# Theoretical Investigation of All Optical Switching by Intersystem Crossing
## 1 Introduction  
Occurring on numerous ferri [1; 2; 3; 4; 5; 6] and ferromagnetic [7; 8; 9; 10] materials, all-optical switching (AOS) is considered as a universal phenomenon. The sub-picosecond time scale within which this magnetization switching is accomplished shows the potential of the ultrafast magnetic data storage. However, the evolution of the magnetization reversal in AOS scenario remains an open question. Two-temperature model [11; 12] and the inverse Faraday Effect (IFE) [13] are the two most accepted models describing the dynamics of AOS.  
Taking the ultrafast heating as a dominant factor, the two-temperature model works well in the magnetic orientation reversal of anti-ferromagnetic coupled materials [14; 15]. However, it fails to explain the transient ferromagnetic state in which the magnetic orientation of the transient metal sublattice reverses and gets parallel to that of the rare earth after demagnetization of the transient metal sublattice.  
Proposed by [PERSON] et al. [16], IFE is interpreted as a thermodynamic potential describing the internal energy of a system. In their later work [17], [PERSON] et al. deduced the effective Hamiltonian to quantify IFE, separating the effective Hamiltonian into two parts: diagonal matrix elements and off-diagonal ones. The former is responsible for the eigenstates of the unperturbed Hamiltonian, and the latter describes the process of the stimulated Raman scattering (SRS). [PERSON] et al. [18; 19] further developed SRS for AOS, and quantified SRS induced magnetization. However, the selection rule requiring theconservation of electrons' angular momentum, suggests that electron transition during SRS should keep the orientation of spins unchanged. Even though results, [PERSON] proposed, show the imbalance of spin-up and spin-down in a few hundred femtoseconds, it should be a temporary state that electrons stagnate at excited states, and finally spins will return to their original states.  
Though the two mentioned models ignore the conservation of the angular momentum of the system, some researchers argued that spins could be reversed by spin-orbit coupling (SOC) [20; 21; 22; 23; 24; 25]. The SOC model assumes that the variation of the angular momentum of spins are balanced by their orbital counterparts. Electron transitions from singlet state to triplet state are used to explain the process of the ultrafast demagnetization [26]. However, experimental evidences [27; 28] show that the angular momentum of spins and orbits both quench to zero during demagnetization, meaning that orbits cannot be the reservoir of the spin angular momentum. Therefore, the angular momentum of spins should flow to lattice [29; 30].  
Intersystem crossing (ISC) is a non-radiative process, which can be described by vibrational SOC. Because of following reasons, ISC can be a candidate to explain the mechanism of a specific AOS that is induced by one linear polarized laser shot. (1) ISC solves the deficiency in [PERSON]'s work that spins can be reserved with no need to meet selection rule; (2) ISC could be a channel to transfer angular momentum from spins to lattices by vibrational SOC [31]; (3) the time scale of AOS is comparable with ISC. (Recent researches [32; 33] on the ultrafast ISC show that the transition time spans from 100 fs to a few picoseconds.)  
Our model is based on a four-level system, but some of its parameters are \"borrowed\" from Fe. The model is in the formulation of Rabi oscillation. The calculation of the model shows that ISC could be a plausible approach leading AOS. The impact of temperature and the intensity of the incident laser pulse to the final state of AOS are also discussed.