The transmon is a type of superconducting qubit designed to be insensitive to charge noise. Its Hamiltonian is:
$$ \hat{H}_{\text{transmon}} = 4E_C(\hat{n} - n_g)^2 - E_J\cos(\hat{\varphi}) $$
where $E_C = e^2/2C$ is the charging energy (the energy cost of adding a single electron), $E_J$ is the Josephson energy, $\hat{n}$ is the Cooper pair number operator, and $n_g$ is the offset charge. The ratio $E_J / E_C \gg 1$ defines the transmon regime: the junction energy dominates, the spectrum becomes weakly anharmonic, and sensitivity to charge fluctuations is strongly suppressed.
The anharmonicity — the difference between the $|0\rangle \to |1\rangle$ and $|1\rangle \to |2\rangle$ transition energies — is approximately $-E_C$.
An easy way to understand the transmon and how its Hamiltonian works is to look at its predecessor, the Cooper Pair Box (CPB).
The first demonstration of a superconducting qubit (to the best of the authors’ knowledge) was the Cooper pair box by Yasunobu Nakamura in 1999 (see paper).
The key idea of a charge qubit is that the qubit is encoded in different physical charge states. A Cooper pair box is a superconducting island shunted (connected in parallel) to a larger ground plane of superconductor through a Josephson junction. Since there is a finite charge on the island, you can tunnel Cooper pairs into and out of the island using a bias voltage, and use different charge states as your two states for a qubit.
Figure and caption from David Schuster’s thesis [1]
In the circuit above, the bias charge on the island (green) can be written as
$$ n_g = C_gV_g/2e $$
where the $2e$ comes from the Cooper pair tunneling across Josephson junctions. Sometimes the 2 for the Cooper pair is written separately from $n_g$, but these are describing the same circuit.
The charging energy $E_C$ , written as
$$ E_C = \frac{e^2}{2(C_j + C_g)} $$