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18
Localization of Classical Waves I: Acoustic Waves.
 Commun. Math. Phys
, 1996
"... We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existe ..."
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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the selfadjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...
Quantum Chaos, Transport, and Decoherence in Atom Optics
, 2001
"... Experimental research is often a collaborative endeavor, and the work presented in this dissertation is certainly no exception. During the past six years I have had the pleasure of working with a number of bright and enthusiastic people that I would like to mention here. First of all, I would like t ..."
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Experimental research is often a collaborative endeavor, and the work presented in this dissertation is certainly no exception. During the past six years I have had the pleasure of working with a number of bright and enthusiastic people that I would like to mention here. First of all, I would like to thank my advisor, Mark Raizen. Mark is always brimming with intriguing new ideas, and he has an exceptional sense for interesting physics problems. Mark has provided an exciting and supportive research environment for his students. I have truly enjoyed and greatly benefited from spending the past few years under his guidance. I have collaborated with Windell Oskay on all of the research in this dissertation. I cannot imagine having done the experiments in this dissertation without Windell’s remarkable productivity and superior technical prowess. This is especially true of the chaosassisted tunneling experiments in Chapter 6, where the two of us managed an enormously complicated experiment and took enough data to literally choke our computer. Windell’s rocksolid and extensive LabVIEW code (which featured its own web
Supersymmetric Analysis of a Simplified Two Dimensional Anderson Model at Small Disorder
, 2002
"... Abstract This work proposes a very simple random matrix model, the Flip Matrix Model, liable to approximate the behavior of a two dimensional electron in a weak random potential. Its construction is based on a phase space analysis, a suitable discretization and a simplification of the true model. Th ..."
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Abstract This work proposes a very simple random matrix model, the Flip Matrix Model, liable to approximate the behavior of a two dimensional electron in a weak random potential. Its construction is based on a phase space analysis, a suitable discretization and a simplification of the true model. The density of states of this model is investigated using the supersymmetric method and shown to be given, in the limit of large size of the matrix by the usual Wigner’s semicircle law. I
A Renormalization Group Approach to the Cosmological Constant Problem,” arXiv:0708.4374 [hepth
"... Abstract: In an earlier paper, it is proposed that, due to resonance tunneling effect, tunneling from a large cosmological constant Λ site in the stringy comic landscape can be fast, while tunneling from a small Λ site may take exponentially long time. If there is a sharp transition at a small criti ..."
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Abstract: In an earlier paper, it is proposed that, due to resonance tunneling effect, tunneling from a large cosmological constant Λ site in the stringy comic landscape can be fast, while tunneling from a small Λ site may take exponentially long time. If there is a sharp transition at a small critical value Λc from fast tunneling for Λ> Λc to suppressed tunneling for Λc> Λ> 0, we may have a qualitative understanding why today’s dark energy is so small. Here, the arguments for fast tunneling and the subsequent sharp transition to exponentially slow tunneling are strengthened by directly borrowing the renormalization group analysis of the conductance in the Anderson localization transition. As an illustration, we
Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review
, 2009
"... A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromod ..."
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A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic JahnTeller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non–Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier–Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasitriangular, quasiHopf algebras, bialgebroids, GrassmannHopf algebras and Higher Dimensional Algebra. On the one hand, this quantum
A Strudy On Mode Localization: Analitucal Approach
"... . The mode localization phenomena in simply supported two span beams of arbitrary span lengths are theoretically investigated. When localization occurs, the free vibration amplitude of a normal mode becomes confined to a local region of the structure, with serious implication for the control problem ..."
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. The mode localization phenomena in simply supported two span beams of arbitrary span lengths are theoretically investigated. When localization occurs, the free vibration amplitude of a normal mode becomes confined to a local region of the structure, with serious implication for the control problem. In many structures, some modes that are not localized become localized seriously by small structural changes. It is well known that the weakly coupled periodic structures are sensitive to certain types of periodicitybreaking disorder, resulting in the mode localization. In the previous researches, perturbation methods are used to discuss the phenomenon and periodic structures are mainly considered. In this study, however, the mode localization phenomenon is discussed with an analytical approach and it is shown that the mode localization can occur in nonperiodic structures also by small structural changes. The results of this study show that the coupling strength plays the important role ...
Eur. Phys. J. H DOI: 10.1140/epjh/e2010000077 THE EUROPEAN
, 2010
"... Longtime behavior of macroscopic quantum systems Commentary accompanying the English translation of John von Neumann’s 1929 article on the quantum ergodic theorem ..."
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Longtime behavior of macroscopic quantum systems Commentary accompanying the English translation of John von Neumann’s 1929 article on the quantum ergodic theorem
and stochastic approaches
, 2011
"... displayed a sharp Bragg type diffraction image with perfect icosahedral point symmetry. This challenged the understanding of solids at the time, and needed a while to be accepted for publication [16]. Ultimately, it led to a paradigm shift for what we understand as ‘order ’ in a solid or in a more g ..."
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displayed a sharp Bragg type diffraction image with perfect icosahedral point symmetry. This challenged the understanding of solids at the time, and needed a while to be accepted for publication [16]. Ultimately, it led to a paradigm shift for what we understand as ‘order ’ in a solid or in a more general structure. For this discovery, Shechtman was awarded the Nobel Prize in Chemistry in 2011, just a few days after our meeting took place. The field of Aperiodic Order is the mathematical counterpart to the physics and chemistry of quasicrystals, and the meeting was concerned with all aspects of it that relate to almost periodicity. In fact, the experimental discovery by Shechtman had many precursors in mathematics, ranging from ornaments via Kepler’s famous fivefold pattern and Penrose’s tiling to modern tiling theory, or from Bohr’s theory of almost periodic functions via Meyer’s book [13] to the present day development of almost periodic measures. 2 Recent Developments and Open Problems The field has seen substantial recent developments in several directions. following three: Specifically, we single out the ∙ The theory of point processes has transformed the field of mathematical diffraction theory (see [3] for a recent review). This has allowed for treatment of various random models as well as a solution to homometry problem for pure point diffraction. ∙ In the spectral theory of associated Schrödinger operators, an in depth analysis of fractal spectral features of the Fibonacci Hamiltonian has been accomplished. ∙ Various methods from dynamical systems have provided new perspectives on old problems. These topics have been a special focus of attention at the conference. They will be discussed in further detail in the next section.
unknown title
, 2008
"... Controllable diffusion of cold atoms in a harmonically driven and tilted optical lattice: Decoherence by spontaneous emission ..."
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Controllable diffusion of cold atoms in a harmonically driven and tilted optical lattice: Decoherence by spontaneous emission
CONTINUOUSTIME QUANTUM WALKS AND TRAPPING
, 903
"... Recent findings suggest that processes such as the electronic energy transfer through the photosynthetic antenna display quantal features, aspects known from the dynamics of charge carriers along polymer backbones. Hence, in modeling energy transfer one has to leave the classical, masterequationtyp ..."
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Recent findings suggest that processes such as the electronic energy transfer through the photosynthetic antenna display quantal features, aspects known from the dynamics of charge carriers along polymer backbones. Hence, in modeling energy transfer one has to leave the classical, masterequationtype formalism and advance towards an increasingly quantummechanical picture, while still retaining a local description of the complex network of molecules involved in the transport, say through a tightbinding approach. Interestingly, the continuous time random walk (CTRW) picture, widely employed in describing transport in random environments, can be mathematically reformulated to yield a quantummechanical Hamiltonian of tightbinding type; the procedure uses the mathematical analogies between timeevolution operators in statistical and in quantum mechanics: The result are continuoustime quantum walks (CTQWs). However, beyond these formal analogies, CTRWs and CTQWs display vastly different physical properties. In particular, here we focus on trapping processes on a ring and show, both analytically and numerically, that distinct configurations of traps (ranging from periodical to random) yield strongly different behaviours for the quantal mean survival probability, while classically (under ordered conditions) we always find an exponential decay at long times.