417 Pages·2000·2.72 MB·English

Lecture Notes in Physics EditorialBoard R.Beig,Wien,Austria J.Ehlers,Potsdam,Germany U.Frisch,Nice,France K.Hepp,Zu¨rich,Switzerland W.Hillebrandt,Garching,Germany D.Imboden,Zu¨rich,Switzerland R.L.Jaffe,Cambridge,MA,USA R.Kippenhahn,Go¨ttingen,Germany R.Lipowsky,Golm,Germany H.v.Lo¨hneysen,Karlsruhe,Germany I.Ojima,Kyoto,Japan H.A.Weidenmu¨ller,Heidelberg,Germany J.Wess,Mu¨nchen,Germany J.Zittartz,Ko¨ln,Germany 3 Berlin Heidelberg NewYork Barcelona HongKong London Milan Paris Singapore Tokyo TheEditorialPolicyforProceedings TheseriesLectureNotesinPhysicsreportsnewdevelopmentsinphysicalresearchandteaching–quickly, informally,andatahighlevel.Theproceedingstobeconsideredforpublicationinthisseriesshouldbelimited toonlyafewareasofresearch,andtheseshouldbecloselyrelatedtoeachother.Thecontributionsshouldbe ofahighstandardandshouldavoidlengthyredraftingsofpapersalreadypublishedorabouttobepublished elsewhere.Asawhole,theproceedingsshouldaimforabalancedpresentationofthethemeoftheconference includingadescriptionofthetechniquesusedandenoughmotivationforabroadreadership.Itshouldnot beassumedthatthepublishedproceedingsmustreﬂecttheconferenceinitsentirety.(Alistingorabstracts ofpaperspresentedatthemeetingbutnotincludedintheproceedingscouldbeaddedasanappendix.) 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Westronglyrecommendthatyoumakeuseofthisoffer,sincetheresultwillbeabookofconsiderably improvedtechnicalquality.Toavoidmistakesandtime-consumingcorrespondenceduringtheproduction periodtheconferenceeditorsshouldrequestspecialinstructionsfromthepublisherwellbeforethebeginning oftheconference.Manuscriptsnotmeetingthetechnicalstandardoftheserieswillhavetobereturnedfor improvement. ForfurtherinformationpleasecontactSpringer-Verlag,PhysicsEditorialDepartmentII,Tiergartenstrasse17, D-69121Heidelberg,Germany Serieshomepage–http://www.springer.de/phys/books/lnpp Michel Planat (Ed.) Noise, Oscillators and Algebraic Randomness From Noise in Communication Systems to Number Theory Lectures of a School Held in Chapelle des Bois, France, April 5–10, 1999 1 3 Editor MichelPlanat LaboratoiredePhysiqueetMetrologie desOscillateursduCNRS 32Avenuedel’Observatoire 25044Besanc¸onCedex,France LibraryofCongressCataloging-in-PublicationData. Noise,oscillators,andalgebraicrandomness:fromnoiseincommunicationsystemsto numbertheory:lecturesofaschoolheldinChapelledesBois,France,April5-10,1999/ MichelPlanat(ed.). p.cm.–(Lecturenotesinphysics,ISSN0075-8540;vol.550) Includesbibliographicalreferences. ISSN3540675728(alk.paper) 1.Electronicnoise–Mathematicalmodels–Congresses.2.Oscillators, Electric–Congresses.3.Numericalanalysis–Congresses.4.Algebra–Congresses.5. Telecommunication–Mathematics–Congresses.I.Planat,Michel,1951-II.Lecture notesinphysics;550. TK7867.5.N6272000 621.382’24–dc21 00-032966 ISSN0075-8450 ISBN3-540-67572-8Springer-VerlagBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthe materialisconcerned,speciﬁcallytherightsoftranslation,reprinting,reuseofillustra- tions, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storageindatabanks.Duplicationofthispublicationorpartsthereofispermittedonly undertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrent version,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violations areliableforprosecutionundertheGermanCopyrightLaw. Springer-VerlagisacompanyintheBertelsmannSpringerpublishinggroup. ©Springer-VerlagBerlinHeidelberg2000 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublication doesnotimply,evenintheabsenceofaspeciﬁcstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Typesetting:Camera-readybytheauthors/editor Coverdesign:design&production,Heidelberg Printedonacid-freepaper SPIN:10719295 57/3144/du-543210 Preface This volume presents most contributions given at the School “Noise of fre- quenciesinoscillatorsandthedynamicsofalgebraicnumbers”whichwasheld in Chapelle des Bois, Jura (France) from 5 to 10 April 1999. The event was madepossiblebythefullsupportofthethematicschoolprogramoftheCen- tre National de la Recherche Scientiﬁque in France. Noise is ubiquitous in nature and in man-made systems. For example, noise inoscillatorsperturbshightechnologydevicessuchastimestandardsordig- ital communication systems. The understanding of its algebraic structure is thus of vital importance to properly guide the human activity. The book addresses these topics in three parts. Several aspects of classical and quantum noise are covered in Part I, both from the viewpoint of quan- tum physics and from the nonlinear or fractal viewpoint. Part II is mainly concerned with noise in oscillating signals, that is phase or frequency noise and 1/f noise. From a careful analysis of the experimental noise attached to the carrier the usefulness of the number theoretical based method is demon- strated.ThisviewisexpandedinPartIII,whichismathematicallyoriented. In conclusion, the noise concept proved to be a very attractive one gathering peoplefromatleastthreescientiﬁccommunities:electronicengineering,the- oretical physics and number theory. They represented an original mixture of talentsandthepresenteditoracknowledgesallauthorsfortheirpatienceand open-mindednessduringtheschool.Mostmanuscriptsarecomprehensibleto a large audience and should allow readers to discover new bridges between the ﬁelds. We ourselves have identiﬁed but a few. ThemeetingwasfollowedbyasmallworkshopsponsoredbyM.Waldschmidt attheInstitutHenriPoincar´einParison3and4December1999:‘Th´eoriedes nombres,bruitdesfr´equencesett´el´ecommunications’.Thepurposeherewas to emphasize the newly discovered link between the Riemann zeta function and communication systems. Some papers and related material are available at the address: http://www.archetypo.web66.com, a new URL site built by Matthew Watkins and devoted to the relationship between prime number theory and physics. Besan¸con, April 2000 Michel Planat Contents Introduction Michel Planat .................................................. 1 Mathemagics (A Tribute to L. Euler and R. Feynman) Pierre Cartier .................................................. 6 Part I Classical and Quantum Noise Thermal and Quantum Noise in Active Systems Jean-Michel Courty, Francesca Grassia, Serge Reynaud .............. 71 Dipole at ν =1 V. Pasquier .................................................... 84 Stored Ion Manipulation Dynamics of Ion Cloud and Quantum Jumps with Single Ions Fernande Vedel ................................................ 107 1/f Fluctuations in Cosmic Ray Extensive Air Showers E. Faleiro, J.M.G. Go´mez ....................................... 125 Stochastic Resonance and the Beneﬁt of Noise in Nonlinear Systems Franc¸ois Chapeau-Blondeau ...................................... 137 Time is Money Marcel Ausloos, Nicolas Vandewalle, Kristinka Ivanova .............. 156 Part II Noise in Oscillators, 1/f Noise and Arithmetic Oscillators and the Characterization of Frequency Stability: an Introduction Vincent Giordano, Enrico Rubiola................................. 175 Phase Noise Metrology Enrico Rubiola, Vincent Giordano................................. 189 VIII Contents Phonon Fine Structure in the 1/f Noise of Metals, Semiconductors and Semiconductor Devices Mihai N. Mihaila ............................................... 216 The General Nature of Fundamental 1/f Noise in Oscillators and in the High Technology Domain Peter H. Handel ................................................ 232 1/f Frequency Noise in a Communication Receiver and the Riemann Hypothesis Michel Planat .................................................. 265 Detection of Chaos in the Noise of Electronic Oscillators by Time Series Analysis Methods C. Eckert, M. Planat ............................................ 288 Geometry and Dynamics of Numbers Under Finite Resolution Jacky Cresson, Jean-Nicolas D´enari´e ............................. 305 Diophantine Conditions and Real or Complex Brjuno Functions Pierre Moussa, Stefano Marmi ................................... 324 Part III Algebraic Randomness Algebraic and Analytic Randomness Jean-Paul Allouche.............................................. 345 From Symbolic Dynamics to a Digital Approach K. Karamanos.................................................. 357 Algebraic Dynamics and Transcendental Numbers Michel Waldschmidt............................................. 372 Dynamics of Some Contracting Linear Functions Modulo 1 Yann Bugeaud, Jean-Pierre Conze ................................ 379 On the Modular Function and Its Importance for Arithmetic Paula B. Cohen................................................. 388 On Generalized Markoﬀ Equations and Their Interpretation Serge Perrine................................................... 398 Introduction Michel Planat (cid:1) Laboratoire de Physique et M´etrologie des Oscillateurs du CNRS, 32 Avenue de l’Observatoire, 25044 Besan¸con Cedex, France In the classical realm the best known type of noise is thermal noise. Due to thermal agitation, free electrons in a metallic conductor are moving around continuously causing collisions with the atoms and a continuous exchange of energybetweenthemodes.ThiswasﬁrstinvestigatedexperimentallybyJ.B. Johnson and H. Nyquist in 1928. Nyquist’s theorem states that the power spectraldensity(psd)ofvoltageﬂuctuationsthrougharesistorR attemper- ature T is S (f) = 4kRT, with k the Boltzman’s constant. The quantum V approachisalsoaveryeﬃcientwaytounderstandthelimitationintheaccu- racy of measurements performed in thermal equilibrium. The paper by J.M. Courty et al. examines the ﬂuctuations which are expected in an operational ampliﬁer from a quantum network approach. The ultimate sensitivity of a cold damped accelerometer designed for space applications is calculated as well. QuantumphysicsisalsousedtounderstandthequantumHalleﬀect,that is the behaviour of charged interacting electrons in the presence of a strong magnetic ﬁeld. The indivisibility of the electron charge e may be demon- strated from a measurement of the current power spectral density (psd) S (f)=2eI whichisknownasshotnoise(Schottky,1918).Suchnoiseresults I from the random emission of electrons from the cathode to the anode in a diode or a semiconductor. Similarly, partition noise is added to the measure- mentwheneveracurrentisdistributedrandomlybetweentwoelectrodes.Its net eﬀect is to introduce an extra multiplicative factor in the relation for the shot noise psd. Partition noise measurements in small size quantum conduc- tors have recently revealed the existence of fractional charges ep (p and q q integers) associated to quasi particle tunneling states. The paper presented attheschoolbyC.Glattliwaspublishedelsewhere(Phys.Rev.Lett.79,2526 (1997)andin“Topologicalaspectsoflowdimensionalsystems”,Proceedings ofLesHouches1998SummerSchool).Thetheoryofthefractionalquantum Hall eﬀect is still very open to debate, as shown in the paper by V. Pasquier whichadressestheproblemofbosonicparticlesinteractingrepulsivelyatthe ﬁlling factor ν =1. InhisearlystudyofthermalnoiseJ.B.Johnsonalsofoundalargeamount of voltage noise S (f) ∼ KV2/f at low Fourier frequencies f. From many V experiments it was found that the noise intensity is inversely proportional to the number of carriers N in the sample, that is K ∼ γ/N, with γ in (cid:1) [email protected] M.Planat(Ed.):LNP550,pp.1–5,2000. (cid:1)c Springer-VerlagBerlinHeidelberg2000 2 Michel Planat the range 10−3 to 10−8. This was generally attributed to diﬀerent scatter- ing mechanisms, by the crystal lattice or the impurities, leading to mobility ﬂuctuationsoftheelectrons.Theseﬁndingspointtoanonlinearoriginofthe 1/f noise.Finestructuresrevealingtheinteractionofelectronswithbulkand surface phonons in several solid-state physical systems are given in Mihaila’s paper in Part II. On the theoretical side, a quantum electrodynamical the- ory was developed by P. Handel in the seventies based on infrared divergent coupling of the electrons to the electromagnetic ﬁeld in the scattering pro- cess.Thebasicresultfortheγ parameterinvolvestheﬁnestructureconstant α ∼ 1/137 times the square of the ratio between the change of the velocity oftheacceleratedchargeoverthelightvelocity(seetheintroductionofHan- del’spaper).Thisso-called(byhim)conventionalquantum1/f eﬀectapplies to small solid-state devices with K of the order 10−7. For large samples K increases to 2α/π ∼4.6×10−3 which is the value predicted by the coherent state approach of the quantum 1/f eﬀect (see Fig. 2 in Handel’s paper). Chargedparticlescanbekeptfreeandinterrogatedforverylongtimesin aminiaturetrapasshowninthepaperbyF.Vedel.Synchronizedandchaotic states of the oscillating ions are studied in such a set-up. Using laser cooling with a few ions, the technique also allows the study of quantum jumps and the design of a very accurate clock. Highenergyparticlesandnucleireachingearthfromspacearecalledcos- mic rays. Their energy spectrum is very broad (from 109 to 1020 eV); they are emitted from multiplicative cascades (cosmic showers) with a variety of disintegrationsasshowninthepaperbyJ.Gomez.Atthegroundlevelparti- cle densities show ﬂuctuations with a 1/f power spectrum in the polar angle coordinate.Thisanewexampleofthedeeprelationshipbetweennonlinearity and 1/f noise. The paper by F. Chapeau-Blondeau studies the intrinsically nonlinear link between signal and noise in a variety of systems. The word stochastic resonancehasbeencoinedtodescribesituationsinwhichthenoisecanbeneﬁt the information-carrying signal. The ability to increase the signal to noise ratiofromnoiseenhancementinanonlinearinformationchanneloranimage is conclusively demonstrated. It is not so well known that the ﬁrst mathematical study of Brownian motion, which is due to Bachelier (prior to Einstein) in 1900 concerned the pricing of options in speculative markets. Anomalous (non-Gaussian) distri- butionsaretheruleinstockmarketdataasshowninthepaperbyM.Ausloos. ThepapersbyV.Giordano(G),E.Rubiola(R),M.Planat(P),C.Eckert (E) and J. Cresson (C) in Part II are closely related. They mainly concern the understanding of the building block of an electronic oscillator, that is a resonant cavity (a quartz crystal) and a sustaining ampliﬁer. Time and frequency metrology was born in an eﬀort to improve the design and perfor- manceofsuchoscillatorsusedasaccurateclocks(G).Instrumentstomeasure

Noise is ubiquitous in nature and in man-made systems. Noise in oscillators perturbs high-technology devices such as time standards or digital communication systems. The understanding of its algebraic structure is thus of vital importance. The book addresses both the measurement methods and the unde

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