Cavity QED with superconductors and its application to the Casimir effect Harald Richard Haakh Diploma thesis Institute of Physics and Astronomy University of Potsdam Second corrected edition June 2009 Supervisor and first assessor Second assessor PD Dr. Carsten Henkel Prof. Dr. Martin Wilkens This work has been accepted as a diploma thesis at the University of Potsdam, Germany in April 2009. The present version features some minor changes with respect to the original one, concerning mostly typographic errors, and does not contain the declaration of authorship. Parts of this work have been presented at • FRISNO-10, Ein Gedi (Israel), February 2009, • DPG Frühlingstagung, Hamburg (Germany), March 2009. Harald R. Haakh Cavity QED with superconductors and its application to the Casimir effect Diploma thesis, University of Potsdam, 2009. Second corrected edition. Contact: [email protected] Published online at the Institutional Repository of the University of Potsdam: URL http://opus.kobv.de/ubp/volltexte/2009/3256/ URN urn:nbn:de:kobv:517-opus-32564 [http://nbn-resolving.org/urn:nbn:de:kobv:517-opus-32564] Zusammenfassung Diese Diplomarbeit untersucht den Casimir-Effekt zwischen normal- und supraleitenden Platten über einen weiten Temperaturbereich, sowie die Casimir-Polder-Wechselwirkung zwischen einem Atom und einer solchen Ober- fläche. Hierzu wurden vorwiegend numerische und asymptotische Rechnungen durchgeführt. Die optischen Eigen- schaften der Oberflächen werden dann aus dielektrischen Funktionen oder optischen Leitfähigkeiten erhalten. Wichtige Modellen werden vorgestellt und insbesondere im Hinblick auf ihre analytischen und kausalen Eigen- schaften untersucht. Es wird vorgestellt, wie sich die Casimir-Energie zwischen zwei normalleitenden Platten berechnen lässt. Frühere Arbeiten über den in allen metallischen Kavitäten vorhandenen Beitrag von Oberflächenplasmonen zur Casimir- Wechselwirkung wurden zum ersten mal auf endliche Temperaturen erweitert. Für Supraleiter wird eine analytis- che Fortsetzung der BCS-Leitfähigkeiten zu rein imaginären Frequenzen, sowohl innerhalb wie außerhalb des schmutzigen Grenzfalles verschwindender mittlerer freier Weglänge vorgestellt. Es wird gezeigt, dass die aus dieser neuen Beschreibung erhaltene freie Casimir-Energie in bestimmten Bereichen der Materialparameter hervorragend mit der im Rahmen des Zwei-Fluid-Modells für den Supraleiter berechneten übereinstimmt. Die Casimir-Entropie einer supraleitenden Kavität erfüllt den Nernstschen Wärmesatz und weist einen charakteristischen Sprung beim Erreichen des supraleitenden Phasenübergangs auf. Diese Effekte treten ebenfalls in der magnetischen Casimir- Polder-Wechselwirkung eines Atoms mit einer supraleitenden Oberfläche auf. Es wird ferner gezeigt, dass die magnetische Dipol-Wechselwirkung eines Atomes mit einem Metall sehr stark von den dissipativen Eigenschaften und insbesondere von den Oberflächenströmen abhängt. Dies führt zu einer starken Unterdrückung der magnetischen Casimir-Polder-Energie bei endlichen Temperaturen und Abständen oberhalb der thermischen Wellenlänge. Die Casimir-Polder-Entropie verletzt in einigen Modellen den Nernstschen Wärmesatz. Ähnliche Effekte werden für den Casimir-Effekt zwischen Platten kontrovers diskutiert. In den entsprechenden elektrischen Dipol-Wechselwirkungen tritt keiner dieser Effekte auf. Die Ergebnisse dieser Arbeit legen nahe, das bekannte Plasma-Modells als Grenzfall eines Supraleiters bei niedri- gen Temperaturen (bekannt als London-Theorie) zu betrachten, statt als Beschreibung eines normales Metalles. Supraleiter bieten die Möglichkeit, die Dissipation der Oberflächenströme in hohem Maße zu steuern. Dies kön- nte einen experimentellen Zugang zu den optischen Eigenschaften von Metallen bei niedrigen Frequenzen er- lauben, die eng mit dem thermischen Casimir-Effekt verknüpft sind. Anders als in entsprechenden Mikrowellen- Experimenten sind hierbei die Energien und Impulse unabhängige Größen. Die Messung der Oberflächenwech- selwirkung zwischen Atomen und Supraleitern ist mit den heute verfügbaren Atomfallen auf Mikrochips möglich und der magnetische Anteil der Wechselwirkung sollte spektroskopischen Techniken zugänglich sein. Abstract This thesis investigates the Casimir effect between plates made of normal and superconducting metals over a broad range of temperatures, as well as the Casimir-Polder interaction of an atom to such a surface. Numerical and asymptotical calculations have been the main tools in order to do so. The optical properties of the surfaces are described by dielectric functions or optical conductivities, which are reviewed for common models and have been analyzed with special weight on distributional properties and causality. The calculation of the Casimir energy between two normally conducting plates (cavity) is reviewed and previous work on the contribution to the Casimir energy due to the surface plasmons, present in all metallic cavities, has been generalized to finite temperatures for the first time. In the field of superconductivity, a new analytical continuation of the BCS conductivity to to purely imaginary frequencies has been obtained both inside and outside the extremely dirty limit of vanishing mean free path. The Casimir free energy calculated from this description was shown to coincide well with the values obtained from the two fluid model of superconductivity in certain regimes of the material parameters. The Casimir entropy in a superconducting cavity fulfills the third law of thermodynamics and features a characteristic discontinuity at the phase transition temperature. These effects were equally encountered in the Casimir-Polder interaction of an atom with a superconducting wall. The magnetic dipole coupling of an atom to a metal was shown to be highly sensible to dissipation and especially to the surface currents. This leads to a strong quenching of the magnetic Casimir-Polder energy at finite temperature. Violations of the third law of thermodynamics are encountered in special models, similar to phenomena in the Casimir-effect between two plates, that are debated controversely. None of these effects occurs in the analog electric dipole interaction. The results of this work suggest to reestablish the well-known plasma model as the low temperature limit of a superconductor as in London theory rather than use it for the description of normal metals. Superconductors offer the opportunity to control the dissipation of surface currents to a great extent. This could be used to access experi- mentally the low frequency optical response of metals, which is strongly connected to the thermal Casimir-effect. Here, differently from corresponding microwave experiments, energy and momentum are independent quantities. A measurement of the total Casimir-Polder interaction of atoms with superconductors seems to be in reach in today’s microchip-based atom-traps and the contribution due to magnetic coupling might be accessed by spectro- scopic techniques. Acknowledgments I would like to thank my supervisor PD Dr. Carsten Henkel who would answer all questions at any time, share his outstanding knowledge of physics, literature and mathematical formalism and never failed to come back to the physically relevant and Dr. Francesco Intravaia, who has become a good friend, helped me with all technical and physics questions and has spend a good share of his time discussing with me. Salvo Spagnolo’s visit to potsdam initiated the work on the Casimir-Polder forces, which - thought first as a small complementary facet - has taken more and more space in this work ever since. Also I want to thank Prof. Giuseppe Bimonte and Simen Ellingsen for sharing their knowledge, research results and numerical tricks. The German-Israeli science foundation (GIF) has funded my participation in the FRISNO 10 symposium and made it possible for me to meet and discuss with Dr. Daniel Rohrlich, Prof. Baruch Horovitz, Prof. Ron Folman and Dr. Valery Dikovsky. During all my studies I was supported by the German national academic foundation (Studien- stiftung des deutschen Volkes). I would also like to mention my friends and fellow students, who took part in the seminar we held on mondays and who had always an open ear and unbiased ideas on any physics problem. The special people who have shared my life during this year, who have supported and accompanied me all the time, stopped me from working too much on the weekends and tolerated if I still did are Ana Sanfrutos Cano and my family. Cavity QED with superconductors and its application to the Casimir effect 1 1.8.3 Reflectivity revisited . 36 1.9 Conclusions . 37 2 Casimir interaction 38 2.1 Casimir interaction . 38 Contents 2.1.1 Vacuum interactions . 38 2.1.2 Vacuum expectation values in bosonic fields . 39 2.1.3 Casimir interaction of ideal Table of contents 1 mirrors . 39 Introduction 2 2.1.4 General boundary conditions . 41 2.1.5 Experiments measuring the 1 Electrodynamics of superconductors 5 Casimir force . 44 1.1 Electrodynamics of solids . 5 2.1.6 The thermal Casimir effect . 44 1.1.1 Basic equations of electrody- 2.1.7 Numerics . 48 namics . 5 2.2 Casimir interaction in the Drude and 1.1.2 Harmonic analysis . 6 plasma model . 50 1.1.3 Electromagnetic waves in 2.2.1 Thermal surface plasmons . 50 media . 6 2.2.2 Plasma and Drude model . 54 1.1.4 Fields at boundaries . 7 2.2.3 Casimir entropy . 57 1.1.5 Electric conductivity . 8 2.2.4 Diamagnetism and diffusive 1.1.6 Temperature dependence of modes . 61 Drude’s relaxation . 9 2.3 Two-fluid superconductors .