Microwave control of superconducting cavity and qubit

Group members

LI Yifan e0653565@u.nus.edu

Zhao Luheng

Introduction

circuit QED is the study of the interaction between light confined in a cavity or resonator and artificial atoms. Usually, the artificial atom is denoted as the qubit which is an essential element in the superconducting circuit. For the qubit, we try to fully control its ground state and the first excited state, which provides a well-defined two-level system. In our setup, we have two aluminum 3D superconducting cavity samples A and B, both embedded with transmon qubit chips inside. This project is intended to characterize the properties of superconducting cavities and the transmon qubits and perform the measurement of qubits. In this characterization project, we will obtain the coupling strength between transmon qubit and cavity <\math>\chi$,theanharmonicityofqubits,coherenttimeofthequbitWewillbenefitfromthischaracterizationwhenwetrytoaccuratelycontrolthequbitstatebythemicrowavepulse.=Setup===Hamiltonian==Thequantumsystemiscomposedofatransmonqubitandareadoutresonator,withthefollowingHamiltonian<math>H=\frac{\hslash }{2}{\omega }_q{\sigma }_z+\hslash {\omega }_r{a}^{†}a+\hslash g(a^{†}{\sigma }^{-}+a{\sigma }^{+}).$

The quantum system works in the dispersive regime where the qubit is strongly detuned from the oscillator with $|\Delta |\gg g$. There, the Hamiltonian is approximated as

$H_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}}\approx \hslash {\omega }_r{\hat{a}}^{†}\hat{a}+\frac{\hslash }{2}({\omega }_q+\chi ){\hat{\sigma }}_z+\hslash \chi {\hat{a}}^{†}\hat{a}{\hat{\sigma }}_z,$

where the third term, i.e. dispersive coupling term, represents the qubit-state dependent shift of oscillator frequency, or equivalently, a photon-number dependent frequency shift in the qubit spectroscopy, which is also denoted as the ac-Stark shift. Due to the dispersive coupling, the qubit frequency is dressed by a Lamb shift corresponding to the second term in this equation. There, the cross-Kerr nonlinearity strength $\chi$ is derived from the coupling strength $g$ as

$\chi =\frac{g^2}{\Delta }.$