Localization for Random Unitary Operators

We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_\omega=D_\omega S$ where $S$ is a deterministic unitary and $D_\omega$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of $U_\omega$ is pure point almost surely for all values of the parameter of $S$. We provide similar results for unitary operators defined on $l^2(\N)$ together with an application to orthogonal polynomials on the unit circle. We get almost sure localization for polynomials characterized by Verblunski coefficients of constant modulus and correlated random phases

Correlated Markov Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on $\Z^d$ performed by a particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in $\Z^d$ for random updates of the coin states of the following form. The random sequences of unitary updates are given by a site dependent function of a Markov chain in time, with the following properties: on each site, they share the same stationnary Markovian distribution and, for each fixed time, they form a deterministic periodic pattern on the lattice. We prove a Feynman-Kac formula to express the characteristic function of the averaged distribution over the randomness at time $n$ in terms of the nth power of an operator $M$. By analyzing the spectrum of $M$, we show that this distribution posesses a drift proportional to the time and its centered counterpart displays a diffusive behavior with a diffusion matrix we compute. Moderate and large deviations principles are also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. An example of random updates for which the analysis of the distribution can be performed without averaging is worked out. The random distribution displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix whose law we compute. We complete the picture by presenting an uncorrelated example.Comment: 37 pages. arXiv admin note: substantial text overlap with arXiv:1010.400

The prediction of macrophyte species occurrence in Swiss ponds

The study attempted to model the abundance of aquatic plant species recorded in a range of ponds in Switzerland. A stratified sample of 80 ponds, distributed all over the country, provided input data for model development. Of the 154 species recorded, 45 were selected for modelling. A total of 14 environmental parameters were preselected as candidate explanatory variables. Two types of statistical tools were used to explore the data and to develop the predictive models: linear regression (LR) and generalized additive models (GAMs). Six LR species models had a reasonable predictive ability (30-50% of variance explained by the selected predictors). There was a gradient in the quality of the 45 GAM models. Ten species models exhibited both a good fit and statistical robustness: Lemnaminor, Phragmitesaustralis, Lysimachiavulgaris, Galiumpalustre, Lysimachianummularia, Irispseudacorus, Lythrumsalicaria, Lycopuseuropaeus, Phalarisarundinacea, Alismaplantago-aquatica, Schoenoplectuslacustris, Carexnigra. Altitude appeared to be a key explanatory variable in most of the species models. In some cases, the degree to which the shore was shaded, connectivity between water bodies, pond area, mineral nitrogen levels, pond age, pond depth, and the extent of agriculture or pasture in the catchment were selected as additional explanatory variables. The species models demonstrated that it is possible to predict species abundance of aquatic macrophytes and that each species responded individually to distinct environmental variable

A novel wideband 140 GHz gyrotron amplifier

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 143-156).The theory, design and experimental results of a wideband 140 GHz, 1 kW pulsed gyro-traveling wave amplifier are presented. The gyro- TWA operates in the HE(0,6) mode of a novel cylindrical confocal waveguide using a gyrating electron beam. The electromagnetic theory, interaction theory, design processes and experimental procedures have been described in detail. The experiment has produced over 820 W peak power, 34 dB linear gain, and a -3 dB bandwidth of over 1.5 GHz (1.1%) from a 37 kV, 2.7 A electron beam having a beam pitch factor of 0.6, radius of 1.9 mm and calculated perpendicular momentum spread of approximately 9%. The gyro-amplifier was nominally operated at a pulse length of 2 microseconds, but was tested to amplify pulses as short as 4 nanoseconds with no noticeable pulse broadening. Internal reflections in the amplifier were identified using these short pulses by time-domain reflectometry. A novel internal mode converter was designed for this device that transforms the confocal HE(0,6) fields into a fundamental Gaussian beam for ultra-low loss transmission of the millimeter wave output power through a corrugated transmission line. The demonstrated performance of this amplifier shows that it can be applied to Dynamic Nuclear Polarization (DNP) and Electron Paramagnetic Resonance (EPR) spectroscopy.by Colin D. Joye.Ph.D

Design of a wideband, 100 W, 140 GHz gyroklystron amplifier

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (p. 105-110).The design study of a 140 GHz, 100 W continuous wave gyroklystron amplifier is presented. The device is intended for use in Dynamic Nuclear Polarization (DNP) enhanced Nuclear Magnetic Resonance (NMR) spectroscopy experiments. The gyroklystron has five cavities and operates in the TE(0,2) mode with a low power electron beam. The design was performed using MAGY, a nonlinear code for modelling gyrotron devices. The design process of the gyroklystron starting from the linear theory to the optimization of the final design in MAGY has been described in detail. Stagger tuning was employed to broadband the device. The design yields 130 W peak power, 36 dB saturated gain, and a -3 dB bandwidth of over 1 GHz (0.75%) with a 15 kV, 150 mA electron beam having a beam pitch factor of 1.5, radius of 0.64 mm and calculated perpendicular momentum spread of 4%. Preliminary designs of the Magnetron Inject Gun (MIG), the input and output couplers, and the mode converter to transform the TE(0,2) operating mode to the HE(1,1) mode for low loss transmission of the output power are also presented. The design meets the specifications for the DNP experiment.by Colin D. Joye.S.M

Temporal Interferometry: A Mechanism for Controlling Qubit Transitions During Twisted Rapid Passage with Possible Application to Quantum Computing

In an adiabatic rapid passage experiment, the Bloch vector of a two-level system (qubit) is inverted by slowly inverting an external field to which it is coupled, and along which it is initially aligned. In twisted rapid passage, the external field is allowed to twist around its initial direction with azimuthal angle $\phi (t)$ at the same time that it is inverted. For polynomial twist: $\phi (t) \sim Bt^{n}$. We show that for $n \geq 3$, multiple avoided crossings can occur during the inversion of the external field, and that these crossings give rise to strong interference effects in the qubit transition probability. The transition probability is found to be a function of the twist strength $B$, which can be used to control the time-separation of the avoided crossings, and hence the character of the interference. Constructive and destructive interference are possible. The interference effects are a consequence of the temporal phase coherence of the wavefunction. The ability to vary this coherence by varying the temporal separation of the avoided crossings renders twisted rapid passage with adjustable twist strength into a temporal interferometer through which qubit transitions can be greatly enhanced or suppressed. Possible application of this interference mechanism to construction of fast fault-tolerant quantum CNOT and NOT gates is discussed.Comment: 29 pages, 16 figures, submitted to Phys. Rev.