Continued fraction digit averages an Maclaurin's inequalities

A classical result of Khinchin says that for almost all real numbers $\alpha$, the geometric mean of the first $n$ digits $a_i(\alpha)$ in the continued fraction expansion of $\alpha$ converges to a number $K = 2.6854520\ldots$ (Khinchin's constant) as $n \to \infty$. On the other hand, for almost all $\alpha$, the arithmetic mean of the first $n$ continued fraction digits $a_i(\alpha)$ approaches infinity as $n \to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1 \leq k \leq n$. On the left end (when $k=n$) we have the geometric mean, and on the right end ($k=1$) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves $f(n)$ steps away from either extreme. We prove sufficient conditions on $f(n)$ to ensure to ensure divergence when one moves $f(n)$ steps away from the arithmetic mean and convergence when one moves $f(n)$ steps away from the geometric mean. For typical $\alpha$ we conjecture the behavior for $f(n)=cn$, $0<c<1$. We also study the limiting behavior of such means for quadratic irrational $\alpha$, providing rigorous results, as well as numerically supported conjectures.Comment: 32 pages, 7 figures. Substantial additions were made to previous version, including Theorem 1.3, Section 6, and Appendix

Coherent backscattering of Bose-Einstein condensates in two-dimensional disorder potentials

We study quantum transport of an interacting Bose-Einstein condensate in a two-dimensional disorder potential. In the limit of vanishing atom-atom interaction, a sharp cone in the angle-resolved density of the scattered matter wave is observed, arising from constructive interference between amplitudes propagating along reversed scattering paths. Weak interaction transforms this coherent backscattering peak into a pronounced dip, indicating destructive instead of constructive interference. We reproduce this result, obtained from the numerical integration of the Gross-Pitaevskii equation, by a diagrammatic theory of weak localization in presence of a nonlinearity.Comment: 4 pages, 4 figure

Partial nonlinear reciprocity breaking through ultrafast dynamics in a random photonic medium

We demonstrate that ultrafast nonlinear dynamics gives rise to reciprocity breaking in a random photonic medium. Reciprocity breaking is observed via the suppression of coherent backscattering, a manifestation of weak localization of light. The effect is observed in a pump-probe configuration where the pump induces an ultrafast step-change of the refractive index during the dwell time of the probe light in the material. The dynamical suppression of coherent backscattering is reproduced well by a multiple scattering Monte Carlo simulation. Ultrafast reciprocity breaking provides a distinct mechanism in nonlinear optical media which opens up avenues for the active manipulation of mesoscopic transport, random lasers, and photon localization.Comment: 5 pages, 4 figure

Emission of photon echoes in a strongly scattering medium

We observe the two- and three-pulse photon echo emission from a scattering powder, obtained by grinding a Pr$^{3+}$:Y$_2$SiO$_5$ rare earth doped single crystal. We show that the collective emission is coherently constructed over several grains. A well defined atomic coherence can therefore be created between randomly placed particles. Observation of photon echo on powders as opposed to bulk materials opens the way to faster material development. More generally, time-domain resonant four-wave mixing offers an attractive approach to investigate coherent propagation in scattering media

Binegativity and geometry of entangled states in two qubits

We prove that the binegativity is always positive for any two-qubit state. As a result, as suggested by the previous works, the asymptotic relative entropy of entanglement in two qubits does not exceed the Rains bound, and the PPT-entanglement cost for any two-qubit state is determined to be the logarithmic negativity of the state. Further, the proof reveals some geometrical characteristics of the entangled states, and shows that the partial transposition can give another separable approximation of the entangled state in two qubits.Comment: 5 pages, 3 figures. I made the proof more transparen

Brillouin propagation modes in optical lattices: Interpretation in terms of nonconventional stochastic resonance

We report the first direct observation of Brillouin-like propagation modes in a dissipative periodic optical lattice. This has been done by observing a resonant behavior of the spatial diffusion coefficient in the direction corresponding to the propagation mode with the phase velocity of the moving intensity modulation used to excite these propagation modes. Furthermore, we show theoretically that the amplitude of the Brillouin mode is a nonmonotonic function of the strength of the noise corresponding to the optical pumping, and discuss this behavior in terms of nonconventional stochastic resonance

Virtuality in human supervisory control: Assessing the effects of psychological and social remoteness

Virtuality would seem to offer certain advantages for human supervisory control. First, it could provide a physical analogue of the 'real world' environment. Second, it does not require control room engineers to be in the same place as each other. In order to investigate these issues, a low-fidelity simulation of an energy distribution network was developed. The main aims of the research were to assess some of the psychological concerns associated with virtual environments. First, it may result in the social isolation of the people, and it may have dramatic effects upon the nature of the work. Second, a direct physical correspondence with the 'real world' may not best support human supervisory control activities. Experimental teams were asked to control an energy distribution network. Measures of team performance, group identity and core job characteristics were taken. In general terms, the results showed that teams working in the same location performed better than team who were remote from one another

Separable approximations of density matrices of composite quantum systems

We investigate optimal separable approximations (decompositions) of states rho of bipartite quantum systems A and B of arbitrary dimensions MxN following the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261 (1998)]. Such approximations allow to represent in an optimal way any density operator as a sum of a separable state and an entangled state of a certain form. For two qubit systems (M=N=2) the best separable approximation has a form of a mixture of a separable state and a projector onto a pure entangled state. We formulate a necessary condition that the pure state in the best separable approximation is not maximally entangled. We demonstrate that the weight of the entangled state in the best separable approximation in arbitrary dimensions provides a good entanglement measure. We prove in general for arbitrary M and N that the best separable approximation corresponds to a mixture of a separable and an entangled state which are both unique. We develop also a theory of optimal separable approximations for states with positive partial transpose (PPT states). Such approximations allow to decompose any density operator with positive partial transpose as a sum of a separable state and an entangled PPT state. We discuss procedures of constructing such decompositions.Comment: 12 pages, 2 figure

Motional effects on the efficiency of excitation transfer

Energy transfer plays a vital role in many natural and technological processes. In this work, we study the effects of mechanical motion on the excitation transfer through a chain of interacting molecules with application to biological scenarios of transfer processes. Our investigation demonstrates that, for various types of mechanical oscillations, the transfer efficiency is significantly enhanced over that of comparable static configurations. This enhancement is a genuine quantum signature, and requires the collaborative interplay between the quantum-coherent evolution of the excitation and the mechanical motion of the molecules; it has no analogue in the classical incoherent energy transfer. This effect may not only occur naturally, but it could be exploited in artificially designed systems to optimize transport processes. As an application, we discuss a simple and hence robust control technique.Comment: 25 pages, 11 figures; completely revised; version accepted for publicatio