## 3 Results and Discussion
It can be seen from Figure 3a that when the pump light is 1.55 eV, the amplitude at zero time delay increases with temperature, which corresponds to the increase of charge transfer transition probability of Mn\({}^{+3}\) ions. Because of [PERSON]'s rule, the transition probability of this single particle charge transfer transition under the influence of spin interactions can be divided into two series, high-spin transition and low-spin transition [19]. As the name implies, high-spin (HS) transition means that the total spin number of ions increases after charge transfer transition, and the corresponding spectral weight is \(S_{1}S_{2}S\to S_{1}\pm\frac{1}{2}S_{2}\mp\frac{1}{2}S\). Low-spin (LS) transition, \(S_{1}S_{2}S\to S_{1}-\frac{1}{2}S_{2}-\frac{1}{2}S\), means that the total spin number of ions decreases after charge transfer [16]. For B site Mott insulators, the dependence of low and high spin transitions on temperature is opposite, so 1.55 eV and 3.1 eV correspond to two different transitions [25]. For the Fe\({}^{3+}\) ion-based oxides we deal only with the LS transitions \(\frac{5}{2}\frac{5}{2}\to 22\); therefore, the 1.55 eV pump for the Mn\({}^{3+}\) d-d transition is corresponding to the HS transition, which means that the temperature dependent energy shift in manganite should be related to the spin structure [16]. The evolution of the spin structure could be reflected in the ions' optical properties if their magnetism is connected with their electronic properties. Therefore, we can except that the spectral weight transfer is strongly correlated to the magnetic phase transition for each 3d cation configuration, which is wavelength selected in Bi\({}_{2}\)FeMnO\({}_{6}\) compounds. A possible origin of the difference that could be considered is their different exchange energy. The absorption of a 1.55 eV photon creates a pair of Mn\({}^{2+}\) and Mn\({}^{4+}\) ions for the short range ordered Mn cations. The blue shift for the spectral weight transfer for the 3.1 eV pump indicates that the creation of Fe\({}^{2+}\) and Fe\({}^{4+}\) ions are suppressed for the increasing of the super-exchange interaction intensity. The low-spin transition is dominated in this Fe 3d charge transfer transition because of the Fe-O-Fe configuration and the anti-ferromagnetic structure. Thus, the charge transfer transition in Bi\({}_{2}\)FeMnO\({}_{6}\) for 3.1 eV is dominated by the d-d transition for the Fe 3d cation, which is also sensitive to the Neel temperature in other ferrites such as LuFe\({}_{2}\)O\({}_{4}\)[18]. The interpretation of different degrees of freedom in Bi\({}_{2}\)FeMnO\({}_{6}\) are also consisted with the complicated phase transitions. As it's shown in Figure 4a,b, the temperature is dependent on the time constant of the electron-phonon coupling and the spin-lattice coupling process. Except the enhancement of the magnetization as shown in Figure 3b, there are two apparent transitions in the d(M/H)/dT curve of the sample. Although the electro-phonon interaction time varies slightly with the temperature, it does not change abruptly with the phase transition temperature. It indicates that the structure and ferroelectric properties of Bi\({}_{2}\)FeMnO\({}_{6}\) samples do not change when the magnetic properties of the samples change [16]. The temperature dependence of spin-lattice coupling time under 3.1 eV pumping and 1.55 eV pumping is obviously different. For 1.55 eV pumping, the slow process occurs at the Mn-O-Mn antiferromagnetic order temperature of 240 K, indicating that the interaction between the magnetic order and the lattice occurs. While at 175 K, ferromagnetic phase transition temperature of Fe-O-Mn changes from \(>\)10 ps to about 5 ps, which indicates that the interaction intensity between spin and lattice increases with the increase of magnetic ordering in Mn.