# Charge basis

A charge basis represents a continuous U(1) degree of freedom $e^\{i\varphi\}$ in its fourier (irrep) basis $|n\rangle$, where $n$ is an integer. It may be used to represent quantum objects with integer charge, such as superconducting circuit elements.

The charge basis is a natural representation for circuit-QED elements such as the "transmon", which has a hamiltonian of the form

```
ncut = 20
E_C = 1.0
E_J = 10.0
n_g = 0.0
b = ChargeBasis(ncut)
H = 4E_C * (n_g * identityoperator(b) + chargeop(b))^2 - E_J * cosφ(b)
```

with energies periodic in the charge offset `n_g`

. See e.g. https://arxiv.org/abs/2005.12667.

**QuantumOptics.jl** provides the `ChargeBasis`

and `ShiftedChargeBasis`

basis types, representing a truncated (and optionally shifted) fourier basis, with `ncut`

(or `nmin`

and `nmax`

in the shifted case) specifying the range of available fourier modes.