• Gravitational-waves detectors
• Quantum noise

Context and motivations

The interplay between optical and mechanical modes in a system produces a wealth of interesting effects which have been investigated and exploited in many different areas. In the context of interferometric gravitational waves detectors the opto-mechanical coupling which originates between mirrors and laser by means of radiation pressure has been proposed to be used, for example, to reduce the thermal noise of the mirrors by cooling their motion. We are particularly interested in its application to reduce the fundamental noise which stems from the quantum nature of light and which will eventually limit the sensitivity of new generation detectors. In this noise, known as “quantum noise”, two different contributions can be individuated: the first one is shot noise, caused by the fluctuations of the number of photons in laser beams, which is higher when the laser power is low; the second one is radiation pressure noise, caused by quantum fluctuations of the laser amplitude, whose effect increases for higher values of the power and usually is more relevant at lower frequencies. It has been proposed to circumvent the lower bound set by quantum noise, usually referred to as standard quantum limit (SQL), by using light with particular noise features. In an ordinary laser, described by a coherent state, the amplitude quadrature uncertainty equals the phase quadrature uncertainty and their product is the minimum possible allowed by the Heisenberg's uncertainty principle. However, quantum physics does not prevent us from modifying quantum fluctuation and new quantum states can be obtained, so that phase and amplitude quadrature uncertainty are no longer equal. For these states, referred to as “squeezed”, the phase uncertainty can be reduced at the expense of amplitude uncertainty which is bound to increase in order to make the uncertainty principle hold true. If the information we need to measure is kept in the phase of the field (as it happens for the mirrors displacement in gravitational waves detectors), we can have an intuitive idea of how, for these states, noise can go below SQL. Shot noise and radiation-pressure noise together need not to adhere to the SQL if they are correlated: in effect squeezed light is generated by inducing correlations between amplitude and phase fluctuations. So far the most effective techniques for producing squeezed light make use of non-linear means to create this correlation. However, in theory, the mere interaction of the laser light with a movable mirror is enough: indeed an amplitude fluctuation of the input field generates a displacement of the suspended mirror which in turn will affect the phase fluctuations. This effect is referred to as ponderomotive squeezing and it is particularly interesting as it naturally arises in interferometric detectors, but due to the intrinsic weakness of radiation pressure fluctuation with respect to other noise sources, it is extremely difficult to observe.

Goals and structure of the thesis

This thesis focuses on the generation of ponderomotive squeezing and arises within the collaboration “Progetto PRIN di Ponderomotive Squeezing” (see www.ppps.it) which aims to build a dedicated interferometer to observe directly this squeezing effect. The main goal of this work is the development of a code able to simulate opto-mechanical effects in optical systems. In order to improve the accuracy of the simulation, optical fields are not regarded as plane waves but are decomposed in Hermite-Gauss (HG) modes. This allows to take into account tilting effects on the mirrors due to a non-zero torque of the radiation pressure. For each HG mode, the field is split in a classical part and in a fluctuating part. The first is evaluated using a classical model and it defines the equilibrium point for the mirrors, around which the dynamics is linearized. In order to describe the quantum fluctuations around a classical carrier we used the two photon formalism, where such fluctuations are treated as amplitude and phase fluctuations for sidebands of the carrier. It provides an effective way to describe the correlations introduced by the interaction of the field with optical elements like movable mirrors. We generalized this formalism, which was developed for plane waves, to the case of HG modes.

In the first part of the work we describe classical opto-mecanical effects in a Fabry-Perot cavity with movable mirrors where the circulating light is decomposed in HG modes. Afterward the light fields are quantized and also the opto-mechanical effects are presented from a quantum point of view.

In the second part we illustrate the operating principles of the simulation and present some results for the squeezing which is possible to obtain both in the case of a large interferometric gravitational wave detector and in a smaller dedicated cavity such as the one proposed in. We also included the possibility of simulating thermal noise in the system in order to compare its modulation effect on mirrors with those of quantum fluctuations.

Lastly we briefly discuss the possibility of a further development of the simulation in order to take into account effects which appears beyond the linear approximation used.