In the Cooper Pair box, the conjugate momentum $\hat{n}$ of the reduced superconducting phase variable $\hat{\varphi}$ takes only discrete, integer values due to the $2\pi$ periodicity of the phase. As this conjugate momentum is the relative charge number on the island, this corresponds to the discrete charge on the island.

enter image description here

In the inductively shunted CPB, the inductance destroys the periodicity of the potential in $\varphi$, we have a $\frac{1}{2}E_L \varphi^2$ term additional to the usual $\cos{(\varphi)}$ term of the Josephson Junction.

Now Girvin (http://www.capri-school.eu/lectureres/master_cqed_les_houches.pdf, after 4.41) or Koch et al. (PRL 103, 217004 (2009)) write that this leads to the charge variable becoming continuous as opposed to integer-valued like in the CPB.

Formally, this makes sense to me: As we do not have periodic boundary conditions for $\varphi$ anymore, we will not have discreteness of the conjugate momentum, i.e. the charge anymore.

I wonder, however, how to interpret this physically: After all, the transferred charge must still be quantized in the form of Cooper pairs. So what is meant by the continuity of the charge here?

  • 2
    $\begingroup$ Hi there, nice question but this might be too "physical" for QC.SE, you might have more luck with it over on Physics.SE. $\endgroup$
    – Condo
    Dec 8, 2021 at 20:35
  • $\begingroup$ Hi thanks for the comment. I will post it on Physics.SE if I won't get a good answer here then! The tags I used for my question are more specific to it than those I could find on Physics.SE though. $\endgroup$ Dec 8, 2021 at 20:51
  • $\begingroup$ I asked this question on the physics.SE now and I understand that following the guidelines this question should be closed. I cannot do this myself as I have less than 50 reputation apparently. $\endgroup$ Dec 12, 2021 at 21:44
  • 5
    $\begingroup$ I’m voting to close this question because the OP wishes to close the question after moving the question to Physics SE. $\endgroup$
    – Condo
    Dec 12, 2021 at 23:27
  • 2
    $\begingroup$ Here is the physics.SE post of this question. $\endgroup$ Dec 13, 2021 at 19:07