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from the Kapitza Institute.

As a new theory effort, we hired Tero Heikkilä in 2003 as a postdoc to work in the area of quantum electron transport. In August 2006 he was awarded by the Academy of Finland a five-year position as Research Fellow working in the LTL. At the same time he established his own THEORY team in the LTL. Nikolai Kopnin is also dedicating about half of his time to mesoscopic superconductivity. The increased theory support in low temperature quantum electronics is already bearing fruits.

### NOnequilibrium properties of mesoscopic superconductors

N.B. Kopnin

Collaborators: Yu.M. Galperin (University of Oslo, Norway), V.M. Vinokur (Argonne National Laboratory, US), A.S. Mel'nikov (Institute of Microstructures, Nizhny Novgorod, Russia), and F. Taddei and F. Giazotto (Scuola Normale Superiore, Pisa, Italy)

The electronic heat conduction along a vortex in a long ballistic mesoscopic superconducting cylinder with a radius of order of several coherence lengths has been studied theoretically. The vortex heat conduction in a mesoscopic superconductor is considerably enhanced as compared to its value in a bulk superconductor due to giant mesoscopic oscillations of the energy levels resulting from the interplay between the Andreev (at the vortex core boundary) and the normal (at the cylinder surface) reflection processes.

Charge transport and noise in a single-mode long SINIS junction (S stands for superconductor, I is an insulator, and N is a normal metal) has been investigated in the limit of low bias voltages and low temperatures. The kinetic equation for the quasiparticle distribution on the Andreev levels is derived taking into account both inelastic relaxation and voltage-driven Landau-Zener transitions between the levels. For a long junction when the number of levels is large, the transitions enhance the action of each other and lead to a drastic increase of the dc current. The Fano factor is also enhanced, being proportional to the number of times which particle flies before it escapes from the level due to inelastic scattering.

The influence of an AC drive on heat transport in a hybrid normal metal - superconductor tunnel junction in the photon-assisted tunneling regime has been investigated theoretically. The heat flux out from the normal metal is shown to be always reduced as compared to its magnitude under the static and quasi-static drive conditions. These results are useful for predicting the operative conditions of AC driven superconducting electron refrigerators.

### SPIN SUPERFLUIDITY and Q-ball

G. Volovik

Collaborator: Yuriy Bunkov (Institut Néel, CNRS, Grenoble, France)

Q-ball is another representative of a state exhibiting the spin superfluidity, i.e. the phase coherent precession, similar to the well known Homogeneously Precessing Domain in 3He-B which we interpret as Bose condensation of spin waves – magnons, see reviews. A Q-ball is a non-topological soliton solution in field theories containing a complex scalar field. Q-balls are stabilized due to the conservation of the global

Annual Report 2007