Superconductors with Magnetic Impurities: Instantons and Sub-gap States

When subject to a weak magnetic impurity potential, the order parameter and quasi-particle energy gap of a bulk singlet superconductor are suppressed. According to the conventional mean-field theory of Abrikosov and Gor'kov, the integrity of the energy gap is maintained up to a critical concentration of magnetic impurities. In this paper, a field theoretic approach is developed to critically analyze the validity of the mean field theory. Using the supersymmetry technique we find a spatially homogeneous saddle-point that reproduces the Abrikosov-Gor'kov theory, and identify instanton contributions to the density of states that render the quasi-particle energy gap soft at any non-zero magnetic impurity concentration. The sub-gap states are associated with supersymmetry broken field configurations of the action. An analysis of fluctuations around these configurations shows how the underlying supersymmetry of the action is restored by zero modes. An estimate of the density of states is given for all dimensionalities. To illustrate the universality of the present scheme we apply the same method to study `gap fluctuations' in a normal quantum dot coupled to a superconducting terminal. Using the same instanton approach, we recover the universal result recently proposed by Vavilov et al. Finally, we emphasize the universality of the present scheme for the description of gap fluctuations in d-dimensional superconducting/normal structures.Comment: 18 pages, 9 eps figure

Invariant sets for discontinuous parabolic area-preserving torus maps

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure and we give bounds on the measure. For several examples we find expressions for the measure of the invariant set but we leave open the question as to whether there are parameters for which this measure is zero.Comment: 19 pages in Latex (with epsfig,amssymb,graphics) with 5 figures in eps; revised version: section 2 rewritten, new example and picture adde

Precise calibration of LIGO test mass actuators using photon radiation pressure

Precise calibration of kilometer-scale interferometric gravitational wave detectors is crucial for source localization and waveform reconstruction. A technique that uses the radiation pressure of a power-modulated auxiliary laser to induce calibrated displacements of one of the ~10 kg arm cavity mirrors, a so-called photon calibrator, has been demonstrated previously and has recently been implemented on the LIGO detectors. In this article, we discuss the inherent precision and accuracy of the LIGO photon calibrators and several improvements that have been developed to reduce the estimated voice coil actuator calibration uncertainties to less than 2 percent (1-sigma). These improvements include accounting for rotation-induced apparent length variations caused by interferometer and photon calibrator beam centering offsets, absolute laser power measurement using temperature-controlled InGaAs photodetectors mounted on integrating spheres and calibrated by NIST, minimizing errors induced by localized elastic deformation of the mirror surface by using a two-beam configuration with the photon calibrator beams symmetrically displaced about the center of the optic, and simultaneously actuating the test mass with voice coil actuators and the photon calibrator to minimize fluctuations caused by the changing interferometer response. The photon calibrator is able to operate in the most sensitive interferometer configuration, and is expected to become a primary calibration method for future gravitational wave searches.Comment: 13 pages, 6 figures, accepted by Classical and Quantum Gravit

Precise calibration of LIGO test mass actuators using photon radiation pressure

Precise calibration of kilometer-scale interferometric gravitational wave detectors is crucial for source localization and waveform reconstruction. A technique that uses the radiation pressure of a power-modulated auxiliary laser to induce calibrated displacements of one of the ~10 kg arm cavity mirrors, a so-called photon calibrator, has been demonstrated previously and has recently been implemented on the LIGO detectors. In this article, we discuss the inherent precision and accuracy of the LIGO photon calibrators and several improvements that have been developed to reduce the estimated voice coil actuator calibration uncertainties to less than 2 percent (1-sigma). These improvements include accounting for rotation-induced apparent length variations caused by interferometer and photon calibrator beam centering offsets, absolute laser power measurement using temperature-controlled InGaAs photodetectors mounted on integrating spheres and calibrated by NIST, minimizing errors induced by localized elastic deformation of the mirror surface by using a two-beam configuration with the photon calibrator beams symmetrically displaced about the center of the optic, and simultaneously actuating the test mass with voice coil actuators and the photon calibrator to minimize fluctuations caused by the changing interferometer response. The photon calibrator is able to operate in the most sensitive interferometer configuration, and is expected to become a primary calibration method for future gravitational wave searches.Comment: 13 pages, 6 figures, accepted by Classical and Quantum Gravit

Discretized rotation has infinitely many periodic orbits

For a fixed k in (-2,2), the discretized rotation on Z^2 is defined by (x,y)->(y,-[x+ky]). We prove that this dynamics has infinitely many periodic orbits.Comment: Revised after referee reports, and added a quantitative statemen

Gravitational waves from Sco X-1: A comparison of search methods and prospects for detection with advanced detectors

The low-mass X-ray binary Scorpius X-1 (Sco X-1) is potentially the most luminous source of continuous gravitational-wave radiation for interferometers such as LIGO and Virgo. For low-mass X-ray binaries this radiation would be sustained by active accretion of matter from its binary companion. With the Advanced Detector Era fast approaching, work is underway to develop an array of robust tools for maximizing the science and detection potential of Sco X-1. We describe the plans and progress of a project designed to compare the numerous independent search algorithms currently available. We employ a mock-data challenge in which the search pipelines are tested for their relative proficiencies in parameter estimation, computational efficiency, robust- ness, and most importantly, search sensitivity. The mock-data challenge data contains an ensemble of 50 Scorpius X-1 (Sco X-1) type signals, simulated within a frequency band of 50-1500 Hz. Simulated detector noise was generated assuming the expected best strain sensitivity of Advanced LIGO and Advanced VIRGO ($4 \times 10^{-24}$ Hz$^{-1/2}$). A distribution of signal amplitudes was then chosen so as to allow a useful comparison of search methodologies. A factor of 2 in strain separates the quietest detected signal, at $6.8 \times 10^{-26}$ strain, from the torque-balance limit at a spin frequency of 300 Hz, although this limit could range from $1.2 \times 10^{-25}$ (25 Hz) to $2.2 \times 10^{-26}$ (750 Hz) depending on the unknown frequency of Sco X-1. With future improvements to the search algorithms and using advanced detector data, our expectations for probing below the theoretical torque-balance strain limit are optimistic.Comment: 33 pages, 11 figure

Accurate calibration of test mass displacement in the LIGO interferometers

We describe three fundamentally different methods we have applied to calibrate the test mass displacement actuators to search for systematic errors in the calibration of the LIGO gravitational-wave detectors. The actuation frequencies tested range from 90 Hz to 1 kHz and the actuation amplitudes range from 1e-6 m to 1e-18 m. For each of the four test mass actuators measured, the weighted mean coefficient over all frequencies for each technique deviates from the average actuation coefficient for all three techniques by less than 4%. This result indicates that systematic errors in the calibration of the responses of the LIGO detectors to differential length variations are within the stated uncertainties.Comment: 10 pages, 6 figures, submitted on 31 October 2009 to Classical and Quantum Gravity for the proceedings of 8th Edoardo Amaldi Conference on Gravitational Wave