Exact solution of the Schrodinger equation for photoemission from a metal

We solve rigorously the time dependent Schrodinger equation describing electron emission from a metal surface by a laser field perpendicular to the surface. We consider the system to be one-dimensional, with the half-line $x<0$ corresponding to the bulk of the metal and $x>0$ to the vacuum. The laser field is modeled as a classical electric field oscillating with frequency $omega$, acting only at $x>0$. We consider an initial condition which is a stationary state of the system without a field, and, at time $t=0$, the field is switched on. We prove the existence of a solution $psi(x,t)$ of the Schrodinger equation for $t>0$, and compute the surface current. The current exhibits a complex oscillatory behavior, which is not captured by the simple three step scenario. As $ttoinfty$, $psi(x,t)$ converges with a rate $t^{-frac32}$ to a time periodic function with period $frac{2pi}{omega}$ which coincides with that found by Faisal, Kaminski and Saczuk (Phys Rev A 72, 023412, 2015). However, for realistic values of the parameters, we have found that it can take quite a long time (over 50 laser periods) for the system to converge to its asymptote. Of particular physical importance is the current averaged over a laser period $frac{2pi}omega$, which exhibits a dramatic increase when $hbaromega$ becomes larger than the work function of the metal, which is consistent with the original photoelectric effect.