In the normal metal surrounded by superconductors an interesting thing can happen. Electrons (charged negative excitations of the ground state) move in this metal and occasionly bounce at the interface of superconductor where it enters the superconductor with it taking another electron under the Fermi level (like a shadow) in the end leaving a hole. Such process repeats with holes and forms a closed cycle. Such trajectory happens at certain energies which we call Andrew Bound spectrum and are highly dependant on all connected superconducting order parameters (phases). This dependance of spectum on phases determine the suppercurrent in Josephson junctions, and as well for multiterminal superconducting junctions. The later ones can be tuned to a Weyl singularities where two Andrew bands croses which quantizes conductance.

However the life for experimentalist might not be so easy. The tuning of phases could be affected by a quantum fluctuations due to always present capacities and geometry of the superconductor terminals. To model such problem we promoted superconducting phases at the vicinity of the Weyl point as dynamical variables which we coupled with external phases by a concave potential. What we found surprised us: in a soft constraint limit when the fluctuations in phase can be large two of three directions (in the space defined by superconducting phases) becomes almost degenerate forming subspaces in place of Weyl points which we call Weyl Discs Erdmanis & Lukacs & Nazarov (2018) (with Julia code in here).

A fascinating application of one such subspace would be in quantum computation. Experimentally a qubit of made of Andrew bound states in Jospehson junction had been demonstrated Urbina et. al. (2015) where excitation of the state and readout had already been developed. The Weyl discs in addition would allow to perform hllonomic computation by changing external superconductor phases and cllecting Berry phases. Particularly one could consider a circular orbit on the disc. Due to Berry curvature a phase difference would be acquired between the states and due to mixing of the lowest levels, the basis (between which a relative phase would be acquired) would depend on the angular velocity Aharanov & Anandan (1986). Thus effectively allowing us to implement any rotation in the Bloch sphere and so any single qubit gate (that follows because time dependence of the Weyl disc Hamiltonian for circular orbit can be factored out as exponentials acting on time independent part). However, it is still unknown how effective (for example how many Rabbi oscillations one could expect to get) the collection of phase difference would be in comparison with parametric noise (which would also affect decoherence time).

Like optical fibers for photons, Quantum Hall effect carry electrons along its edge channels. With this similarity it seems possible to make electron interferometry experiments in a solid state device. Although now it is an ambitious setting because of many difficulties, in future knowledge how electron emission protocol from ondemand electron source affects its quantum state and its quantumness could become useful.

For this purpose we did consider toy model Hamiltonian
$$ H = \epsilon |d\rangle \langle d| + \sum_k E_k |k\rangle \langle k| + \sum_k V_k |k\rangle \langle d| + h.c. $$
where \(|d\rangle\) is state of quantum dot and \(|k\rangle\) are evenly spaced energy states on quantum Hall edge channel. In the experiments it is posible obtain exponential tunneling rate time dependence. But with wide band approximation we may say that process is described with a couplings \(\sqrt{2 \pi \rho} V_k(t) = \sqrt{\hbar/\tau} e^{t/2\tau}\). The corresponding emission when energy of quantum dot state remains steady is shown in animation to the left. For more detailed analysis see my *BSc thesis summary*.

In a more realistic case quantum dot coupling depends on energy which could be deduced for example by considering a parabolic potential barrier. In case of static barrier emission can be initiated by driving the quantum dot potential up until the electron tunnels. Analysis of this case can be found in Kashcheyevs & Samuelsson (2017).

In my masters project, I considered properties of magnetic fluid droplets created by induced phase separation. Under fast rotating magnetic field these droplets takes oblate, three-axial and starfish shape depending on magnetic field intensity. It was my task to understand if this rich behavior for large magnetic fields could be understood with a simple magnetic drop model, where the equilibrium is only between scalar surface tension and linear magnetic force \(({\bf M} \cdot \nabla) {\bf H}\) (\({\bf B} = \mu {\bf H}\)). To answer this question the task of calculating equilibrium figures and direct comparison with experiment was set. (see video on the right for starfish equilibrium calculation).

With equilibrium figure calculations, we validated ellipsoidal model for calculation of critical field value, when transition from oblate to three axial figure happens, and for calculation of equilibrium figure axes even when droplet becomes nonellipsodial. This knowledge allowed us to find \(\mu\) and \(\gamma/R_0\) from droplet behavior in rotating field experiment alone by fitting theory to the experiment.

We compared obtained values with elongation experiment and found good agreement for \(\mu\), but significant difference for surface tension \(\gamma/R_0\). Also slight discrepancy was found in critical field value obtained experimentally from critical slowing down effect, where we also validated method with simulation. Thus we may conclude that surface tension is affected by magnetic field in a unknown way.

One way we expected for these effects to become present was deviations between simple magnetic drop model and experiment at large magnetic fields. Qualitatively we saw the same behavior - forming of sharp tips for three-axial figure and the shape widening in one direction. Also we did direct comparison of three-axial and starfish equilibrium figures at \(\mu\) and \(\gamma/R_0\) values obtained previously. Very good agreement was observed thus excluding anisotropic surface tension effects when droplet is in rotating field.

We also did study transition from three axial figure to starfish. Although we could not manage to induce transition numerically, we were able to demonstrate that starfish becomes energetically more favorable for large magnetic fields. For small magnetic fields, we were able to show the existence for minimal field, when higher order perturbation does not grow. This result is also in agreement qualitatively with previously derived analytic result from linear perturbation theory for two dimensions. The research can be found in Erdmanis et al. (2017).

Although our aim was to make viscous magnetic droplet simulation for calculation of equilibrium figures, we found astonishing hole in boundary element literature. Let's say that you would like calculate force between magnet and magnetizable body. The usual way considered in the literature is to sum up all \(({\bf M} \cdot \nabla) {\bf H}\) contributions over all volume. But in case \({\bf B} = \mu {\bf H}\) the magnetic force can also be evaluated only from surface magnetic field values $${\bf F} = \int ({\bf M} \cdot \nabla) {\bf H} dV = ... \int B_n^2 {\bf n} dS + ... \int H_t^2 {\bf n} dS $$ where \(B_n\) and \(H_t\) are normal component of magnetic induction and tangential component modulus of magnetic intensity. We found no literature using this formula for force calculation!

Essentially our main issue for making full 3D viscous droplet simulation to calculate equilibrium figures was the same. Although literature was perfected for making 2D and axialsymetric simulations, despite our hopes was not possible to generalize due to presence of strong singularity in boundary integral equations, where we found no consensus how to integrate them numerically in 3D. To overcome this difficulty we developed a new way to calculate magnetic field on the magnetizable body surface.

Firstly we calculated magnetic potential and then with numerical differentiation tangential components of magnetic field intensity. The normal components of field induction was further recalculated with Biot-Savarat integral, which we derived with pure boundary integral formalism and with physical assumptions in Erdmanis et al. (2017).

The physical way makes use of principle which states that perturbed magnetic field of linear magnetic body is equivalent of field as it would come from surface currents only. This allows us to write formula for the surface current \(4 \pi {\bf K}/c = - (\mu -1) {\bf n} \times {\bf H}\), which we can now use in the Biot-Savart integral for calculating magnetic field perturbation everywhere from tangential filed components. It is very fascinating that the result is valid also for electric field where this derivation does not apply!

Due to simplicity, generality, efficiency and many analytic tests found in article we believe that combined field calculation algorithm will find its use in the industry, where our approach offers precise magnetic field calculation near the magnets and iron parts, calculation of magnetic force for iron body near the magnet and electric force calculation in electric field. These are examples for industrial applications, which gains a new tool as a result for pursuing fundamental science.