The Study Of Geometrical Phase And Topological Properties In Superconducting Circuit System

Geometry has always played a key role in the formulation of physical theories and in solving physical problems.Such as,geometrical phase in quantum mechanics plays an extraordinary role in broadening our understanding of fundamental significance of geometry in nature.One of the best known examples is the Berry phase,which naturally emerges in quantum adiabatic evolution.Topology,with its abstract mathematical constructs,often manifests itself in physics and has a pivotal role in our understanding of natural phenomena.The history of topology in quantum physics begins in gauge theories,with the emergence of new phenomena like the Aharonov-Bohm effect,magnetic monopoles and so on.As is well known,Einstein failed to realize his dream of finding a geometric unified theory of all of physics.However,we can still get inspiration from the failure.For example,from the perspective of geometry and topology,we could find some certain spatial characters of quantum gravity to pave the way to solve the contradiction between quantum mechanics and general relativity.Up to now,researchers still have not confirmed the existence of magnetic monopoles,experimenters also not yet observed the phenomena of quantum gravity under the Planck scale.Therefore,it is particularly important to execute auxiliary study on these phenomena through the approach of quantum simulation.As a kind of macroscopic solid-state artificial atomic qubit(as compared to atomic qubits),superconducting qubits are easily to manipulate and to control.Also,they are easy to be scalable.Therefore,it can be used as an effective physical system to perform quantum simulation.In the dissertation,we simulate the monopoles in parameter space and the gravitational-like waves in minisuperspace with a superconducting transmon qubit,and then make an analysis from the point of both geometry(by use of the Berry curvature)and topology(by use of the first Chem number).The details are as follows:In Chapter 1,we first introduce the mathematical knowledge of differential geometry and topology involved in the dissertation.Otherwise,we talk about the connection among geometry,topology and physics.At the end of this chapter,we give the major research subjects and the organization of the dissertation.In Chapter 2,we first introduce several kinds of common superconducting qubits.Through the theory of one dimensional transmission line,we introduce the quantization of the superconducting transmission line.After that,we introduce the circuit quantum electrodynamics(QED)by using the superconducting transmon qubit.Then we introduce the mechanism of decoherence in superconducting qubits.Finally,we talk about the current situation of quantum simulation with superconducting qubits.In Chapter 3,with the help of the Berry curvature and the first Chern number,we both analytically and numerically investigate and simulate Abelian Wu-Yang monopoles formed in parameter space of the Hamiltonian of a superconducting transmon qubit.We find that we can control the evolution of quantum states by moving the degenerate point in parameter space.This provides a new freedom in manipulating quantum states.In addition to this,we also find the flip of quantum states are asymmetric during the topological transition of the quantum state manifolds,and the corresponding fidelity of quantum states experience a fluctuation.In Chapter 4,we investigate the geometrical and topological structure in the theory of quantum gravity by using the path integral method and half-classical approximation.Based on the phenomenon of fluctuation of the fidelity that emerged in the Chapter 3,we set up the connection between the ripples characterized by the fidelity of quantum states in Hilbert space and gravitational-like waves in minisuperspace.This could open a window for the study of geometrical and topological properties of quantum gravity with the help of quantum physical systems.Finally,we summarize the results at the end of the dissertation.