How is the superconducting coherence length measured experimentally?

From Ginzburg-Landau theory we have

$mu_0 H_{c2} = frac{hbar}{2 e zeta(0)^2} frac{T_C-T}{T_c} $

where $mu_0 = 4 pi times 10^{-7} H/m$, $H_{c2}$ is the upper critical field of the sample, $hbar$ is Dirac's constant and $T_c$ is the critical temperature. You can measure $H_{c2}$ as a function of temperature and fit your data to the above equation to find the coherence length $zeta(0)$.

Ref. : J.F. Annett, Superconductivity, Superfluidity and Condensates, Oxford UP,2004

I think it is important to distinguish between the different length scales that are used in superconductivity. One can distinguish three fundamental length scales in superconductivity:

• London penetration depth $lambda_L (T)$: the typical distance it takes to screen the magnetic field inside a superconductor.
• Coherence BCS length $xi_0$: characterizes the length scale over which the correlation between the electrons in a Cooper pair is active. It is the typical size/diameter of a Cooper pair (BCS theory)
• Ginzburg-Landau (GL) coherence length $xi_{GL}(T)$: gives the minimal distance over which the superconducting order parameter can vary (Ginzburg-Landau theory)

Now for your question I will focus on how the two latter length scales can be measured. First, the Ginzburg-Landau coherence length can be measured in scattering experiments where we probe microscopic fluctuations. The scattering amplitude is given $$A(textbf{q}) propto sigma(textbf{q})int d^dtextbf{x} e^{itextbf{q}cdottextbf{x}}rho(textbf{q})$$ where $sigma(textbf{q})$ is the (local) form factor, it tells us something about the scattering of an individual element, whereas $rho(textbf{q})$ is the global density of scatterers, which contains more global info about the material. Now what we measure is the scattering intensity $$S(textbf{q})propto <|A(textbf{q})|^2> propto <|rho(textbf{q})|^2>$$ Hence, from the dependence on the wave vector $textbf{q}$ of this intensity at the critical temperature one can obtain an idea about $xi_{GL}$. To fully comprehend how this work consult chapter 3 of 'Statistical physics of fields' by Kardar M. However, there are many other ways to obtain an estimate of $xi_{GL}$, e.g. in type-II superconductors, the second critical magnetic field can also give an estimate of $xi_{GL}.

Next, we discuss the experimental method to determine the BCS coherence length $xi_0$, which is a less common length scale to be determined. More often, one is interested in $xi_{GL}$. As you will study superconductivity one will often use the term 'superconducting coherence length' where they are actually referring to the GL coherence length. However, one can estimate $xi_0$ from the measurement of $xi_{GL}$ or by studying the electron mean free path. To see how it relates to these quantities, I refer to section 4.2 of the standard text 'Introduction to superconductivity' of Thinkham.

I would disagree here. The coherence length is the length scale where the electrons stay in their coherent, superconducting state. This gets important on boundaries of a superconductor (i.e. the proximity effect) or at vortices of a type II superconductor in the mixed phase. In both examples, you can measure the coherence length.

This is done by fitting the decay of a parameter of the superconducting condensate, such as the magnitude of the energy gap. For this you have to do some math, because the relationships are far from trivial. But first, you have to define your system under consideration and then solve the right set of equations, such as Usadels equations for the diffusive proximity effect.