# Low-Frequency Communication Based on Rydberg-Atom Receiver
## 3 Theory of Signal Detection  
The EIT and AC Stark shift are the primary quantum optical phenomena used in Rydberg atoms to detect low-frequency signals. When a weak probe laser beam (852 nm) traverses the vapor cell containing alkali metal gas, the atoms typically absorb the probe laser, potentially exciting them from the ground state \(|6S_{1/2},F=4>\) to the intermediate state \(|6P_{3/2},F^{\prime}=5>\). Simultaneously, a more intense coupling laser (509 nm) induces resonance between the intermediate state and the Rydberg state \(|63D_{5/2}>\), thereby re  
Figure 1: Schematic of the experimental setup. An 852 nm probe laser and an identical reference beam propagated in parallel through the vapor cell, which incorporated a pair of copper electrode plates spaced 18 mm apart. A 509 nm Rydberg laser counter-propagated and overlapped with the probe laser but not the reference beam. The transmissions of the probe and reference beams were directed into a differential photodetector to measure the transmission difference. The experimental process commenced with the generation of low-frequency signals using a whip antenna connected to one of the copper plates. These signals were coupled into the vapor cell, where they interacted with the Rydberg atoms. The EIT effect was generated by the probe and Rydberg lasers, enabling the detection of low-frequency signals through changes in the probe laser transmission. The detected signals were subsequently processed using a differential photodetector and an oscilloscope for data acquisition and analysis. ducing the probe laser's absorption and creating a transparent region for the probe beam. This phenomenon is known as the EIT effect. The EIT spectrum, as shown in Figure 2a, was obtained by locking the frequency of the probe laser and scanning the coupling laser. The peak of the spectrum corresponds to the transition from the intermediate state \(|6P_{3/2},F^{\prime}=5>\) to the Rydberg state \(|63D_{5/2}>\). The three-level Rydberg EIT scheme is shown on the insert figure of Figure 2a. When the low-frequency electromagnetic wave irradiates to the vapor cell via the whip antenna and electrode plates, the signal will be reflected on the specific area of EIT spectrum, shown as Figure 2b. The EIT spectrum shifts due to the electric field generated by the 100 kHz signal, a phenomenon known as AC Stark shift. The total electric field \(E_{tot}\) in vapor cell is given by:  
\[E_{tot}=E_{DC}+E_{sig}\big{(}cos\omega_{sig}+\varphi_{sig}\big{)} \tag{1}\]  
It is evident from Figure 2b that the intensity, frequency, and phase of the signal are fully reflected in the probe transmission. Consequently, information can be modulated onto the low-frequency carrier using digital modulation methods such as OOK, BPSK, and 2 FSK. The 100 kHz signal can be easily detected by locking the frequency of the coupling laser at the most sensitive point. When a strong DC field is applied to the parallel electrode plates inside the vapor cell, the EIT spectrum splits into three components, corresponding to different \(m_{j}\) magnetic substates (\(m_{j}=1/2\), \(3/2\), \(5/2\)) due to the stark shift, shown in Figure 2c. When transmitting the low-frequency signal, it becomes more apparent from Figure 2d that the signal is reflected in the EIT spectrum, particularly near the \(m_{j}=1/2\) region. Under these conditions, \(E_{tot}\) is modified as follow [23]:  
\[E_{tot}^{2}=E_{DC}^{2}+E_{sig}^{2}+2E_{DC}E_{sig}\big{(}cos\omega_{sig}+ \varphi_{sig}\big{)} \tag{2}\]  
The DC field acts as an amplifier, enhancing the signal and providing a higher signal-to-noise ratio (SNR) and sensitivity compared to detection methods without a DC field. As highlighted by [PERSON] et al. [25], the quantum internet depends on the ability to transmit quantum information over long distances with high fidelity.