Wigner function for tunneling in a uniform static electric field

The Wigner function is used to study a simple model system for strong-field induced ionization: an electron tunneling out of a zero-range potential in the presence of a uniform static electric field. We derive an analytic expression for an approximate Wigner function describing a stationary situation where the part lost to ionization is continuously replenished. This approach is well justified by comparison with the true time dependent Wigner function obtained by numerically solving the one-dimensional problem. The three- and one-dimensional Wigner functions both suggest that the electron leaves the tunnel with a non-zero velocity.