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We study dynamics of fermions loaded in an optical lattice with a superimposed parabolic trap potential. In the recent Hamburg experiments [J.Heinze et.al., Phys. Rev. Lett. 110, 085302 (2013)] on quantum simulation of photoconductivity, a modulation pulse on the optical lattice transferred part of the population of the lowest band to an excited band, leaving a hole in the particle distribution of the lowest band. Subsequent intricate dynamics of both excited particles and holes can be explained by a semiclassical approach based on the evolution of Wigner function. Here we provide a more detailed analysis of the dynamics taking into account the dimensionality of the system and finite temperature effects, aiming at reproducing experimental results on longer timescales. A semiclassical wave packet is constructed more accurately than in the previous theory. As a result, semiclassical dynamics indeed reproduces experimental data and full quantum numerical calculations with much better accuracy. In particular, fascinating phenomenon of collapse and revival of holes is investigated in a more detail. We presume the experimental setup can be used for deeper exploration of nonlinear waves in fermionic gases.

We consider dynamical stabilization of Bose-Einstein condensates (BEC) by time-dependent modulation of the scattering length. The problem has been studied before by several methods: Gaussian variational approximation, the method of moments, method of modulated Townes soliton, and the direct averaging of the Gross-Pitaevskii (GP) equation. We summarize these methods and find that the numerically obtained stabilized solution has different configuration than that assumed by the theoretical methods (in particular a phase of the wavefunction is not quadratic with $r$). We show that there is presently no clear evidence for stabilization in a strict sense, because in the numerical experiments only metastable (slowly decaying) solutions have been obtained. In other words, neither numerical nor mathematical evidence for a new kind of soliton solutions have been revealed so far. The existence of the metastable solutions is nevertheless an interesting and complicated phenomenon on its own. We try some non-Gaussian variational trial functions to obtain better predictions for the critical nonlinearity $g_{cr}$ for metastabilization but other dynamical properties of the solutions remain difficult to predict.

We determined the eigenstates of a single electron in a parabolic anisotropic 2D quantum dot in a magnetic field. Using obtained expressions for these eigenstates, we study the spin coupling J between two electrons located in two laterally coupled anisotropic quantum dots (QD). The exhange coupling J is calculated using the Heitler-London and Hund-Mulliken approaches. We found that the exhange J changes sign at certain values of parameters of the system, in particular at certain anisotropy of the QD. Therefore, we present a new method to switch on and off the spin coupling between QD: switching by means of changing shape of the QD. Switching the spin coupling is essential for quantum computation using electronic spins as qubits. We note that our calculations can be applied to the system of vertically coupled QD as well., Comment: 6 pages, 3 figures, submitted for publication