Michael Bonitz Quantum Kinetic Theory Second Edition Quantum Kinetic Theory Michael Bonitz Quantum Kinetic Theory Second Edition 123 Michael Bonitz Institut für Theoretische Physik UniversitätKiel Kiel Germany Originally published byB.G. Teubner,Leipzig, 1998 ISBN978-3-319-24119-7 ISBN978-3-319-24121-0 (eBook) DOI 10.1007/978-3-319-24121-0 LibraryofCongressControlNumber:2015950456 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland1998,2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) Preface to the Second Edition Quite quickly after its publication, this book was unavailable calling for a second edition. However, with my move to Christian-Albrechts-Universität Kiel in 2003 there was no time to tackle this task. The good thing about this time gap is that a critical and a less-biased look at the field and its coverage in the book became possible.Itissatisfactory torealize that the(very enthusiastic)perspectivesforthe subjectoffield–matterinteractionandnonequilibriummany-bodytheoryexpressed inthefirsteditionturnedouttobecompletelyjustified.Certainly,someofthefields that were in the focus before, such as semiconductor optics or laser–plasma interaction, have reached some maturity. At the same time, new fields that are facingverysimilarproblemshaveundergonearapiddevelopmentthatisdrivenby theamazingprogressinattosecondlasertechnology[1,2].Thisincludesthefields ofattosecondatomicandmolecularphysicsandthedynamicsofultracoldatomsin traps and optical lattices, to name just two. The challenges formany-body physics ingeneral and short-timeapproaches, in particular, have not faded. The opposite is true: The need for a time-dependent description ofcorrelated quantumsystes hasgrownsteadily. Fortunately, therapid growth in computational resources, combined with brilliant young scientists including K. Balzer, N.E. Dahlen, M. Garny, S. Hermanns, F. Jahnke, R. van Leeuwen, G. Stefanucci, C. Verdozzi, and many others entering the field, has allowed for an unexpected burst of activity in computational studies of ultrafast quantum dynamics. In fact, non-Markovian quantum kinetic equations and, even more,nonequilibrium Green's functions have now become notjust nice theoretical concepts but standard practical tools in many fields.1 Some of the progress can be seen in recent text books by Kremp et al. [8], Rammer [9], Stefanucci et al. [10], and Balzer et al. [11]. Some overview on the numerical activity can be found in [12]. 1ThedevelopmentsinnonequilibriumGreen’sfunctionsarealsoreflectedintheproceedingsofthe conferences“ProgressinNonequilibriumGreenFunctions”thathavebeentakingplacesince1999 [3–7]. v vi PrefacetotheSecondEdition In this active environment this book still seems to have its place. The central topics—the methods of nonequilibrium density operators and nonequilibrium Green’s functions—have not lost their importance. An analysis of the connections of both has become even more important. This issue is now presented in more detail. Also, to improve the understandability, introductory discussions have been extended, in particular, in Chaps. 1, 6 and 13. In this spirit, a new chapter “Properties of the quantum kinetic equation” has been included. Obviously, many otherapproachestothenonequilibriumquantumdynamicshaveappearedorgrown to importance since the first edition was published. This includes density matrix renormalization group approaches, time-dependent density functional theory or wavefunction-based approaches of quantum chemistry. Here only brief comments have been found appropriate, together with references to the dedicated literature. Over the last 15 years, this book has been used actively in my lectures in Rostock and Kiel on short-time physics and quantum statistical theory. This practical test has led not only to the correction of errors but also to many improvements, such as the inclusion of more and better examples. Some technical details and derivations have been taken out of the main text and moved to new problemsandsolutionssection.IthankKarstenBalzer,AlfredLeitenstorfer,Hauke Thomsen,andTorbenOttforprovidingfigures.Iamgratefultomanystudentsand colleagues,inparticulartoKarstenBalzer,SebastianBauch,TobiasDornheim,Jim Dufty, Alexei Filinov, Frank Graziani, Simon Groth, Sebastian Hermanns, Christopher Hinz, David Hochstuhl, Frank Jahnke, Hanno Kählert, Nai-Hang Kwong, Denis Lacroix, Patrick Ludwig, Torben Ott, Hartmut Ruhl, Niclas Schlünzen,Tim Schoof, DirkSemkat,GianluccaStefanucci, Robert vanLeeuwen, ClaudioVerdozzi,andJanVorbergerforvaluablefeedback.Finally,IthankHauke Thomsen for assistence with editing the final version. Kiel Michael Bonitz Preface to the First Edition We are at the beginning of a new revolution in science and technology, which is fueled by the tremendous progress in short-pulse laser technology over the last 10–15years.Fundamentalphysicalprincipleswhichhavebeenknownbeforeonly from abstract theory, are suddenly becoming accessible to direct experimental observation.Thisincludesthequantumstatesofsingleatoms,Heisenberg’suncer- taintyprinciple,lowtemperaturephenomenasuchasBosecondensationandWigner crystallization, as well as ultraintense electromagnetic fields, and the possibility to create,insmalluniversityfacilities,relativisticelectronsorhotdenseplasmaswhich eventually will lead to thermonuclear fusion. In the view of technological applica- tions promised by these developments, many countries have established special research projects, including, the National Ignition Facility in the U.S. or the new Schwerpunkt“Laserfelder”oftheDeutscheForschungsgemeinschaftinGermany. These developments are a major challenge for Theoretical Physics—to under- standandpredicttheinteractionofmatterwithelectromagneticfieldsrangingfrom milliwatt to petawatt (10(cid:1)3(cid:1)1015 W) powers and lasting from seconds to femto- seconds (10(cid:1)15 s). Within this very complex frame, the current monograph is devoted to quantum many-particle systems under extreme nonequilibrium condi- tions. It attempts to answer the question how these systems behave immediately after and also during their creation, thereby focusing on a general approach. The statistical description of many-particle systems in nonequilibrium began withL.Boltzmann’sfamouskineticequation[13].Sincethen,numeroustheoretical methods have been developed to describe nonequilibrium processes in various fields, including fluids, dense plasmas, solids, and nuclear matter, which are often veryefficientbut,atthesametime,sospecializedthattheyareunderstoodonlybya few immediate specialists. And this, despite the fact that the underlying physics is often very similar if it is uncovered from the shell of (certainly needed) field-specificjargonandnotationaswellassystem-inherentdetailsandparameters. Moreover, many of these specialized theories have common roots, and a return to them, from time to time, can be extremely fruitful, even for an experienced vii viii PrefacetotheFirstEdition specialist,andleadtonewideas.Ontheotherhand,forstudentsorresearchersnew in the field, the best choice is to start from the roots. Thisisparticularlytruefortheproblemsrelatedtoultrafastrelaxation.Hereexist three main branches of the theory: the two statistical methods based on density operators and nonequilibrium Green’s functions, respectively, and the mechanical approachofmoleculardynamics.Allthreeofthemarediscussedandcomparedin this book. The simplest and the most intuitive one is the density operator theory which, for this reason, is chosen as the basic concept. A very general and com- prehensive introduction to this approach is given in Chaps. 2 and 3. First, appli- cations to many-particle effects are given in Chaps. 4–6, to collective phenomena (plasmons and instabilities) and to correlations. Students with basic knowledge in quantummechanicsandstatisticalphysicswillhavenoproblemsinfollowingthis treatment.2 More advanced applications follow in Chaps. 7–11. It is demonstrated that the density operator approach is very efficient in deriving generalized non-Markovian quantumkinetic equationswith memory andinitial correlations,andthat it further allows to incorporate self-energy, dynamical screening, strong coupling effects, as well as the interaction with electromagnetic fields. An introduction to the method of nonequilibrium Green’s functions is given in Chap. 12, starting from a fully relativistic formulation. Here, the main results are coupled Kadanoff–Baym equations for carriers and photons for which various approximationsareconsidered,includingthenonrelativisticlimitandtheextension tomulti-bandsystems.Moreover,recentnumericalresultsarepresentedwhichwill be of interest also for experienced readers. Finally, Chap. 13 is devoted to the molecular dynamics approach, where the modern developments in the field quan- tumMD arealsodiscussed.Furthermore,adetailed comparisonoftheconceptsof quantumstatisticsandmoleculardynamicsisperformed,andtheideaofnumerical comparisons of the two is developed. For illustration of the theoretical methods, a variety of recent numerical results on quantum kinetic equations, non-Markovian effects, and correlation phenomena have been included. Numerical analysis is an important part of modern kinetic theory. Therefore, throughout this book, the theoretical results are cast in a form best suitable for numerical evaluation. Furthermore, to enable the reader to do numerical work by himself, an extensive introductionto theconceptsof numerical solution of quantum kinetic equations is supplied in Appendix F. Thus, I hope the reader will gain broad fundamental knowledge in quantum kinetictheoryingeneral,andonthetheoreticaldescriptionofultrafastrelaxationin particular,whichshouldallowhimtocreativelyadapttheseconceptstoanyfieldof many-particle physics. 2Sectionswhicharenotnecessaryfortheunderstandingofthebasicconceptsandwhichmaybe skipped on the first reading are marked with an asterisk. Readers interested in derivations and details beyond the basic material can find the main ideas outlined in footnotes and appendices. Extensive references are provided to the specialized literature of various fields and to classical worksaswell. PrefacetotheFirstEdition ix Physics would not even be half as exciting and rewarding without continuous discussions and sharing of ideas with colleagues—so I am grateful to Martin Axt, Lazi Banyai, Karim ElSayed, Hartmut Haug, Klaus Henneberger, Frank Jahnke, Tilman Kuhn, Günter Manzke, Klaus Morawetz, Ronald Redmer, Hartmut Ruhl, Wilfried Schäfer, Chris Stanton, Heinrich Stolz, and Günter Zwicknagel. I very much enjoyed working together with Rolf Binder, Thomas Bornath, Jim Dufty, AndreasFörster,DirkGericke,YuriL’vovichKlimontovich,StephanKoch,Sigurd Köhler, Sylvio Kosse, Wolf Kraeft, Dietrich Kremp, Nai Kwong, Thomas Ohde, Manfred Schlanges, Don Scott, and Dirk Semkat. It is my great pleasure to thank myremarkableteachersYuriL’vovichKlimontovich,DietrichKremp,andStephan Koch, who guided me through various fields of physics and life, and Werner Ebeling and Wilfried Schäfer for continuous encouragement of my work. The results which are the basis for this book would not have been possible without the generous support from the Deutscher Akademischer Austauschdienst, grants from the Deutsche Forschungsgemeinschaft and grants for CPU–time at the HLRZ Jülich and the CCIT of the University of Arizona. This book greatly ben- efited from comments of Nai Kwong and Dirk Semkat who also assisted me in preparing several figures for Chap. 12, as did Renate Nareyka and my wife. Finally, I thank my father for raising my interest in physics, Christine for her tremendous patience and encouragement, and Sebastian and Martin for their will- ingness to give up our PC for this book. Rostock Michael Bonitz June 1998 Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Correlated Many-Particle Systems . . . . . . . . . . . . . . . . . . . . . 2 1.2 Thermodynamic Properties of Correlated Systems . . . . . . . . . . 9 1.3 Ultrafast Nonequilibrium Phenomena. . . . . . . . . . . . . . . . . . . 12 1.3.1 Dynamics of Isolated Systems . . . . . . . . . . . . . . . . 13 1.3.2 Interaction of Matter with Short Laser Pulses. . . . . . 14 1.3.3 Overview of Relaxation Processes. . . . . . . . . . . . . . 18 1.4 The Boltzmann Equation–Successes and Failure . . . . . . . . . . . 20 1.4.1 An Elementary Introduction to the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4.2 Unphysical Ultrafast Relaxation in Charged Particle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 Improved Theoretical Concepts. . . . . . . . . . . . . . . . . . . . . . . 24 1.5.1 Outline of this Book . . . . . . . . . . . . . . . . . . . . . . . 26 1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 The Method of Reduced Density Operators. . . . . . . . . . . . . . . . . . 29 2.1 Density Operator. Von Neumann Equation. . . . . . . . . . . . . . . 29 2.2 BBGKY-Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1 Reduced Density Operators. Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 Basic Representations of the Hierarchy . . . . . . . . . . . . . . . . . 41 2.3.1 Coordinate Representation . . . . . . . . . . . . . . . . . . . 41 2.3.2 Wigner Representation. . . . . . . . . . . . . . . . . . . . . . 43 2.3.3 Classical Limit and Quantum Corrections. . . . . . . . . 45 2.3.4 Spatially Homogeneous Systems. Momentum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 xi