# Influence of Magnetic Field on Amplification without Inversion Induced by a Femtosecond Pulse Train
## 2 Theoretical Model  
To simplify the analysis, a two-level system with \(m_{F}=\pm 1\) is adopted, which can be easily found in atoms, such as in the \(5^{2}S_{1/2}-5^{2}P_{1/2}\) transition of \({}^{87}\)Rb, as shown in Figure 1. Under a static external magnetic field, a two-level system splits into a three-level \(\Lambda\) system because of the Zeeman effect, which is the simplest structure for our study. Assuming that the probe field is weak enough not to affect the preparation of the atomic state, we then consider this three-level \(\Lambda\) system interacting with the fs pulse train. The pulse train is assumed to be linearly polarized, which can be decomposed into left-circularly andright-circularly polarized components coupled with the transitions from \(|1\rangle\) to \(|2\rangle\) and \(|3\rangle\) to \(|2\rangle\), respectively. The Hamiltonian of this system is  
\[H=H_{0}+H_{int} \tag{1}\]  
where \(H_{0}\) is the intrinsic Hamiltonian of the atomic system and \(H_{int}=-\mu E(t)\), representing the interacting Hamiltonian where \(\mu\) is the electric dipole moment and \(E(t)\) is the electric field. Here, \(E(t)\) can be expressed as:  
\[E(t)=\left[\sum_{n=0}^{N}\mathcal{E}(t-nT_{R})e^{in\varphi_{R}}\right]e^{i \omega_{L}t} \tag{2}\]  
where \(N\) is the number of pulses, \(T_{R}\) is the pulse repetition period, \(\varphi_{R}=0\) is the phase shift, and \(\omega_{L}\) is the central laser frequency. We assume the envelope of the pulse train to be Gaussian, \(\varepsilon(t)=\hat{\varepsilon}E_{0}e^{-\pi\left(\frac{t}{\varphi}\right)^{2}}\), where \(\hat{\varepsilon}\) is the polarization vector, \(E_{0}\) is the peak electric field intensity, \(\sigma=200\) fs is the pulse duration, and the Fourier limit of the frequency bandwidth is 2.205 THz. The terms \(|1\rangle\) and \(|3\rangle\) are Zeeman sublevels, and the shifts caused by the magnetic field B are (with \(m_{F}=\pm 1\)) \(\Delta_{B}=\pm\frac{\mu_{B}m_{F}g_{F}B}{2}\), Bohr magneton \(\mu_{B}=9.274\times 10^{-24}\) J/T, Lande factor \(g_{F}=-\frac{1}{2}\), \(\hbar=1.054\times 10^{-34}\) J \(\cdot\) s, so \(\Delta_{B}=(2\pi\cdot 0.7\) MHz/G\()\times B\). Under the rotating-wave approximation, the Hamiltonian of this system can be written as:  
\[H=\left( \begin{array}{ccc}-\hbar(\Delta+\Delta_{B})&-\frac{\hbar}{2}\Omega_ {21}^{*}&0\\ -\frac{\hbar}{2}\Omega_{21}&0&-\frac{\hbar}{2}\Omega_{23}\\ 0&-\frac{\hbar}{2}\Omega_{23}^{*}&-\hbar(\Delta-\Delta_{B})\end{array} \right) \tag{3}\]  
where \(\Delta\) is the detuning of the central laser frequency from the atomic transition frequency and \(\Omega_{ij}=\frac{\hbar_{ij}E_{0}}{\hbar}\) (\(i\
eq j\)) is the Rabi frequency of the corresponding transition of levels \(|i\rangle\leftrightarrow|j\rangle\). Assuming that the Rabi frequency is a real parameter, it can be considered that \(\Omega_{21}=\Omega_{21}^{*}=\Omega_{23}=\Omega_{23}^{*}=\Omega\). The pulse area is expressed as \(\theta=\int_{-\infty}^{+\infty}\Omega\frac{E(t)}{E_{0}}dt\). To describe the dynamic evolution of this system, we used the density matrix equation [18]:  
\[i\hbar\frac{\partial\rho}{\partial t}=[H,\rho]+\mathcal{L}\rho, \tag{4}\]  
where \(\mathcal{L}\rho\) is the damping term in Langevin form. Figure 1: Theoretical scheme of two-level \({}^{87}\)Rb sublevels under Zeeman splitting. The nuclear spin quantum number of \({}^{87}\)Rb is \(I=3/2\). The spontaneous relaxation time of \({}^{87}\)Rb is \(T_{1}=28\) ns. The Zeeman splitting of the F = 2 sublevels is not shown. For F = 1, \(g_{F}=-1/2\); when B = 100 G, \(\Delta_{B}=2\pi\cdot\)70 MHz. Combining Equations (3) and (4), the dynamical evolution of this system is given as:  
\[