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We introduce an analytical approximation scheme to diagonalize parabolicallyconfined two dimensional electron systems with both the Rashba and Dresselhausspin-orbit interactions. The starting point of our perturbative expansion is azeroth-order Hamiltonian for an electron confined in a quantum wire with aneffective spin-orbit induced magnetic field along the wire, obtained byproperly rotating the usual spin-orbit Hamiltonian. We find that thespin-orbit-related transverse coupling terms can be recast into two parts W andV, which couple crossing and non-crossing adjacent transverse modes,respectively. Interestingly, the zeroth-order Hamiltonian together with W canbe solved exactly, as it maps onto the Jaynes-Cummings model of quantum optics.We treat the V coupling by performing a Schrieffer-Wolff transformation. Thisallows us to obtain an effective Hamiltonian to third order in the couplingstrength k_Rl of V, which can be straightforwardly diagonalized via anadditional unitary transformation. We also apply our approach to other types ofeffective parabolic confinement, e.g., 2D electrons in a perpendicular magneticfield. To demonstrate the usefulness of our approximate eigensolutions, weobtain analytical expressions for the n^th Landau-level g_n-factors in thepresence of both Rashba and Dresselhaus couplings. For small values of the bulkg-factors, we find that spin-orbit effects cancel out entirely for particularvalues of the spin-orbit couplings. By solving simple transcendental equationswe also obtain the band minima of a Rashba-coupled quantum wire as a functionof an external magnetic field. These can be used to describe Shubnikov-de Haasoscillations. This procedure makes it easier to extract the strength of thespin-orbit interaction in these systems via proper fitting of the data.

Energy spectra for quantum wires and 2DEGs in magnetic fields with Rashba and Dresselhaus spin-orbit interactions