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Quantum limits of cooling based on un-conventional optomechanical forces

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Quantum limits of cooling based on un-conventional optomechanical forces

The coupling between light and mechanical motion can take different forms. The most elementary way to couple photons with a mechanical oscillator is the standard radiation pressure phenomenon : a photon bouncing back on a moving mirror imparts a momentum to the motion of the mirror, providing a mechanical force.

Other optomechanical forces are however of interest in optomechanics. In the photothermal force case, the photon is absorbed by the mirror and produces its thermal distortion, giving rise to an effective mechanical displacement of the mirror. In a semiconductor movable structure, photons can also be absorbed and generate an out-of-equilibrium population of electron-hole pairs. By means of electron-phonon coupling mechanisms, this population gives part of its energy to the structure motion, producing an effective optomechanical force. These un-conventional optomechanical forces are much larger in amplitude than the radiation pressure that would be produced by the same amount of photons.

The standard radiation pressure optomechanical coupling has been used in many experiments to optically cool the mechanical motion of a movable element placed in an optical cavity. This cooling can be understood simply using the following scheme, where a mono-frequency laser (central red line) pumps an optical cavity resonance (black lorentzian).

Because of optical/mechanical coupling to the mechanical motion of frequency, two sidebands are generated. The Stokes sideband (left red line in the scheme) corresponds to elementary processes where the pump photon enters the cavity, gives some energy to the mechanical motion and leaves the cavity with a red-shifted energy. The anti-Stokes sideband (right red line in the scheme) corresponds to the converse process where the photon is blue-shifted after having absorbed energy (phonons) from the mechanical motion. In the figure above, we have red-detuned the pump laser with respect to the optical cavity resonance : this results in a disequilibrium between Stokes and anti-Stokes processes that produces cooling of the mechanical motion. Phonons of the mechanical oscillator are absorbed by photons and exit the cavity in form of optical fluctuations.

The quantum theory predicts that radiation pressure cooling down to the ground-state of mechanical motion is only feasible if the Stokes process is completely cancelled-out. This necessitates a mechanical frequency (spacing between two red lines in the scheme) superior to the width of the cavity resonance and implies to detune the laser line out of cavity resonance to make the anti-Stokes sideband resonant with the cavity (see figure below). This is the situation known as sideband cooling, where a limited number of photons effectively enters the cavity.

Now let us consider the case of other optomechanical forces like photo-thermal or semiconductor opto-electronic forces discussed above. The dynamical behavior of these forces is different : their response time "tau" is no longer given by the photon lifetime in the cavity (as in the radiation pressure case), but by other slower physical mechanisms like thermal relaxation (for the photothermal case) or electron-hole pairs radiative relaxation (in case of an opto-electronic force in the GaAs material for example). This slower dynamical behavior results in a novel spectral signature : the dynamical optomechanical coupling possesses a resonance (represented by a dashed line in the figure below), which becomes now narrower (with a width of one over tau) than the optical cavity resonance .

This provides us with an original optomechanical cooling scenario : the laser-line can be placed on a flank of the optical cavity resonance, to allow efficient injection of photons in the cavity, but at the same time the Stokes process can be completely squashed to allow cooling down to the quantum regime of mechanical motion.

See the details in our recent long publication on the topic :

J. Restrepo, J. Gabelli, C. Ciuti and I. Favero. ’Classical and quantum theory of photothermal cavity cooling of a mechanical oscillator’. Comptes Rendus Physique doi:10.1016/j.crhy.2011.02.005 (2011). 11 pages.