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Non-Markovian Open Quantum Dynamics from Dissipative Few-Level Systems to Quantum Thermal Machines Michael Wiedmann 2019

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Non-Markovian Open Quantum Dynamics from Dissipative Few-Level Systems to Quantum Thermal Machines Michael Wiedmann 2019 Non-Markovian Open Quantum Dynamics from Dissipative Few-level Systems to Quantum Thermal Machines kumulative Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult¨atf¨ur Naturwissenschaften der Universit¨atUlm vorgelegt von Michael Wiedmann aus Hambach bei Schweinfurt 2019 Amtierender Dekan: Prof. Dr. Thorsten Bernhardt Erstgutachter: Prof. Dr. Joachim Ankerhold Zweitgutachter: Prof. Dr. Matthias Freyberger Tag der Promotion: 20. Dezember 2019 List of publications This cumulative thesis is composed of the following peer-reviewed publications which will be discussed in chronological order: 2016 Time-correlated blip dynamics of open quantum systems, M. Wiedmann, J. T. Stockburger, J. Ankerhold, Phys. Rev. A, 052137 2018 Rectification of heat currents across nonlinear quantum chains: A versatile approach beyond weak thermal contact, T. Motz, M. Wiedmann, J. T. Stockburger, J. Ankerhold, New J. Phys. 20, 113020 2019 Out-of-equilibrium operation of a quantum heat engine: The cost of thermal coupling control, M. Wiedmann, J. T. Stockburger, J. Ankerhold, submitted: https://arxiv.org/abs/1903.11368 v Contents 1 Introduction1 2 Non-Markovian open quantum systems5 2.1 Caldeira-Leggett model and generalized Langevin equation.......6 2.2 Path integral approach and stochastic unraveling............9 2.3 Ohmic reservoirs with frequency cutoff - the SLED........... 15 2.3.1 Time-dependent reservoir coupling................ 16 2.3.2 Adjoint equation of motion..................... 19 2.3.3 System-reservoir correlations in SLED probability space..... 20 2.3.4 SLED with momentum-coupled z(t)-noise............ 25 2.4 Projection operator techniques....................... 27 3 Finite-memory, stochastic reduced state propagation 31 3.1 Spin-boson system............................. 34 3.2 Dissipative three-level system....................... 39 4 Non-Markovian real-time simulations of quantum heat engines 45 4.1 Driven oscillator with time-dependent thermal contact - SLED formalism 46 4.2 The four-stroke quantum Otto cycle.................... 47 4.2.1 Heat, heat flux and work...................... 49 4.2.2 Squeezing, entropy and coherence dynamics........... 54 5 Time-correlated blip dynamics of open quantum systems 57 5.1 Summary Phys. Rev. A, 052137, (2016)................. 58 5.2 Author's contribution............................ 60 6 Rectification of heat currents across nonlinear quantum chains 73 6.1 Summary New J. Phys. 20, 113020, (2018)................ 74 6.2 Author's contribution............................ 76 7 Out-of-equilibrium operation of a quantum heat engine 99 7.1 Summary arXiv: 1903.11368, (2019).................... 100 7.2 Author's contribution............................ 101 8 Summary and Outlook 117 9 Epilogue 133 Danksagung.................................... 133 vii Contents Curriculum vitae................................. 134 Publications and conference contributions................... 135 Declaration of Academic Honesty........................ 136 viii 1 Introduction As the most fundamental theory to our understanding of matter among all natural sci- ences, quantum mechanics evolved during the last century from a mainly mathematical and theoretical concept, to a comprehensive framework with strong implications for ac- tual experiments that enter the level of individual atoms. It's intrinsically probabilistic character requires a meticulous, statistical interpretation of any quantum measurement or observation that inevitably provokes a consecutive collapse to the observable's eigen- state. The notion of an open quantum system1 emerged from the classical analogue of a realistic system which is inescapably exposed to an uncontrollable environment. Any attempt that aims at a complete microscopic description of such a system-plus-reservoir complex fails due to the sheer abundance of involved degrees of freedom and motivates the demand for techniques that can cope with the reservoir's influences on a quantum system without compromising the dynamics by essential components. A pervasive class of dynamical open system techniques applies to the case of weak environmental interaction and short correlation times. Time-local master equations of Lindblad structure [Kos72b, Kos72a, GKS76, Lin76] satisfy fundamental concepts like trace preservation and complete positivity of the dynamical maps and are omnipresent in the field of quantum optics and information. However, as soon as environmental decay times (thermal timescale ~β) increase compared to the system's relaxation, de- coherence or timescales of external driving, the perturbative character of a description by dynamical semigroups constitutes a major weakness of the procedure. Open systems that evolve subject to strong environmental coupling, low temperatures or entanglement in the initial state are conventionally described in terms of non- Markovian processes [BLPV16, dVA17] and require techniques that take the full reser- voir influence on the respective system into account. While the Lindblad formalism can be applied beyond the Markov assumption, at least within certain boundaries of non- Markovian generalizations [Bre07, CC14, TSHP18], a rigorous treatment of inherently non-Markovian quantum memory processes [Gra06] requires a different approach. An exact formulation of the reservoirs influence action is provided by the Feynman-Vernon influence functional [FV63, GS98]. As there exists no ad-hoc solution to the involved, non-local path integrals, a number of numerical methods adopts an indirect approach of Monte Carlo methods [EM94, MAE04, KA13], discrete time quasiadiabatic tensor state propagation [MM94] or auxiliary density matrices [TK89, Tan90]. Alternative stochastic methods include the diffusion of quantum states [DS97], reservoir decou- pling through stochastic fields [Sha04, YYLS04] and the stochastic unraveling of the 1For a detailed, topical introduction readers may be referred to the seminal books [BP10, Wei12]. 1 1 Introduction influence functional [SM98, SG02, Sto03]. This dissertation comprises three publications that build on the non-perturbative and time-local stochastic Liouville-von Neumann (SLN) framework that is based on the ex- act mapping of the Feynman-Vernon influence to a c-number probability space [SG02]. Its applicability to arbitrary Hamiltonian systems in notoriously challenging parame- ter regimes for strong coupling and low temperatures, without the necessity of a re- derivation of the dissipative dynamics, comes at the cost of an inherently non-unitary propagator. In reference to the famous geometric Brownian motion [Gar09], the SLN is, in general, exposed to a non-preservation of quantum state norms, which leads to a high computational complexity for long-time simulations that was recently addressed in [Sto16, WSA16]. Over the past few years, SLN-related strategies have been success- fully used in various fields from optimal control [SNA+11, SSA13, SSA15], semiclassical [KGSA08, KGSA10] and spin dynamics [Sto04] to bio-molecules and structured spec- tral densities [IOK15], up to work and heat in discrete systems and harmonic chains [SCP+15, MAS17]. The ambition to further refine numerically exact open system techniques and adapt them to complex systems in various fields is more than a mere academic exercise. Solid-state implementations of superconducting quantum circuits [BHW+04, WS05] including qubits [VAC+02, KTG+07, GCS17], quantum networks and biological light harvesting complexes [PH08, CDC+10, HP13], raise questions on how quantum me- chanics modifies classical concepts and how quantum phenomena can advance future technologies. Can quantum effects maybe even boost the performance of thermal ma- chines to an unprecedented level? At least for thermal reservoirs, the Carnot effi- ciency limit still holds in the quantum case [Ali79, CF16], fluctuation theorems do not need to be altered [CTH09, CHT11a, CHT11b, Cam14, CPF15] and nonadia- batic coupling in finite-time protocols brings about the quantum analogue of classi- cal friction [KF02, PAA+14]. Experiments with trapped ions and solid state circuits [ARJ+12, KMSP14, RDT+16] have not crossed the line to the deep quantum regime, though recent efforts [ULK15, FL17, TPL+17, RKS+18, vLGS+19] that implement e.g. squeezed thermal reservoirs [KFIT17], thermodynamic cycles in NV centers [KBL+19] or non-thermal engines in transmon qubits [CBD19] challenge classical thermodynam- ical concepts [GMM09, EHM09, LJR17, GNM+19] and may even reach beyond the Carnot limit at maximum power [CA75, RASK+14, Def18]. The latter search for traces of thermodynamic quantum supremacy as opposed to classical engines and in- clude recent studies on many particle effects and quantum statistics [JBdC16]. Re- lated questions attempt to fathom the quantum nature of heat transport [SHWR00] and essential implications for heat valves, switches and thermal current transistors [SSJ15, LRW+12, VRD+15, JDEOM16]. In contrast to the classical separability of time and length scales, a complete, microscopic description of non-equilibrium dynam- ics of thermal devices is, however, exceptionally challenging, yet urgently required. As a natural extension to the existing SLN-framework, this thesis presents new advances in reducing the order of numerical complexity due to the stochastic state propagation 2 (chapter 5, [WSA16]), provides benchmark data for a generalization of a deterministic heat transfer model to nonlinear quantum chains (chapter 6, [MWSA18]) and offers a non-perturbative, microscopical platform to explore the out-of-equilibrium
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