# Number and phase operators in superconductors

It is stated in many texts that the number operator $$N$$ which counts the number of Cooper pairs and the phase operator $$\phi$$ which counts the superconducting order parameter's phase $$\text{Arg}(\Delta)=\phi$$ are canonically conjugate and thus satisfy $$\begin{equation} N = -i\frac{\partial}{\partial \phi} \iff \phi = i\frac{\partial}{\partial N} \end{equation}$$ This makes sense when one looks at the BCS ground state $$\begin{equation} |BCS\rangle = \prod_k (u_k+v_kc^\dagger_{k\uparrow}c^\dagger_{-k\downarrow})|0\rangle \end{equation}$$ and since $$u^*_kv_k=|u_k||v_k|e^{i\phi}$$ this can be written as \begin{align} |BCS\rangle &= \prod_k (|u_k|+|v_k|e^{i\phi}c^\dagger_{k\uparrow}c^\dagger_{-k\downarrow})|0\rangle \end{align} up to some irrelevant global phase.

We see that every Cooper pair comes with a phase $$e^{i\phi}$$ so differentiating with respect to $$\phi$$ gives the same result as counting the number of Cooper pairs.

Due to this well-defined phase $$\phi$$, $$|BCS \rangle$$ is a coherent superposition of states with different particle number, and there does not have a well defined $$N$$. But now let's consider what happens when we project the BCS state on a the subspace with fixed particle number $$N$$ as explained in https://canvas.harvard.edu/courses/79258/files/folder/Problem%20Sets?preview=12301649. We find that $$\begin{equation} |BCS_N\rangle\equiv \int_0^{2\pi} \frac{d\phi}{2\pi} e^{-iN\phi} |BCS\rangle = C \bigg(\sum_k \frac{|v_k|}{|u_k|} c^\dagger_{k\uparrow}c^\dagger_{-k\downarrow}\bigg)^N |0\rangle \end{equation}$$ which looks a lot like a BEC state formed by $$N$$ Cooper pairs. This state clearly has $$N$$ cooper pairs. However when I use the definition $$N = -i\frac{\partial}{\partial \phi}$$, since there is no $$\phi$$ dependence in this state (it got integrated out) I am expected to get $$N=0$$?!

So what's wrong with this reasoning?

The identification of $$N$$ with $$-i\partial_\phi$$ is mathematicaly inconsistent as $$N$$ cannot take negative values. As a consequence it is not surprising that there are some paradoxes.

For example, from $$[\phi, \hat N]=i$$ we can derive an uncertainly relation $$\Delta N \Delta \phi\ge 1/2$$, but when $$\Delta N$$ is small (zero in your fixed number state) this implies $$\Delta \phi>2\pi$$, which is not possible as $$\phi$$ is an angle with bounded range.

• Your statement " ... Δϕ>2π ...is not possible as ϕ is an angle with bounded range" is closely related to univalence superselection, which is not uncontroversial. Apr 25, 2022 at 12:47
• @Andrea Alciato What is"univalence superselection"? Apr 25, 2022 at 13:13
• See en.wikipedia.org/wiki/Superselection#Examples for something directly related to your statement and pirsa.org/14030114 for the full story on univalence superselection. Apr 25, 2022 at 15:18

It is useful to first review the physics of a quantum planar rotor. Consider the Hilbert space spanned by $$|N\rangle$$. They are eigenstates of the number operator $$\hat{N}|{N}\rangle=N|{N}\rangle$$. The phase eigenstates are defined as $$|\phi\rangle=\sum_N e^{iN\phi}|N\rangle$$. The $$|\phi\rangle$$'s also form an orthonormal basis. The meaning of $$N=-i\partial_\phi$$ is that for a given state written in the $$|\phi\rangle$$ basis:

$$|\psi\rangle=\int_0^{2\pi}d\phi\, \psi(\phi)|\phi\rangle$$

We have

$$\hat{N}|\psi\rangle=\int_0^{2\pi}d\phi\, \Big(-i\partial_\phi\psi(\phi)\Big)|\phi\rangle$$

Now, the BCS case is not really like a rotor, since $$N$$ can not be negative. In this case, it is better to think of $$N$$ as measuring the deviation from the average number of Cooper pairs, which is supposed to be huge, and approximately consider the range $$N$$ to be all integers.