This thesis focuses on the simulation of the physics of a charged particle under an external magnetic field by using discrete-time quantum walks of a spin-1/2 particle in a two-dimensional lattice. By Floquet-engineering the quantum-walk protocol, an Aharonov–Bohm geometric phase is imprinted onto closed-loop paths in the lattice, thus realizing an abelian gauge field—the analog of a magnetic flux threading a two-dimensional electron gas. I show that in the strong-field regime, i.e. when the flux per plaquette of the lattice is a sizable fraction of the flux quantum, magnetic quantum walks give rise to nearly flat energy bands. I demonstrate that the system behaves like a Chern insulator by computing the Chern numbers of the energy bands and studying the excitation of the midgap topologically protected edge modes. These modes are extended all along the boundaries of the magnetic domains and remain robust against perturbations that respect the gap closing conditions. Furthermore, I discuss a possible experimental implementation of this scheme using neutral atoms trapped in two dimensional spin-dependent optical lattices. The proposed scheme has a number of unique features, e.g. it allows one to generate arbitrary magnetic-field landscapes, including those with sharp boundaries along which topologically protected edge states can be localized and probed. Additionally, I introduce the scattering matrix approach in discrete-time quantum walks to probe the Hofstadter spectrum and compute its topological invariants. By opening up a discrete-time quantum walk system and connecting it to metallic leads, I demonstrate that the reflection/transmission probabilities of a particle from the scattering region give information on the energy spectrum and topological invariants of the system. Although the work presented here focuses on the physics of a single particle in a clean system, it sets the stage for studies of many-body topological states in the presence of interactions and disorder.
Im Rahmen dieser Arbeit soll die Intensität eines Lasers über eine Feedback-Schleife stabilisiert werden, und dabei soll eine Bandbreite von über einem MHz erreicht werden, da Rauschen und Schwankungen bei diesen und höheren Frequenzen von den Atomen nicht mehr "wahrgenommen" werden. Es ist bereits möglich eine Stabilisierung der Intensität des Lasers mit einer Bandbreite bis zu ungefähr 100kHz aufzubauen. Dieser Aufbau soll in dieser Arbeit optimiert werden, um Rauschen bis zu mindestens einem MHz zu unterdrücken. Dabei soll hauptsächlich ein geeigneter Verstärker gebaut und so angepasst werden, dass er eine optimale Rauschunterdrückung gewährleistet.
In the Discrete Quantum Simulator (DQSIM) experiment in Bonn it has been planned to study topologically protected transport of atoms along the edges separating distinct topological phases. To realize sharp edges, structured intensity patterns will be used and are to be holographically projected onto the caesium atoms trapped in a two-dimensional (2D) state-dependent optical lattice. A robust technique to reproduce sharp-edged diffracted patterns has been accomplished and is presented in this thesis. This technique is based on a Gerchberg-Saxton-like algorithm, which has overcome the well-known stagnation problem and is able to suppress speckles induced by random phase vortices, and has been further extended to create sharp, diffraction-limited edges in the reconstructed intensity pattern. Computer-generated holograms (CGHs) corresponding to the desired intensity patterns can be calculated with high computational efficiency (∼ 100 iterations) and the intensity patterns can be reconstructed with high fidelity (relative RMS 3.9%).
This thesis describes the theoretical and experimental work for reaching fast, high fidelity transport operations of single cesium atoms in a state-dependent optical lattice. By applying optimal control theory to position and depth of the optical lattice potential and using a computer simulation judging the fidelity, fast transport sequences preserving the internal atomic quantum state and preventing any motional excitation can be identified. To allow transport times down to a few microseconds the feedback control system used for steering depth and position of the optical lattice deterministically is overdriven in a controlled way. Transport induced motional excitations are measured experimentally by means of a special microwave sideband spectroscopy, which is improved to reliably detect any excitation and allows a full tomography of the vibrational states of the anharmonic optical lattice potential. Optimal control sequences allowing single site transport of atoms in the oscillation period of the trapping potential are believed to reach the fundamental quantum speed limit of the system.
In our laboratory, we use cesium atoms, which are trapped in the optical lattices. For the practice of quantum walks, atoms must be well isolated from the noisy environment so that long decoherence time can be achieved. It has been analyzed that fluctuations of the lattice depth originated from intensity fluctuations is one mechanism of decoherence. To suppress the intensity noise of optical lattices, we implement an intensity stabilization control loop based on a field-programmable gate array (FPGA) digital platform (Keysight AIO-H3336F). With the advantages of its integratability and flexibility, the application of digital control opens more possibilities for light intensity modulation. In addition to the intensity stabilization, a feedforward control of light intensity becomes feasible with the use of digital signal processing function of FPGA. The realization of intensity feedforward control provides us with a high bandwidth of intensity modulation as well as the conveniences of creating arbitrary intensity ramp implementation. Therefore, it plays an important role in our exploration into the physics related with a time-varying optical trap depth.
Der Ansatz, der in dieser Arbeit verfolgt wird, verwendet einen elektrooptischen Modulator, welcher eine Bandbreite im Bereich von 100 MHz hat. Das Ziel dieser Arbeit ist es, eine Intensitätsstabilisierung mit einer Bandbreite von mindestens 10 MHz zu erreichen und die daraus resultierende Verbesserung der Störungsunterdrückung zu demonstrieren. Die nötigen theoretischen Grundlagen sowie die Funktionsweise der Intensitätsstabilisierung mithilfe eines elektrooptischen Modulators werden erarbeitet. Weiterhin werden Aufbauten und Messungen, die zum Erreichen einer hohen Bandbreite erforderlich sind, dargestellt und erklärt.
This master thesis investigates the potential application of a Liquid Crystal on Silicon Spatial Light Modulator for control of the coin flip operation in a two-dimensional quantum walk. First, the phase noise characteristics of two such modulators from different manufacturers are analyzed. Then an optical setup is constructed to perform holography using phase modulation. The evaluation of this setup verifies pattern generation capability and shows further directions for improvement of image quality.
This thesis reports on a novel concept of state-dependent transport, which achieves an unprecedented control over the position of individual atoms in optical lattices. Utilizing this control I demonstrate an experimental violation of the Leggett Garg inequality, which rigorously excludes (i.e. falsifies) any explanation of quantum transport based on classical, well-defined trajectories. Furthermore, I demonstrate the generation of arbitrary low-entropy states of neutral atoms following a bottom-up approach by rearranging a dilute thermal ensemble into a predefined, ordered distribution in a one-dimensional optical lattice. Additionally, I probe two-particle quantum interference effects of two atom trajectories by realizing a microwave Hong-Ou-Mandel interferometer with massive particles, which are cooled into the vibrational ground state.
The first part of this thesis reports on several new experimental tools and techniques: three-dimensional ground state cooling of single atoms, which are trapped in the combined potential of a polarization-synthesized optical lattice and a blue-detuned hollow dipole potential; A high-NA (0.92) objective lens achieving a diffraction limited resolution of 460 nm; and an improved super-resolution algorithm, which resolves the position of individual atoms in small clusters at high filling factors, even when each lattice site is occupied.
The next part is devoted to the conceptually new optical-lattice technique that relies on a high-precision, high-bandwidth synthesis of light polarization. Polarization-synthesized optical lattices provide two fully controllable optical lattice potentials, each of them confining only atoms in either one of the two long-lived hyperfine states. By employing one lattice as the storage register and the other one as the shift register, I provide a proof of concept that selected regions of the periodic potential can be filled with one particle per site.
In the following part I report on a stringent test of the non-classicality of the motion of a massive quantum particle, which propagates on a discrete lattice. Measuring temporal correlations of the position of single atoms performing a quantum walk, we observe a 6 σ (standard deviation) violation of the Leggett-Garg inequality. The experiment is carried out using so-called ideal negative measurements – an essential requisite for any genuine Leggett-Garg test – which acquire information about the atom’s position while avoiding any direct interaction with it. This interaction-free measurement is based on our polarization-synthesized optical lattice, which allows us to directly probe the absence rather than the presence of atoms at a chosen lattice site. Beyond its fundamental aspect, I demonstrate the application of the Leggett-Garg correlation function as a witness of quantum superposition. The witness allows us to discriminate the quantumness of different types of walks spanning from merely classical to quantum dynamics and further to witness the decoherence of a quantum state.
In the last experimental part I will discuss recent results on collisional losses due to inelastic collisions occurring at high two-atom densities and demonstrate a Hong-Ou-Mandel interference with massive particles. Our precise control over individual indistinguishable particles embodies a direct analogue of the original Hong-Ou-Mandel experiment. By carrying out a Monte Carlo analysis of our experimental data, I demonstrate a signature of the two-particle interference of two-atom trajectories with a statistical significance of 4 σ.
In the final part I will introduce several new experiments which can be realized with the tools and techniques developed in this thesis, spanning from the detection of topologically protected edge states to the prospect of building a one-million-operation quantum cellular automaton.
This thesis describes the development of an optical phase lock loop on a digital platform, in order to realize state-dependent transport on a two-dimensional optical lattice. The digital platform consists of a field programmable gate array in combination of a vector generator module, which is used to steer the amplitude and phase of the optical lattice deterministically. The digital system enables the implementation of a feedforward control scheme based on internal model control, which overcomes the bandwidth limitations of feedback systems. The control bandwidth is shown to be increased by more than an order of magnitude, directly improving the number of coherent operations that can be executed with the atoms in the optical lattice. The system is implemented into the optical setup of the experimental apparatus, and the first signatures of state-dependent transport of atoms in the two-dimensional optical lattice is observed and presented.
Diese Arbeit beschäftigt sich mit der Untersuchung der Atomfluoreszenzbilder, die im DQSIM - Experiment aufgenommen worden sind. Die Fluoreszenzmikroskopie bzgl. des zweidimensionalen Vielteilchensystems hat eine wichtige Bedeutung für das messungsbasierte Quantenrechnen mit Clusterzuständen. Anhand der Fluoreszenzaufnahmen wird das Abbildungsprinzip erläutert, das Konzept der Punktspreizfunktion (engl. point spread function, kurz PSF) und der Faltung wird eingeführt und deren mathematische Eigenschaften werden vorgestellt. Durch die Analyse der PSF wird die reale numerische Apertur (kurz NA) zusammen mit dem Strehl-Verhältnis (engl. Strehl ratio) untersucht. Die Anwendung der Zernike-Polynome wird ebenso einen Einblick in die Analyse der optischen Aberrationen eines Abbildungssystems anbieten.
In this thesis, I present single-site detection of neutral atoms stored in a three-dimensional optical lattice using a numerical aperture objective lens (NA_{design} = 0.92). The combination of high-resolution imaging with state-dependent trapping along two-direction of the lattice opens up the path towards quantum simulations via quantum walks. Suppressing the interactions of a quantum system with the environment is essential for all quantum simulation experiments. It demands a precise control of both the external magnetic (stray) fields and the polarization properties of laser beams inside the vacuum chamber. I designed a metal shielding to reduce magnetic field fluctuations and designed, assembled and characterized a novel ultra-high vacuum glass cell. The glass cell consists of special glass material and exhibits an ultra-low birefringence Δn of a few times 10^{−8} to highly suppress polarization disturbances originating from stress birefringence in vacuum windows. Furthermore, anti-reflection coatings avoid reflections on all window surfaces. The cell hosts the assembled vacuum-compatible objective, that exhibits a diffraction limited resolution of up to 453 nm and allows to optically resolve the spacing of the optical lattice. Fluorescence images of single trapped atoms are used to characterize the imaging system. The filling, orientation and geometry of the optical lattice is precisely reconstructed using positions of atoms that can be determined from fluorescence images. Furthermore, I present a scheme to realize state-dependent transport and discuss its robustness against experimental imperfections in a technical implementation. This transport scheme enable the realization of discrete-time quantum walks with neutral atoms in two dimensions. These quantum walks pave the way towards the simulation of artificial magnetic fields and topologically protected edge states.
Den Polarisationszustand elektromagnetischer Strahlung zu kennen, und kontrollieren zu können, ist in der modernen Physik von großer Wichtigkeit. Beispielsweise braucht man zum Laserkühlen von Atomen eine klar definierte Polarisation. Ziel dieser Bachelorarbeit ist es ein Polarimeter aufzubauen, welches die Polarisation eines einfallenden Laserstrahls misst und Auskunft darüber zu geben, wie genau es Polarisationen messen kann. Ferner sollen die wichtigsten systematischen Messfehler untersucht und charakterisiert werden.
Ein Stokes-Polarimeter wurde erfolgreich aufgebaut und getestet. Es wurde gezeigt, dass die Genauigkeit des Polarimeters von der Polarisation abhängig ist, wobei es zirkulare Polarisation besser als auf 0.5% genau messen kann. Über die Möglichkeit, die zirkulare Polarisation genauer zu messen, lässt sich nichts aussagen, da die Polarisation nicht genauer eingestellt werden konnte. Die Güte der Messung der linearen Polarisation wird am stärksten durch die genaue Kenntnis des Winkels zwischen schneller Achse der QWP und der Polarisationsachse des Analysators bestimmt. Mit der Kenntnis dieses Winkels auf 0.2º genau ist es auch möglich, lineare Polarisation mit 1% Genauigkeit zu messen. Eine Ausnahme stellt hier die horizontale Polarisation dar, bei der eine Abweichung von 5% von den erwarteten Werten beobachtet wurde. Mögliche Ursachen hierfür sind eine geringere Verzögerung der QWP und ein Offset der Intensitätsmessung, der weiter untersucht werden muss. Ferner wurde Wave-Plate-Ripple als eine weitere Fehlerquelle identifiziert und eine qualitative Abschätzung seiner Einflüsse gegeben.
This master thesis contains two major parts. On the theoretical side simulating the state-dependent transport of neutral caesium atoms in a two-dimensional optical lattice gives an intuitive picture of the state-dependent potential during transport. The more practical part deals with the task to load the atoms into the lattice since they cannot be trapped directly in the plane of the experiment. A magneto-optical trap in a distance of about ∼2 mm below the lattice traps atoms out of the vacuum background and lift them afterwards into the lattice. This is realised using a precise electronic control of the imbalance of current guided through the two coils for building up the required quadrupole field by diverting a certain amount of current around one coil. Analysis of the step-response signal of the used metal band coils yields mandatory characteristics for achieving a fast and stable control. A galvanic isolated signal transfer protects the computer control against damage due to accidental voltage pulses from the high-power system. The diverted current can change with ∼0.9 A ms^{−1}.
This work reports on the implementation of feedback methods for position control of single Cesium atoms in spin-dependent optical lattices. For this purpose a new control software has been developed. It encapsulates experimental control, fluorescence image acquisition and analysis into a single program. Combining the new software with spin-dependent transport and position-dependent addressing, we have developed a versatile feedback algorithm to deterministically arrange arbitrary patterns of up to 6 atoms. In addition, we have prepared two atoms in a common lattice site. We observed light-assisted collisions between both atoms and thereby characterized the preparation effciency of 83 ± 4 %. The results pave the way for the study of controlled interactions between precisely two atoms in a single well of the optical lattice potential, enabling the realization of a fundamental quantum logic gate.
This master-thesis investigates a new approach for state-dependent transport of atoms in an optical lattice. It is based on a direct synthesis of light polarization by superimposing two circular polarized beams and employing RF sources integrated with acousto-optic modulators for phase control. An interferometrically stable phase between the two beams is achieved by locking them actively with a heterodyne technique. The influence of polarization crosstalk and erroneous components on the optical lattice and the phase locked loop are investigated and the quality of the phase locked loop is analyzed.
Compared to conventional methods [25] the direct synthesis method avoids the need of an electro-optic modulator, where rotations on the Poincare sphere are limited by the applicable voltage and restrictions on manufacturing and crystal quality exist. Overcoming these limitations it is expected to reach higher polarization purity and larger shift distances in the new design.
The experimental realisation of electric quantum walks, i.e. quantum walks that are subject to a force, is presented with individual caesium atoms. Hereby, the behaviour of a charged quantum particle in a static electric eld is simulated in a time as well as space discrete system. Building on previous achievements [1], the demonstration of ordinary quantum walks of up to 100 steps is shown. Further thorough theoretical studies expose the underlying simulator properties of such a quantum walk system experiencing a force. Similarities to the continuous time analogue as well as characteristic features that are indebted to the discrete evolution of the system are presented. The implementation of a direct digital synthesizer allows the experimental application of discrete forces in the system by employing frequency ramps, and thus leads to the realisation of electric walks. Results are given for selected force parameters, showing the phenomenon of Bloch oscillations. Additionally, pure ballistic transport of the electric quantum walk due to strong Landau-Zener tunnelling in the strong force regime is demonstrated.