Transmon Anharmonicity Derivation

This note expands the sketch from Superconducting Qubit Feature Lab.

Starting Hamiltonian

The transmon Hamiltonian is

$$\hat{H}=4E_C(\hat{n}-n_g{)}^2-E_J\cos \hat{\phi },$$

with conjugate variables obeying

$$[\hat{\phi },\hat{n}]=i.$$

In the transmon regime $E_J/E_C\gg 1$, charge dispersion is suppressed and one often starts by setting $n_g\approx 0$ for intuition.

Cosine Expansion

Expand the cosine potential:

$$-E_J\cos \hat{\phi }=-E_J+\frac{E_J}{2}{\hat{\phi }}^2-\frac{E_J}{24}{\hat{\phi }}^4+\cdots .$$

The quadratic Hamiltonian is

$${\hat{H}}_0=4E_C{\hat{n}}^2+\frac{E_J}{2}{\hat{\phi }}^2-E_J.$$

Define zero-point fluctuations

$$\hat{\phi }={\phi }_{\mathrm{zpf}}(a+a^{†}),\hat{n}=i{n}_{\mathrm{zpf}}(a^{†}-a),$$

with

$${\phi }_{\mathrm{zpf}}={\left(\frac{2E_C}{E_J}\right)}^{1/4},n_{\mathrm{zpf}}={\left(\frac{E_J}{32E_C}\right)}^{1/4}.$$

Then

$${\hat{H}}_0=-E_J+\sqrt{8E_J{E}_C}\left(a^{†}a+\frac{1}{2}\right).$$

Quartic Perturbation

The leading correction is

$${\hat{H}}_1=-\frac{E_J}{24}{\hat{\phi }}^4.$$

Using the number-state expectation value

$$⟨m|(a+a^{†}{)}^4|m⟩=6m^2+6m+3,$$

we get the first-order correction

$$E_m^{(1)}=-\frac{E_C}{12}(6m^2+6m+3).$$

So the approximate energy levels are

$$E_m\approx -E_J+\sqrt{8E_J{E}_C}\left(m+\frac{1}{2}\right)-\frac{E_C}{12}(6m^2+6m+3).$$

The transition energy is

$$E_{m+1}-E_m\approx \sqrt{8E_J{E}_C}-E_C(m+1).$$

Therefore

$$\hslash {\omega }_{01}\approx \sqrt{8E_J{E}_C}-E_C,$$

and

$$\hslash {\omega }_{12}\approx \sqrt{8E_J{E}_C}-2E_C.$$

The anharmonicity is

$$\hslash \alpha =\hslash ({\omega }_{12}-{\omega }_{01})\approx -E_C.$$

Summary

Increasing $E_J/E_C$ suppresses charge sensitivity, but decreasing $E_C$ also reduces anharmonicity. Transmon design lives inside that tradeoff.

Related

• Transmon Qubits
• Josephson Junctions
• Superconducting Qubit Feature Lab