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# Dispersive Shifts

**Dispersive shift** is the frequency shift of one qubit when another qubit changes state. In coupled qubits, the interaction Hamiltonian causes energy shifts dependent on the state of the coupled qubit.

For capacitively coupled transmons with coupling $g_{12} n_1 n_2$:

$$\Delta \omega_1 = 2 g_{12} |\alpha|$$

where $\alpha$ is the anharmonicity of qubit 2. The shift is proportional to the anharmonicity, so low-$\alpha$ qubits have small dispersive shifts.

```python
from scqubits import Transmon, HilbertSpace, InteractionTerm

qubit1 = Transmon(EJ=15.0, EC=0.3, ncut=30)
qubit2 = Transmon(EJ=15.0, EC=0.3, ncut=30)

hilbert = HilbertSpace([qubit1, qubit2])
g = 0.01  # Coupling
interaction = InteractionTerm(g=g, op1=qubit1.n_operator(), op2=qubit2.n_operator())
hilbert.add_interaction(interaction)

# Spectrum with coupling
evals = hilbert.hamiltonian().eigenenergies()

# The level splittings reveal dispersive coupling
# |00⟩, |01⟩, |10⟩, |11⟩ levels show the shift
```

Dispersive shifts are used for qubit readout: measure the frequency shift to infer the state. They also cause qubit-qubit dephasing if unaccounted for in pulse design.