M.Sc Thesis


M.Sc StudentLevi Roei
SubjectNonlinear Analysis of Oscillatory Quantum Systems
Experiencing Bifurcation
DepartmentDepartment of Electrical and Computer Engineering
Supervisor ASSOCIATE PROF. Eyal Buks
Full Thesis textFull thesis text - English Version


Abstract

Nonlinear dynamics are essential in many physical systems. One of the properties of nonlinear systems, known as a bifurcation, is a qualitative change in the dynamical behavior, as one changes the system parameters. Nonlinear dynamics manifest also in quantum systems, such as systems consisting of spins and cavities.

In this work, a nonlinear analysis of a family of dynamical systems is carried out. The considered model consists of an oscillator with a high Q-factor, which is weakly coupled to another ancilla system.

Nonlinear analysis of this family of systems reveals the conditions for linear instability and for the onset of bifurcation. Specifically, the family of systems considered in this work is shown to exhibit an Andronov-Hopf bifurcation and obtain a steady state limit cycle solution, under given conditions. Expressions describing the dynamics beyond the bifurcation point are obtained. A relation between g, the oscillator-ancilla coupling constant, and the solution amplitude, is obtained. The results of this analysis are applied to the dynamics of two quantum systems: A system of two spins interacting through dipolar coupling, and a system of a spin coupled to a cavity mode. In both cases, the system can, under suitable conditions, become unstable and exhibit a supercritical Andronov-Hopf bifurcation. In both cases, the effective linear damping is proportional to g2, and is anti-symmetric in the detuning parameter of the systems. The analytic results predict a limit cycle amplitude and show that it is proportional to g-1.

Finally, Numerical simulations are performed to verify the analytic results for these two systems, using Mathematica to numerically solve the equations of motion and using MatCont to obtain the steady state amplitude with a continuation algorithm.