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In the field of molecular electronics, single-molecule junctions, which comprise a molecule contacted by electrodes, represent the ultimate miniaturization of electronic components in an electric circuit. In this context, experiments on single-molecule junctions have revealed interesting transport phenomena, including transistor- and diode-like behavior as well as negative differential resistance. These transport effects mimic the basic functions of conventional semiconductor devices. Additionally, molecules provide intrinsic functionalities which may result in applications like molecular sensors and machines, memory and spintronic devices, as well as optoelectronics. In contrast to rigid solid-state components, the transport characteristics in molecular junctions are strongly influenced by the intricate interplay of electronic and nuclear (vibrational) degrees of freedom due to their small size and flexible structure. A central task in the field of molecular electronics is to attribute the transport phenomena to the structure and the properties of the molecule in the contact. To this end, theoretical model studies are performed, which facilitate the understanding of this complex nonequilibrium transport problem on the nanoscale. In these studies, the analysis of the average current is complemented by the investigation of current fluctuations in order to obtain detailed knowledge on the underlying transport processes and mechanisms.
In this thesis, we investigate electron transport through single-molecule junctions on the basis of numerically exact results. In particular, we focus on transport phenomena induced by electronic-vibrational coupling. To this end, we develop two approaches within the hierarchical quantum master equation (HQME) framework, which differ by the treatment of electronic-vibrational coupling, and thus cover a large spectrum of parameters ranging from the nonadiabatic to the adiabatic regime and including both resonant and nonresonant transport. In particular, we show that the nonequilibrium vibrational excitation significantly influences the transport characteristics of a single-molecule contact in all these regimes by comparison with results where the vibrational mode is equilibrated. In addition, we extend the HQME approach to the evaluation of the full counting statistics, which allows the calculation of higher-order current cumulants beyond the average current. This extension is not limited to the description of vibrationally coupled transport but can be applied to all transport setups which can be treated within the HQME formalism. The numerically exact HQME results are used to benchmark approximate master equation and nonequilibrium Green’s function methods.
Our findings demonstrate that vibrational nonequilibrium effects play an important role in a wide range of parameters, and thus cannot be neglected in the description of this transport problem. In particular in the nonresonant transport regime, the inelastic cotunneling signal is analyzed for a vibrational mode in full nonequilibrium, revealing a complex interplay of different transport processes and deviations from the commonly used G0/2-rule of thumb. The inelastic correction to noise also exhibits strong deviations from the prediction for a thermally equilibrated vibration. Additionally, we investigate how the phenomenon of vibrational instability, that is, the increase of current-induced vibrational excitation for decreasing electronic-vibrational coupling, is influenced by level broadening due to molecule-lead coupling as well as broadening of the Fermi distribution caused by temperature. Results obtained for the first two moments suggest that the vibrational excitation is always described by a geometric distribution in the weak electronic-vibrational coupling limit, which confirms our approximate analytic description. Moreover, we study the influence of cotunneling on avalanche-like transport in the regime of strong electronic-vibrational coupling. We find signatures in the current noise reflecting the complex interplay of inelastic cotunneling processes and resonant avalanches.
In the field of optomechanics, one typically studies the interaction of a single mechanical resonator
and a single optical cavity mode. In this thesis, we investigate collective quantum effects in
optomechanical systems with two or more optical and mechanical modes.
First, we study the generation and destruction of entanglement between two mechanical resonators
effected by optical cavity fields. We also describe how the optomechanical interaction can
be employed to mediate strong coherent coupling and entanglement between a micromechanical
membrane and a single atom.
Secondly, we show that a configuration with two optical and a single mechanical mode can
lead to enhanced nonlinear interactions, which can be exploited for quantum non-demolition
measurements of the phonon and photon numbers.
Finally, we explore the quantum many-body dynamics of optomechanical arrays. In particular,
we research into the transition from a disordered state due to quantum noise to a state with phasecoherent
mechanical oscillations. As an intermediate step towards this study, we also investigate
the nonlinear classical dynamics of single- and multi-mode optomechanical setups and compare
theoretical predictions to experimental data.
In this thesis we investigate nonlinear quantum effects and squeezing in cavity optomechanical systems, where light interacts with mechanical motion.
In the first part of this thesis we analyze how to generate squeezed mechanical states and squeezed output light with state-of-the-art optomechanical setups via dissipation. We predict that arbitrary large steady-state bosonic squeezing can be generated. Furthermore, we show that our dissipative output light squeezing scheme can be used directly to enhance the intrinsic measurement sensitivity of an optomechanical cavity.
In the second part, we explore the so-called “single-photon strong coupling regime” of optomechanics. In this regime, the nonlinear quantum nature of the optomechanical interaction becomes important. We work out the first signatures of this nonlinear quantum interaction. We also propose how to observe these signatures with near-future optomechanical experiments. In the following, we analyze how an even stronger quantum interaction between photons and phonons modifies the statistics of photons which are transmitted through an optomechanical system.
In the last part of this thesis, we discuss how to verify energy quantization of a mechanical degree of freedom. We propose to make use of an optomechanical setup where the position squared of a mechanical degree of freedom is coupled to the light field. We predict that energy quantization could be observable e.g. with nanometer-sized dielectric spheres.
Frequency-tripled photon generation in a nonlinear optical process far beyond the paraxial regime
(2020)
Nonlinear optical experiments are usually performed using weakly focused Gaussian beams, so that these experiments can be described in the context of paraxial approximation.
In contrast, the goal of our study is to investigate a basic nonlinear optical
process by focusing the pump light under conditions far beyond the paraxial regime. In
the experiments presented here, the pump light is focused by means of a parabolic mirror
from an almost full solid angle which resembles the radiation pattern of a linear dipole.
On the basis of the paradigmatic process investigated - frequency tripling in an isotropic
medium with normal dispersion, here argon gas - differences and similarities to the paraxial regime are discussed. The experimental results are compared with simulations, which are also used to interpret our results.
First, an overview of earlier experimental and theoretical studies on frequency-tripled
photon generation in normal dispersive media in the paraxial focusing regime is given.
Then the new focusing regime discussed here will be described in detail. Essential characteristics,
in particular the distribution of the electromagnetic field in the focus of the
parabolic mirror and their influence on the nonlinear process investigated are explained.
We conduct some experiments observing the behaviour of the frequency-tripled beam as a
function of different parameters such as input power, solid angle of focusing and pressure
of the nonlinear medium.
Based on these results, the similarities and differences of the frequency-tripled photon
generation under the conditions investigated in this thesis to the paraxial regime are
highlighted. Particular attention is paid to the role of phase matching and the dependence
of the number of photons generated on the input power of the pump light. From
the latter results, it is deduced with the help of a theoretical consideration that six-wave-mixing is the mechanism behind frequency-tripled photon generation in a normal
dispersive isotropic medium under the conditions investigated here.
Ultracold atoms can be trapped in periodic intensity patterns of light created by counterpropagating laser beams, so-called optical lattices. In contrast to its natural counterpart, electrons in a solid state crystal, this man-made setup is very clean and highly isolated from environmental degrees of freedom. Moreover, to a large extent, the experimenter has dynamical control over the relevant system parameters: the interaction between atoms, the tunneling amplitude between lattice sites, and even the dimensionality of the lattice. These advantages render this system a unique platform for the simulation of quantum many-body dynamics for various lattice Hamiltonians as has been demonstrated in several experiments by now.
The most significant step in recent times has arguably been the introduction of single-site detection of individual atoms in optical lattices. This technique, based on fluorescence microscopy, opens a new doorway for the study of quantum many-body states: the detection of the microscopic atom configuration.
In this thesis, we theoretically explore the dynamics of ultracold atoms in optical lattices for various setups realized in present-day experiments. Our main focus lies on aspects that become experimentally accessible by (realistic extensions of) the novel single-site measurement technique.
The first part deals with the expansion of initially confined atoms in a homogeneous lattice, which is one way to create atomic motion in experiments. We analyze the buildup of spatial correlations during the expansion of a finitely extended band insulating state in one dimension. The numerical simulation reveals the creation of remote spin-entangled fermions in the strongly interacting regime. We discuss the experimental observation of such spin-entangled pairs by means of a single-site measurement.
Furthermore, we suggest studying the impact of observations on the expansion dynamics for the extreme case of a projective measurement in the spatial occupation number basis realized by a single-site detection. The analysis of the resulting quantum Zeno physics shows regimes for which the initial many-particle configurations are stabilized or destabilized, depending on the observation time interval and the interaction strength.
In the second part, the measurement of the local current operator in an optical lattice is discussed. We propose a measurement protocol that combines single-site detection with already existing optical superlattices. The measurement outcomes can even be used to calculate spatial current-current correlations since the local currents are simultaneously measured at various positions. We illustrate the prospects of this new sensing method by a numerical study of the current statistics for interacting bosons in one and two dimensions. In the latter case, we discuss how the on-site interactions affect the equilibrium currents of bosons in an artificial magnetic field. We substantiate the feasibility of the protocol by considering possible error sources, restrictions in currently used single-site detection, and its applicability in experimental setups used to create artificial gauge fields.
In this thesis we investigate two different aspects of nonlinear dynamical behaviour: First, we study quantum synchronization which is based on nonlinear, so-called limit-cycle oscillators. Second, we investigate the influence of nonlinear interactions on topological transport
in bosonic systems.
We begin with the investigation of the synchronization of two optomechanical systems in
the presence of quantum noise. With numerical simulations of the full quantum and a semiclassical model, we identify different phase synchronization regimes, although exact phase locking is prevented due to the inevitable quantum noise. Our main results are noise-induced
transitions between different synchronization states, as well as noise-induced bistability, i.e., the appearance of a second synchronization state in the quantumregime, even if in the classical, noiseless limit only a single stable synchronization state exists. We give an overview about the classical-to-quantumtransition and also compare our findings to synchronization in the presence of classical, thermal noise.
Subsequently, we study the quantum synchronization dynamics of a quantum Van der
Pol oscillator coupled to an external periodic driving. This paradigm system for limit-cycle oscillators allows us to derive an effective, analytical quantum model and identify the different
dynamical regimes, like overdamped, underdamped, and notably also quantum-coherent
phase motion. We explore the distinct signatures of these different dynamical regimes, e.g., in the spectrum or the squeezing. Moreover, we show that in the remarkable regime of quantum-coherent
phase dynamics quantum states with negative Wigner densities (like Schrödinger cat states) are preserved for many oscillations of the system. Our effective model gives insight into how to achieve quantum coherence in synchronization dynamics, while numerical simulations of the full model confirm the predicted behaviour.
Finally, we turn to the classical dynamics of a Chern topological insulator in a nonlinear bosonic system. To this end, we study the half Bernevig-Hughes-Zhang model on a square lattice in the presence of a local Kerr nonlinearity. We investigate the stability of a linear
edge state in the nonlinear model and present a linear stability analysis to gain insight into the unstable scattering processes enabled by the nonlinearity. In a time evolution of the full
nonlinear model, we can show that these processes lead to a spatially periodic modulation of the edge channel on an intermediate timescale. However, eventually we observe that strong nonlinear processes lead to more complicated behaviour and a significant radiation into the bulk.
Coupled limit-cycle oscillators exhibit interesting collective phenomena, like synchronization and pattern formation. Effective models of the classical phase dynamics in these systems have been very successful in describing these effects. Important examples are the canonical Kuramoto model and the Kuramoto-Sakaguchi model. In this thesis, we study a closely related, slightly more general effective phase model, which we call Hopf-Kuramoto model. We focus on the phase dynamics in one-dimensional and two-dimensional lattices.
One central topic is the pattern formation in the deterministic model. As a main result, we present the pattern phase diagram for two-dimensional arrays. This diagram illustrates which patterns are relevant in the long-time dynamics, after starting from random initial conditions, in dependence on the parameters of the model. We examine details of important stationary and non-stationary patterns. This includes the shape and movement of spiral structures, as well as their influence on correlations. Regarding one-dimensional systems, we find smooth stationary patterns with characteristic defects and solitary-wave-like structures in different limiting cases of our model.
Subsequently, we discuss the stochastic dynamics. We study the effects of noise on some of the patterns found in the deterministic case. We then continue with an analysis of a limiting case of the Hopf-Kuramoto model, the noisy Kuramoto-Sakaguchi model. For smooth phase fields, this model is related to the Kardar-Parisi-Zhang model of surface growth. This enables us to explain scaling properties of the phase field with time, as well as a sudden desynchronization process which we find in simulations.
As an example of a system where our model is applicable, we discuss future optomechanical arrays. Moreover, we show that the derivation of the Hopf-Kuramoto model is based on very general assumptions about the dynamics of nonlinear oscillators close to their limit cycle. Hence, our results are relevant for a large class of experiments on arrays of locally coupled oscillators.
Most of the recent experimental achievements in the field of optomechanics, e.g. ground state cooling of a vibrational mode, were realized in single-mode optomechanical systems comprising one vibrational degree of freedom that interacts with one optical degree of freedom via radiation pressure. In the present thesis, we theoretically study multi-mode optomechanical systems that are made of multiple optical and multiple vibrational modes.
We first propose and analyze a scalable scheme in which linear quantum operations can be performed on a set of vibrational modes by simply modulating the amplitude of the laser beam that drives the system. The generation of entanglement between distinct vibrational modes and squeezed mechanical states is analyzed. Moreover, we investigate the feasibility of state transfers between distinct vibrational modes.
Next, we focus on the many-body dynamics of photons and phonons in a lattice with optomechanical on-site interaction and introduce the optomechanical band structure as a means to describe the linearized dynamics of the many-body non-equilibrium system. We then turn to describe its main features, i.e. the formation of photon-phonon polaritions and an array version of self-induced mechanical oscillations, and discover a novel entangling instability that is only present in multi-mode systems. Finally, we provide a universal stability diagram which is valid for all symmetric multi-band lattices.
Subsequently, we turn to optomechanical arrays with a honeycomb geometry and investigate the emergence of photon-phonon Dirac polaritons. In contrast to the usual case, Dirac polaritons feature dispersive edge states, which consequently gives rise to polarition transport along the edge, which is tunable via the driving laser. Putting the large external tunability of optomechanical arrays to use, we predict a version of Klein tunneling for photons. A peculiarity of this effect is that photons can be interconverted into phonons while tunneling through the barrier that is solely created by shaping the beam of the driving laser.
As a next step, we propose two schemes to all-optically create gauge fields for photons in optomechanical arrays and analyze their feasibility numerically. Moreover, we study photon transport and generalize the photonic gauge field to include the optomechanical coupling to phonons. To this end, we introduce a synthetic photon-phonon dimension that provides a universal description of optomechanical gauge fields.
Finally, we investigate the interplay of the surface modes of a levitated fluid droplet with its rotation. For this purpose, we extend the liquid drop model of nuclear physics to include the rotation of the droplet. We derive a Lagrangian to leading order, using the Euler angles and the amplitudes of the lowest energy surface modes as generalized coordinates. Finally, we display the lengthy and highly non-linear equations of motion.
Quantum control has recently been a problem of great relevance and is gaining even more traction. In many fields, including quantum technologies, quantum metrology, and quantum computing, control of a system is essential.
In physics, quantum control has been studied both from the theoretical and the experimental point of view. Its tasks include purification of quantum states, optimal control, quantum feedback, and quantum state preparation. Each task is essential for studying and developing scalable and reliable quantum systems.
While quantum control was only approached through theoretical analytical solutions and gradient-based techniques in the past, in recent years, there has been considerable interest in applying machine learning (ML) techniques to such problems. One of the most promising branches of ML to do so is called Reinforcement Learning (RL).
RL was applied successfully to solve complex problems that required finding control strategies in computer games and robotics. Reinforcement learning is a very general technique, and its framework can be applied with little effort to various tasks, so physicists saw it as a powerful tool for their quantum control tasks. Among the most notable applications of reinforcement learning in quantum control are state preparation, state transfer, parameter estimation, gate and circuit design, and quantum error correction.
This thesis will define the main concepts of machine learning and display the general framework of Reinforcement Learning and a few state-of-the-art algorithms. Several essential concepts of quantum control will also be presented, focusing on two RL applications in the field.
One of the applications is a deep reinforcement learning scheme used for preparing and stabilizing quantum Fock states and superpositions. The studied system is a cavity subject to quantum-non-demolition detection of photon number and controlled by only a simple linear drive, with RL reaching high fidelities in a multitude of tasks without any prior knowledge of the physical system. This work was inspired on an experimental application of Prof. Benjamin Huard’s experimental group at ENS Lyon, who also collaborated on the manuscript.
The second application showcases an extension of the popular GRAPE algorithm. GRAPE is a gradient-based technique primarily used to optimize quantum systems’ control sequences. Our implementation incorporates discrete and continuous strong stochastic measurements into an RL-inspired approach
This powerful technique is illustrated on a Jaynes-Cummings model with feedback. The strategies found by RL can prepare states and stabilize them in the presence of noise and are also human-interpretable.
All of the work in this thesis has been conducted under the supervision of Prof. Dr. Florian Marquardt at the Max Planck Institute for the Science of Light from January 2019 to March 2022.
As new machine learning technologies are conceived, endless possibilities for breakthrough applications in science open up. We wonder how far we are from developing an artificial scientist that acts like a human one. It would be capable of making observations, analyzing data, building a model, deriving hypotheses, and designing experiments to falsify them. After an overview of common machine learning techniques, we review recent work that attempts to achieve some of these goals. We then present the main ideas that we believe are necessary for future artificial scientific discovery. In particular, we take the perspective of a physicist, and show examples from physical domains that could benefit from the presented approaches. We develop these ideas in various works. First, we focus on learning to make a high-level description of a physical system, e.g. by extracting collective variables. Therefore, we present the idea of using information theory to describe them and introduce the concept of “Renormalized Mutual Information”. Second, we study the problem of experimental design and investigate how an artificial scientist should choose which experiments to perform. We show applications of a deep learning approximation technique to quantum many-body systems. We then generalize our discussion to the problem of scientific exploration, diving into reinforcement learning techniques for intrinsic motivation as a framework for creating “curious” artificial scientists. Furthermore, we imagine ways in which an algorithm could learn concepts from experience and reuse them in different settings. We adopt the program synthesis approach and explore applications to the synthesis of quantum circuits given the associated unitary matrix. Finally, we discuss additional proof-of-concept examples and future developments in the field.