Internal cyclotron probe of the vertical polarizability of a surface-state electron on helium

We study a surface-state electron on liquid helium in a tilted magnetic field. The tilt couples the cyclotron ladder to the vertical Rydberg-like subbands of the same electron and thereby realizes a self-coupled Landau-subband Hamiltonian. In the far-detuned regime, the cyclotron line acquires a state-dependent shift described by standard second-order perturbation theory. The resulting expression identifies the cyclotron resonance as a direct probe of the dynamical polarizability of the vertical ladder. The theory is intrinsically multilevel. Exact diagonalization confirms a broad microwave window in which sizable state-dependent cyclotron shifts are expected while the system remains dispersive.