Optomechanical Cooling of a Macroscopic Oscillator by Homodyne Feedback

The problem of considering a macroscopic oscillator in terms of Quantum Mechanics is usually avoided because one can obtain the right results without using any quantum mechanical hypothesys. When, however, one whishes to use it as a device to detect extremely small displacements due to very weak forces, as in the gravitational wave detectors, one has to be careful in considering it as a mere macroscopic object. Should one consider a macroscopic oscillator as a quantum oscillator, once all other possible noise sources were eliminated by using filters, screens, insulators etc., the ultimate criterion one has to satisfy is the one associated with the thermal noise [1,2]. For the harmonic oscillator it means kBT < h̄ωm/2, where kB is the Boltzmann constant, ωm the mechanical angular frequency and T the temperature of the environment in which the oscillator lives. This prohibitive limit, for macroscopic massive oscillators, is however only valid when the measurement time τ is of the order of the mechanical relaxation time τm. The actual limit can be expressed as 2kBTτ/Qm < h̄ [1]. In this case it is possible to consider a macroscopic mechanical oscillator as a quantum oscillator even at liquid He temperature [3], but very high mechanical Qm = ωmτm factors and also short observation times should be considered. To have better results and, for instance, to detect millisecond duration bursts of gravitational waves from supernovae, one should measure out of resonance, as in VIRGO or LIGO proposals [4], or at lower temperatures, as in massive bar detector schemes [5]. The thermal fluctuations are, however, the fundamental limitations and, in order to reduce their effects, one usually should lower the environment’s temperature. In this letter we present an alternative way of cooling the oscillator’s observed quadrature which could be experimentally accessible. We consider an empty Fabry-Perot cavity with one fixed mirror with transmittivity Tr and one perfectly reflecting end mirror. The completely reflecting mirror can move, undergoing harmonic oscillations damped by the coupling to a thermal bath in equilibrium at temperature T . The cavity resonances are calculated in the absence of the impinging field, hence, if L is the equilibrium cavity length, the resonant frequency of the cavity will be