## 3 Results and Discussion  
Figure 2a shows the 1.55 eV pump-probe signal of Bi\({}_{2}\)FeMnO\({}_{6}\) (BFMO) and its temperature dependence. It is different from the nonlinear two-photon absorption of BiFeO\({}_{3}\) with a 1.55 eV pump described elsewhere. The signal of the sample is negative and proportional to the pump power, so they represent the charge transfer transition probabilities of 1.55 eV and 3.1 eV for Bi\({}_{2}\)FeMnO\({}_{6}\) samples, respectively [22]. Figure 2: **(a)** The time resolved \(\Delta\)R of Bi\({}_{2}\)FeMnO\({}_{6}\) film with a 1.55 eV pump and 1.55 eV probe setup. The vertical direction is shift with equal magnitude for clarity. **(b)** The time resolved \(\Delta\)R of Bi\({}_{2}\)FeMnO\({}_{6}\) film with 3.1 eV pump and 1.55 eV probe setup, the vertical direction is shift with equal magnitude for clarity. Figure 1: **(a)** The reflection spectrum of Bi\({}_{2}\)FeMnO\({}_{6}\) film. A reflection spectrum of Si was used as a reference, and the sample signal is multiplied by 5 times for a visual guide. **(b)** Schematically shows the electronic band structure of Bi\({}_{2}\)FeMnO\({}_{6}\). A convolution of the Gaussian pulse G(\(t\)) with a bi-exponential decay function is utilized to analyze the electron dynamics phenomenologically,  
\[\vartriangle\mathrm{R}=\left[A_{1}\exp(-\frac{t}{\tau_{e-ph}})+A_{2}\exp(-\frac{t }{\tau_{s-ph}})+A_{3}\right]\otimes G(t) \tag{1}\]  
where \(A_{1}\) and \(A_{2}\) are the amplitudes of the fast and slow components mentioned above, respectively. \(\tau_{e-ph}\) and \(\tau_{s-ph}\) are the relaxation time constants of the electron-phonon and spin-lattice interactions, respectively. \(A_{3}\) is a flat background, which recovers to the ground state after a much longer time delay (hundreds of picoseconds-several nanoseconds) [17]. Figure 2b shows the temperature dependent 3.1 eV pump, 1.55 eV probe time resolved reflectivity. For the 1.55 eV pump, the peak \(\Delta\)R signal increased gradually when the temperature decreased from 290 K to 120 K, which is similar to the blue shift effect of d-d transition energy gap in YMnO\({}_{3}\) thin film material [23]. The blue-shift of the absorption band with the temperature is related to the emergence of the magnon and the optical phonon side bands, the emergence of the phonon side bands results in the effect of the temperature-dependent shift of the absorption band [24]. For 3.1 eV pumping, the peak \(\Delta\)R signal decreases with the decrease of temperature, which is similar to that of d-d transition signal in LuFe\({}_{2}\)O\({}_{4}\) sample with the change of Neel temperature [18]. Therefore, the corresponding transition of 3.1 eV and 1.55 eV in Bi\({}_{2}\)FeMnO\({}_{6}\) samples corresponds to the d-d transition of different element ions. The negative response of the transient reflectivity shows optically induced absorption behavior of both the 1.55 eV and the 3.1 eV pump scenarios. Thus, the instantaneous reflectivity represents the excited state population of the carriers. Figure 3a shows the temperature dependent reflectivity values at zero delay time for both 1.55 eV and 3.1 eV pump pulses. Figure 3b shows the temperature dependent magnetization of the film, which indicated that the total sample magnetization become stronger when the temperature went down. From the d (M/H)/dT signal, which is the derivative of the (M/H)/T signal of the film, which is sensitive to the magnetic phase transition temperature. It can be concluded that with the decrease of temperature, two magnetic phase transition processes took place. The phase transition temperatures of 200 K and 150 K are considered to correspond with the ordered phase transition of the Fe-O-Mn ferromagnetic phase and Mn-O-Mn antiferromagnetic phase respectively [8]. The ordered temperature of Fe-O-Fe is 440 K. The possible reasons for the magnetic phase transition are the random distribution of elements in the lattice, the growth environment of the sample and the strain induced by the substrate [8].