Microwave control of superconducting cavity and qubit

Group members

LI Yifan e0653565@u.nus.edu

Zhao Luheng

Setup

Hamiltonian

The quantum system is composed of a transmon qubit and a readout resonator, with the following Hamiltonian

$H=\frac{\hslash }{2}{\omega }_q{\sigma }_z+\hslash {\omega }_r{a}^{†}a+\hslash g(a^{†}{\sigma }^{-}+a{\sigma }^{+}).$

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

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