Universidade de São Paulo Instituto de Física Distribuições espectrais e angulares da radiação síncrotron no âmbito da teoria quântica Anastasia Burimova Orientador: Prof. Dr. Dmitri Maximovitch Guitman Tese de doutorado apresentada ao Instituto de Física para a obtenção do título de Doutor em Ciências Banca Examinadora: Prof. Dr. Dmitri M. Guitman (IFUSP) Prof. Dr. Josif Frenkel (IFUSP) Prof. Dr. Renato Higa (IFUSP) Prof. Dr. Anatoliy Shabad (LPI) Prof. Dr. Viktor V. Dodonov (UnB) São Paulo 2014 University of São Paulo Institute of Physics Spectral and angular distributions of synchrotron radiation in quantum theory Anastasia Burimova Adviser: Prof. Dr. Dmitri Maximovitch Guitman Commission: Prof. Dr. Dmitri M. Guitman (IFUSP) Prof. Dr. Josif Frenkel (IFUSP) Prof. Dr. Renato Higa (IFUSP) Prof. Dr. Anatoliy Shabad (LPI) Prof. Dr. Viktor V. Dodonov (UnB) São Paulo 2014 Acknowledgement I am using this opportunity to express my gratitude to everyone who supported me throughout my Doctorate course at IF USP. Immeasurable appreciation and deepest gratitude are extended to my adviser,ProfessorDmitriiMaximovitchGuitman,aswellastoallthemembersof"Quanta"group. Iam thankful for their aspiring guidance, patience and friendly advice during the project work. This thesis was written as a summary of research project which was generally supported by FAPESP foundation. Iamsincerelygratefultoourcollaborator, ProfessorVladislavGavriilovitchBagrov,forhisconstruc- tive criticism and fruitful advice. MysincerethankalsogoestothemembersoftheComissionofQuali(cid:12)cationExamfortheirremarks, adviceandobservations. Thesewereofgreathelpduringtherevisionofparticularchaptersofthisthesis. I kindly appreciate the support and patience of Post-Graduate Comission of IF USP and it is my greatest pleasure to thank Eber de Patto Lima personally. Special gratitude goes to Nelson Yokomizo, the former member of "Quanta" group and Thais Silva, Doctorate student of IAG. My family and friends, who had been helping me throughout all these years of work deserve special thanks. Thank you. i ii Resumo Consideramos as caracter(cid:19)(cid:16)sticas da radia(cid:24)c~ao s(cid:19)(cid:16)ncrotron (RS) no ^ambito da teoria qu^antica. Para simpli(cid:12)car a descri(cid:24)c~ao te(cid:19)orica do processo de radia(cid:24)c~ao restringimos (cid:18)a considera(cid:24)c~ao da emiss~ao de u(cid:19)nico f(cid:19)oton. Paratransi(cid:24)c~oesqu^anticasarbitr(cid:19)arias,asdistribui(cid:24)c~oesespectraiseangularesdapot^enciadaRSs~ao dadas de forma anal(cid:19)(cid:16)tica exata. Tratamos separadamente part(cid:19)(cid:16)culas escalares (b(cid:19)osons) e com spin ~=2 (el(cid:19)etrons). Aten(cid:24)c~ao especial (cid:19)e dada (cid:18)as transi(cid:24)c~oes particulares, a saber, as transi(cid:24)c~oes ao primeiro estado excitado e estado fundamental. E(cid:19) mostrado que os componentes de polariza(cid:24)c~ao linear da radia(cid:24)c~ao de el(cid:19)etron se trocam em rela(cid:24)c~ao (cid:18)a orienta(cid:24)c~ao de spin quando o el(cid:19)etron passa para o estado fundamental. Este fato pode ser considerado como uma comprova(cid:24)c~ao anal(cid:19)(cid:16)tica para a presenca de (cid:25)-componente da radia(cid:24)c~ao qu^antica no plano de movimento. Analisamos minuciosamente a radia(cid:24)c~ao emitida pela part(cid:19)(cid:16)cula fracamente excitada. V(cid:19)arias fun(cid:24)c~oes s~ao introduzidas para descrever a evoluc(cid:24)~ao dos per(cid:12)s de distribui(cid:24)c~oes angulares para sistemas de dois e tr^es n(cid:19)(cid:16)veis. Para transi(cid:24)c~oes qu^anticas do primeiro estado excitado ao estado fundamental a an(cid:19)alise comparativadaradia(cid:24)c~aodeb(cid:19)osonseel(cid:19)etrons(cid:19)erealizada,eissoajudaaestimarain(cid:13)u^enciadespinesua dire(cid:24)c~ao sobre as caracter(cid:19)(cid:16)sticas da RS. A radia(cid:24)c~ao de el(cid:19)etrons n~ao polarizados (cid:19)e considerada separada- mente. Observando o comportamento dos ^angulos efetivos,(cid:19)e f(cid:19)acil perceber a inconsist^encia da conclus~ao cl(cid:19)assica bem conhecida sobre a concentra(cid:24)c~ao de radia(cid:24)c~ao ultra-relativista total no plano do movimento. Mostramosqueos^angulosefetivosdaradia(cid:24)c~aoqu^anticatendemaosvalores(cid:12)nitosen~aodesaparecemna regi~ao ultrarelativista. Umarevis~aobrevedateoriacl(cid:19)assicaincluiaintrodu(cid:24)c~aodoconceitonovo,isto(cid:19)ean-partedoespectro. A (cid:12)m de encontrar um an(cid:19)alogo cl(cid:19)assico adequado para a radia(cid:24)c~ao das part(cid:19)(cid:16)culas fracamente excitadas, a ideia de reduzir o espectro cl(cid:19)assico foi desenvolvida. Constatamos que as caracteristicas da radia(cid:24)c~ao calculadasparaoespectrocl(cid:19)assicoreduzidopermanecememboaconcord^ancia,tantoquantitativaquanto qualitativa, com os seus an(cid:19)alogos qu^anticos, pelo menos no que diz respeito aos espectros qu^anticos de uma ou duas harm^onicas. Neste sentido, a teoria cl(cid:19)assica do espectro reduzido pode ser chamada de representativa. A evolu(cid:24)c~ao do m(cid:19)aximo no espectro da radia(cid:24)c~ao(cid:19)e considerada em cap(cid:19)(cid:16)tulo separado. A aproxima(cid:24)c~ao, comumenteconsideradanateoriaclassicaparafrequ^enciacr(cid:19)(cid:16)tica,(cid:19)einv(cid:19)alidaquandoascorre(cid:24)c~oesqu^anticas entram em cena. Mas existe uma possibilidade de encontrar as condi(cid:24)c~oes para o m(cid:19)aximo transferir-se (cid:18)a harm^onica maior do espectro qu^antico. E(cid:19) mostrado que as transfer^encias ocorrem sucessivamente, comecando com a harm^onica principal no caso n~ao relativ(cid:19)(cid:16)stico, e este resultado permanece v(cid:19)alido, inde- pendentementedespin. Paraumapart(cid:19)(cid:16)culaescalarexisteumconjunto(cid:12)xodosvalorescr(cid:19)(cid:16)ticosdocampo iii externo, de tal modo que a transfer^encia do m(cid:19)aximo da radia(cid:24)c~ao entre duas harm^onicas espec(cid:19)(cid:16)(cid:12)cas pode acontecer somente quando a intensidade do campo externo (cid:19)e maior do que o valor cr(cid:19)(cid:16)tico associado com essas harm^onicas. Se essa condi(cid:24)c~ao n~ao for satisfeita, a posi(cid:24)c~ao do m(cid:19)aximo permanece inalterada. Veri(cid:12)- camosqueapresencadespinperturbaestacondi(cid:24)c~ao,nocasodoel(cid:19)etronosvalorescr(cid:19)(cid:16)ticosdaintensidade do campo dependem de nu(cid:19)mero do n(cid:19)(cid:16)vel inicial. iv Abstract Intheframeworkofquantumtheorythecharacteristicsofsynchrotronradiation(SR)areconsidered. Inordertosimplifytheoreticaldescriptiontheprocessofradiationisrestrictedtosingle-photonemission. For arbitrary quantum transitions the spectral-angular distributions of SR power are given in exact analytical form. Scalar particles (bosons) and particles with spin ~=2 (electrons) are treated separately. Special attention is given to the particular transitions, namely, to the transitions to (cid:12)rst excited and ground states. It is shown that the components of linear polarization of radiation from electron switch places due to the orientation of spin when the electron jumps to the ground state. This fact can be considered an analytical proof for the presence of (cid:25)-component of quantum radiation in the plane of motion. Theradiationemittedfromweaklyexcitedparticlesisthoroughlyanalysed. Todescribetheevolution of the pro(cid:12)les of angular distributions various functions are introduced both for two- and three-level systems. Forquantumtransitionsfromthe(cid:12)rstexcitedstatetothegroundstatethecomparativeanalysis of radiation from bosons and electrons is performed, which helps to estimate the in(cid:13)uence of spin and its direction on the characteristics of radiation. The radiation from unpolarized electron is considered separately. Tracking the behavior of effective angles allows to discover the inconsistency of well-known classical conclusion about the concentration of total (summed over spectrum) ultrarelativistic radiation in the plane of motion. It is shown that the effective angles of quantum radiation tend to (cid:12)nite values and do not vanish in ultrarelativistic region. A brief review of classical theory includes an introduction of the new concept, n-part of spectrum. In order to (cid:12)nd an adequate classical analogue for the radiation from weakly excited particles, the idea to reduce classical spectrum was developed. It turns out that the characteristics of radiation calculated for reduced classical spectrum stay in good quantitative and qualitative agreement with their quantum analogues, at least for single-harmonic and two-harmonic quantum spectra, and classical theory of a reduced spectrum can be claimed representational in this sense. The evolution of maximum in radiation spectrum is considered in separate chapter. A well-known approximationobtainedforcriticalfrequencyintheframeworkofclassicaltheoryisinvalidwhenquantum corrections enter the picture. But there appears a possibility to (cid:12)nd the conditions for the maximum to shift to the highest harmonic of (cid:12)nite quantum spectrum. It is shown that the shifts occur successively starting with primary harmonic in non-relativistic case, and this result remains valid independently of spin. For a scalar particle there exists a (cid:12)xed set of numbers, which are the critical values of external (cid:12)eld, such that the shift of radiation maximum in the spectrum of boson can only happen when the v intensity of external (cid:12)eld is greater than certain critical value related to corresponding harmonic. If this condition is not satis(cid:12)ed, the position of maximum remains unchanged. It turns out that the presence of spin perturbs this picture, so that the critical values of (cid:12)eld intensity depend on the number of initial level. vi Contents Introduction 1 1 Classical Theory of Synchrotron Radiation 7 1.1 Spectral and Angular Distributions of Synchrotron Radiation in Classical Theory . . . . 7 1.2 First Harmonic of Classical Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Polarization Properties of Radiation Associated with the First Harmonic . . . . . 15 1.2.2 Partial Contribution of the First Harmonic . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Classical Spectrum of Synchrotron Radiation and its Primary Harmonics (cid:23) =1 and (cid:23) =2 20 2 An Introduction to Quantum Theory of Synchrotron Radiation 25 2.1 The Power of Radiation in Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.2 Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3 Spontaneous Transitions to the First Excited State and to the Ground State . . . . . . . 39 3 Single-harmonic Spectrum of Synchrotron Radiation in Quantum Theory 45 3.1 The Characteristics of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.1 Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.1.2 Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.3 Comparative Analysis of the Angular Distributions of Radiation from Bosons and Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 The Difference Between Classical and Quantum Results . . . . . . . . . . . . . . . . . . . 55 3.2.1 Linear and Circular Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.2 Effective Angles of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4 Quantum Two-harmonic Spectrum 67 4.1 Polarization Properties of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Spectral-Angular Distributions of Power. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 Effective Angles and Deviation Angles of Radiation . . . . . . . . . . . . . . . . . . . . . . 77 vii 5 The Evolution of Maximum in the Quantum Spectrum of the Radiation 83 5.1 The Evolution of Maximum in Classical Spectrum . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Quantum Approach to the Spectral Distributions of Synchrotron Radiation . . . . . . . . 87 5.2.1 Position of the Radiation Maximum in the Quantum Spectrum of a Scalar Particle 88 5.2.2 The Role of Spin in the Quantum Picture . . . . . . . . . . . . . . . . . . . . . . . 91 5.2.3 The Evolution of Maximum with Respect to Polarization . . . . . . . . . . . . . . 93 Conclusion 97 Appendix A Laguerre Functions 101 Appendix B Effective angles 103 Bibliography 105 viii