DOCSLIB.ORG

The Casimir Energy and Radiative Heat Transfer Between Nanostructured Surfaces Lussange Johann

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Casimir Energy and Radiative Heat Transfer Between Nanostructured Surfaces Lussange Johann The Casimir energy and radiative heat transfer between nanostructured surfaces Lussange Johann To cite this version: Lussange Johann. The Casimir energy and radiative heat transfer between nanostructured surfaces. Quantum Physics [quant-ph]. Université Pierre et Marie Curie - Paris VI, 2012. English. tel-00879989 HAL Id: tel-00879989 https://tel.archives-ouvertes.fr/tel-00879989 Submitted on 5 Nov 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE DE DOCTORAT DE L’UNIVERSITE PIERRE ET MARIE CURIE Sp´ecialit´e: Physique Ecole doctorale ED 107 R´ealis´ee au Laboratoire Kastler Brossel (UPMC-ENS-CNRS) Pr´esent´ee par Johann LUSSANGE Pour obtenir le grade de : DOCTEUR de l’UNIVERSITE´ PIERRE ET MARIE CURIE Sujet de la th`ese : The Casimir energy and radiative heat transfer between nanostructured surfaces Soutenue le 10 Septembre 2012 devant le jury compos´ede : Mme. Astrid LAMBRECHT : Directrice de th`ese M. Karl JOULAIN : Rapporteur M. George PALASANTZAS : Rapporteur M. Mauro ANTEZZA : Examinateur M. Daniel BLOCH : Examinateur M. Jean-Marc FRIGERIO : Examinateur M. Cyriaque GENET : Examinateur M. Serge REYNAUD : Invit´e 2 blank 3 Contents I. Bref r´esum´een Fran¸cais 7 II. Short summary in English 9 III. Acknowledgments 11 IV. Introduction 13 V. Theory : Quantum fields and the vacuum state 19 A. Quantum theory and radiative heat transfer in the classical description 19 1. Quantization of light : Planck’s law, and Einstein’s generalization to photons 19 2. Laws of thermodynamics and Onsager’s reciprocal relations 21 3. Properties of radiative heat transfer 25 4. Thermal radiation through a medium 28 5. Conclusion 31 B. Quantum fields and electrodynamics 32 1. From quantum particles to relativistic fields 32 2. Noether’s theorem and the stress-energy tensor 35 3. Green’s functions and field interactions 37 4. QED equation of motion and quantization 41 5. The vacuum state, zero-point energy, and vacuum fluctuations 47 6. Conclusion 50 VI. Theory : Scattering theory applied to Casimir and near-field heat transfer 51 A. Casimir effect in the plane-plane geometries 51 1. Complex permittivity and fitting models 51 2. Fresnel-Stokes amplitudes and S-matrices 55 3. Quantization and Airy function 60 4. Derivation of the Casimir force between two planes 64 5. Analyticity conditions of the cavity function : causality, passivity, stability 67 6. The Casimir force over imaginary frequencies and Cauchy’s theorem 71 7. Matsubara frequencies and Casimir for non-zero temperatures 75 8. Derjaguin’s Proximity Approximation 79 9. Polariton coupling with surface-quasiparticles 80 10. Conclusion 84 B. Casimir effect in non-planar geometries 86 4 1. The RCWA method and associated reflection matrix for gratings 86 2. The Casimir energy for periodic gratings 95 3. The Casimir energy for arbitrary periodic gratings 98 C. Out-of-thermal equilibrium phenomena 101 1. The Casimir energy for periodic gratings 101 2. Near-field radiative heat transfer for periodic gratings 114 VII. Numerical evaluation : Casimir for zero temperatures 120 A. Casimir energy between planar surfaces 120 B. Casimir energy between corrugated gratings 121 1. Casimir energy between corrugated gratings as a function of the separation distance L 122 2. Casimir energy between corrugated gratings as a function of the grating period d 126 3. Casimir energy between corrugated gratings as a function of the filling factor p 129 4. Casimir energy between corrugated gratings as a function of the corrugation depth a 134 5. Casimir energy between corrugated gratings as a function of the lateral displacement δ 137 C. Casimir force between corrugated gratings 141 D. Lateral Casimir force between corrugated gratings 143 E. Casimir energy between arbitrary periodic gratings 147 1. Casimir energy between periodic profiles shaped as sawteeth 147 2. Casimir energy between periodic profiles shaped as barbed wires 149 3. Casimir energy between periodic profiles shaped as a sinusoid 151 4. Casimir energy between periodic profiles shaped as ellipsoids 154 5. Casimir energy for different compared arbitrary profiles 155 VIII. Numerical evaluation : Casimir for non-zero temperatures 158 A. Casimir at thermal equilibrium 158 1. Casimir energy as a function of the separation distance for T = 0 K and T = 300 K 158 2. Casimir energy as a function of the temperature for L = 100 nm, 500 nm, 1µm, and 10µm 161 B. Casimir out-of-thermal equilibrium 167 1. Comparison between equilibrium and out-of-equilibrium situations 167 2. Dependence on the temperature gradient and temperature average 170 3. Repulsivity and contribution of the non-equilibrium term 171 IX. Numerical evaluation : Radiative heat transfer in near-field 175 A. Radiative heat transfer between planes 175 B. Radiative heat transfer between gratings 179 C. A thermal modulator device for nanosystems 184 5 X. Discussion of the results and conclusion 189 References 205 6 blank 7 I. BREF RESUM´ E´ EN FRANC¸AIS Le sujet de cette th`ese porte sur les calculs num´eriques de deux observables quantiques influents `al’´echelle sub- microm´etrique : le premier ´etant la force de Casimir et le second ´etant le transfert thermique radiatif. En champ proche, ces deux grandeurs physiques sont `al’origine de nombreuses applications potentielles dans le domaine de la nano-ing´enierie. Elles sont th´eoriquement et exp´erimentalement bien ´evalu´ees dans le cas de g´eom´etries simples, comme des cavit´es de Fabry-P´erot form´ees par deux miroirs plans parall`eles. Mais dans le cas des g´eom´etries complexes invariablement rencontr´ees dans les applications nanotechnologiques r´eelles, les modes ´electromagn´etiques sur lesquels elles sont construites sont assujettis `ades processus de diffractions, rendant leur ´evaluation consid´erablement plus complexe. Ceci est le cas par exemple des NEMS ou MEMS, dont l’architecture est souvent non-triviale et hautement d´ependante de la force de Casimir et du flux thermique, avec par exemple le probl`eme de malfonctionnement courant dˆu `al’adh´erence des sous-composants de ces syst`emes venant de ces forces ou flux. Dans cette th`ese, je m’int´eresse princi- palement `ades profils p´eriodiques de forme corrugu´ee —c’est-`a-dire en forme de cr´eneaux— qui posent d’importantes contraintes sur la simplicit´ede calcul de ces observables. Apr`es une revue fondamentale et th´eorique jetant les bases math´ematiques d’une m´ethode exacte d’´evaluation de la force de Casimir et du flux thermique en champ proche centr´ee sur la th´eorie de diffusion, la seconde et principale partie de ma th`ese consiste en une pr´esentation des estimations num´eriques de ces grandeurs pour des profils corrugu´es de param`etres g´eom´etriques et de mat´eriaux diverses. En particulier, j’obtiens les tous premiers r´esultats exacts de la force de Casimir hors-´equilibre-thermique et du flux thermique radiatif entre des surfaces corrugu´ees. Je conclus par une proposition de conception d’un modulateur thermique pour nanosyst`emes bas´ee sur mes r´esultats. 8 blank 9 II. SHORT SUMMARY IN ENGLISH The subject of this thesis is about the numerical computations of two influent quantum observables at the nanoscale: the Casimir force and the radiative heat transfer. In near field, these two physical quantities are at the origin of numerous potential applications in the field of nano-engineering. They are theoretically and experimentally well evaluated in the case of simple geometries such as Fabry-P´erot cavities, which consist in two parallel plane mirrors separated by vacuum. But in the case of the more complex geometries which are unavoidably encountered in practical nanotechnological applications, the electromagnetic modes from which they are derived are subject to scattering processes which make their evaluation considerably more complex. This is for instance the case of NEMS and MEMS, whose general architecture is often non-trivial and highly dependent on the Casimir force and radiative heat flux, with for example the often encountered problem of stiction in these nano-devices. In this thesis I mainly focus on corrugated periodic profiles, which bring important constraints on the simplicity of the computations associated with these observables. After a fundamental review of the mathematical foundations of an exact method of computation of the Casimir force and of the heat flux based on scattering theory, I present in the second part of this thesis the results of the numerical calculations of these quantities for corrugated profiles of various geometrical parameters and for different materials. In particular, I obtain the very first exact results of the out-of-thermal equilibrium Casimir force and of the radiative heat flux between corrugated surfaces. I conclude with a proposal for the design of a thermal modulator device for nanosystems based on my results. 10 blank 11 III. ACKNOWLEDGMENTS Je voudrais tout d’abord remercier chaleureusement Romain, sans qui la plus grande partie de ce travail n’aurait pas pu ˆetre fait. Un grand merci ´egalement `aAstrid, qui m’a guid´ependant ces trois ann´ees de th`ese et qui a toujours su m’encourager dans les diff´erents sujets de recherche, ainsi qu’`aSerge pour tous ses conseils et son aide. Merci ´egalement `aDiego Dalvit, Jean-Jacques Greffet, Felipe Rosa, et Jean-Paul Hugonin pour tout leur travail, et pour leur fructueuse collaboration.
Load more

Recommended publications
  • Computational Characterization of the Wave Propagation Behaviour of Multi-Stable Periodic Cellular Materials∗
    Computational Characterization of the Wave Propagation Behaviour of Multi-Stable Periodic Cellular Materials∗ Camilo Valencia1, David Restrepo2,4, Nilesh D. Mankame3, Pablo D. Zavattieri 2y , and Juan Gomez 1z 1Civil Engineering Department, Universidad EAFIT, Medell´ın,050022, Colombia 2Lyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA 3Vehicle Systems Research Laboratory, General Motors Global Research & Development, Warren, MI 48092, USA. 4Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX 78249, USA Abstract In this work, we present a computational analysis of the planar wave propagation behavior of a one-dimensional periodic multi-stable cellular material. Wave propagation in these ma- terials is interesting because they combine the ability of periodic cellular materials to exhibit stop and pass bands with the ability to dissipate energy through cell-level elastic instabilities. Here, we use Bloch periodic boundary conditions to compute the dispersion curves and in- troduce a new approach for computing wide band directionality plots. Also, we deconstruct the wave propagation behavior of this material to identify the contributions from its various structural elements by progressively building the unit cell, structural element by element, from a simple, homogeneous, isotropic primitive. Direct integration time domain analyses of a representative volume element at a few salient frequencies in the stop and pass bands are used to confirm the existence of partial band gaps in the response of the cellular material. Insights gained from the above analyses are then used to explore modifications of the unit cell that allow the user to tune the band gaps in the response of the material.
    [Show full text]
  • Curiens Principle and Spontaneous Symmetry Breaking
    Curie’sPrinciple and spontaneous symmetry breaking John Earman Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, PA, USA Abstract In 1894 Pierre Curie announced what has come to be known as Curie’s Principle: the asymmetry of e¤ects must be found in their causes. In the same publication Curie discussed a key feature of what later came to be known as spontaneous symmetry breaking: the phenomena generally do not exhibit the symmetries of the laws that govern them. Philosophers have long been interested in the meaning and status of Curie’s Principle. Only comparatively recently have they begun to delve into the mysteries of spontaneous symmetry breaking. The present paper aims to advance the dis- cussion of both of these twin topics by tracing their interaction in classical physics, ordinary quantum mechanics, and quantum …eld theory. The fea- tures of spontaneous symmetry that are peculiar to quantum …eld theory have received scant attention in the philosophical literature. These features are highlighted here, along with a explanation of why Curie’sPrinciple, though valid in quantum …eld theory, is nearly vacuous in that context. 1. Introduction The statement of what is now called Curie’sPrinciple was announced in 1894 by Pierre Curie: (CP) When certain e¤ects show a certain asymmetry, this asym- metry must be found in the causes which gave rise to it (Curie 1894, p. 401).1 This principle is vague enough to allow some commentators to see in it pro- found truth while others see only falsity (compare Chalmers 1970; Radicati 1987; van Fraassen 1991, pp.
    [Show full text]
  • Vector Wave Propagation Method Ein Beitrag Zum Elektromagnetischen Optikrechnen
    Vector Wave Propagation Method Ein Beitrag zum elektromagnetischen Optikrechnen Inauguraldissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften der Universitat¨ Mannheim vorgelegt von Dipl.-Inf. Matthias Wilhelm Fertig aus Mannheim Mannheim, 2011 Dekan: Prof. Dr.-Ing. Wolfgang Effelsberg Universitat¨ Mannheim Referent: Prof. Dr. rer. nat. Karl-Heinz Brenner Universitat¨ Heidelberg Korreferent: Prof. Dr.-Ing. Elmar Griese Universitat¨ Siegen Tag der mundlichen¨ Prufung:¨ 18. Marz¨ 2011 Abstract Based on the Rayleigh-Sommerfeld diffraction integral and the scalar Wave Propagation Method (WPM), the Vector Wave Propagation Method (VWPM) is introduced in the thesis. It provides a full vectorial and three-dimensional treatment of electromagnetic fields over the full range of spatial frequen- cies. A model for evanescent modes from [1] is utilized and eligible config- urations of the complex propagation vector are identified to calculate total internal reflection, evanescent coupling and to maintain the conservation law. The unidirectional VWPM is extended to bidirectional propagation of vectorial three-dimensional electromagnetic fields. Totally internal re- flected waves and evanescent waves are derived from complex Fresnel coefficients and the complex propagation vector. Due to the superposition of locally deformed plane waves, the runtime of the WPM is higher than the runtime of the BPM and therefore an efficient parallel algorithm is de- sirable. A parallel algorithm with a time-complexity that scales linear with the number of threads is presented. The parallel algorithm contains a mini- mum sequence of non-parallel code which possesses a time complexity of the one- or two-dimensional Fast Fourier Transformation. The VWPM and the multithreaded VWPM utilize the vectorial version of the Plane Wave Decomposition (PWD) in homogeneous medium without loss of accuracy to further increase the simulation speed.
    [Show full text]
  • Quasiparticle Anisotropic Hydrodynamics in Ultra-Relativistic
    QUASIPARTICLE ANISOTROPIC HYDRODYNAMICS IN ULTRA-RELATIVISTIC HEAVY-ION COLLISIONS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Mubarak Alqahtani December, 2017 c Copyright All rights reserved Except for previously published materials Dissertation written by Mubarak Alqahtani BE, University of Dammam, SA, 2006 MA, Kent State University, 2014 PhD, Kent State University, 2014-2017 Approved by , Chair, Doctoral Dissertation Committee Dr. Michael Strickland , Members, Doctoral Dissertation Committee Dr. Declan Keane Dr. Spyridon Margetis Dr. Robert Twieg Dr. John West Accepted by , Chair, Department of Physics Dr. James T. Gleeson , Dean, College of Arts and Sciences Dr. James L. Blank Table of Contents List of Figures . vii List of Tables . xv List of Publications . xvi Acknowledgments . xvii 1 Introduction ......................................1 1.1 Units and notation . .1 1.2 The standard model . .3 1.3 Quantum Electrodynamics (QED) . .5 1.4 Quantum chromodynamics (QCD) . .6 1.5 The coupling constant in QED and QCD . .7 1.6 Phase diagram of QCD . .9 1.6.1 Quark gluon plasma (QGP) . 12 1.6.2 The heavy-ion collision program . 12 1.6.3 Heavy-ion collisions stages . 13 1.7 Some definitions . 16 1.7.1 Rapidity . 16 1.7.2 Pseudorapidity . 16 1.7.3 Collisions centrality . 17 1.7.4 The Glauber model . 19 1.8 Collective flow . 21 iii 1.8.1 Radial flow . 22 1.8.2 Anisotropic flow . 22 1.9 Fluid dynamics . 26 1.10 Non-relativistic fluid dynamics . 26 1.10.1 Relativistic fluid dynamics .
    [Show full text]
  • A Discussion on Characteristics of the Quantum Vacuum
    A Discussion on Characteristics of the Quantum Vacuum Harold \Sonny" White∗ NASA/Johnson Space Center, 2101 NASA Pkwy M/C EP411, Houston, TX (Dated: September 17, 2015) This paper will begin by considering the quantum vacuum at the cosmological scale to show that the gravitational coupling constant may be viewed as an emergent phenomenon, or rather a long wavelength consequence of the quantum vacuum. This cosmological viewpoint will be reconsidered on a microscopic scale in the presence of concentrations of \ordinary" matter to determine the impact on the energy state of the quantum vacuum. The derived relationship will be used to predict a radius of the hydrogen atom which will be compared to the Bohr radius for validation. The ramifications of this equation will be explored in the context of the predicted electron mass, the electrostatic force, and the energy density of the electric field around the hydrogen nucleus. It will finally be shown that this perturbed energy state of the quan- tum vacuum can be successfully modeled as a virtual electron-positron plasma, or the Dirac vacuum. PACS numbers: 95.30.Sf, 04.60.Bc, 95.30.Qd, 95.30.Cq, 95.36.+x I. BACKGROUND ON STANDARD MODEL OF COSMOLOGY Prior to developing the central theme of the paper, it will be useful to present the reader with an executive summary of the characteristics and mathematical relationships central to what is now commonly referred to as the standard model of Big Bang cosmology, the Friedmann-Lema^ıtre-Robertson-Walker metric. The Friedmann equations are analytic solutions of the Einstein field equations using the FLRW metric, and Equation(s) (1) show some commonly used forms that include the cosmological constant[1], Λ.
    [Show full text]
  • Superconducting Metamaterials for Waveguide Quantum Electrodynamics
    ARTICLE DOI: 10.1038/s41467-018-06142-z OPEN Superconducting metamaterials for waveguide quantum electrodynamics Mohammad Mirhosseini1,2,3, Eunjong Kim1,2,3, Vinicius S. Ferreira1,2,3, Mahmoud Kalaee1,2,3, Alp Sipahigil 1,2,3, Andrew J. Keller1,2,3 & Oskar Painter1,2,3 Embedding tunable quantum emitters in a photonic bandgap structure enables control of dissipative and dispersive interactions between emitters and their photonic bath. Operation in 1234567890():,; the transmission band, outside the gap, allows for studying waveguide quantum electro- dynamics in the slow-light regime. Alternatively, tuning the emitter into the bandgap results in finite-range emitter–emitter interactions via bound photonic states. Here, we couple a transmon qubit to a superconducting metamaterial with a deep sub-wavelength lattice constant (λ/60). The metamaterial is formed by periodically loading a transmission line with compact, low-loss, low-disorder lumped-element microwave resonators. Tuning the qubit frequency in the vicinity of a band-edge with a group index of ng = 450, we observe an anomalous Lamb shift of −28 MHz accompanied by a 24-fold enhancement in the qubit lifetime. In addition, we demonstrate selective enhancement and inhibition of spontaneous emission of different transmon transitions, which provide simultaneous access to short-lived radiatively damped and long-lived metastable qubit states. 1 Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA. 2 Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA. 3 Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA. Correspondence and requests for materials should be addressed to O.P.
    [Show full text]
  • Electromagnetism As Quantum Physics
    Electromagnetism as Quantum Physics Charles T. Sebens California Institute of Technology May 29, 2019 arXiv v.3 The published version of this paper appears in Foundations of Physics, 49(4) (2019), 365-389. https://doi.org/10.1007/s10701-019-00253-3 Abstract One can interpret the Dirac equation either as giving the dynamics for a classical field or a quantum wave function. Here I examine whether Maxwell's equations, which are standardly interpreted as giving the dynamics for the classical electromagnetic field, can alternatively be interpreted as giving the dynamics for the photon's quantum wave function. I explain why this quantum interpretation would only be viable if the electromagnetic field were sufficiently weak, then motivate a particular approach to introducing a wave function for the photon (following Good, 1957). This wave function ultimately turns out to be unsatisfactory because the probabilities derived from it do not always transform properly under Lorentz transformations. The fact that such a quantum interpretation of Maxwell's equations is unsatisfactory suggests that the electromagnetic field is more fundamental than the photon. Contents 1 Introduction2 arXiv:1902.01930v3 [quant-ph] 29 May 2019 2 The Weber Vector5 3 The Electromagnetic Field of a Single Photon7 4 The Photon Wave Function 11 5 Lorentz Transformations 14 6 Conclusion 22 1 1 Introduction Electromagnetism was a theory ahead of its time. It held within it the seeds of special relativity. Einstein discovered the special theory of relativity by studying the laws of electromagnetism, laws which were already relativistic.1 There are hints that electromagnetism may also have held within it the seeds of quantum mechanics, though quantum mechanics was not discovered by cultivating those seeds.
    [Show full text]
  • Matter Wave Speckle Observed in an Out-Of-Equilibrium Quantum Fluid
    Matter wave speckle observed in an out-of-equilibrium quantum fluid Pedro E. S. Tavaresa, Amilson R. Fritscha, Gustavo D. Tellesa, Mahir S. Husseinb, Franc¸ois Impensc, Robin Kaiserd, and Vanderlei S. Bagnatoa,1 aInstituto de F´ısica de Sao˜ Carlos, Universidade de Sao˜ Paulo, 13560-970 Sao˜ Carlos, SP, Brazil; bDepartamento de F´ısica, Instituto Tecnologico´ de Aeronautica,´ 12.228-900 Sao˜ Jose´ dos Campos, SP, Brazil; cInstituto de F´ısica, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, RJ, Brazil; and dUniversite´ Nice Sophia Antipolis, Institut Non-Lineaire´ de Nice, CNRS UMR 7335, F-06560 Valbonne, France Contributed by Vanderlei Bagnato, October 16, 2017 (sent for review August 14, 2017; reviewed by Alexander Fetter, Nikolaos P. Proukakis, and Marlan O. Scully) We report the results of the direct comparison of a freely expanding turbulent Bose–Einstein condensate and a propagating optical speckle pattern. We found remarkably similar statistical properties underlying the spatial propagation of both phenomena. The cal- culated second-order correlation together with the typical correlation length of each system is used to compare and substantiate our observations. We believe that the close analogy existing between an expanding turbulent quantum gas and a traveling optical speckle might burgeon into an exciting research field investigating disordered quantum matter. matter–wave j speckle j quantum gas j turbulence oherent matter–wave systems such as a Bose–Einstein condensate (BEC) or atom lasers and atom optics are reality and relevant Cresearch fields. The spatial propagation of coherent matter waves has been an interesting topic for many theoretical (1–3) and experimental studies (4–7).
    [Show full text]
  • 1 the LOCALIZED QUANTUM VACUUM FIELD D. Dragoman
    1 THE LOCALIZED QUANTUM VACUUM FIELD D. Dragoman – Univ. Bucharest, Physics Dept., P.O. Box MG-11, 077125 Bucharest, Romania, e-mail: [email protected] ABSTRACT A model for the localized quantum vacuum is proposed in which the zero-point energy of the quantum electromagnetic field originates in energy- and momentum-conserving transitions of material systems from their ground state to an unstable state with negative energy. These transitions are accompanied by emissions and re-absorptions of real photons, which generate a localized quantum vacuum in the neighborhood of material systems. The model could help resolve the cosmological paradox associated to the zero-point energy of electromagnetic fields, while reclaiming quantum effects associated with quantum vacuum such as the Casimir effect and the Lamb shift; it also offers a new insight into the Zitterbewegung of material particles. 2 INTRODUCTION The zero-point energy (ZPE) of the quantum electromagnetic field is at the same time an indispensable concept of quantum field theory and a controversial issue (see [1] for an excellent review of the subject). The need of the ZPE has been recognized from the beginning of quantum theory of radiation, since only the inclusion of this term assures no first-order temperature-independent correction to the average energy of an oscillator in thermal equilibrium with blackbody radiation in the classical limit of high temperatures. A more rigorous introduction of the ZPE stems from the treatment of the electromagnetic radiation as an ensemble of harmonic quantum oscillators. Then, the total energy of the quantum electromagnetic field is given by E = åk,s hwk (nks +1/ 2) , where nks is the number of quantum oscillators (photons) in the (k,s) mode that propagate with wavevector k and frequency wk =| k | c = kc , and are characterized by the polarization index s.
    [Show full text]
  • Wave-Whirl-Particle Gudrun Kalmbach H.E University of Ulm, MINT Verlag, Germany
    OSP Journal of Nuclear Science Short Communication Wave-Whirl-Particle Gudrun Kalmbach H.E University of Ulm, MINT Verlag, Germany Corresponding Author: Gudrun Kalmbach H.E, University of Ulm, MINT Verlag, Germany Received: December 19, 2019; Accepted: December 26, 2019; Published: January 02, 2019 In earlier articles the author suggested to extend the wave For dark energy and matter are suggested a pinched and a particle duality of physics to a triple, adding whirls. Whirls Horn torus with a singularity. Below are two models for this: are mathematically computed differently than waves. Parti- dark energy has inverted frequency helix lines inside and cles can be considered as a kind of bubble formation in other aggregate phase state. Wave equations have a double Minkowski cone. They have higher speeds than light 1-dimensional space expansion in direction where a wave is their cylindrical location is closed at projective infinity by a travelling. Harmonic waves for a vibrating string miss this are inverted at the Schwarzschild radius to 1-dimensional coordinate. Sound expands through pressure on the other lemniscates,inside the pinched drawn torus.with twoIn figure red or 1 greenis shown or bluethat quarkswings. hand in a sound cone. Measured is its power as intensity per All quarks are joined in this kind of pinched torus at their free carrying one of the three olor charges r, g, b,in nucle- area (figure 1). center as the Horn torus singularity. (figure 2) They are set - onsons, arebut whirls,confined often in nucleons drawn as by stretchable the two color springs charged between gluon twoquasiparticles quarks as (fieldend points; quantums) different as an geometries energy exchange.
    [Show full text]
  • A Photomechanics Study of Wave Propagation John Barclay Ligon Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1971 A photomechanics study of wave propagation John Barclay Ligon Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Applied Mechanics Commons Recommended Citation Ligon, John Barclay, "A photomechanics study of wave propagation " (1971). Retrospective Theses and Dissertations. 4556. https://lib.dr.iastate.edu/rtd/4556 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 72-12,567 LIGON, John Barclay, 1940- A PHOTOMECHANICS STUDY OF WAVE PROPAGATION. Iowa State University, Ph.D., 1971 Engineering Mechanics University Microfilms. A XEROX Company, Ann Arbor. Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED A photomechanics study of wave propagation John Barclay Ligon A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject; Engineering Mechanics Approved Signature was redacted for privacy. Signature was redacted for privacy. Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1971 PLEASE NOTE; Some pages have indistinct print.
    [Show full text]
  • Coupled Wire Model of Z4 Orbifold Quantum Hall States
    Coupled Wire Model of Z4 Orbifold Quantum Hall States Charles L. Kane Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104 Ady Stern Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4 parafermion Read Rezayi state. The Z4 orbifold quantum Hall states occur at filling factors ν = 2=(2m − p) for odd integers m and p, and have a topological order with a neutral sector characterized by the orbifold conformal field theory with central charge c = 1 at radius p R = p=2. When p = 1 the state is Abelian. The state with p = 3 is the Z4 Read Rezayi state, and the series of p ≥ 3 defines a sequence of non-Abelian states that resembles the Laughlin sequence. Our model is based on clustering of electrons in groups of four, and is formulated as a two fluid model in which each wire exhibits two phases: a weak clustered phase, where charge e electrons coexist with charge 4e bosons and a strong clustered phase where the electrons are strongly bound in groups of 4. The transition between these two phases on a wire is mapped to the critical point of the 4 state clock model, which in turn is described by the orbifold conformal field theory. For an array of wires coupled in the presence of a perpendicular magnetic field, strongly clustered wires form a charge 4e bosonic Laughlin state with a chiral charge mode at the edge, but no neutral mode and a gap for single electrons.
    [Show full text]