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 Failed to parse (unknown function "\math"): {\displaystyle |\Delta|\gg g<\math>. There, the Hamiltonian is approximated as <math> 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} \ , }

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 Failed to parse (unknown function "\math"): {\displaystyle \chi<\math> is derived from the coupling strength <math>g<\math> as <math> \chi = \frac{g^{2}}{\Delta} \ . }