Charge Noise in Superconducting Qubits

The dominant decoherence mechanism in charge-sensitive superconducting qubits. Random fluctuations in background charge (from two-level systems in oxide layers, charge traps at interfaces) modulate the offset charge $n_g$ and dephase the qubit.

This note is about the noise source itself and how it enters the Hamiltonian. For the broader cross-platform design pattern of operating at protected extrema, see charge-noise-sweet-spot.

The Problem

The Cooper pair box Hamiltonian $H=4E_C(n-n_g{)}^2-E_J\cos \hat{\phi }$ has energy levels that depend on $n_g$. At generic operating points, charge noise $\delta n_g$ causes dephasing at rate:

${\Gamma }_{\phi }\propto {\left|\frac{\partial E_{01}}{\partial n_g}\right|}^2{S}_{n_g}(\omega \to 0)$

where $S_{n_g}$ typically has $1/f$ spectral density.

Solutions

1. Sweet spots: Operate at $n_g=1/2$ where $\partial E_{01}/\partial n_g=0$ (first-order insensitivity). Used in the “quantronium” design.

2. Transmon regime: Increase $E_J/E_C\gg 1$. Energy bands flatten exponentially, making $E_{01}$ insensitive to $n_g$ at ALL operating points. Cost: reduced anharmonicity $\sim E_C$.

3. Fluxonium: Uses superinductance to create a different energy landscape with charge-insensitive sweet spots.

Historical Arc

• cooper-pair-box-charge-qubit (1999): First qubit, heavily charge-sensitive → $T_2\sim$ ns
• Quantronium (2002): Sweet-spot operation → $T_2\sim$ μs
• transmon (2007): Exponential suppression → $T_2\sim$ 10–100 μs
• Modern transmons: $T_1,T_2\sim$ 100+ μs (charge noise no longer dominant)

References

• nakamura-1999-cooper-pair-box
• koch-2007-transmon