# Transmons

**Transmon** (transmission line shunted plasma oscillation qubit) is the most common superconducting qubit design. It consists of a Josephson junction shunted (paralleled) by a large capacitor. The capacitor reduces charge noise sensitivity compared to bare charge qubits, making transmons robust and practical.

Transmon Hamiltonian:

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

where $\hat{n}$ is the charge (number of Cooper pairs) operator, $E_C$ is charging energy, $E_J$ is Josephson energy, and $\hat{\phi}$ is the phase.

```python
from scqubits import Transmon

# Create a transmon
transmon = Transmon(
    EJ=15.0,    # Josephson energy (GHz)
    EC=0.3,     # Charging energy (GHz)
    ng=0.1,     # Offset charge (0 to 1)
    ncut=30     # Cutoff for charge basis (|n⟩ up to |ncut⟩)
)

# Eigenvalues and eigenstates
evals = transmon.eigenvals(n=5)  # First 5 eigenvalues
evecs = transmon.eigenvecs(n=5)

# Anharmonicity: frequency difference between transitions
anh = transmon.anharmonicity()
print(f"Anharmonicity: {anh:.4f} GHz")

# Matrix elements for transitions
m01 = transmon.matrixelement_table(operator=transmon.n_operator, evec_num=5)
```

The ratio $E_J / E_C$ controls the qubit's properties: large ratios suppress charge noise; small ratios make the qubit more charge-sensitive but easier to control.

Transmons are the workhorse of superconducting quantum computing. IBM, Google, and most others use transmons.