Superadiabatic optical forces on a dipole: exactly solvable model for a vortex field

The forces exerted by light on a small particle are modified by the particle's motion, giving a series of superadiabatic corrections to the lowest-order approximation in which the motion is neglected. The correction forces can be calculated recursively for an electric dipole modelled as a damped oscillator. In lowest order, there is, as is known, a non-potential though non-dissipative 'curl force', in addition to the familiar gradient force. In the next order, there are forces of geometric magnetism and friction, related to the geometric phase 2-form and the metric of the driving field. For the paraxial field of an optical vortex, the hierarchy of superadiabatic forces can be calculated explicitly, revealing a four-sheeted Riemann surface on which fast and slow dynamics are connected. This leads to an exact 'slow manifold', on which the dipole is driven without oscillations by the same forces as in the first two adiabatic orders, but with frequency-renormalized strengths.