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.

nonequilibrium dynamics

nonlinear dynamics




superconducting circuits.


dispersive coupling


Brownian motion

PJ-Salen, Kemigården 1.
Opponent: Steven Shaw, Florida Institute of Technology, USA.


Christin Rhen

Chalmers, Physics, Condensed Matter Theory

Particle number scaling for diffusion-induced dissipation in graphene and carbon nanotube nanomechanical resonators

Physical Review B: covering condensed matter and materials physics,; Vol. 93(2016)

Journal article

Diffusion-induced dissipation and mode coupling in nanomechanical resonators

Physical Review B - Condensed Matter and Materials Physics,; Vol. 90(2014)p. Art. no. 155425-

Journal article

Stepwise relaxation and stochastic precession in degenerate oscillators dispersively coupled to particles

Physical Review B - Condensed Matter and Materials Physics,; Vol. 96(2017)p. 104302-

Journal article

Rhén, C., Isacsson, A., Stepwise relaxation and stochastic precession in degenerate oscillators dispersively coupled to particles.

The speed of the vibrations of a guitar string depends on how many times one turns the tuner screw - how hard the string is pulled. A musical person, who can tell the exact pitch of the string, would from the pitch be able to figure out the tension the string is under (maybe with a bit of math). In this way, the sound of the guitar can be used to measure the elastic force it is under. The same idea can be used to create sensors that measure a wide range of forces: electrostatic forces, magnetic forces, and even the force of gravity. In each case, electronics keep track of how fast something vibrates, and measures a force by checking to see if the vibration speed changes.

In science, we use guitar strings so small that hundreds of them can fit on a single hair from your head. They vibrate a million times faster than a regular guitar string, and can measure impossibly tiny things, like weighing a single atom. The fact that these "nanostrings" are so sensitive means that they can have a hard time measuring the right thing - what if the dust floating in the air gets stuck on the string? Scientists avoid this problem by putting the nanostring in vaccum, where it is protected from all kinds of disturbances. However, it is hard to isolate the nanostring like this if it is to be used in new electronics, like contact lenses for augmented reality or touch screens tattooed on your skin.

In my research, I study how nanostrings react to disturbances, so that other scientists will know what to look for when they want to use nanostring sensors in new technology.

Driving Forces

Sustainable development

Areas of Advance

Nanoscience and Nanotechnology (2010-2017)

Subject Categories

Other Physics Topics

Other Electrical Engineering, Electronic Engineering, Information Engineering

Condensed Matter Physics



Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4273


Chalmers University of Technology

PJ-Salen, Kemigården 1.

Opponent: Steven Shaw, Florida Institute of Technology, USA.

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