M.Sc Thesis | |

M.Sc Student | Levi Roei |
---|---|

Subject | Nonlinear Analysis of Oscillatory Quantum Systems Experiencing Bifurcation |

Department | Department of Electrical and Computer Engineering |

Supervisor | ASSOCIATE PROF. Eyal Buks |

Full Thesis text |

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 g^{2},
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.