Coarse-Grained Picture for Controlling Complex Quantum Systems

We propose a coarse-grained picture to control ``complex'' quantum dynamics, i.e., multi-level-multi-level transition with a random interaction. Assuming that optimally controlled dynamics can be described as a Rabi-like oscillation between an initial and final state, we derive an analytic optimal field as a solution to optimal control theory. For random matrix systems, we numerically confirm that the analytic optimal field steers an initial state to a target state which both contains many eigenstates.Comment: jpsj2.cls, 2 pages, 3 figure files; appear in J. Phys. Soc. Jpn. Vol.73, No.11 (Nov. 15, 2004

Turbulent mixing of a slightly supercritical Van der Waals fluid at Low-Mach number

Supercritical fluids near the critical point are characterized by liquid-like densities and gas-like transport properties. These features are purposely exploited in different contexts ranging from natural products extraction/fractionation to aerospace propulsion. Large part of studies concerns this last context, focusing on the dynamics of supercritical fluids at high Mach number where compressibility and thermodynamics strictly interact. Despite the widespread use also at low Mach number, the turbulent mixing properties of slightly supercritical fluids have still not investigated in detail in this regime. This topic is addressed here by dealing with Direct Numerical Simulations (DNS) of a coaxial jet of a slightly supercritical Van der Waals fluid. Since acoustic effects are irrelevant in the Low Mach number conditions found in many industrial applications, the numerical model is based on a suitable low-Mach number expansion of the governing equation. According to experimental observations, the weakly supercritical regime is characterized by the formation of finger-like structures-- the so-called ligaments --in the shear layers separating the two streams. The mechanism of ligament formation at vanishing Mach number is extracted from the simulations and a detailed statistical characterization is provided. Ligaments always form whenever a high density contrast occurs, independently of real or perfect gas behaviors. The difference between real and perfect gas conditions is found in the ligament small-scale structure. More intense density gradients and thinner interfaces characterize the near critical fluid in comparison with the smoother behavior of the perfect gas. A phenomenological interpretation is here provided on the basis of the real gas thermodynamics properties.Comment: Published on Physics of Fluid

Positivity and strong ellipticity

We consider second-order partial differential operators $H$ in divergence form on \Ri^d with a positive-semidefinite, symmetric, matrix $C$ of real $L_\infty$-coefficients and establish that $H$ is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds.Comment: 9 page

Second-order operators with degenerate coefficients

We consider properties of second-order operators $H = -\sum^d_{i,j=1} \partial_i \, c_{ij} \, \partial_j$ on \Ri^d with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost everywhere, but allow for the possibility that $C$ is singular. We associate with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. The semigroup extends to a positive contraction semigroup on the $L_p$-spaces with $p \in [1,\infty]$. We establish that it conserves probability, satisfies $L_2$~off-diagonal bounds and that the wave equation associated with $H_0$ has finite speed of propagation. Nevertheless $S^{(0)}$ is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor H\"older continuous. In particular one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in C^{2-\varepsilon}(\Ri^d) with $\varepsilon > 0$.Comment: 44 page

Topological Characterization of Non-Abelian Moore-Read State using Density-Matrix Renormailzation Group

The non-Abelian topological order has attracted a lot of attention for its fundamental importance and exciting prospect of topological quantum computation. However, explicit demonstration or identification of the non-Abelian states and the associated statistics in a microscopic model is very challenging. Here, based on density-matrix renormalization group calculation, we provide a complete characterization of the universal properties of bosonic Moore-Read state on Haldane honeycomb lattice model at filling number $\nu=1$ for larger systems, including both the edge spectrum and the bulk anyonic quasiparticle (QP) statistics. We first demonstrate that there are three degenerating ground states, for each of which there is a definite anyonic flux threading through the cylinder. We identify the nontrivial countings for the entanglement spectrum in accordance with the corresponding conformal field theory. Through inserting the $U(1)$ charge flux, it is found that two of the ground states can be adiabatically connected through a fermionic charge-$\textit{e}$ QP being pumped from one edge to the other, while the ground state in Ising anyon sector evolves back to itself. Furthermore, we calculate the modular matrices $\mathcal{S}$ and $\mathcal{U}$, which contain all the information for the anyonic QPs. In particular, the extracted quantum dimensions, fusion rule and topological spins from modular matrices positively identify the emergence of non-Abelian statistics following the $SU(2)_2$ Chern-Simons theory.Comment: 5 pages; 3 figure