Polarization-controlled effective Rabi dynamics in driven Graphene: A Floquet-Magnus approach

Polarization ellipticity $\beta$ and the relative angle $\Delta$ between electron momentum and driving field act as independent control parameters for coherent dynamics in periodically driven Dirac systems. In this work, we analyze the dynamics of resonantly driven Dirac electrons in graphene under elliptically polarized electromagnetic radiation using the Floquet-Magnus expansion. Working in the interaction picture and applying a rotating-wave-type transformation, we derive an effective two-level Hamiltonian that governs the macromotion at resonance ($\omega = \Omega/2$). The resulting quasienergy splitting depends nontrivially on $\beta$ and $\Delta$ through interference between the Bessel harmonics $J_0(\zeta)$ and $J_2(\zeta)$. Circular polarization ($\beta = \pm 1$) restores rotational symmetry and yields a $\Delta$-independent effective Rabi frequency, whereas elliptical and linear polarizations produce anisotropic responses with a $\pi$-periodic angular modulation. Beyond spectral properties, we identify a polarization-induced phase that acts as an effective initial Floquet kick, shifting the effective initial conditions and producing measurable shifts in the timing of occupation oscillations, whose sign depends on both helicity and relative orientation. Through an explicit Fourier decomposition of the time-evolution operator, we separate macromotion from micromotion contributions and validate the zeroth-order Magnus approximation via numerical simulations, achieving root-mean-square errors of $\sim 1\%$ over 100 driving periods in the weak-field regime. These results establish polarization ellipticity and relative orientation as tunable and experimentally accessible knobs for quantum control in two-dimensional Dirac materials, with direct implications for time-resolved spectroscopy.