Is there Johnson noise in superconductor? For conductor, the Johnson Noise is 
$v_n = \sqrt { 4 k_B T R \Delta f }$.
It is clear that the noise depends on $R$. I'm curious whether this noise will appear in supercondutor? That is for conductor the leading order of the noise is $v_n = \sqrt { 4 k_B T R \Delta f }$, and for superconductor the leading order is zero but it has subleading order term.
 A: Since the superconductor is not dissipative (at least for very low frequencies), it will not generate thermal noise the same way a resistor does. However, every superconductor of finite length forming a loop of non-zero area has an inductivity L. 
Just like capacitors held at a finite temperature will generate charge fluctuations for a charge measurement of finite aperture (i.e. measurement time), which are proportional to $\sqrt{kTC}$, inductors have to generate a similar flux noise term. Capacitor "sampling" noise is of importance for many devices (e.g. CCD readout), flux noise matters in squid sensors and bolometer applications, see e.g. http://iopscience.iop.org/0953-2048/26/7/075022.
A simple physical way of thinking about this is that inductors forming a finite loop couple to the electromagnetic field inside the loop. If the temperature of that field (i.e. the temperature of the inductor) is >0, there has to be a fluctuating current in response to the (magnetic flux) field fluctuations. 
Now, whether you want to assign these fluctuations to the superconductor itself, is a matter of taste, since they do not arise as a consequence of the superconductivity but are just as well present in ordinary inductors (but they are, to the best of my knowledge, usually far smaller than resistive noise). 
If you are looking for noise sources that are intimately related to the mechanism of superconductivity, the coupling between phonons and Cooper pairs comes to mind, and unsurprisingly there are phonon noise sources in bolometers and other devices: http://en.wikipedia.org/wiki/Phonon_noise.
