Trees and the dynamics of polynomials

The basin of infinity of a polynomial map f : {\bf C} \arrow {\bf C} carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial tree $(T,\eta)$, with limiting dynamics F : T \arrow T. In this paper we characterize the trees that arise as limits, and show they provide a natural boundary \PT_d compactifying the moduli space of polynomials of degree $d$. We show that $(T,\eta,F)$ records the limiting behavior of multipliers at periodic points, and that any divergent meromorphic family of polynomials \{f_t(z) : t \mem \Delta^* \} can be completed by a unique tree at its central fiber. Finally we show that in the cubic case, the boundary of moduli space \PT_3 is itself a tree. The metrized trees $(T,\eta,F)$ provide a counterpart, in the setting of iterated rational maps, to the ${\bf R}$-trees that arise as limits of hyperbolic manifolds.Comment: 60 page

Rapidly Rotating Fermi Gases

We show that the density profile of a Fermi gas in rapidly rotating potential will develop prominent features reflecting the underlying Landau level like energy spectrum. Depending on the aspect ratio of the trap, these features can be a sequence of ellipsoidal volumes or a sequence of quantized steps.Comment: 4 pages, 1 postscript fil

Random-phase approximation study of collective excitations in the Bose-Fermi mixed condensate of alkali-metal gases

We perform Random Phase Approximation (RPA) study of collective excitations in the bose-fermi mixed degenerate gas of Alkali-metal atoms at T=0. The calculation is done by diagonalization in a model space composed of particle-hole type excitations from the ground state, the latter being obtained from the coupled Gross-Pitaevskii and Thomas-Fermi equations. We investigate strength distributions for different combinations of bose and fermi multipole ($L$) operators with $L=0,1,2,3$. Transition densities and dynamical structure factors are calculated for collective excitations. Comparison with the sum rule prediction for the collective frequency is given. Time dependent behavior of the system after an external impulse is studied.Comment: 28 pages, 13 figures, submitted to Phys. Rev.

Quantum information processing with trapped ions

Experiments directed towards the development of a quantum computer based on trapped atomic ions are described briefly. We discuss the implementation of single qubit operations and gates between qubits. A geometric phase gate between two ion qubits is described. Limitations of the trapped-ion method such as those caused by Stark shifts and spontaneous emission are addressed. Finally, we describe a strategy to realize a large-scale device.Comment: Article submitted by D. J. Wineland ([email protected]) for proceeding of the Discussion Meeting on Practical Realisations of Quantum Information Processing, held at the Royal Society, Nov. 13,14, 200

Effective s- and p-Wave Contact Interactions in Trapped Degenerate Fermi Gases

The structure and stability of dilute degenerate Fermi gases trapped in an external potential is discussed with special emphasis on the influence of s- and p-wave interactions. In a first step an Effective Contact Interaction for all partial waves is derived, which reproduces the energy spectrum of the full potential within a mean-field model space. Using the s- and p-wave part the energy density of the multi-component Fermi gas is calculated in Thomas-Fermi approximation. On this basis the stability of the one- and two-component Fermi gas against mean-field induced collapse is investigated. Explicit stability conditions in terms of density and total particle number are given. For the single-component system attractive p-wave interactions limit the density of the gas. In the two-component case a subtle competition of s- and p-wave interactions occurs and gives rise to a rich variety of phenomena. A repulsive p-wave part, for example, can stabilize a two-component system that would otherwise collapse due to an attractive s-wave interaction. It is concluded that the p-wave interaction may have important influence on the structure of degenerate Fermi gases and should not be discarded from the outset.Comment: 18 pages, 11 figures (using RevTEX4

Pauli Blocking of Collisions in a Quantum Degenerate Atomic Fermi Gas

We have produced an interacting quantum degenerate Fermi gas of atoms composed of two spin-states of magnetically trapped $^{40}$K. The relative Fermi energies are adjusted by controlling the population in each spin-state. Measurements of the thermodynamics reveal the resulting imbalance in the mean energy per particle between the two species, which is as large as a factor of 1.4 at our lowest temperature. This imbalance of energy comes from a suppression of collisions between atoms in the gas due to the Pauli exclusion principle. Through measurements of the thermal relaxation rate we have directly observed this Pauli blocking as a factor of two reduction in the effective collision cross-section in the quantum degenerate regime.Comment: 11 pages, 4 figure

On Evidence-based Risk Management in Requirements Engineering

Background: The sensitivity of Requirements Engineering (RE) to the context makes it difficult to efficiently control problems therein, thus, hampering an effective risk management devoted to allow for early corrective or even preventive measures. Problem: There is still little empirical knowledge about context-specific RE phenomena which would be necessary for an effective context- sensitive risk management in RE. Goal: We propose and validate an evidence-based approach to assess risks in RE using cross-company data about problems, causes and effects. Research Method: We use survey data from 228 companies and build a probabilistic network that supports the forecast of context-specific RE phenomena. We implement this approach using spreadsheets to support a light-weight risk assessment. Results: Our results from an initial validation in 6 companies strengthen our confidence that the approach increases the awareness for individual risk factors in RE, and the feedback further allows for disseminating our approach into practice.Comment: 20 pages, submitted to 10th Software Quality Days conference, 201

Transport of a quantum degenerate heteronuclear Bose-Fermi mixture in a harmonic trap

We report on the transport of mixed quantum degenerate gases of bosonic 87Rb and fermionic 40K in a harmonic potential provided by a modified QUIC trap. The samples are transported over a distance of 6 mm to the geometric center of the anti-Helmholtz coils of the QUIC trap. This transport mechanism was implemented by a small modification of the QUIC trap and is free of losses and heating. It allows all experiments using QUIC traps to use the highly homogeneous magnetic fields that can be created in the center of a QUIC trap and improves the optical access to the atoms, e.g., for experiments with optical lattices. This mechanism may be cascaded to cover even larger distances for applications with quantum degenerate samples.Comment: 7 pages, 8 figure