We do not measure the Transmon directly. Instead, every qubit is capacitively coupled to its own personal microwave readout resonator. The interaction between the qubit (a two-level artificial atom) and the resonator (a quantized electromagnetic cavity) is described by the Jaynes-Cummings Hamiltonian:
\( \hat{H}_{JC} = \frac{1}{2} \hbar \omega_q \hat{\sigma}_z + \hbar \omega_r \hat{a}^\dagger \hat{a} + \hbar g (\hat{\sigma}^+ \hat{a} + \hat{\sigma}^- \hat{a}^\dagger) \)Here, \( \omega_q \) and \( \omega_r \) are the qubit and resonator frequencies, \( \hat{a}^\dagger \) and \( \hat{a} \) create and annihilate cavity photons, and \( g \) is the transverse capacitive coupling strength. The operators \( \hat{\sigma}^+ \) and \( \hat{\sigma}^- \) are the qubit raising and lowering operators, and \( \hat{\sigma}_z \) is the Pauli-Z operator.
Jaynes-Cummings Dispersive Shift
Observe how the transmon state (|0⟩, |1⟩, or |2⟩) shifts the resonance frequency of the coupled microwave cavity. This is the foundation of dispersive qudit readout.
Cavity Frequency: ωr,0 + χ0
The Dispersive Shift & Schrieffer-Wolff Derivation
We operate in the dispersive regime, where the detuning between the qubit and the resonator is much larger than their coupling strength:
\( \Delta = \omega_q - \omega_r \gg g \)Under this condition, the physical exchange of real excitations between the qubit and the cavity is blocked. However, they can still interact virtually. To find the effective Hamiltonian in this regime, we perform a Schrieffer-Wolff (SW) transformation—a canonical perturbation theory method that diagonalizes the Hamiltonian to first order in \( g/\Delta \).
The transformation is defined by the unitary operator \( \hat{U} = e^{\hat{S}} \), where we choose the generator \( \hat{S} \) to satisfy the condition \( [\hat{H}_0, \hat{S}] = -\hat{H}_{\text{int}} \), where \( \hat{H}_0 \) is the uncoupled Hamiltonian and \( \hat{H}_{\text{int}} \) is the interaction term. This yields:
\( \hat{S} = \frac{g}{\Delta} (\hat{a} \hat{\sigma}^+ - \hat{a}^\dagger \hat{\sigma}^-) \)Applying the transformation \( \hat{H}_{\text{eff}} = e^{\hat{S}} \hat{H}_{JC} e^{-\hat{S}} = \hat{H}_{JC} + [\hat{S}, \hat{H}_{JC}] + \frac{1}{2}[\hat{S}, [\hat{S}, \hat{H}_{JC}]] + \dots \) and expanding to order \( O(g^2/\Delta) \), we get the dispersive Hamiltonian:
\( \hat{H}_{\text{eff}} \approx \hbar \left(\omega_r + \chi \hat{\sigma}_z\right) \hat{a}^\dagger \hat{a} + \frac{1}{2} \hbar \left(\omega_q + \chi\right) \hat{\sigma}_z \)where the dispersive shift \( \chi \) is defined as:
\( \chi = \frac{g^2}{\Delta} \)This reveals that the resonator frequency is effectively shifted by \( +\chi \) if the qubit is in state \( |0\rangle \) (spin down) and by \( -\chi \) if the qubit is in state \( |1\rangle \) (spin up):
\( \omega_r' = \omega_r + \chi \hat{\sigma}_z \)To measure the qubit, we bounce a microwave tone off the resonator. The phase of the reflected microwave tone shifts depending on the cavity's new frequency, which directly correlates to the state of the qubit.
The Critical Photon Number Limit
The dispersive approximation relies on the assumption that the number of photons \( n = \langle \hat{a}^\dagger \hat{a} \rangle \) in the cavity is small. If we drive the resonator too hard to increase the signal-to-noise ratio, the perturbation expansion breaks down. The boundary is defined by the critical photon number:
\( n_{\text{crit}} = \frac{\Delta^2}{4g^2} \)When the photon count in the resonator approaches or exceeds \( n_{\text{crit}} \), the approximation \( \chi \hat{\sigma}_z \hat{a}^\dagger \hat{a} \) collapses. Multi-photon resonances occur, causing strong hybridization between the qubit and resonator, which triggers measurement-induced transmon ionization (promoting the transmon to high-energy non-computational states like \( |2\rangle \), \( |3\rangle \) or higher) and destroys the non-demolition (QND) nature of the measurement.