Tunneling decay in a magnetic field

We provide a semiclassical theory of tunneling decay in a magnetic field and a three-dimensional potential of a general form. Because of broken time-reversal symmetry, the standard WKB technique has to be modified. The decay rate is found from the analysis of the Hamilton trajectories of the particle in complex phase space and time. In a magnetic field, the tunneling particle comes from beneath the barrier with a nonzero velocity. The exit location in the classically allowed region is obtained by matching the decaying and outgoing branches of the WKB wave function on a caustic of the set of the complex trajectories. The slope of the logarithm of the wave function sharply changes on the anti-Stokes surface where there occurs switching between different WKB branches. For potential wells that are parabolic near the minimum, we also provide a bounce-type formulation. The theory is applied to the models that are relevant to tunneling from correlated two-dimensional electron systems in a magnetic field parallel to the electron layer.