Microwave control of superconducting cavity and qubit: Difference between revisions
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== Hamiltonian == | == Hamiltonian == | ||
The quantum system is composed of a transmon qubit and a readout resonator, with the following Hamiltonian | The quantum system is composed of a transmon qubit and a readout resonator, with the following Hamiltonian | ||
<math> | <math> | ||
H=\frac{\hbar}{2} \omega_{q} \sigma_{z}+\hbar \omega_{r} a^{\dagger} a+\hbar g\left(a^{\dagger} \sigma^{-}+a \sigma^{+}\right) \ . | H=\frac{\hbar}{2} \omega_{q} \sigma_{z}+\hbar \omega_{r} a^{\dagger} a+\hbar g\left(a^{\dagger} \sigma^{-}+a \sigma^{+}\right) \ . | ||
</math> | </math> | ||
The quantum system works in the dispersive regime where the qubit is strongly detuned from the oscillator with | 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 | ||
<math> | <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} \ , | 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} \ , | ||
</math> | </math> | ||
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 | 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 this equation. There, the cross-Kerr nonlinearity strength {{\chi}} is derived from the coupling strength {{g}} as | ||
< | |||
<math> | |||
\chi = \frac{g^{2}}{\Delta} \ . | \chi = \frac{g^{2}}{\Delta} \ . | ||
</math> | </math> | ||
Revision as of 14:04, 29 March 2021
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
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
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 this equation. There, the cross-Kerr nonlinearity strength Template:\chi is derived from the coupling strength Template:G as