Accurately simulating nine-dimensional phase space of relativistic particles in strong field

Next-generation high-power laser systems that can be focused to intensity $>10^{23}$ W/cm$^2$ enable new physics and applications. Matter interacting with these lasers is highly nonlinear, relativistic, and involves non-classical processes such as radiation reaction (RR) and quantum effects. The current tool of choice for modeling these interactions is the particle-in-cell (PIC) method. In strong fields, the motion of charged particles and their spin are effected by RR. Standard PIC codes usually use Boris or similar operator-splitting methods to advance particles. These methods have been shown to fail in the strong field regime and it is not straightforward to extend them to include RR. In addition, some problems require tracking the particle spin. Therefore numerical algorithms that enable high-fidelity modeling of 9D phase space (x, p, s) in strong field are required. We present a new particle pusher that works in 9D phase space based on the analytical solutions of momentum, together with the semi-classical form of RR in Landau-Lifshitz, and the Bargmann-Michel-Telegdi equation for the evolution of the spin. Analytic solutions for the position advance are also obtained but these are not amenable to the staggering of space and time in PIC codes. These analytical solutions are obtained by only assuming a locally uniform and constant electromagnetic field during a time step. Owing to the analytical integration of particle trajectory and spin orbit, the constraint on the time step needed to resolve the trajectory in ultra-high fields can be greatly reduced. We present single particle simulations to show the proposed particle pusher can greatly improve the accuracy of the particle trajectory in 9D phase space for given laser fields. We also implemented the new pusher into the state-of-art PIC code OSIRIS to include the effects of the field solver for lasers in vacuum.