## 2 Frequency Shift of Intensity Modulators  
In this section, we will recap the frequency shift of the IMs and test experimentally to show a side channel in the frequency domain. There are several kinds of IMs such as the Mach-Zehnder type electro-optical intensity modulators (EOIMs), electro-absorption modulators (EAMs), and acousto-optical modulators (AOMs). EOIMs, especially LiNbO\({}_{3}\)-based devices, possess excellent performance of wavelength-independent modulation characteristics, excellent extinction performance (typically 20 dB), and low insertion losses (typically 5 dB) [54]. LiNbO\({}_{3}\)-based EOIMs work using the principle of interference, which is controlled by modulating the optical phase. The incoming light is coupled into a waveguide and then split into two paths of a Mach-Zehnder interferometer equally, which finally interfere at an output coupler. The two arms made of lithium-niobate will induce a phase change when the modulation voltages are applied. Accordingly, the intensity and phase of the output light will be modulated after interference depending on the applied modulation voltages. Assuming voltages \(V_{1}(t)\) and \(V_{2}(t)\) are applied to the two arms separately with the input field of intensity \(E_{0}\) and frequency \(\omega_{0}\), the output field can be written as [52]  
\[E_{\text{out}}(t)=E_{0}\text{cos}[\Delta\varphi(t)]e^{i[\omega_{0}t+\varphi(t)]}, \tag{1}\]  
where \(\Delta\varphi(t)=[\gamma V_{1}(t)+\varphi_{1}-\gamma V_{2}(t)-\varphi_{2}]/2\), \(\varphi(t)=[\gamma V_{1}(t)+\varphi_{1}+\gamma V_{2}(t)+\varphi_{2}]/2\). Here, \(\gamma=\pi/V_{\pi}\) is the voltage-to-phase conversion coefficient for two arms, and \(\varphi_{1}\) and \(\varphi_{2}\) are the static phases which we will omit for simplicity. Here, \(V_{\pi}\) is the half-wave voltage that is required to change the phase in one modulator arm by \(\pi\) radians. The output intensity is given by  
\[P_{\text{out}}(t)=|E_{\text{out}}(t)|^{2}=\frac{P_{0}}{2}\big{[}1+\text{cos}[ \gamma V(t)]\big{]}, \tag{2}\]  
where \(V(t)=V_{1}(t)-V_{2}(t)\). Here, \(P_{0}\) is the input optical power. The phase is maintained and the intensity is determined by Equation (2) on the condition that the two modulator arms are driven by the same amount, but in opposite directions (i.e., \(V_{1}(t)=-V_{2}(t)\)), which is known as the balanced driving or a push-pull operation. When \(V(t)\) is constant we will get pure intensity modulation without a frequency shift. However, once \(V(t)\) is not a constant anymore, something unexpected arises in the output field. For example, if \(V_{1}(t)=-V_{2}(t)=V_{0}+kt\), the output field can be expressed as [52]  
\[E_{\text{out}}=\frac{E_{0}}{2}\Big{[}e^{i[(\omega_{0}+\gamma k)t+\gamma V_{0}] }+e^{i[(\omega_{0}-\gamma k)t+\gamma V_{0}]}\Big{]}. \tag{3}\]  
From Equation (3), we can see a frequency shift of the light pulses with \(\pm\omega_{m}=\pm\gamma k\) compared with the original frequency \(\omega_{0}\). The frequency shift of the output field becomes more confusing when the modulation voltages are more complicated. To analyze the spectrum of the output field, the fast Fourier transform method can be used. To evaluate the frequency shift of different intensity pulses, we tested it experimentally. The 33-MHz optical pulses with 1 ns pulse width were produced by a CW laser, modulated by an IM, and measured by an optical-spectrum analyzer. The IM was driven by an arbitrary waveform generator with an electric signal amplified by an electric amplifier. The measurement was taken before the fixed attenuation where the photon number follows the same distribution with the emitting pulses at the single-photon level. Figure 1a illustrates the wavelength spectrum of three signal pulses, where the intensity ratio is taken from the SNS experiment [36] as 0.1: 0.384: 0.447 (\(\mu_{a}=0.1\), \(\mu_{b}=0.384\), \(\mu_{z}=0.447\)), and the original continuous light. The normalized intensity probability distributions are shown in Figure 1b to distinguish the difference.