Role of particle statistics in quantum mechanical decay and Fano resonance

Quantum mechanical decay and quantum interference of metastable states coupled to a continuum is a fundamental problem in many areas of physics. Landmark physical phenomena rooted in the quantum theory of decay and interference include the famous Weisskopf-Wigner theory of decay and Breit-Wigner resonances, deviations of quantum mechanical decay from an exponential law at short and long times, quantum Zeno effect, population trapping and bound states inside the continuum, Fano interference and Fano resonances. In particular, quantum interference and Fano resonances, i.e. the asymmetric deviation from the natural line shape of a resonance in a continuum structured by an embedded state, are found in many physical systems such as nuclei, atoms, molecules, semiconductors, optical and plasmonic structures. Most of previous experimental and theoretical studies on quantum mechanical decay and Fano interference have been focused to single-particle dynamics. For many-particle systems, recent theoretical works have predicted that particle statistics and particle interaction can deeply modify the onset of quantum decay, multi-particles Zeno dynamics and Fano interference. Remarkably, even in the absence of particle interactions, particle statistics solely can deeply modify the single-particle picture of decay and Fano resonance, for example a Fano resonance can fully disappear for fermionic particles but survive for bosonic ones. The aim of this thesis is to study theoretically the role of particle statistics on quantum mechanical decay and Fano interference, and to propose a quantum simulator of two-particle fermionic/bosonic quantum decay and Fano interference based on propagation of entangled photon states in suitably engineered arrays of optical waveguides. The quantum simulator is designed to mimic the quantum decay of two indistinguishable particles, either fermions or bosons, in the framework of the Fano-Anderson (or Freidrich-Lee) Hamiltonian. Numerical simulations are also performed to provide design guidelines of the optical waveguide arrays that are required to perform quantum optical experiments. The thesis is organized as follows. In the first chapter I will introduce the quite general theoretical models of quantum mechanical decay for the single particle problem. After a short discussion of a simple classical model of the decay, the full quantum mechanical theory is presented. I will discuss about the deviation from the exponential trend at short times, as a result of the continuity and differentiability of the probability amplitude and of the Heisenberg uncertainty principle; and deviations at long times according to a Fourier Transform theorem that imposes a particular functional form to the decay. I will then introduce the simplest model of decay, i.e. that of a single discrete state coupled to continuum of (scattered) states, the so-called Fano-Anderson (or Friedrichs-Lee) model, pointing out the role of the memory function and discussing the Weisskopf-Wigner (Markovian) approximation in the weak coupling limit. I will next describe the quantum mechanical decay for the case of a discrete energy level embedded or not not in a continuum of energy levels, explaining the phenomena of Fano resonance and of fractional decay. The case of strong coupling is also presented, including the onset of Rabi oscillations and the James-Cummings model. In the second chapter of the thesis the problem of many particle quantum decay is addressed starting from the multilevel Fano-Anderson model. i.e. in case where two or more discrete states can decay into the same continuum of states. The theory emphasizes the importance of the bound states inside the continuum in the dynamics of fractional decay and the major impact of particle statistics. In particular, it will be shown rather generally that for fermionic particles fractional decay and Fano resonances can fully disappear, whereas they survive for bosonic particles. An explanation of this phenomenon based on the Pauli exclusion principle and dressed state Hamiltonian is discussed. In the third chapter of the thesis I propose a quantum simulator of two-particle fermionic/bosonic quantum decay, which is designed to mimic the quantum decay of two indistinguishable particles, either fermions or bosons, in a two-level Fano-Anderson model with a tight-binding continuum. The optical system is based on a semi-infinite array of evanescently-coupled optical waveguides, where two boundary waveguides mimic the two discrete levels of the model and the semi-infinite array the tight-binding continuum into which they decay via evanescent coupling. Probing the optical array with non-classical light, namely with polarization entangled two-photon states ($HV+e^{i \varphi} VH$), the quantum decay dynamics can be revealed by photon coincidence measurements along the spatial propagation direction. Tuning the phase $\varphi$ between HV and VH states enables to mimic bosonic, fermionic and even anyonic particle decay. A numerical code has been developed to provide design guidelines of the waveguide array system. Finally, at the end of the thesis four Appendices provide some technical details of the theoretical analysis and show the numerical code (in MatLab) developed for the waveguide array design.