Quantum gate design based on the interaction between Orbital momentum of an electron into a magnetic field

The total Hamiltonian of an electron in an atom is expressed in terms of its kinetic energy, its potential energy of interaction into the nucleus and the interaction of its total orbital angular momentum into a control magnetic field. By paring over to the interaction picture we get a second order Dyson series approximation for the evolution operator and then control the magnetic field so that the generator of the evolution operator well approximate a given generator when there is noise in the control magnetic field ,we evaluate the mean square error energy in this approximation