Dispersive regime (Schrieffer‑Wolff)

Dispersive regime (Schrieffer‑Wolff)§

This is a quick short article introducing the dispersive regime as it have the same math of deriving the JC-model. Where now, starting from the JC-model, out goal is to eliminate the interaction term since we now know far detuning light will almost not interact with atoms.

Here is the exact dispersive Hamiltonian in the lab frame (Schrödinger picture) with the Lamb shift explicitly separated:

$$\hat{H}_{\text{disp}} = \hbar\omega_k \hat{a}^\dagger\hat{a} + \frac{\hbar}{2}\omega_{eg}\hat{\sigma}_z - \frac{\hbar g^2}{2\Delta}\hat{\sigma}_z - \frac{\hbar g^2}{\Delta}\hat{a}^\dagger\hat{a}\hat{\sigma}_z$$

If we use the standard definition $\chi \equiv g^2/\Delta$, it is written as:

$$\hat{H}_{\text{disp}} = \hbar\omega_k \hat{a}^\dagger\hat{a} + \frac{\hbar}{2}(\omega_{eg}\underbrace{- \frac{\chi}{2}}_{\text{Lamb Shift}})\hat{\sigma}_z \underbrace{- \hbar\chi\hat{a}^\dagger\hat{a}\hat{\sigma}_z}_{\text{Dispersive Shift (AC Stark)}}$$

Where $\chi$ is the suspectibility.

Properties§

• photon number is preserved
• atomic energy is preserved
• Lamb shift
• Dispersive AC Stark Shift
• All of the above while photon and atom are still coupled
• Photon‑number splitting: the atomic resonance splits into a series of peaks, one for each photon number.
• The cavity resonance is pulled by $\pm\chi$ depending on the qubit state, enabling conditional phase gates and QND readout.

QND§

Quantum non‑demolition (QND) readout Because the Hamiltonian is diagonal, we can measure the atom without disturbing it. Shine a weak probe tone on the cavity and measure its resonance frequency. The frequency shift will tell you whether the atom is in $\ket{e}$ or $\ket{g}$ – without absorbing a single photon of the probe. This is a quantum non‑demolition measurement, the gold standard for qubit readout in circuit QED and cavity QED.

The same principle works in reverse: by measuring the atomic transition frequency (e.g., with a separate laser), we can count how many photons are inside the cavity without destroying them – a photon‑number QND measurement.

perturbative limit§

If we look only at the energy value, we find it is the higher order term of the full JC-model energy upon expansion, thus this is a perturbative limit