Quantum synchronization

Using the methods of quantum trajectories we study numerically the phenomenon of quantum synchronization in a quantum dissipative system with periodic driving. Our results show that at small values of Planck constant $\hbar$ the classical devil's staircase remains robust with respect to quantum fluctuations while at large $\hbar$ values synchronization plateaus are destroyed. Quantum synchronization in our model has close similarities with Shapiro steps in Josephson junctions and it can be also realized in experiments with cold atoms.Comment: 5 pages, 5 figs, 1 fig added, research at http://www.quantware.ups-tlse.f

Anderson transition for Google matrix eigenstates

We introduce a number of random matrix models describing the Google matrix G of directed networks. The properties of their spectra and eigenstates are analyzed by numerical matrix diagonalization. We show that for certain models it is possible to have an algebraic decay of PageRank vector with the exponent similar to real directed networks. At the same time the spectrum has no spectral gap and a broad distribution of eigenvalues in the complex plain. The eigenstates of G are characterized by the Anderson transition from localized to delocalized states and a mobility edge curve in the complex plane of eigenvalues.Comment: 9 pages, 12 figs, revte

Thermoelectricity of Wigner crystal in a periodic potential

We study numerically the thermoelectricity of the classical Wigner crystal placed in a periodic potential and being in contact with a thermal bath modeled by the Langevin dynamics. At low temperatures the system has sliding and pinned phases with the Aubry transition between them. We show that in the Aubry pinned phase the dimensionless Seebeck coefficient can reach very high values of several hundreds. At the same time the charge and thermal conductivity of crystal drop significantly inside this phase. Still we find that the largest values of $ZT$ factor are reached in the Aubry phase and for the studied parameter range we obtain $ZT \leq 4.5$. We argue that this system can provide an optimal regime for reaching high $ZT$ factors and realistic modeling of thermoelecriticy. Possible experimental realizations of this model are discussed.Comment: 7 pages, 9 figs, EPL latex, larger statistics and parameter rang

Quantum Resonances and Regularity Islands in Quantum Maps

We study analytically as well as numerically the dynamics of a quantum map near a quantum resonance of an order q. The map is embedded into a continuous unitary transformation generated by a time-independent quasi-Hamiltonian. Such a Hamiltonian generates at the very point of the resonance a local gauge transformation described the unitary unimodular group SU(q). The resonant energy growth of is attributed to the zero Liouville eigenmodes of the generator in the adjoint representation of the group while the non-zero modes yield saturating with time contribution. In a vicinity of a given resonance, the quasi-Hamiltonian is then found in the form of power expansion with respect to the detuning from the resonance. The problem is related in this way to the motion along a circle in a (q^2-1)-component inhomogeneous "magnetic" field of a quantum particle with $q$ intrinsic degrees of freedom described by the SU(q) group. This motion is in parallel with the classical phase oscillations near a non-linear resonance. The most important role is played by the resonances with the orders much smaller than the typical localization length, q << l. Such resonances master for exponentially long though finite times the motion in some domains around them. Explicit analytical solution is possible for a few lowest and strongest resonances.Comment: 28 pages (LaTeX), 11 ps figures, submitted to PR

Phase diagram for the Grover algorithm with static imperfections

We study effects of static inter-qubit interactions on the stability of the Grover quantum search algorithm. Our numerical and analytical results show existence of regular and chaotic phases depending on the imperfection strength $\epsilon$. The critical border $\epsilon_c$ between two phases drops polynomially with the number of qubits $n_q$ as $\epsilon_c \sim n_q^{-3/2}$. In the regular phase $(\epsilon < \epsilon_c)$ the algorithm remains robust against imperfections showing the efficiency gain $\epsilon_c / \epsilon$ for $\epsilon \gtrsim 2^{-n_q/2}$. In the chaotic phase $(\epsilon > \epsilon_c)$ the algorithm is completely destroyed.Comment: 4 pages, 4 figs, research at http://www.quantware.ups-tlse.f

Synchronization theory of microwave induced zero-resistance states

We develop the synchronization theory of microwave induced zero-resistance states (ZRS) for two-dimensional electron gas in a magnetic field. In this theory the dissipative effects lead to synchronization of cyclotron phase with driving microwave phase at certain resonant ratios between microwave and cyclotron frequencies. This synchronization produces stabilization of electron transport along edge channels and at the same time it gives suppression of dissipative scattering on local impurities and dissipative conductivity in the bulk, thus creating the ZRS phases at that frequency ratios. The electron dynamics along edge and around circular disk impurity is well described by the Chirikov standard map. The theoretical analysis is based on extensive numerical simulations of classical electron transport in a strongly nonlinear regime. We also discuss the value of activation energy obtained in our model and the experimental signatures that could establish the synchronization origin of ZRS.Comment: revtex, 15 pages, 17 fig