On May 18, 2023, Xu Xiao successfully defended the thesis “A Diagrammatic Method to Compute the Effective Hamiltonian of Driven Nonlinear Oscillators” (advisor: Michel Devoret).
Xiao explained, “In the field of quantum information, an essential task is to engineer the native nonlinearity in a physical system, be it an atom, a trapped ion, or a superconducting circuit, so that nontrivial manipulations of quantum information can be achieved. One powerful approach to accomplish “nonlinearity engineering” is by applying periodic forces to modulate a nonlinear system. This creates what we call “driven nonlinear systems,” which have been known to exhibit fascinating nonlinear dynamics for centuries. To effectively control the diverse range of phenomena in driven nonlinear systems, which typically lack simple mathematical descriptions, the key is to develop perturbation theories that provide analytical insights into their dynamics. While low-order perturbation methods like the rotating wave approximation have been successful in describing weakly driven nonlinear systems and have contributed to the advancements in quantum information over the past two decades, they fall short when it comes to nonlinear processes beyond the leading order. These limitations become more pronounced when the drive is moderately strong, which is of the case in the state-of-art quantum information experiments. My thesis focuses on developing perturbation methods to arbitrary orders, which forms the basis for more sophisticated control of quantum information.”
After graduation, he is planning to work on a startup that provides simulation tools for the quantum information industry.
Thesis abstract:
The question answered in this thesis is whether it is possible to construct a rigorous and efficient perturbation method to obtain the effective static Hamiltonian of a nonlinear oscillator driven by a time-dependent oscillatory force. We present in this dissertation such a method, based on Feynman-like diagram. The pictorial structure associated with each diagram corresponds directly to a Hamiltonian term, the prefactor of which involves a simple counting of topologically equivalent diagrams. We also leverage the algorithmic simplicity of our scheme in a readily available computer program that generates the effective Hamiltonian to arbitrary order. At the heart of our diagrammatic method is a novel canonical perturbation expansion developed in phase space to capture the quantum nonlinear dynamics. A merit of this expansion is that it reduces to classical harmonic balance in the limit of ħ -> 0. Our method establishes the foundation of the dynamic control of quantum systems with the precision needed for future quantum machines. We demonstrate its value by treating five examples from the field of superconducting circuits. These examples involve an experimental proposal for the Hamiltonian stabilization of a three-legged Schrödinger cat, modeling of energy renormalization phenomena in superconducting circuits experiments, a comprehensive characterization of multiphoton resonances in a driven transmon, a proposal for an inductively shunted transmon circuit, and a characterization of classical ultra-subharmonic bifurcation in driven oscillators. Lastly, we benchmark the performance of our method by comparing it with experimental data and exact Floquet numerical diagonalization.