Nonequilibrium Aspects of Quantum Thermodynamics Mathias Michel

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Nonequilibrium Aspects of Quantum Thermodynamics Mathias Michel Institut fur¨ Theoretische Physik I Universitat¨ Stuttgart Pfaffenwaldring 57 70550 Stuttgart Nonequilibrium Aspects of Quantum Thermodynamics Von der Fakult¨at Mathematik und Physik der Universit¨at Stuttgart zur Erlangung der Wurde¨ eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung Vorgelegt von Mathias Michel aus Stuttgart Hauptberichter: Prof. Dr. G¨unter Mahler Mitberichter: Prof. Dr. Ortwin Hess Tag der mundlichen¨ Prufung:¨ 28. Juli 2006 Institut fur¨ Theoretische Physik der Universit¨at Stuttgart 2006 D 93 (Dissertation Universit¨at Stuttgart) Alle Rechte vorbehalten c Selbstverlag M. Michel Stuttgart M. Michel • Institut fur¨ Theoretische Physik I • Universit¨at Stuttgart Pfaffenwaldring 57 • 70550 Stuttgart • Germany E-Mail: [email protected] Homepage: itp1.uni-stuttgart.de/mitarbeiter/michel/ Printed in Germany ISBN-10: 3-00-019902-0 ISBN-13: 978-3-00-019902-8 To my parents and Mirjam Acknowledgements It is my pleasure to acknowledge advice and support from several persons who have influenced this work intensively. First of all I would like to thank my supervisor Prof. Dr. G. Mahler for his constant interest in the progress of this work, his constructive criticism, and his support during the last years. It is impossible to honor the impact that Jun.-Prof. Dr. J. Gemmer had on this work. I am indebted to him for suggesting me the very interesting and challenging problem of nonequilibrium quantum thermodynamics, for count- less fruitful discussions, for sharing his immense knowledge and expertise with me, for having time to answer my numerous questions, helping me in any com- plicated situation, and last but not least for his friendship. I thank Prof. Dr. O. Hess (University of Surrey) for writing the second report on my dissertation. Within the Institute of Theoretical Physics I (Universit¨at Stuttgart) I have experienced a stimulating environment for research. I would like to thank my present and former colleagues P. Borowski, Dr. M. Hartmann, M. Henrich, C. Kostoglou, Dr. A. Otte, H. Schmidt, H. Schr¨oder, M. Stollsteimer, J. Teiffel, F. Tonner and P. Vidal for fruitful discussions (not only physical ones). I am grateful for the warm welcome at the Universit¨at Osnabruc¨ k, the successful collaboration and many interesting discussions with Apl.-Prof. Dr. J. Schnack, Dr. M. Exler and R. Steinigeweg. Furthermore, I have benefited a lot from discussions with Dr. H.-P. Breuer (Universit¨at Freiburg) and Dr. F. Heidrich-Meisner (University of Tennessee). Financial support by the Deutsche Forschungsgemeinschaft is gratefully ac- knowledged. I am indebted to Rebekah and Ron Abramski for proofreading parts of the manuscript and thank all my other friends, especially K. Bihlmaier, and my brother for being by my side. Finally, I would like to thank my parents for their extensive support during my studies and especially my father Dr. H. Michel for interesting discussions about physics and my work. Last but not least, I am deeply indebted to v vi Mirjam for her infinite patience, support and love and for proofreading the manuscript. Stuttgart, September 2006 Mathias Michel Contents 1. Introduction 1 I. Relaxation to Equilibrium 5 2. System and Environment 7 2.1. Class of Model Systems . 8 2.2. Time-Evolution of Pure and Mixed States . 10 2.3. Environment or Heat Bath . 12 3. Open System Approach to Relaxation Processes 15 3.1. Nakajima-Zwanzig Equation . 16 3.2. Born Approximation . 18 3.3. Redfield, Markov and the Rotating Wave Approximation . 19 3.4. The Dissipator . 21 4. Global and Local Properties 25 4.1. Global Stationary Equilibrium State . 25 4.2. Approach of Equilibrium . 27 4.3. Temperature . 29 4.4. Local Energy and Temperature . 31 4.5. Open System Equilibrium State . 32 4.6. Relaxation Dynamics in QME . 34 5. Quantum Thermodynamic Environment 37 5.1. Finite Environment . 38 5.2. Beyond the Born Approximation . 39 6. Hilbert Space Average Method 41 6.1. Hamiltonian Model . 41 6.1.1. Projection and Transition Operators . 43 6.1.2. Canonical and Microcanonical Interaction . 43 vii viii Contents 6.1.3. Dyson Short Time Dynamics . 46 6.1.4. Reduced Dyson Short Time Dynamics . 48 6.2. Hilbert Space Average . 49 6.2.1. Definition of the Hilbert Space Average . 49 6.2.2. Approximation Scheme . 49 6.2.3. Hilbert Space Averaged Time Evolution . 51 7. General HAM Rate Equation 53 7.1. Short Time Iteration Scheme . 53 7.2. Closed Differential Equation . 55 7.3. Limits of Applicability . 57 7.3.1. Decay of Environmental Correlations . 57 7.3.2. Truncation of the Dyson Series . 58 7.3.3. Hilbert Space Variance . 60 8. Thermalization 61 8.1. Damping Model . 61 8.1.1. Effective Hamiltonian Model . 62 8.1.2. Correlation Function . 63 8.1.3. Linear Regime . 65 8.2. The HAM Rate Equation . 68 8.3. Comparison to Schr¨odinger Dynamics . 69 8.3.1. Finite Environment Imposing Infinite Temperature . 70 8.3.2. Accuracy of HAM . 72 9. Decoherence 75 9.1. Fermi's Golden Rule . 76 9.2. Advanced Model System . 76 9.3. Several Different Thermalization Times . 79 9.4. Decoherence According to Canonical Couplings . 82 9.5. Additional Microcanonical Coupling . 83 II. Transport 87 10.Heat Transport 89 10.1. Fourier's Law . 90 10.2. Kinetic Gas Theory for Phonons . 92 10.3. Peierls-Boltzmann Equation . 93 10.4. Linear Response Theory . 94 10.4.1. Currents, Forces and Onsager Relation . 95 Contents ix 10.4.2. Transport Coefficient . 96 10.5. Survey of Recent Developments . 97 10.5.1. Heat Conduction in Classical Systems . 98 10.5.2. Experimental Investigations . 99 10.5.3. Quantum Mechanical Approaches . 99 11.Heat Current 101 11.1. Current Operator . 102 11.2. Conserved Quantities . 104 11.3. Quasiparticle Interpretation . 105 11.3.1. Harmonic Chain . 106 11.3.2. Heisenberg Chain . 107 12.An Open System Approach to Heat Conduction 111 12.1. Heat Transport in Low Dimensional Systems . 111 12.2. Fourier's Law for a Heisenberg Chain . 114 12.3. Scaling Behavior . 116 13.Perturbation Theory in Liouville Space 117 13.1. Super Operators . 117 13.2. Unperturbed System . 119 13.3. Local Equilibrium State . 120 13.4. Standard Kubo-Formula . 122 13.5. Kubo-Formula in Liouville Space . 125 13.6. Heat Transport Coefficient . 127 13.7. Heat Conductivity of a Model System . 129 14.Reservoir Perturbation Theory 131 14.1. Perturbation Theory . 131 14.2. Dependence on the Bath Coupling . 132 14.3. Influence of the Bath Coupling . 135 15.Quantum Thermodynamic Approach to Heat Conduction 139 15.1. Mesoscopic Model . 141 15.2. Dyson Time-Evolution Expansion . 143 15.3. HAM for Transport Scenario . 145 16.Diffusive Behavior from Schr¨odinger Dynamics 149 16.1. Decay Behavior of a Model System . 149 16.2. Fluctuations and Size Effects . 151 16.3. Initial States . 153 x Contents 17.Transport Coefficients 155 17.1. Energy Transport . 155 17.2. Heat Transport . 157 17.2.1. Thermal Initial State . 157 17.2.2. Current . 161 17.2.3. Heat Conductivity . 162 17.3. Connection between Heat and Energy Transport . 165 18.Conclusion and Outlook 167 III. Appendices 171 A. Hilbert Space Average of Expectation Values 173 B. Hyperspheres 177 B.1. Surface of a Hypersphere . 177 B.2. Integration of a Function on a Hypersphere . 179 C. Hilbert Space Averages 181 C.1. Definition . 181 C.2. Hilbert Space Average of Expectation Values . 182 C.3. Special Hilbert-Space Averages . 183 D. Hilbert-Space Variance 185 E. German Summary { Deutsche Zusammenfassung 187 E.1. Zerfall ins Gleichgewicht . 188 E.1.1. Offene Quantensysteme . 188 E.1.2. Hilbertraummittelmethode . 189 E.2. Transport . 191 E.2.1. Badkopplung . 193 E.2.2. Hilbertraummittelmethode . 194 E.3. Fazit . 196 List of Symbols 197 Bibliography 201 List of Previous Publications 213 Index 215 1. Introduction Eine Theorie ist desto eindrucksvoller, je gr¨oßer die Einfachheit ihrer Pr¨amissen ist, je verschiedenartigere Dinge sie verknupft¨ und je weiter ihr Anwendungsbereich ist. Deshalb der tiefe Eindruck, den die klassische Thermodynamik auf mich machte. Es ist die einzige physikalische Theorie allgemeinen Inhalts, von der ich ub¨ erzeugt bin, daß sie im Rahmen der Anwendbarkeit ihrer Grundbegriffe niemals umgestoßen werden wird (zur besonderen Beachtung der grunds¨atzlichen Skeptiker). A. Einstein [129] Phenomenological thermodynamics and statistical mechanics are part of the most impressive and far-reaching theories of modern physics: Their implica- tions reach from central technical devices of the contemporary human society, like heat engines and refrigerators etc. to recent physics at almost all length scales, from Bose-Einstein-condensates and superconductors to black holes. After a phenomenological introduction by some early scientists (Celsius, Fahrenheit, etc.) of quantities from our everyday experiences like temperature and pressure, the whole picture dramatically changed with Joule's calculation of the heat equivalent in 1840. This celebrated investigation showed that heat is nothing else but energy. Joule thus opened a connection between early thermodynamics and (by this time already advanced) classical Hamiltonian mechanics. In 1866 Boltzmann was able to reduce thermodynamics entirely to classical mechanics by identifying the so far phenomenological entropy with the volume of a certain region in phase space [7]. Finally, this conjecture led to our modern understanding of thermodynamics. Besides Bolzmann, such famous physicists as Gibbs [43], Ehrenfest [25], Birkhoff [5] and von Neumann [110] tried to prove the celebrated postulate, but did not succeed without using some further assumptions like ergodicity, quasi-ergodicity, molecular chaos etc. which could neither be proven, in general. Nevertheless, thermodynamics 1 PSfrag replacements 2 1. Introduction Quantum Mechanics Classical Mechanics of composite systems classical of many particles limit dρ^ q_i = fqi; Hg i~ = [H^ ; ρ^] dt p_i = fpi; Hg Quantum Thermodynamics Gibbsian Ensemble Theory (Partitioning) (Ergodicity) Nano- Thermodynamics thermo- of large systems dynamics dU = T dS − pdV of small dS > 0 systems Figure 1.1.: Foundation of thermodynamics on quantum mechanics (left) or classical mechanics (right), respectively.
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