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Identical Emitters, Collective Effects and Dissipation in Quantum Optics

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Identical Emitters, Collective Effects and Dissipation in Quantum Optics Identical Emitters, Collective Efects and Dissipation in Quantum Optics Novel numerical approaches for quantum master equations vorgelegt von M.Sc. Michael Gegg geb. in München von der Fakultät II – Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften – Dr.rer.nat. – genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Thomas Möller Gutachter: Prof. Dr. Andreas Knorr Gutachter: Prof. Dr. Jan Wiersig Gutachter: Dr. Marten Richter Tag der wissenschaftlichen Aussprache: 6. Oktober 2017 Berlin 2017 ii iii Abstract In this thesis a formalism for indistinguishable multi-level quantum emitters in quantum master equations is developed. The complexity of the approach scales only polynomially in the number of quantum emitters. Complexity here means the number of coupled equations or rather the dimension of the Liouville space. This approach opens new possibilities for calculating open quantum systems. The method is implemented in the PsiQuaSP library, which allows to setup and solve arbitrary master equations, in particular master equations with the reduced polyno- mial scaling. The introduced tools are utilized to study subradiance in the open Dicke model and various cQED lasers. One of the main current research goals in quantum optics and quantum information science is the generation and stabilization of quantum coherence between distinct quantum systems. These systems can for instance be photons or quantum emitters, qubits. From a theoretical stance the difculty with these systems is that their complexity scales in general exponentially with the system size, like the number of quantum emitters. The reduced scaling of the formalism introduced in this thesis stems from the permutation symmetry of the indistinguishable emit- ters. The method is derived in the context of Lindblad quantum master equations. Lindblad quantum master equations for many identical multi-level systems have been used to study funda- mental quantum optical systems, such as lasing and laser like action, various phase transitions, optical bistability, cooperative resonance fuorescence, entanglement, quantum light generation, quantum to classical transitions, super- and subradiance etc. In most of these cases the single emitter, the many emitter and the weak correlation limit can be satisfactorily treated with ex- isting techniques, such as direct integration, phase space methods or cluster expansion. However these techniques are not well suited for the few emitter case with strong correlations. The the- ory developed in this thesis allows to study all these systems for moderate numbers of quantum emitters and arbitrary correlation strengths, thus flling the gap left open by conventional meth- ods. Even though the theory is developed around Lindblad quantum master equations it is not limited to this case – the requirement of indistinguishability is used to construct symmetrized Liouville space basis states and operators which can also be used to construct other master equations. In this thesis the general methodology for the polynomial scaling of the many emitter master equation is derived by two diferent approaches. Generalized permutation symmetric Liouville space states and operators are introduced that allow for the construction of arbitrary mas- ter equations and observables. These Liouville space states and operators are translated into a graphical representation that greatly facilitates their usage. This representation is found to have a close connection to Lie algebras as well as graph theory. The whole method is implemented in a general and modular way in a C++ library called PsiQuaSP, which allows to simulate arbitrary master equations based on a number state representation. The design of the library heavily relies on the introduced graphical representation, which greatly facilitates the setup of a simulation. The treatment of non-identical systems with PsiQuaSP is illustrated and further techniques for complexity reduction and computational speedup are discussed. These tools are applied to the two main cases of cQED lasers/spasers and super-, subradiance. In the latter case a new type of phase transition in the open Dicke model is predicted that leads to a deterministic generation of subradiant steady state coherences, with applications in quantum information science. Furthermore the interaction of quantum dots with the dissipative modes at metal interfaces is investigated in order to explain experimental fndings. iv v Zusammenfassung In dieser Arbeit wird ein Formalismus für ununterscheidbare Multi-Niveau Quantenemitter in Quanten Mastergleichungen entwickelt. Die Komplexität dieses Ansatzes skaliert nur poly- nomiell und nicht exponentiell in der Anzahl der Emitter. Komplexität bedeutet hier die Anzahl der gekoppelten Gleichungen bzw. die Dimension des Liouville Raumes. Dieser Ansatz erlaubt neuartige Betrachtungen in ofenen, quantenoptischen Systemen. Die Methode wurde in der Psi- QuaSP Programmbibliothek implementiert. Diese erlaubt es generelle Quanten-Mastergleichungen numerisch zu lösen, insbesondere Mastergleichungen mit der reduzierten polynomiellen Skalierung. Mit Hilfe der eingefürten Methoden werden Subradianz im ofenen Dicke Modell und ver- schiedene cQED Laser untersucht. Eines der zentralen, aktuellen Forschungsziele in der Quantenoptik und der Quanteninforma- tions Technologie ist die Generierung und Stabilisierung von Quantenkorrelationen zwischen unterschiedlichen Quantensystemen. Beispiele für solche Quantensysteme sind Photonen oder Quantenemitter, Qubits. Aus theoretischer Sicht liegt die Schwierigkeit mit diesen Systemen in der Tatsache, dass im Allgemeinen ihre Komplexität exponentiell mit der Systemgröße, wie etwa der Anzahl der Quantenemitter, skaliert. Die Reduzierung der Komplexität in dem hier einge- fürten Formalismus resultiert aus der Vertauschungssymmetrie der ununterscheidbaren Emit- ter. Die Methode wird im Kontext von Lindblad Quanten-Mastergleichungen entwickelt. Lind- blad Quanten-Mastergleichungen für identische Multi-Niveau Systeme werden verwendet um fundamentale quantenoptische Systeme zu untersuchen, wie etwa Lasing und laserartiges Ver- halten, verschiedene Phasenübergänge, optische Bistabilität, Kooperative Resonanzfuoreszenz, Verschränkung, Quantenlicht, Übergang von der Quanten- zur klassischen Physik und Super-, Subradianz. In den meisten dieser Fälle können die Grenzfälle von einzelnen Emittern, vie- len Emittern und schwachen Korrelationen mit bestehenden Methoden zufriedenstellend gelöst werden. Diese Methoden sind etwa direkte Integration, Phasenraummethoden oder Cluster Entwicklung. Diese Methoden sind jedoch nicht geeignet für den Fall von mittleren Emitter- Anzahlen mit starken Korrelationen. Der hier entwickelte Formalismus erlaubt es all diese Sys- teme mit mittleren Emitter-Anzahlen und beliebigen Korrelationsstärken zu berechnen. Somit füllt der Formalismus die Lücke welche von herkömmlichen Methoden zurückgelassen wird. In dieser Arbeit wird der Formalismus anhand von zwei verschiedenen Betrachtungsweisen en- twickelt. Generelle Liouville Raum Operatoren und symmetrisierte Liouville Raum Basiszustände werden eingeführt mit denen beliebige Mastergleichungen und Observablen konstruiert wer- den können. Diese Zustände und Operatoren werden in eine graphische Darstellung übersetzt welche die Benutzung stark vereinfacht. Die graphische Darstellung hat Verknüpfungen zu Lie Algebren und der Graphentheorie. Der gesamte Formalismus ist in der Bibliothek PsiQuaSP implementiert, welche Simulationsrechnungen für beliebige Mastergleichungen auf Basis von Nummernzuständen erlaubt. Die Bibliothek basiert auf der graphischen Darstellung, was das Implementieren von Simulationen stark vereinfacht. Des weiteren werden die Behandlung von nicht-identischen Emittern in PsiQuaSP sowie zusätzliche Methoden zur Reduzierung der Kom- plexität vorgestellt. Der Formalismus und die Bibliothek werden auf die zwei Gebiete cQED Laser und Super-, Subradianz angewendet. Im letzteren Fall wird ein neuartiger Phasenüber- gang im ofenen Dicke Modell vorhergesagt, der zur Ausbildung von subradianten Kohärenzen im stationären Zustand führt. Des weiteren wird die Wechselwirkung von einzelnen Quantenpunkten mit den dissipativen op- tischen Moden an Metallgrenzfächen untersucht um experimentelle Befunde zu erklären. vi Contents 1 Introduction 1 I Theoretical Background 5 2 Model systems 7 2.1 Quantum emitters . .8 2.1.1 Two-level Systems . .8 2.1.2 Three and more levels . 10 2.2 Electromagnetic modes in quantum optics . 11 2.2.1 Electromagnetic modes in vacuum . 11 2.2.2 Dissipative modes – Quantum optics in media . 12 3 Open system dynamics 15 3.1 Density matrix . 15 3.2 Closed quantum systems - von Neumann equation . 16 3.3 Open quantum systems - Lindblad equation . 17 3.4 Steady states and the Liouvillian spectrum . 19 II Formalism 21 4 Many emitters – Complex quantum systems 23 4.1 Identical two-level emitters . 24 4.1.1 The product state basis . 25 4.1.2 The Dicke basis . 25 vii viii CONTENTS 4.2 Identical multi-level systems . 28 4.2.1 Product state basis . 29 4.2.2 The Lie algebras su(d) ............................. 30 4.2.3 Dicke states for multi-level systems – su(d) multiplets . 31 4.3 Conclusion . 32 5 Permutation symmetry in quantum master equations 33 5.1 Symmetries, size and averages . 33 5.2 Dissipation vs. symmetry . 35 5.3 Time evolution of identical multi-level systems . 37 5.3.1 Time evolution of a simple master
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