Towards an optimal control system of an opto-mechanical resonator for quantum noise reduction in GW interferometers

The first direct detection of a gravitational wave has given further evidence for Einstein’s theory of General Relativity and it has placed new perspectives on the investigation of the Universe. The detection was made possible thanks to the upgrades of the second generation detectors. The limit in sensitivity of gravitational wave (GW) detectors is set by different noise sources. In particular, in the high frequency band (above 200 Hz) it is dominated by shot noise. This noise is due to the quantum phase fluctuation of the coherent vacuum field entering the interferometer through its output port. As a solution to this limitation, the current generation ofgravitational wave detectors adopted a technique based on the injection of phase squeezed state, an idea initially proposed by Caves in 1981. A coherent state is a minimum uncertainty state, which means it has minimum quantum fluctuations on the two orthogonal quadratures. A squeezed state belongs to another class of minimum uncertainty states, with non-classical distribution of noise, for which the fluctuations on one quadrature are lower than those of a coherent state, whereas the fluctuations on the orthogonal quadrature are higher. This class of states can be produced by degenerate parametric down-conversion processes in second-order nonlinear crystals and they were first observed in 1985 in the radio-frequency band. The first sensitivity enhancement of gravitational wave detectors by means of injection of squeezed vacuum was first demonstrated by the British-German GEO interferometer. Later on, the effectiveness of this technique has been largely demonstrated in gravitational wave interferometers thanks to which, in the last two years, many new GW detections have been realized. During the last observing run O3, the injection of phase-squeezed vacuum from the dark port of the GW interferometers (LIGO and Virgo) demonstrated the quantum noise reduction (shot noise) in the high-frequency region of the detection band. Nevertheless, frequency-independent squeezing with phase-squeezed vacuum has the counter-effect of increasing radiation pressure noise at low frequency (below 100 Hz). Indeed, the second generation of GW detectors, due to the sensitivity improvement,are facing the limit imposed by the quantum nature of light: the Standard Quantum Limit (SQL). Hence, in order to enhance the detector sensitivity, also at low frequencies, the injection of squeezed states with a frequency-dependent squeezing angle is necessary, meaning that the ellipse representing the noise fluctuation is frequency dependent and, at low frequency reduce the noise on amplitude, while at high frequency on phase quadrature. This scientific reason lays below the realization of table-top experiments for the production of frequency dependent squeezing (FDS) in the audio frequency band of GW detectors. The FDS technique used in current gravitational wave advanced detectors consists in the filtering of the Frequency Independent Squeezing (FIS) by means of suitable detuned cavity, so that the squeezing ellipse will be rotated as a function of the frequency inside its linewidth. One of the most interesting alternative techniques for FDS generation is the ponderomotive technique. In this method, squeezing is generated by exploiting the radiation pressure effect on suspended mirrors inside an optical cavity. The coupling between the fluctuations of the optical field of the laser beam (coherent light) and the mechanical motion of the mirrors, due to the radiation pressure from the laser light, the so-called optical spring effect, creates a phase shift, in the light reflected from the mirrors, which depends on the intensity of the laser light incident on the suspended mirror. This results in a quantum correlation between phase (shot noise) and amplitude noise (radiation pressure noise), that is called ponderomotive squeezing. We have designed a tabletop suspended interferometer with low dissipation mechanical suspensions (monolithic suspensions) of the main optics, named SIPS (Suspended Interferometer for Ponderomotive Squeezing), that will be sensitive to radiation pressure noise in the audio frequency band of GW detectors. The proposal of this thesis is to study and design a highly sophisticated mechanical control for an optomechanical resonator such as SIPS, based on Pontryagin’s optimal control theory. The optimal control problem is analysed considering opto-mechanical interaction models developed from the model already described in literature for acoustic waves reflection in a wave guide. Indeed, a crucial point for this type of device is the mirror motion due to external mechanical disturbances, such as vibration and acoustic noise, which can bring the interferometer out of its working point. Moreover, the nonlinear optomechanical coupling is expected to generate the emergence of spurious frequencies in the reflected light spectrum, with respect to the monochromatic incoming laser spectrum. We consider how the control on suspended mirror position could take into account such non-linear effects. To this aim, we will apply Pontryagin’s approach to develop an integro-differential model that can be used to build an adequate and optimized control system, which can be implemented and tested in our small-scale suspended interferometer. As a first application, SIPS interferometer will be used to test broadband quantum noise reduction through the injection of entangled vacuum states, i.e. Einstein Podolsky Rosen (EPR) states, generated by a not degenerate Optical Parametric Oscillator (OPO), into a radiation pressure noise limited interferometer. The integration of these two experiments is an important test bench for EPR squeezing technique applied to an interferometer before any possible integration in the next future in large scale detectors. Moreover, SIPS experiment will provide both a source of squeezed states by ponderomotive effect, for a broadband quantum noise reduction, and a suitable test bench for the optimal control algorithm we propose