to $T = \Delta E/k _{B} \approx 0.001 \ K$ in a mesoscopic ring with a radius $r \approx 500 \ nm = 5 \ 10^{-7} \ m$. Therefore the quantum phenomena connected with the discrete spectrum, such as the persistent current of electrons, can be observed in nano-rings with a real radius $r > 300 \ nm$ only at very low temperature [@PCScien09; @PCPRL09]. The energy difference of superconducting loop $\Delta E \approx N _{s}\hbar ^{2}/2mr^{2} $ is much larger due to the impossibility for all $N _{s}$ Cooper pairs in the loop to change their quantum state $n$ individually [@FPP2009]. This impossibility of individual motion of quantum particle was postulated first by Lev Landau as far back as 1941 [@Landau41] for the description of superfluidity of $^{4}He$ liquid.

All $N _{s}$ pairs in the loop with the volume $V = ls$, the length $l$ and the section area $s$ are described with the wave function $\Psi = |\Psi |\exp{i\varphi }$, according to the Ginzburg-Landau theory [@GL1950]: $|\Psi |^{2} = n_{s}$ is the density of Cooper pairs and $\int_{V}dV |\Psi |^{2} = \oint_{l}dl s|\Psi |^{2} = N_{s} \gg 1$ is the total number of Cooper pairs in the loop. 