A new spectroscopic approach to collective excitations in solids: pump-probe quantum state tomography

In a classical description the displacement of the atoms along the vibrational eigenmodes of a crystal can be measured with unlimited precision. Conversely, in the quantum formalism positions and momenta of the atoms can be determined simultaneously only within the boundary given by the Heisenberg uncertainty principle. For this reason, in real materials, in addition to the thermal disorder, the atomic displacements are subject to fluctuations which are intrinsic to their quantum nature. Because a crystalline solid has symmetries, these vibrations can be analyzed in terms of collective modes of motion of the atoms. These modes correspond to collective excitations called phonons. The aim of this thesis is to study the quantum fluctuations of the atoms involved in such collective vibrations. The motivation of studying the quantum proprieties of phonons in crystals comes from various evidences, recently reported in the literature, suggesting that quantum fluctuations of the atoms in solids may be of relevance in determining the onset of intriguing and still not completely understood material properties, such as quantum para-electricity, charge density waves, or high temperature superconductivity. The time evolution of phonons in crystals is usually addressed in the framework of ultrafast optical spectroscopy by means of pump-probe experiments. In these experiments the phonon dynamics is driven by an intense ultrashort laser pulse (the pump), and then the collective excitation is investigated in time domain through the interaction with a weaker pulse (the probe). Unfortunately this method typically provides information only about the average position of the atoms and an intense scientific debate is on-going about the possibility to have access also to the fluctuations of such positions measured with respect to a bound level for the shot-noise limit (intrinsic quantum noise limit). In this research activity a new approach to investigate quantum fluctuations of collective atomic vibrations in crystals is proposed. It combines time resolved optical spectroscopy techniques (pump and probe experiments) and quantum optics techniques (balanced homodyne detection). The novel spectroscopic tool, pumpprobe quantum state tomography, consists in the time domain characterization of the quantum state of probing light pulses after the interaction with the photo-excited material. The approach has been tested by investigating quantum fluctuations of the atomic positions in α-quartz, which serves as a case study for transparent materials. However, it can be in principle generalized to the study any collective excitations in crystals.