On the Uniformity of (3/2)n(3/2)^n Modulo 1

It has been conjectured that the sequence $(3/2)^n$ modulo $1$ is uniformly distributed. The distribution of this sequence is signifcant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we describe an algorithm to compute $(3/2)^n$ modulo $1$ to $n = 10^8$. We then statistically analyze its distribution. Our results strongly agree with the hypothesis that $(3/2)^n$ modulo 1 is uniformly distributed.Comment: 12 pages, 2 figure

Optical and Near-Infrared UBVRIJHK Photometry for the RR Lyrae stars in the Nearby Globular Cluster M4 (NGC 6121)

We present optical and near-infrared UBVRIJHK photometry of stars in the Galactic globular cluster M4 (NGC 6121) based upon a large corpus of observations obtained mainly from public astronomical archives. We concentrate on the RR Lyrae variable stars in the cluster, and make a particular effort to accurately reidentify the previously discovered variables. We have also discovered two new probable RR Lyrae variables in the M4 field: one of them by its position on the sky and its photometric properties is a probable member of the cluster, and the second is a probable background (bulge?) object. We provide accurate equatorial coordinates for all 47 stars identified as RR Lyraes, new photometric measurements for 46 of them, and new period estimates for 45. We have also derived accurate positions and mean photometry for 34 more stars previously identified as variable stars of other types, and for an additional five non-RR Lyrae variable stars identified for the first time here. We present optical and near-infrared color-magnitude diagrams for the cluster and show the locations of the variable stars in them. We present the Bailey (period-amplitude) diagrams and the period-frequency histogram for the RR Lyrae stars in M4 and compare them to the corresponding diagrams for M5 (NGC 5904). We conclude that the RR Lyrae populations in the two clusters are quite similar in all the relevant properties that we have considered. The mean periods, pulsation-mode ratios, and Bailey diagrams of these two clusters show support for the recently proposed "Oosterhoff-neutral" classification.Comment: 33 pages, 16 figures, 7 table

Energy decay and frequency shift of a superconducting qubit from non-equilibrium quasiparticles

Quasiparticles are an important decoherence mechanism in superconducting qubits, and can be described with a complex admittance that is a generalization of the Mattis-Bardeen theory. By injecting non-equilibrium quasiparticles with a tunnel junction, we verify qualitatively the expected change of the decay rate and frequency in a phase qubit. With their relative change in agreement to within 4% of prediction, the theory can be reliably used to infer quasiparticle density. We describe how settling of the decay rate may allow determination of whether qubit energy relaxation is limited by non-equilibrium quasiparticles.Comment: Main paper: 4 pages, 3 figures, 1 table. Supplementary material: 8 pages, 3 figure

Microwave Dielectric Loss at Single Photon Energies and milliKelvin Temperatures

The microwave performance of amorphous dielectric materials at very low temperatures and very low excitation strengths displays significant excess loss. Here, we present the loss tangents of some common amorphous and crystalline dielectrics, measured at low temperatures (T < 100 mK) with near single-photon excitation energies, using both coplanar waveguide (CPW) and lumped LC resonators. The loss can be understood using a two-level state (TLS) defect model. A circuit analysis of the half-wavelength resonators we used is outlined, and the energy dissipation of such a resonator on a multilayered dielectric substrate is considered theoretically.Comment: 4 pages, 3 figures, submitted to Applied Physics Letter

Improving the Coherence Time of Superconducting Coplanar Resonators

The quality factor and energy decay time of superconducting resonators have been measured as a function of material, geometry, and magnetic field. Once the dissipation of trapped magnetic vortices is minimized, we identify surface two-level states (TLS) as an important decay mechanism. A wide gap between the center conductor and the ground plane, as well as use of the superconductor Re instead of Al, are shown to decrease loss. We also demonstrate that classical measurements of resonator quality factor at low excitation power are consistent with single-photon decay time measured using qubit-resonator swap experiments.Comment: 3 pages, 4 figures for the main paper; total 5 pages, 6 figures including supplementary material. Submitted to Applied Physics Letter

Deterministic entanglement of photons in two superconducting microwave resonators

Quantum entanglement, one of the defining features of quantum mechanics, has been demonstrated in a variety of nonlinear spin-like systems. Quantum entanglement in linear systems has proven significantly more challenging, as the intrinsic energy level degeneracy associated with linearity makes quantum control more difficult. Here we demonstrate the quantum entanglement of photon states in two independent linear microwave resonators, creating N-photon NOON states as a benchmark demonstration. We use a superconducting quantum circuit that includes Josephson qubits to control and measure the two resonators, and we completely characterize the entangled states with bipartite Wigner tomography. These results demonstrate a significant advance in the quantum control of linear resonators in superconducting circuits.Comment: 11 pages, 11 figures, and 3 tables including supplementary materia

Generation of Three-Qubit Entangled States using Superconducting Phase Qubits

Entanglement is one of the key resources required for quantum computation, so experimentally creating and measuring entangled states is of crucial importance in the various physical implementations of a quantum computer. In superconducting qubits, two-qubit entangled states have been demonstrated and used to show violations of Bell's Inequality and to implement simple quantum algorithms. Unlike the two-qubit case, however, where all maximally-entangled two-qubit states are equivalent up to local changes of basis, three qubits can be entangled in two fundamentally different ways, typified by the states $|\mathrm{GHZ}> = (|000> + |111>)/\sqrt{2}$ and $|\mathrm{W}> = (|001> + |010> + |100>)/\sqrt{3}$. Here we demonstrate the operation of three coupled superconducting phase qubits and use them to create and measure $|\mathrm{GHZ}>$ and $|\mathrm{W}>$ states. The states are fully characterized using quantum state tomography and are shown to satisfy entanglement witnesses, confirming that they are indeed examples of three-qubit entanglement and are not separable into mixtures of two-qubit entanglement.Comment: 9 pages, 5 figures. Version 2: added supplementary information and fixed image distortion in Figure 2

NASA Space Flight Vehicle Fault Isolation Challenges

The Space Launch System (SLS) is the new NASA heavy lift launch vehicle in development and is scheduled for its first mission in 2018.SLS has many of the same logistics challenges as any other large scale program. However, SLS also faces unique challenges related to testability. This presentation will address the SLS challenges for diagnostics and fault isolation, along with the analyses and decisions to mitigate risk.