Reading Ref. 1, in the section about the authors discuss the simplest examples of qubits, among which there is the charge qubit, built from a Cooper-pair box (CPB). After writing the CPB Hamiltonian in the number/charge basis \begin{equation} H_{CPB}=\sum_{n\in\mathbb{Z}}\bigg[4E_C(\hat{n}-n_g)^2\lvert n\rangle\langle n\rvert-\frac{E_J}{2}(\lvert n+1\rangle\langle n\rvert+\lvert n-1\rangle\langle n\rvert)\bigg]. \tag{20} \label{20} \end{equation} The notation is pretty standard: $\hat{n}$ is the cooper pair number operator, $n_g=CV_g/2|e|$ is tunable tuning the gate potential. $E_C=e^2/2C$ and $E_J$ are respectively the charging energy and the Josephson energy. At the end of the sections there are some bullet points characterizing the system described by \eqref{20}. More specifically, they are related to the dependence of the spectrum on $n_g$. The half-integer values $n_g=m+\frac{1}{2}, m\in\mathbb{Z}$ play a special role, since as they say in the second bullet point:
For the two charge states $\lvert m\rangle$ and $\lvert m+1\rangle$, the effective charging energies $4E_C(n-n_g)^2$ are degenerate at these points.
Which is quite evident from the Hamiltonian \eqref{20}. On the other hand, the first bullet point is not as clear to me:
For these values of $n_g$, the eigenstates of the system have well-defined parities.
First, how do we know that? Ref. 1 has no other mention of this and I can't find it elsewhere$^{(\ast)}$. Also, what parity operator are we talking about? Reflection around what (cf. footnote)? I'd be grateful if you could reply to my questions and/or point at some relevant references.
References
- Quantum bits with Josephson junctions, sec. 3.1.
$^{(\ast)}$ I've seen another reference that I can't quote here (course notes) mention a similar point, but the computations were quite vague; also, in that case they mention a reflection with respect to this point, which makes me more confused.
