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# Transition Frequencies

**Transition frequencies** are energy differences between levels: $\omega_{i \to j} = (E_j - E_i) / \hbar$. The qubit frequency is $\omega_{01} = (E_1 - E_0) / \hbar$.

For a transmon, the qubit frequency is approximately:

$$\omega_{01} \approx \sqrt{8 E_J E_C} - E_C$$

(first-order approximation in $E_C / E_J$).

```python
from scqubits import Transmon
import numpy as np

transmon = Transmon(EJ=15.0, EC=0.3, ncut=30)

# Qubit frequency
f_01 = transmon.f_01()  # GHz
print(f"Qubit frequency: {f_01:.4f} GHz")

# All transition frequencies
evals = transmon.eigenvals(n=5)
print(f"Transitions:")
for i in range(len(evals) - 1):
    print(f"  {i} → {i+1}: {evals[i+1] - evals[i]:.4f} GHz")
```

## Parameter Dependence

For a transmon, $\omega_{01} \propto \sqrt{E_J}$ (roughly)—increasing Josephson energy increases frequency. For fluxoniums, $\omega_{01}$ is flux-dependent, enabling tunable couplings.

Transition frequencies must be known precisely for gate design, resonance conditions, and multi-qubit couplings. scqubits computes them numerically.