Zeeman shift of an electron trapped near a surface

Boundary-dependent corrections to the spin energy eigenvalues of an electron in a weak magnetic field and confined by a harmonic trapping potential are investigated. The electromagnetic field is quantized through a normal-mode expansion obeying the Maxwell boundary conditions at the material surface. We couple the electron to this photon field and a classical magnetic field in the Dirac equation, to which we apply the unitary Foldy-Wouthuysen transformation in order to generate a nonrelativistic approximation of the Hamiltonian to the desired order. We obtain the Schrödinger eigenstates of an electron subject to double confinement by a harmonic potential and a classical magnetic field, and then use these within second-order perturbation theory to calculate the spin energy shift that is attributable to the surface-modified quantized field. We find that a pole at the eigenfrequency of a set of generalized Landau transitions gives dominant oscillatory contributions to the energy shift in the limit of tight harmonic confinement in a weak magnetic field, which also make the energy shift preferable to the magnetic moment for a physically meaningful interpretation.

Figure 1

Physical setup, with the horizontal axis representing the $z$ coordinate (solid lines) and the potential (dashed lines).

Figure 2

Lower complex $k_z$ plane for the integrals in Eqs. (58) and (59). The poles at $k_z^2={\Delta }_i^2-k_{\parallel }^2$ can appear in one of three positions depending on the relative values of $k_{\parallel }$, ${\Delta }_i$, and $n{\Delta }_i$. These positions are at equal and opposite points on the real axis (shown as positions 1), and on the positive and negative imaginary axis, either to the side of the cut due to $k_z^d$ (2), or between the two cuts (3). Poles in the upper half plane are not shown as they are irrelevant for the calculation at hand.