Polynominals related to powers of the Dedekind eta function

The vanishing properties of Fourier coefficients of integral powers of the Dedekind eta function correspond to the existence of integral roots of integer-valued polynomials Pn(x) introduced by M. Newman. In this paper we study the derivatives of these polynomials. We obtain non-vanishing results at integral points. As an application we prove that integral roots are simple if the index n of the polynomial is equal to a prime power pm or to pm + 1. We obtain a formula for the derivative of Pn(x) involving the polynomials of lower degree

Hudson's Theorem for finite-dimensional quantum systems

We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson's Theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counter-example. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.Comment: 17 pages, 3 figures; References updated. Title changed to match published version. See also quant-ph/070200

Rotational periods of very young brown dwarfs and very low-mass stars in ChaI

We have studied the photometric variability of very young brown dwarfs and very low-mass stars (masses well below 0.2 M_sun) in the ChaI star forming region. We have determined photometric periods in the Gunn i and R band for the three M6.5-M7 type brown dwarf candidates ChaHa2, ChaHa3 and ChaHa6 of 2.2 to 3.4 days. These are the longest photometric periods found for any brown dwarf so far. If interpreted as rotationally induced they correspond to moderately fast rotational velocities, which is fully consistent with their v sini values and their relatively large radii. We have also determined periods for the two M5-M5.5 type very low-mass stars B34 and CHXR78C. In addition to the Gunn i and R band data, we have analysed JHK_s monitoring data of the targets, which have been taken a few weeks earlier and confirm the periods found in the optical data. Upper limits for the errors in the period determination are between 2 and 9 hours. The observed periodic variations of the brown dwarf candidates as well as of the T Tauri stars are interpreted as modulation of the flux at the rotation period by magnetically driven surface features, on the basis of a consistency with v sini values as well as (R-i) color variations typical for spots. Furthermore, the temperatures even for the brown dwarfs in the sample are relatively high (>2800K) because the objects are very young. Therefore, the atmospheric gas should be sufficiently ionized for the formation of spots on one hand and the temperatures are too high for significant dust condensation and hence variabilities due to clouds on the other hand.Comment: 18 pages, 6 figures, accepted for publication in Ap

First optical images of circumstellar dust surrounding the debris disk candidate HD 32297

Near-infrared imaging with the Hubble Space Telescope recently revealed a circumstellar dust disk around the A star HD 32297. Dust scattered light is detected as far as 400 AU radius and the linear morphology is consistent with a disk ~10 degrees away from an edge-on orientation. Here we present the first optical images that show the dust scattered light morphology from 560 to 1680 AU radius. The position angle of the putative disk midplane diverges by 31 degrees and the color of dust scattering is most likely blue. We associate HD 32297 with a wall of interstellar gas and the enigmatic region south of the Taurus molecular cloud. We propose that the extreme asymmetries and blue disk color originate from a collision with a clump of interstellar material as HD 32297 moves southward, and discuss evidence consistent with an age of 30 Myr or younger.Comment: 5 pages; Accepted for publication in ApJ Letter

High-order time-splitting Hermite and Fourier spectral methods

In this paper, we are concerned with the numerical solution of the time-dependent Gross-Pitaevskii Equation (GPE) involving a quasi-harmonic potential. Primarily, we consider discretisations that are based on spectral methods in space and higher-order exponential operator splitting methods in time. The resulting methods are favourable in view of accuracy and efficiency; moreover, geometric properties of the equation such as particle number and energy conservation are well captured. Regarding the spatial discretisation of the GPE, we consider two approaches. In the unbounded domain, we employ a spectral decomposition of the solution into Hermite basis functions: on the other hand. restricting the equation to a sufficiently large bounded domain, Fourier techniques are applicable. For the time integration of the GPE, we study various exponential operator splitting methods of convergence orders two, four, and six. Our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting Fourier and Hermite pseudospectral methods for the time evolution of the GPE. Furthermore, we illustrate the effectiveness of higher-order time-splitting methods compared to standard integrators in a long-term integration

Raman cooling and heating of two trapped Ba+ ions

We study cooling of the collective vibrational motion of two 138Ba+ ions confined in an electrodynamic trap and irradiated with laser light close to the resonances S_1/2-P_1/2 (493 nm) and P_1/2-D_3/2 (650 nm). The motional state of the ions is monitored by a spatially resolving photo multiplier. Depending on detuning and intensity of the cooling lasers, macroscopically different motional states corresponding to different ion temperatures are observed. We also derive the ions' temperature from detailed analytical calculations of laser cooling taking into account the Zeeman structure of the energy levels involved. The observed motional states perfectly match the calculated temperatures. Significant heating is observed in the vicinity of the dark resonances of the Zeeman-split S_1/2-D_3/2 Raman transitions. Here two-photon processes dominate the interaction between lasers and ions. Parameter regimes of laser light are identified that imply most efficient laser cooling.Comment: 8 pages, 5 figure