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Cavity Quantum Electrodynamics with Ultracold Atoms

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Cavity Quantum Electrodynamics with Ultracold Atoms Cavity Quantum Electrodynamics with Ultracold Atoms Dissertation Tesis doctoral zur Erlangung des Grades des Doktors der Naturwissenschaften del Departament de Física der Naturwissenschaftlich-Technischen Fakultät II - Physik und Mechatronik - de la Universitat Autònoma de Barcelona der Universität des Saarlandes von por Hessam HABIBIAN Saarbrücken, Barcelona 2013 Universität des Saarlandes Universitat Autònoma de Barcelona Theoretische Physik Departament de Física Betreuerin der Doktorarbeit: Director de la Tesis: Prof. Dr. Giovanna MORIGI Prof. Dr. Ramón CORBALÁN YUSTE Cavity Quantum Electrodynamics with Ultracold Atoms Dissertation Tesis doctoral zur Erlangung des Grades des Doktors der Naturwissenschaften del Departament de Física der Naturwissenschaftlich-Technischen Fakultät II - Physik und Mechatronik - de la Universitat Autònoma de Barcelona der Universität des Saarlandes von por Hessam HABIBIAN This thesis is dedicated to my kind-hearted father, open-armed mother, and my lovely patient wife Naeimeh Behbood... Abstract In this thesis we investigate the interactions between ultracold atoms confined by a periodic potential and a mode of a high-finesse optical cavity whose wavelength is incommensurate with the potential periodicity. The atoms are driven by a probe laser and can scatter photons into the cavity field. When the von-Laue condition is not satisfied, there is no coherent emission into the cavity mode. We consider this situation and identify conditions for which different nonlinear optical processes can occur. We characterize the properties of the light when the system can either operate as a degenerate parametric amplifier or as a source of antibunched light. Moreover, we show that the stationary entanglement between the light and spin- wave modes of the array can be generated. In the second part we consider the regime in which the zero-point motions of the atoms become relevant in the dynamics of atom-photon interactions. Numerical calculations show that for large parameter regions, cavity backaction forces the atoms into clusters with a local checkerboard density distribution. The clusters are phase-locked to one another so as to maximize the number of intracavity photons. Zusammenfassung Die vorliegende Arbeit befasst sich mit der Wechselwirkung ultrakalter Atome mit der Mode eines optischen Resonators hoher Güte. Die Atome sind dabei in einem periodischen Potenzial gefangen, dessen Periodizität nicht kommensurabel mit der Wellenlänge des Resonators ist. Ein Laser regt die Atome an und sie streuen Pho- tonen in die Resonatormode, wobei die Emission inkohärent ist, falls die Laue- Bedingung nicht erfüllt ist. Dieser Fall wird betrachtet und es werden Bedingungen ermittelt, für welche nichtlineare optische Prozesse auftreten können. Die Eigen- schaften des Lichtes werden untersucht, wenn sich das System entweder wie ein parametrischer Verstärker verhält oder wie eine Lichtquelle mit "Antibunching"- Statistik. Weiterhin kann eine stationäre Verschränkung zwischen Licht und Spin- wellen der Atome erzeugt werden. Im zweiten Teil wird die Situation betrachtet, in der die Nullpunktsbewegung der Atome für die Atom-Licht-Wechselwirkung rel- evant ist. Für große Parameterbereiche zeigen numerische Berechnungen, dass die Rückwirkung des Resonators die Formierung eines lokalen Schachbrettmusters in der atomaren Dichteverteilung erzeugt. Die einzelnen Atomgruppe dieses Musters stehen zueinander in fester Phasenbeziehung, was zur Erhöhung der Zahl der Res- onatorphotonen führt. Contents Introduction 1 1 Atom-photon interactions inside a cavity: Basics 5 1.1 Coherent dynamics of an atom coupled to a cavity field . 6 1.1.1 The cavity field ...................... 7 1.1.2 Atom-cavity field interaction: Jaynes-Cumming model 7 1.1.3 An external pump: a laser driving the atoms . 8 1.2 Dissipative dynamics ....................... 9 1.3 The system of this thesis: an atomic array in a cavity . 11 Part I: Pointlike atoms in a periodic array inside a cavity 15 2 Quantum light by an atomic array in a cavity 17 2.1 Some properties of nonclassical light .............. 18 2.2 Atomic array in a cavity: effective dynamics .......... 22 2.2.1 Weak excitation limit .................. 23 2.2.2 Linear response: polaritonic modes ........... 26 2.2.3 Effective Hamiltonian .................. 27 2.2.4 Discussion ......................... 29 2.2.5 Cavity input-output formalism ............. 31 2.3 Results ............................... 33 2.4 Summary and outlook ...................... 37 3 Two-mode squeezing by an atomic array in a cavity 41 3.1 Non-degenerate parametric amplifier and Entanglement . 42 3.2 Parametric amplifier based on an atomic array in a cavity . 43 3.3 Results: stationary entanglement between matter and light . 47 3.4 Summary and outlook ...................... 56 i ii CONTENTS Part II: Quantum ground state of atoms due to cavity backaction 59 4 Quantum ground state of ultracold atoms in a cavity 61 4.1 Bose-Hubbard model and disorder ............... 62 4.2 Trapped atoms in a cavity: effective dynamics ......... 67 4.2.1 Coherent dynamics .................... 67 4.2.2 Heisenberg-Langevin equation and weak-excitation limit 70 4.2.3 Adiabatic elimination of the cavity field . 72 4.2.4 Effective Bose-Hubbard Hamiltonian .......... 73 4.2.5 Discussion ......................... 77 4.3 Results ............................... 78 4.3.1 One-dimensional lattice ................. 79 4.3.2 Two-dimensional lattice ................. 85 4.4 Experimental Parameters .................... 91 4.5 Anderson glass .......................... 95 4.6 Summary and outlook ...................... 96 5 Concluding remarks 101 Appendices 103 A Derivation of the effective Hamiltonian (2.40) 105 B Positivity 107 C Gaussian dynamics 108 D Covariance matrix and logarithmic negativity 110 E Two-mode squeezing spectrum of the emitted field 112 E.1 Spectral properties of the emitted field .............112 E.2 Measurement of the squeezing spectrum ............114 F Wannier function for a periodic potential 117 Bibliography 135 Introduction Ultracold atoms in cavity quantum electrodynamics (CQED) setups offer possibilities to investigate basic processes in the interaction of atoms and electromagnetic fields [1–3]. For example, Rabi oscillations with a single photon are observed in a so-called strong coupling regime in which atom and cavity can exchange a single photon many times before the photon is lost from the cavity by dissipative processes. A high-finesse cavity mode inter- acting with ultracold atoms may enhance Bragg scattering of light into one spatial direction, and increase the collection efficiency and thereby suppress a diffusion related to photon scattering [3,4]. In this thesis, we focus on vari- ous physical phenomena emerging from scattering of light into a high-finesse cavity mode, in particular, we study quantum properties of a light emit- ted outside the cavity, an stationary entanglement between the scattered light and a collective excitation mode of the atoms, and quantum ground state properties of the medium when the light scattering into the cavity is enhanced. Bragg diffraction of light by ultracold atoms in optical lattices may reveal the microscopic crystalline structures of the medium [5,6]. For a regular array of the atoms, at the solid angles for which the von-Laue condition is not satisfied [5], the light is scattered inelastically [7–10]. It has been shown that in this case the scattered light in far field can exhibit vacuum squeezing [10]. The nonlinear response of the atomic medium can be enhanced when the atoms of the array strongly interact with a mode of a high-finesse cavity. In this case, the nonlinearity of the light can be controlled by the angle between laser and the cavity fields wave-vectors and by the intra-atomic distance. When the geometry of the setup is such that the von-Laue condition is not satisfied, photons can only be inelastically scattered into the cavity mode. The smaller system size for which coherent scattering is suppressed, is found 1 2 Introduction for two atoms inside the resonator. The properties of the light at the cavity output for this specific case have been studied in Refs. [11, 12]. To the best of our knowledge, however, the scaling of the dynamics with the number of atoms N is still largely unexplored in this regime. In Chapter 2 of this thesis, we characterize the coherence properties of the light at the cavity output when the light is scattered from a laser into the resonator by a periodic array of atoms and the geometry of the system is such that coherent scattering is suppressed. For the phase-matching conditions, at which in free space the light is in a squeezed-vacuum state [10], we find that inside a resonator and at large N the system behaves as an optical parametric oscillator, which in certain regimes can operate above threshold [13]. For a small number of atoms N, on the contrary, the medium can act as a source of antibunched light. In this case it can either behave as single-photon or, for the saturation parameters here considered, two-photon “gateway” [14]. The latter behaviour is found for a specific phase-matching condition. We identify the parameter regimes which allow one to control the specific nonlinear optical response of the medium. Following the famous gedanken experiment by Einstein, Podolsky, and Rosen (EPR) in 1935 [15] on the completeness of quantum mechanics, it has been realized by Schrödinger [16, 17] that the EPR paradox was closely related to the concept of entanglement. A realization of the EPR
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