Relaxation and Resonance in Brownian Motion-Coupled Resonators

Doctoral thesis, 2017

In physics, there exists a number of paradigm systems – exactly solvable models that can represent a wide variety of physical realizations. My research is concerned with two of these paradigm systems: the harmonic oscillator and Brownian motion. In this thesis, I investigate the dynamics of coupled oscillator-diffusion systems, in different contexts. A general Hamiltonian model illustrates the rich dynamics of a particle-like degree of freedom coupled to two degenerate oscillators. The interplay of the three creates unpredictable particle behaviour, switching between trapped, regular oscillations, and untrapped, possibly chaotic motion. The trapping-retrapping dynamics of the particle corresponds to switching between high and low dissipation of oscillator energy, resulting in an unusual, stepwise relaxation. Motivated by the rapid strides made in recent years in the field of nanomechanical sensing, in particular mass sensing, I consider particles loosely adsorbed on a one- or twodimensional carbon nanomechanical resonator. The particles are allowed to diffuse across the surface of the resonator, and fluctuations in their positions induce dissipation of vibrational resonator energy. I show that depending on vibration amplitude, the motion of the resonator-particle system separates into different regimes. In each I describe the particle motion and characterize the resonator relaxation towards equilibrium. An immediately experimentally attainable diffusion-resonator system is a superconducting LC-resonator inductively coupled to a superconducting quantum interference device (SQUID). The superconducting phase of the SQUID takes the role of a Brownian variable. I find that the resonant response of the circuit is multistable, an effect that becomes more pronounced the weaker the noise is; the severity of the circuit's nonlinearity can be tuned by the level of noise in the system. With superconducting circuit quantum electrodynamics being routinely done in the lab, experimental verification of my results concerning the LC-resonator coupled to a SQUID should be possible. For the nanomechanical particle-resonator system, experimental interest in room-temperature applications, and in adsorbate-induced anomalous dynamics is growing. My work in this area functions as a record of possible diffusion-induced ringdown effects.