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Casimir-Polder Interaction in Second Quantization

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Casimir-Polder Interaction in Second Quantization Universitat¨ Potsdam Institut fur¨ Physik und Astronomie Casimir-Polder interaction in second quantization Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) in der Wissen- schaftsdisziplin theoretische Physik. Eingereicht am 21. Marz¨ 2011 an der Mathematisch – Naturwissen- schaftlichen Fakultat¨ der Universitat¨ Potsdam von Jurgen¨ Schiefele Betreuer: PD Dr. Carsten Henkel This work is licensed under a Creative Commons License: Attribution - Noncommercial - Share Alike 3.0 Germany To view a copy of this license visit http://creativecommons.org/licenses/by-nc-sa/3.0/de/ Published online at the Institutional Repository of the University of Potsdam: URL http://opus.kobv.de/ubp/volltexte/2011/5417/ URN urn:nbn:de:kobv:517-opus-54171 http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-54171 Abstract The Casimir-Polder interaction between a single neutral atom and a nearby surface, arising from the (quantum and thermal) fluctuations of the electromagnetic field, is a cornerstone of cavity quantum electrodynamics (cQED), and theoretically well established. Recently, Bose-Einstein condensates (BECs) of ultracold atoms have been used to test the predic- tions of cQED. The purpose of the present thesis is to upgrade single-atom cQED with the many-body theory needed to describe trapped atomic BECs. Tools and methods are developed in a second-quantized picture that treats atom and photon fields on the same footing. We formulate a diagrammatic expansion using correlation functions for both the electromagnetic field and the atomic system. The formalism is applied to investigate, for BECs trapped near surfaces, dispersion interactions of the van der Waals-Casimir-Polder type, and the Bosonic stimulation in spontaneous decay of excited atomic states. We also discuss a phononic Casimir effect, which arises from the quantum fluctuations in an interacting BEC. Zusammenfassung Die durch (quantenmechanische und thermische) Fluktuationen des elektromagnetischen Feldes hervorgerufene Casimir-Polder-Wechselwirkung zwischen einem elektrisch neu- tralen Atom und einer benachbarten Oberflache¨ stellt einen theoretisch gut untersuchten Aspekt der Resonator-Quantenelektrodynamik (cavity quantum electrodynamics, cQED) dar. Seit kurzem werden atomare Bose-Einstein-Kondensate (BECs) verwendet, um die theoretischen Vorhersagen der cQED zu uberpr¨ ufen.¨ Das Ziel der vorliegenden Arbeit ist es, die bestehende cQED Theorie fur¨ einzelne Atome mit den Techniken der Vielteilchen- physik zur Beschreibung von BECs zu verbinden. Es werden Werkzeuge und Methoden entwickelt, um sowohl Photon- als auch Atom-Felder gleichwertig in zweiter Quantisie- rung zu beschreiben. Wir formulieren eine diagrammatische Storungstheorie,¨ die Korrela- tionsfunktionen des elektromagnetischen Feldes und des Atomsystems benutzt. Der Formalismus wird anschließend verwendet, um fur¨ in Fallen nahe einer Ober- flache¨ gehaltene BECs Atom-Oberflachen-Wechselwirkungen¨ vom Casimir-Polder-Typ und die bosonische Stimulation des spontanen Zerfalls angeregter Atome zu untersuchen. Außerdem untersuchen wir einen phononischen Casimir-Effekt, der durch die quantenme- chanischen Fluktuationen in einem wechselwirkenden BEC entsteht. i I would like to thank my supervisor PD Dr. Carsten Henkel for the countless hours of illuminating discussions and explanations during the last three years. I am also indebted to the members of the quantum optics group at Potsdam, most notably Harald Haakh and Francesco Intravaia, for sharing their knowledge of physics and providing help and encouragement; and to Vanik Mkrtchian and Diego Dalvit for helpful discussions. The project received financial support of Deutsche Forschungsgemeinschaft (DFG). For support in all the aspects of life not directly related to theoretical physics, I would like to thank my family, Dagmara, Ewa, Ryszard, Tadeusz and, above all, Konstancja and Lew. iii Contents Abstract . i Zusammenfassung . i Acknowledgments . iii 1 Introduction 1 The Casimir effect, cavity quantum electrodynamics, and Bose-Einstein condensates 2 Weakly interacting, dilute BECs 7 2.1 Field operators and the Bogoliubov approximation . 8 2.2 Critical temperature and long-range order in an ideal Bose gas . 9 2.3 Effective Hamiltonian for interacting atomic BECs . 12 2.4 Perturbative treatment of weak interactions . 13 2.5 The interacting uniform system . 15 2.6 Trapped gases . 18 3 Phononic Casimir effect 23 3.1 Free energy in a weakly interacting, dilute BEC at T = 0 . 25 3.2 The Casimir energy as an integral over a mode density . 26 3.3 Large distance expansion . 29 3.4 A formal analogy to finite temperature systems . 30 3.5 The non-interacting limit . 32 3.6 Summary . 33 4 Quantum field theory of the CP interaction 35 4.1 Perturbative framework . 36 4.2 Atom-field coupling . 37 4.3 Single atom self energies . 38 4.4 Correlation functions of the electric field . 42 4.5 Attractive atom-surface potential . 43 5 Casimir-Polder force on a BEC 47 5.1 Second-order energy shift . 47 v CONTENTS 5.2 Ideal Bose gas in a surface trap . 48 5.3 Interacting Bose gas near a surface . 51 5.4 Summary . 56 6 Bosonic enhancement of spontaneous emission 59 6.1 Decay of an excited wavepacket . 60 6.2 Bose enhancement near a surface . 64 6.3 Virtual excited atoms produced by laser absorption . 67 6.4 Summary . 69 7 Self-energy processes in higher orders 71 7.1 Single atom processes . 71 7.2 Two atom processes . 73 7.3 Generalization to many-body systems . 75 7.4 Perspectives for future work . 77 8 Summary 79 Appendices 81 A Correlation functions for the electric field 83 A.1 Integral identities in the complex frequency plane . 83 A.2 Explicit expressions for Gαβ .................... 84 B Mathematical details 87 B.1 One-body correlation function for an ideal BEC in a harmonic trap 87 B.2 Approximating the error function integral . 87 C Feynman rules for pointlike atoms 89 Bibliography 93 vi CHAPTER 1 Introduction The Casimir effect, cavity quantum electrodynamics, and Bose-Einstein condensates The name of the Dutch physicist Hendrik Casimir is associated with several phe- nomena where detectable forces between macroscopic bodies or single atoms arise from a –quoting Casimir and Polder (1948)– ‘somewhat academic exercise in quantum electrodynamics’. In its strict sense, the term ‘Casimir effect’ refers to Casimir’s prediction of an attractive force acting between two neutral, electrically conducting plates placed in the vacuum (Casimir, 1948). From the standpoint of classical electromagnetism, there is no interaction between the two neutral plates, but Casimir took into ac- count the vacuum-fluctuations of the electromagnetic (em) field. The plates im- pose boundary conditions on the em field, and the spectrum of the vacuum energy contained in the resonator volume between the plates changes upon varying their separation L. Casimir found a resulting force which varies as L−4, at a distance of 1 µm, two plates of area 1 mm2 attract each other with a force of 10−9 N. An early attempt by Sparnaay (1958) to experimentally verify this effect did not yield con- clusive results, but a series of precision experiments in the late 1990s, beginning with Lamoreaux (1997), could establish the Casimir force as a direct manifestation of the electromagnetic zero-point energy.1 For Casimir’s argument, it is, however, not essential that the fluctuating medium is the em field in its vacuum state. Mehra (1967) showed that the effect of ther- mal fluctuations of the em field on the interaction of the plates actually dominates the quantum Casimir force for large plate separations. At room temperature, this thermal contribution to the Casimir effect changes the distance-dependence of the 1For a comprehensive review of the Casimir effect, see for example Bordag et al. (2001); Lam- oreaux (2005); Milton (2001); Mostepanenko and Trunov (1997). 1 1. INTRODUCTION 3 force into a kBT=L power-law for distances larger than 3 µm, an effect observed in the recent experiment by Sushkov et al. (2011). The so-called critical Casimir effect does not deal with the em field at all, but describes forces that arise in adsorbed films of liquids near a second order phase transition. The fluctuating quantity is the order parameter, the resulting forces are observable in the thickness of the film (see Gambassi et al. (2009) and the references therein). Two other effects involving neutral atoms with electric dipole moments were explained within the framework of quantum electrodynamics by Casimir and Polder in their 1948 seminal paper: The Casimir-Polder-van der Waals (CPvdW) interac- tion between two atoms, and the Casimir-Polder (CP) interaction between an atom and a macroscopic body. (For simplicity, we will only consider the case of the macroscopic body being a plane, infinitely extended surface.) As in the well- known Lamb shift in atomic hydrogen (Bethe, 1947), the energy level structure of an atom is disturbed by the zero-point fluctuations of the em field. If the presence of an additional atom or a surface alters the spectrum of the zero-point fluctua- tions, the first atom’s energy shift depends on the inter-atomic distance or the atom surface distance, respectively, resulting in an interaction potential. For two atoms in their internal ground state (separated by a distance r), Casimir and Polder found an attractive CPvdW-potential which behaves as r−6 for inter- atomic distances smaller than typical atomic transition wavelengths from the ground state, and as r−7 for larger distances. Historically, the r−6 potential had al- ready been found in 1930 by Eisenschitz and London2 in an attempt to quantum- mechanically explain the weak intermolecular forces which were introduced heuris- tically half a century earlier by van der Waals (see Maxwell, 1874) in his equation of state. While London’s calculation is based on the interaction energy of fluctu- ating dipole moments, Casimir focused on the local action of fields at the position of the atoms. This includes the effect of retardation due to the finite speed of light, which results in the modification of the power-law behavior at large distances.
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