# Influence of Magnetic Field on Amplification without Inversion Induced by a Femtosecond Pulse Train
## 3 Results and Discussion  
The optical properties of the atomic system can be controlled by changing the magnetic field strength. As magnetic field B increases, the population is redistributed between Zeeman sublevels \(|1\rangle\) and \(|3\rangle\), resulting in destructive quantum interference between the ground and excited states [24]. The absorption and population variation of the system are shown in Figures 2 and 3, respectively. The repetition rate of the pulse train is assumed to be 0.36 GHz, corresponding to \(T_{R}=2.8\) ns. The repetition rate of the absorption with detuning is shown in Figure 2 and is dependent on the pulse repetition rate. Without the magnetic field, the AWI phenomenon also occurs when the fs pulse train interacts with the atomic system, but the system has a smaller gain. However, with the presence of the magnetic field, it is clearly shown that the AWI effect can be significantly enhanced by the magnetic field. Because of the assistance of the magnetic field, a larger population is transferred from level \(|3\rangle\) to level \(|1\rangle\) during the detuning where the gain (G\({}_{21}\)\(\propto\) Im\(\rho_{21}\)) is maximized, and thus the population inversion (\(\rho_{22}-\rho_{11}\)) is reduced, but the gain improves, as shown in Figure 3. Since Im\(\rho_{21}\) in Figure 2a and Im\(\rho_{23}\) in Figure 2b are mirror-symmetric, the same conclusion for levels \(|1\rangle\) and \(|2\rangle\) are also available for levels \(|3\rangle\) and \(|2\rangle\). In order to describe the influence caused by the magnetic field more clearly, the variation of amplification and population inversion (\(\rho_{22}-\rho_{11}\)) for different magnetic field strengths and laser detuning is illustrated in Figure 4. The magnetic field and detuning jointly affect the atomic system gain. When the magnetic field changes, the detuning at which the gain achieves its maximum also varies. As the maximum gain at different magnetic fields was not obtained at the same detuning value, a projection was used to verify the effects induced by the magnetic field (see Figure 4b,d). It can be concluded that as the magnetic field increases, the system gain improves while the population inversion decreases during the detuning where the gain is maximized. The number of pulses is also an important parameter. The influence of whether the magnetic field is added to the system on the gain variation with the pulse number is shown in Figure 5. When there is no magnetic field, the system's gain gradually decreases as  
Figure 3: Variation of the population inversion of (**a**) (\(\rho_{22}-\rho_{11}\)) and (**b**) (\(\rho_{22}-\rho_{33}\)) with detuning at different magnetic fields. Variation of the population (\(\rho_{22},\ \rho_{33},\ \rho_{11}\)) with detuning when (**c**–**e**) are B = 0 G, 30 G, 100 G, respectively. Other parameters are the same as those in Figure 2.  
Figure 2: Variation of (**a**) Im\(\rho_{21}\) and (**b**) Im\(\rho_{23}\) with detuning for different magnetic fields. Other parameters are chosen as follows: N = 400, \(T_{R}\) = 2.8 ns and \(\theta=\pi/15\). the pulse number increases. It should be noted that the system at a steady state has no gain, and all populations are equally distributed on ground state levels \(|1\rangle\) and \(|3\rangle\). In this case, the system no longer absorbs and amplifies the probe field, which can be considered the EIT phenomenon. However, with the presence of a magnetic field, the gain of the three-level system increases rapidly as the pulses accumulate, and it reaches its maximum after several cycles, while the population of the system remains almost completely in its ground state \(|1\rangle\). Then, we consider the influence of the pulse repetition period. As the pulse repetition period is reduced, the population inversion decreases, while the gain of the system improves.