Abstract
An object whose position is measured in real time, but which can be accelerated by a time-varying external force that responds to the position signal via a filter, provides a simple example of a system for which the control theory covered in the 2011 lecture can be applied. In the fourth lesson, we dealt with the cooling of a moving mirror from this point of view. We first introduced a simplified model of the system, which nevertheless includes all the essential control elements (force sources and position sensors). This is an electromechanical system based on an LC oscillator acting as an electromagnetic resonator. One of its capacitor plates is movable and connected to a spring. This movable plate constitutes the mechanical oscillator. The radiation pressure here is the square of the electric field present at the plate, but the Hamiltonian of the coupling between electromagnetic and mechanical resonators is identical to that of the optomechanical resonator. We can easily write the equations of motion for this system, which are those of two damped and forced harmonic oscillators, coupled by a cubic term of order 2 in the amplitude of the first oscillator and of order 1 in the amplitude of the second. When we linearize the problem by looking for solutions in the vicinity of the operating point imposed by the oscillating source forcing the electrical resonator, we obtain two coupled linear equations. These can be interpreted by noting that the vibrational state of the mechanical oscillator renormalizes the frequency and damping of the electrical oscillator, and vice versa.
