Dynamics of interacting oscillators in the presence of environments

Abstract

This thesis explores the dynamics of linearly coupled quantum and classical oscillators interacting with environments. In the quantum regime, such dynamics is described in terms of the Liouvillian superoperator, which leads to non-unitary time evolution of the system, specifically of the density matrix. In particular, when the correlation time of the environment is significantly shorter than the characteristic timescale of the system, the dynamics are classified as Markovian. Therefore, the equation of motion for the density matrix that fulfils this property is known as the Lindblad master equation. Systems of coupled cavities and waveguides in the presence of gain and loss can be described by this equation and, more importantly, by non-Hermitian Hamiltonians. In particular, when two coupled cavities exhibit balance between gain and loss, the system is PT -symmetric. However, for certain values in the parameter space, this symmetry is broken and the non-Hermitian Hamiltonian is no longer diagonalizable. Such values are denoted as exceptional points. In the case of large arrays of coupled quantum oscillators, the dynamics description becomes more challenging since the dimensions of the density matrix grow exponentially. To address this, we study the dynamics of these arrays via the quantum state diffusion approach. In such approach, the Lindblad master equation is unraveled by a stochastic Schrödinger equation. Then, we decouple this equation by diagonalizing the non-Hermitian Hamiltonian that describes the system. Such diagonalization leads to a transformation of a system of coupled oscillators into a decoupled one, offering a great reduction in the basis size. For systems of classical oscillators with dissipative nearest-neighbor coupling, the dynamics leads to synchronized motion. We study how an array of such oscillators reaches the synchronized state. This behavior is found when the system is dissipation free, which implies that all the eigenvectors of the dynamical matrix are damped, except one. To measure the degree of synchronization, we introduce the inverse of participation ratio, whose maximum value is associated with a delocalized state and, therefore, fully synchronized motion. For larger arrays of oscillators, we examine the case where the frequency fluctuations of each oscillator are obtained from a Gaussian distribution with zero mean and a given standard deviation. Then, we derive a scaling law for the largest value of the mentioned standard deviation that still allows the oscillators fully synchronize their motion.:1 Introduction 2 Open quantum systems 3 Dynamics of two coupled cavities in the presence of losses 4 Unraveling of Lindblad equation for system of oscillators 5 Synchronization of dissipatively coupled harmonic oscillators 6 Conclusions and outlook A Derivation of equations of motion for the amplitudes B Quantum state diffusion unraveling for N decoupled oscillators C Model for disspative coupling D Perturbation theory for disorder in the frequencies References