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 \begin{equation}\label{eq:hamiltonian_jc}

H=\frac{\hbar}{2} \omega_{q} \sigma_{z}+\hbar \omega_{r} a^{\dagger} a+\hbar g\left(a^{\dagger} \sigma^{-}+a \sigma^{+}\right) \ .

\end{equation} 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 \begin{equation}\label{eq:dispersive}

   H_{\mathrm{disp}}\approx \hbar\omega_{r} \hat{a}^{\dagger}\hat{a} + \frac{\hbar}{2}(\omega_{q}+ \chi)\hat{\sigma}_{z} +\hbar \chi \hat{a}^{\dagger}\hat{a}\hat{\sigma}_{z} \ ,

\end{equation} 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 with a Lamb shift corresponding to the second term in Eq. \ref{eq:dispersive}. There, the cross-Kerr nonlinearity strength $\chi$ is derived from the coupling strength $g$ \\ \begin{equation}

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

\end{equation}