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Out-Of-Equilibrium Dynamics of Open Quantum Many-Body Systems

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Out-Of-Equilibrium Dynamics of Open Quantum Many-Body Systems Out-of-equilibrium dynamics of open quantum many-body systems vorgelegt von Leon Janek Droenner, M. Sc. geb. in Achim 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. Michael Kneissl, TU Berlin 1. Gutachter: Prof. Dr. Andreas Knorr, TU Berlin 2. Gutachter: Prof. Dr. Peter Rabl, TU Wien Tag der wissenschaftlichen Aussprache: 17.12.2018 Berlin, 2019 Abstract Many-body localization, which prevents a many-body quantum systems from reaching thermal equilibrium has opened the possibility to study out-of-equilibrium dynamics of isolated quantum systems, leading to new phases of matter such as the discrete time crystal. However, little is known about these effects when the quantum many-body system is coupled to an external environment. An intuitive assumption is that out-of-equilibrium dynamics and quantum coherences would not survive due to thermalization with external degrees of freedom. The present work focuses on open quantum many-body systems and aims at the question how an external reservoir can preserve such out-of-equilibrium effects. In the first part, common generic reservoirs are assumed to focus on many-body effects of the systems of interest. As a first many-body system, an optically driven many-emitter phonon laser is investigated. The full quantum mechanical treatment reveals additional resonances caused by collective effects of the many-emitter setup. Optically addressing these resonances results in an enhancement of phonon intensities within the acoustic cavity. The Heisenberg spin-chain has achieved the role of a standard model to study many-body localization. As a second many-body system, the model is generalized to an open quantum system via boundary reservoirs which are driving the system out-of equilibrium and a spin-current is induced. When driven far from equilibrium, the transport is very much dependent on the external reservoirs. In contrast, it is demonstrated that a setup with long-range coupling between the spins shows transport behavior independent of the external environment. In the second part of this work, the coupling to the reservoir is structured by assuming a boundary condition. Experimentally this is achieved by placing a distant mirror close to the system. A common Born-Markov approximation leading to factorizing system and reservoir states is insufficient to describe these feedback dynamics. In this work, an approach is used to treat the reservoir states as part of the many-body system which is based on matrix product states using the quantum stochastic Schrödinger equation. Focusing on the reservoir statistics of a single two-level system, it is demonstrated that by combining time-dependent optical excitation with time-delayed feedback via structuring the reservoir, individual photon probabilities are controlled by changing external degrees of freedom. Finally, this method is generalized to a many-body system including a coupling to an external reservoir beyond Born-Markov approximation. It is shown that feedback dynamics stabilize a discrete time crystal against dissipation to the environment. Out-of-equilibrium dynamics and many-body localization of the discrete time crystal survive for long times even in the presence of an external environment and become independent of the coupling to the reservoir. The developed method benefits from small entanglement entropy of the many-body localized system as well as from a suppression of entanglement between system and reservoir due to the feedback dynamics. This allows to consider up to forty spins with individual structured reservoirs in a numerically exact manner. III Kurzfassung Das Verständnis von Lokalisierung in Vielteilchensystemen unter Berücksichtigung von Wechselwirkungen hat neue Möglichkeiten eröffnet, insbesondere die Erforschung von Nichtgleichgewichtsdynamiken in isolierten Quantensystemen. Dies hat zur Entdeckung neuer Nichtgleichgewichtsphasen, wie zum Beispiel den diskreten Zeitkristall, geführt. Wenn das Vielteilchensystem mit einem externen Reservoir wechselwirkt, ist wenig bekannt darüber, ob etwaige Nichtgleichgewichtszustände erhalten bleiben. Eine intuitive Annahme wäre, dass Nichtgleichgewichtszustände und Quantenkohärenzen durch Thermalisierung mit externen Freiheitsgraden zerstört werden. Diese Arbeit berücksichtigt offene quanten- mechanische Vielteilchensysteme und zielt darauf ab, Nichtgleichgewichtszustände unter Berücksichtigung von externen Reservoirs zu untersuchen. Der erste Teil dieser Arbeit berücksichtigt einfache allgemeine Reservoirs, um genauer auf Vielteilcheneffekte im jeweiligen System einzugehen. Ein Phononlaser, bestehend aus mehreren Emittern als aktives Medium, wird als erstes Vielteilchensystem untersucht. Eine voll quantenmechanische Beschreibung weist kollektive Effekte auf, welche zusätzliche optisch adressierbare Resonanzen hervorrufen. Durch optis- che Anregung dieser kollektiven Resonanzen, werden die erreichbaren Phonon-Intensitäten innerhalb des akustischen Resonators deutlich verbessert. Die Heisenberg Spin-Kette ist ein bekanntes Modell, welches zur Beschreibung von Vielteilchenlokalisierung herangezogen wird. Als zweites untersuchtes Vielteilchensys- tem wird dieses Modell zu einem offenen Quantensystem verallgemeinert. Durch zwei Nichtgleichgewichts-Reservoirs an den Rändern der Kette wird ein Strom durch Spin-flips induziert. Im starken Nichtgleichgewicht ist der Transport abhängig von den Eigenschaften der externen Reservoirs. Dies ist nicht der Fall bei einer langreichweitigen Wechselwirkung zwischen den Spins, welche den Transport unabhängig von den Reservoir Parametern werden lässt. Im zweiten Teil der Arbeit wird eine Randbedingung in der Kopplung ans Reservoir berücksichtigt. Dies kann experimentell zum Beispiel durch einen Spiegel realisiert werden. Das hat zur Folge, dass eine allgemeine Beschreibung des Reservoirs durch eine Born- Markov-Näherung nicht mehr durchgeführt werden kann, da für Rückkopplungseffekte ein Gedächtnis des Reservoirs angenommen werden muss. In dieser Arbeit wird das Reservoir innerhalb des Vielteilchenproblems beschrieben, basierend auf Matrix-Produkt-Zuständen und der Quantenstochastischen Schrödinger Gleichung. Mit Blick auf die Reservoirstatistik eines einzelnen Zweiniveausystems wird gezeigt, dass durch die Kombination von zeitab- hängiger Anregung mit zeitverzögerter Rückkopplung einzelne Photonwahrscheinlichkeiten durch externe Kontrollparameter manipuliert werden. Im letzten Teil wird diese Methode auf ein Vielteilchensystem verallgemeinert. Es wird gezeigt, dass Rückkopplungsdynamik einen Zeitkristall gegenüber Dissipation ins Reservoir stabilisiert. Nichtgleichgewichtszustände und Vielteilchenlokalisierung bleiben für lange Zeit erhalten, auch mit Ankopplung an externe Reservoirs. Die diskreten Oszillationen V VI des Zeitkristalls werden unabhängig von der Kopplung ans Reservoir. Die entwickelte Methode profitiert sowohl von kleiner Verschränkungsentropie des lokalisierten Vielteilchen- systems, als auch von einer Unterdrückung der Verschränkung mit dem Reservoir durch zeitverzögerte Rückkopplung. Diese Kombination erlaubt eine Untersuchung von bis zu vierzig Spins mit individuellen strukturierten Reservoirs innerhalb einer numerisch exakten Beschreibung. Contents Abstract III KurzfassungV 1. Introduction1 1.1. Motivation . .1 1.2. Structure of the thesis . .3 I. Theoretical background5 2. Basics 7 2.1. Quantum mechanics in the Schrödinger picture . .7 2.2. Spin operator . .8 2.3. Quantization of the Maxwell field . .9 2.3.1. Hamiltonian . .9 2.3.2. Quantization . 10 2.4. Two-level system in a classical electromagnetic field coupled to a continuum 13 2.4.1. Quantization . 15 3. Quantum stochastic Schrödinger equation (QSSE) 17 3.1. Quantum-noise operators . 18 3.2. Itô calculus . 21 3.3. Lindblad form of the reduced density matrix . 23 4. Ergodicity versus many-body localization in closed quantum systems 27 4.1. Quantum thermalization . 28 4.2. Many-body localization . 30 II. Factorized system-reservoir dynamics 33 5. Many-emitter phonon lasing 35 5.1. Model . 35 5.2. Collective phonon processes . 38 5.2.1. Effective Hamiltonian approach . 41 5.2.2. Collective resonances . 45 5.2.3. Two-phonon resonances . 46 5.3. Non-identical emitters . 47 5.4. Quantum yield . 49 5.5. Conclusion . 52 VII Contents VIII 6. Boundary-driven Heisenberg spin-chain 55 6.1. Model . 56 6.2. Characterizing spin-transport . 59 6.2.1. Weak driving . 60 6.2.2. Maximal driving . 61 6.2.3. Reservoir dependency . 62 6.3. Absence of negative differential conductivity . 64 6.4. Effect of disorder . 66 6.5. Conclusion . 70 III. Entangled system-reservoir dynamics 71 7. Introduction to matrix product states 73 7.1. Singular-value decomposition (SVD) . 74 7.2. Diagrammatic tensor representation . 75 7.3. Canonical form of a matrix product state . 77 7.4. Matrix product operators . 80 7.5. Expectation values . 81 8. Feedback controlled two-photon purification 85 8.1. Theoretical model . 87 8.1.1. Higher-order time-evolution operator . 89 8.1.2. Feedback algorithm in the QSSE picture . 91 8.1.3. Computing photon probabilities from the matrix product state . 94 8.2. Controlling photon statistics . 95 8.2.1. Effect of the time-dependent pulse . 95 8.2.2. Effect of time-delayed feedback . 97 8.2.3. Controlling individual photon probabilities . 99 8.3. Conclusion . 100 9. Feedback-stabilized time crystal 103 9.1.
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