }{2}-\Delta}\;\hat{v}_{s}(t) \tag{1}\] \[\hat{a}_{4}(t) =\sqrt{\frac{1}{2}-\Delta}\;\hat{a}_{\text{LO}}(t)+i\sqrt{\frac{1 }{2}+\Delta}\;\hat{v}_{s}(t).\]  
Subsequently, the output signals are attenuated by two VOAs, modeled as two lossless BS, and detected by two photodiodes. Here, the VOAs are introduced to allow for balancing of the HD, since neither the BS nor the photodetectors are ideal in a realistic application. In this framework, the efficiency of the photodetectors is modeled by imposing a virtual BS before a theoretical perfect photodetector. As illustrated, one of the output signals of the virtual BS will be lost while the other, \(\hat{d}_{3,4}(t)\), is directly received by the detector. Here, \(\hat{v}_{3,4}(t)\) and \(\hat{v}_{\text{VOA3},\text{VOA4}}(t)\) denote, respectively, the vacuum states of the VOA\({}_{3,4}\) and the respective photodiodes. Analogously to Equation (1):  
\[\hat{d}_{3}(t) =i\sqrt{\left(\frac{1}{2}+\Delta\right)\eta_{3}\eta_{\text{VOA3}} }\;\hat{a}_{\text{LO}}(t)+\sqrt{\left(\frac{1}{2}-\Delta\right)\eta_{3}\eta_{ \text{VOA3}}}\;\hat{v}_{s}(t) \tag{2}\] \[+i\sqrt{\eta_{3}(1-\eta_{\text{VOA3}})}\;\hat{v}_{\text{VOA3}}(t )+i\sqrt{1-\eta_{3}}\;\hat{v}_{3}(t)\]  
\[\hat{d}_{4}(t) =i\sqrt{\left(\frac{1}{2}-\Delta\right)\eta_{4}\eta_{\text{VOA}} }\;\hat{a}_{\text{LO}}(t)+\sqrt{\left(\frac{1}{2}+\Delta\right)\eta_{4}\eta_{ \text{VOA}}}\;\hat{v}_{s}(t)\] \[+i\sqrt{\eta_{4}(1-\eta_{\text{VOA4}})}\;\hat{v}_{\text{VOA4}}(t )+i\sqrt{1-\eta_{4}}\;\hat{v}_{4}(t),\]  
where \(\eta_{3,4}\) denote the quantum efficiency of each photodetector, and \(\eta_{\text{VOA3},\text{VOA4}}\) express the transmissivities of each VOA [33]. The photocurrents yielded in each photodetector, \(\hat{i}_{3,4}(t)\), are posteriorly subtracted and amplified by a transimpedance amplifier (TIA). Thus, in an ideally balanced scenario, any fluctuations common to both detectors are cancelled and only fluctuations in photon arrivals are measured, the so-called shot noise [33, 35]. In reality, the presence of excess noise is unavoidable, and the output signal will always contain at least electronic noise. The resultant output voltage, \(\hat{V}_{H}(t)\), is given by the convolution, \(\otimes\), between the difference current, \(\hat{i}_{H}\), and the detector's impulse response function, \(h(t)\):  
\[ \begin{cases}\hat{V}_{H}(t)=G_{\text{TIA}}\big{[}\hat{i}_{e}(t)+\hat{i}_{H}(t) \big{]}\otimes h(t)\\ \hat{i}_{H}(t)=q\big{[}\hat{d}_{3}^{\intercal}(t)\hat{d}_{3}(t)-\hat{d}_{4}^{ \intercal}(t)\hat{d}_{4}(t)\big{]}\end{cases} , \tag{3}\]  
where \(\hat{i}_{e}\) is the amplifier electronic noise, \(G_{\text{TIA}}\) is the transimpedance amplifier gain, \(q\) is the charge of the electron, and \(\hat{d}_{3,4}^{\intercal}(t)\hat{d}_{3,4}(t)\) corresponds to the photon flux of each output beam. Here, the normalized impulse response function, \(h(t)\), should consider both the impact of finite bandwidth of the photodetectors and the TIA's time response.