Acceleration-gauge Hamiltonian for a laser-driven particle with a position-dependent mass

We address theoretically the problem of laser-induced dynamics of an electron, whose effective mass is position dependent (e.g., due to an effect of a semiconductor nanostructure environment). We derive the associated classical acceleration-gauge Hamiltonian ${H}_{\mathrm{AG}}(t)$ under the most general conditions, even without imposing the dipole approximation. It is shown that ${H}_{\mathrm{AG}}(t)$ possesses an intriguing structure arising due to the spatial dependence of the electronic mass. Subsequently, we restrict ourselves to a weak-field intensity regime, and derive the corresponding quantum mechanical acceleration-gauge Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\mathrm{AG}}(t)$ which differs from its classical counterpart by an extra quantum term, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{QM}}$. Our theoretical findings are illustrated numerically by calculating the probability ${|T(E)|}^{2}$ of the resonance transmission of an electron through a model semiconductor nanostructure. It is demonstrated that the quantum mechanical $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{QM}}$ term of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\mathrm{AG}}(t)$ does often affect crucially the profile of ${|T(E)|}^{2}$.