Nanomechanical resonators (2)

Abstract

The second lesson was devoted to the modeling of light-matter coupling for the elementary system, representative of the problem of controlling a quantum mechanical oscillator, that is the Fabry-Pérot resonator, one mirror of which is fixed and the other, mobile, spatially confined by a spring. This minimal system can be treated by considering a single mechanical mode of vibration, that of the moving mirror, as well as the modes of a field in one dimension of space, assumed to be of the transverse electromagnetic type, with the electric field parallel to the plane of the mirrors. These mirrors impose the boundary conditions of the field: cancellation of the electric field at the mirror location. At the level of the moving mirror, this very fundamental time-dependent constraint leads to coupling between the field excitations and those of the mechanical oscillator. By calculating the total energy of the Fabry-Pérot resonator as a function of the position of the moving mirror, we discover that the field exerts a pressure on the mirror, the instantaneous value of which is precisely the square of the magnetic field at the mirror. This pressure corresponds to the radiation pressure exerted on a mirror by a travelling wave, which is not stationary, as in the present calculation.

This radiation pressure can be interpreted in particle terms : the photons of the electromagnetic field bounce off the mirror, each transferring an impulse equal to twice their energy divided by the speed of light. The square of the magnetic field also appears naturally in this way, when the Poynting vector giving the field's energy flux is involved. If the field exerts pressure on the mirror, the latter, by moving, reciprocally exerts an influence on the field. This feedback action can be seen in the Doppler effect, where the radiation reflected by the moving mirror is shifted in frequency relative to the incident radiation. Here too, the phenomenon can be seen in both wave and particle form. All the above considerations enable us to formulate the classical field-coupled oscillator Lagrangian and to proceed to the usual quantization of the system via the Hamiltonian. This leads to a general expression, in second quantization, for the coupling term between the field and oscillator modes. This expression contains, among other things, the Casimir force exerted between the mirrors by quantum fluctuations in the field. By introducing the adiabatic approximation, which involves treating the ratio between the frequency of the mechanical oscillator and that of the fundamental field mode as an infinitesimal, we arrive at the general three-wave Hamiltonian that describes both the optomechanical and electromechanical systems covered in the lecture. This term, which corresponds to a linear coupling between the number of photons in the field mode and the position of the mirror, very generally involves three quantum amplitude operators : two for the electromagnetic field and one for the mechanical oscillator