• feedbak Markoviano
• informatione quantistica
• misure continuuee
• sistemi mesoscopici
• squeezing generation
• stati Gaussiani

In Quantum Mechanics (QM) measurements of observables are traditionally introduced as instantaneous, but experimentally this assumption is often simply not true. So a theory for measurements performed continuously in time has been developed and its applications have became increasingly important mainly due to the growing interest in the application of feedback control in quantum systems, especially in quantum information and computation for continuous variable systems.

This Thesis develops applications of the theory for continuous measurement to experimentally feasible bosonic systems and explores the resulting advantages.

The first chapter of this Thesis consists of a review about basic facts in QM. Because the main subject is continuous-measurements, measurement theory is treated with particular attention: besides postulating projective measurements, the theory for postive-operator valued measurements is developed. In order to set the scenario for the applications chapter, in the second part of chapter 1 Gaussian states for bosonic systems are introduced. The importance of Gaussian states relies in the fact that they are easily generated in laboratory with quantum optics and bilinear interaction Hamiltonians and as such, are a commonly studied quantum systems. So the main characteristics of Gaussian states and the unitary evolution generated by bilinear Hamiltonians are presented.

The second chapter develops the theory of continuous-measurement with Markovian feedback. Firstly, in section 2.1, the general form of a Markovian master equation describing an open quantum system is presented. This is followed by an explicit example of a single bosonic mode interacting with a bosonic Markovian bath. In section 1.1.4 the notions introduced in the first chapter are used to study the details of two examples of quantum measurement for bosonic quantum system that are experimentally very important, namely the photodetection and the homodyne detection. Finally in section 2.3 we study continuous-measurements starting with two examples in order to present the problem in a physically concrete setting. In the first case a quantum system subjected to the continuous monitoring of the position observable is described. In this simpler case we also study the effects of multiple and inefficient observers. In the second case we derive the
stochastic master equation for a bosonic mode interacting with an environment which is continuously monitored through a homodyne detection. In section 2.3.3, we present the general mathematical formalism describing continuous-measurement and feedback control for linear quantum system. This formalism allows to introduce important results and features of continuous-measurements. In particular we focus on the stabilization of unstable systems by means of continuous monitoring, the possibility to stabilize the stochastic dynamics of Gaussian states and to generate steady-state squeezing and entanglement, by using a simple Markovian feedback protocol.

The third chapter contains the original results of the thesis. Here we focus on specific bosonic systems and we investigate the possibilities offered by continuous measurement in order to deal with unstable dynamics and to generate squeezed states, i.e. states with fluctuations below the vacuum level, that are considered resources both for quantum information and quantum metrology.

In the first case we study a degenerate parametric amplifier. It is a commonly used optical device to generate squeezed states of light. It is known in literature that there exist a 3dB limit on the achievable squeezing for a steady-state, but in the previous chapter it is shown that, in the case of a continuously monitored system, a new ultimate bound exists. We study in detail the evolution of the parametric amplifier when continuous homodyne detection is performed and we also show that the new optimal bound can actually be achieved when the system is monitored with maximum efficiency. We also study the achievable steady-state squeezing for a generic efficiency and calculate the relation between the physical parameters in order to obtain a squeezing amplification larger than the old 3dB bound.

Eventually an opto-mechanical system is considered: a nano-dielectric trapped in a harmonic potential by optical tweezers and interacting with a single cavity mode. This is an experimentally achievable system and several research groups are working on the possibility to cool the centre of mass motion of the dielectric to its quantum ground state in the harmonic potential that confines it. We thus have a two-mode system consisting of a mechanical oscillator and a single cavity mode and two possible mode on which to apply continuous measurement. Studying the corresponding master equation, it is possible to show that the dynamics of the system is not stable, due to the decoherence introduced by the mechanical oscillator and therefore there is not a steady-state. In this thesis we prove that a monitoring with any efficiency on any mode is sufficient to generate a stable dynamics. In particular, when a continuous measurement on the position of the mechanical oscillator is performed, the purity of the dielectric is studied when varying the efficiency of
the measurement and it is shown that a pure state is obtained for maximum efficiency. It is also shown that there exist a minimum value for the efficiency in order to obtain sub-shot noise squeezing for the mechanical oscillator. Finally we explore how the variations of the oscillator's frequency and the coupling strength between the mechanical and the optical mode change the achievable squeezing for the oscillator, in order to provide indications for future experimental improvement of the implementation of the system.