What is the physical meaning of phase in a superconductor?

Question

I am trying to understand the physical meaning of phase in the order parameter of a superconductor. In particular,I was looking at this article that states the phase $e^{i\theta}$ in the BCS ground state

$$|\psi_G\rangle=\prod_k (u_k +v_k e^{i\theta}c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)|\phi_0\rangle $$

is not a gauge invariance, but a global $U(1)$ phase rotation symmetry. However, the author (on page 6) then says that

The most significant difference between an antiferromagnet and a superfluid or superconductor with regard to the order parameter is that the broken rotational symmetry in the former case is much more evident to us, as all the macroscopic objects in our daily life experience violate rotational symmetry at one level or another...In the case a superconductor, we need a second superconductor to have a reference direction for the phase, and an interaction between the order parameter in both superconductors to detect a relative difference in the phases.

I can easily visualize this breaking of $U(1)$ symmetry in a antiferromagnet. However, I am still confused as to what, exactly, is the physical meaning of phase in the superconducting order parameter? In particular, how can I prescribe a physical meaning to the phase on the Cooper pairing term in the BCS ground state? As with my other post on a similar question I had, any explanations or resources at the level of Tinkham would be greatly appreciated.

Answer 1

The phase of a superconductor is closely related to its quantum mechanical nature. Typically when people ask about the physical meaning of quantum mechanical notions they want a classical (i.e. non quantum mechanical) analogue.

For a phase the obvious analogue is waves. Optics is good, but one can even discuss water waves. Alternatively one can also think on a collection of harmonic oscillators (can be pendulums, masses on a spring, etc.). When there is no coherence all the components are moving uncorrelated. One cannot see any peaks or valley or notice any special feature at a specific time, so we call this cacophony "symmetry". Now imagine that some correlation appear, all oscillators reach the extrema point at the same time, or that the water at some point are lowest exactly when they are highest in another point and vice versa. Now we can see a structure and we say the symmetry is broken. The phase how we describe where the repeated motion start.

But, nothing is moving in the superconductor, so how does this apply? Well, as it is usually with classical analogues, they cannot be fully applied. Indeed the phase of a superconductor is a quantum mechanical phase and just like the phase of a quantum state can only be observed via comparison to another phase. In this case one would need another superconductor and observe the Josephson effect