In this thesis observations on the application of parametric drives to superconducting quantum circuits in disparate parameter regimes are presented. By the nonlinear inductance of the Josephson junction, a variety of interactions in circuit quantum electrodynamical systems comprised of strongly, moderately, and weakly nonlinear oscillators are realized.

Chapter 1 contains an introduction to classical and quantum information and introduces superconducting circuits as a platform for quantum information processing. An outline of the contents of the thesis is also provided.

In Chapter 2 a theoretical foundation for the later chapters is established, spanning from the classical harmonic oscillator to circuit quantum electrodynamical systems and parametric driving. The transmon qubit, junction-embedded coplanar waveguide, tunable coupler, and Josephson junction array resonator are introduced and some methods for realizing parametrically activated interactions in such systems are discussed.

Chapter 3 focuses on the steps necessary for constructing a superconducting quantum circuit. The design, simulation, and fabrication methods necessary for creating the experimental devices of later chapters are discussed.

In Chapter 4 results of the parametrically activated interactions between two tunably coupled transmon qubits by flux modulation of a SQUID are presented. When the coupling SQUID is modulated at the sum or difference frequencies of the transmons, level repulsion and attraction are observed spectroscopically. The viability of the platform for analog quantum simulations is discussed and the experimental results are compared to analytical models and numerical simulations of the quantum master equation.

In Chapter 5 spectroscopic signatures of a few-photon Kerr parametric oscillator are observed upon the application of an all-microwave bichromatic drive to a Josephson junction-embedded coplanar waveguide resonator. Semiclassical analytical, numerical, and quantum master equation simulations are performed and compared with the experimental results. An effective model based on semiclassical methods proves insufficient in modelling the behaviour of the system, indicating the presence of quantum effects.

In Chapter 6 a weakly nonlinear Josephson junction array resonator is bichromatically driven into a parametric phase state. Stochastic switching between the two non-equilibrium stationary states of the system is observed and the time between stochastic switching events is determined for a range of drive strengths. An additional microwave drive resonant with the frequency of parametric response is applied and the system is biased into one of the phase states. The biasing and change in switching time as a function of drive power and phase is shown. The contributions of classical and quantum effects to the occurrence of switching events is discussed.

In Chapter 7 measurements of a strongly parametrically driven Duffing oscillator are presented. As the system is strongly driven at a variety of large negative detunings, signatures of chaotic behaviour are observed in the output field spectrum and quadrature histograms. The observed features are discussed and compared to known markers of chaotic behaviour in classical parametrically driven Duffing oscillators.

Chapter 8 concludes the thesis, providing a review of the contents and findings of the previous chapters. The thesis ends with an outlook and suggestions for potential future topics of study.
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This thesis presents the design, implementation, and evaluation of the RF (Radio Frequency) section of an Open-Hardware Vector Network Analyzer (VNA) intended for quantum research applications. The project aims to create a cost-effective, modular VNA system that fulfills the functional requirements necessary for qubit readout and other quantum measurements.
In the initial chapters, the overall architecture of the VNA is outlined, with specific attention to the power budget and system requirements. The RF generation principles are examined, and a range of RF generators are tested to ensure they meet the signal quality standards, such as spurious emissions and harmonic content. The performance of various RF mixers is also evaluated and found to be sufficient for the RF system.
Experimental results demonstrate the system’s capability to measure the S21 parameter of a resonator cavity, comparable to commercial VNAs. This validates that the RF system meets the specified requirements and can be effectively used in quantum research.
Future work suggested includes the measurement of generator frequency/phase stability over time and exploring the feasibility of implementing power sweeps to enhance the system’s functionality. The findings of this thesis contribute to the development of accessible and flexible tools for quantum technology research, promoting further advancements in the field.

For microwave qubit readout in research applications, a Vector Network Analyser has been designed. The objective of this project was to design and build a modular, extensible VNA, containing open hard- ware and implemented in open-source software.
This thesis discusses the Python implementation of an interface between the digital signal processing step, taking place inside an FPGA, and the output of data to the user, being in graphical form and as systematical data structure to be stored on a PC. The interface is split up into a server, responsible for communicating with the FPGA on the same chip, and a client, which receives the measurement data from the server via the Transmission Control Protocol and controls the radio frequency signal genera- tors that serve as stimulus for the device under test and as local oscillator for downconversion.
An overview of VNAs and their application in this project is given in the first chapter. The programme of requirements and implementation overview are discussed next, followed by detailed explanations of the Python implementation of the server and client software. The achieved results satisfy the requirements for throughput, extensibility and data transfer overhead time. The thesis concludes with recommendations for future developments and extensions to this project.

Galaxy clusters are some of the largest known structures in the universe. Studying them observationally and theoretically can provide a lot of information on how these clusters form and are structured. One way to study them is through the so-called Sunyaev-Zel’dovich (SZ) effect, which is an interaction between the cosmic microwave background (CMB) and hot electrons in the cluster medium. The SZ effect can be further broken down into a thermal component (tSZ) arising from the random motion of the electrons, and a kinematic component (kSZ) arising from the bulk motion of the cluster medium, making it a good probe for several properties of the cluster. The SZ effect can be observed as a distortion of the CMB spectrum using submillimeter spectrometry. However, at many submillimeter frequencies radiation is absorbed strongly by the atmosphere. This makes it hard to interpret the measured SZ signal, and measurements require long observation times in order to reach a sufficient signal-to-noise ratio. In this thesis, we present a framework that simulates a submillimeter spectrometer observation of the tSZ effect including noise factors. It then fits a model tSZ signal to the noisy signal. This allows us to investigate the relation between observation time, noise and retrievability of cluster properties. We simulate a galaxy cluster with an electron temperature 𝑇𝑒 = 15.3 keV and central optical depth 𝜏𝑒 = 0.0172 with two simulated DESHIMA-type filterbanks spanning different frequency ranges. For each filterbank we perform 20 simulations with an observation time of 16 hours each, and 20 simulations of 32 hours. We fit every simulation separately, but average over simulations to obtain an expectation value for 𝑇𝑒 and 𝜏𝑒 given a filterbank and observation time. We also repeat each fit over rebinned copies of the noisy spectra, combining 7 data points into each bin. All tested combinations of filterbanks and observation times produce fits with results that are consistent with the input parameters. The 160-320 GHz filterbank consistently gives lower errors than the 220-440 GHz filterbank. From rebinning, we do not find any significant improvement or degradation of the quality of the fits. The estimates obtained from rebinned data deviate very little from the original estimates, by at most 5%, and show no change in consistency. From this result, we conclude that SZ observations using DESHIMA 2.0 could provide estimates on cluster parameters. These estimates are already consistent after 16 or 32 hours of observation time. However, we recommend a new filterbank design that covers 160-320 GHz since the error on estimates using this range are smaller than the errors obtained using the original 220-440 GHz filterbank. This is likely due to the atmosphere absorbing much less radiation at this frequency range. Additionally, the results from rebinning show that this new filterbank could contain fewer filters with a lower resolving power without degradation of fit quality.

Producing arbitrary quantum states in mechanical oscillators is an essential part of the research con- cerning combining the theory of quantum mechanics with general relativity. In recent years, a lot of progress was made by the development of optomechanics and circuit quantum electro dynamics using which a quantum mechanical interaction between an LC oscillator and a mechanical oscillator can be created. This only left the need for the ability to create arbitrary desired states in an LC oscillator while keeping its properties as a linear resonator in tact. The interaction needed for this was recently designed in the group and is called the photon-pressure interaction. Using this interaction, effectively a Jaynes-Cummings interaction between a qubit and a LC oscillator was created which can truly be turned on and off, keeping the linear properties of the LC oscillator while the interaction is turned off. In this thesis a protocol that makes use of the Jaynes-Cummings interaction and qubit drives to create arbitrary states in the LC oscillator is developed. To show that the desired oscillator state has been created a protocol is also developed to perform Wigner tomography on the LC oscillator. Both protocols have been tested using simulations with loss effects corresponding to the ones encountered in our lab setting. The simulation results show that using the current lab system settings, states can successfully be produced in the LC oscillator and measured using the tomography protocol. This paves the way for arbitrary state generation and state measurement experimentally in the lab.

Superconducting qubits require a low-noise, high-gain parametric amplifier to overcome the added noise during their readout. Josephson junction arrays are suitable candidates, as they have quantum-limited added noise, large saturation power and a tunable resonance frequency. Dimer Josephson Junction Amplifier Arrays (DJJAAs) have a mode structure, which is engineered in such a way that it exploits the wave mixing processes in JPAs when amplifying incoming signals. The resonant modes in this device form pairs, referred to as dimers. The behaviour of two resonant modes in a DJJAA when a drive tone is applied at an intermediate frequency, is analyzed. A maximum gainGmax = 43.6 dB and saturation power P1dB = 105.9 dBm is measured. It is demonstrated that the mode resonance frequency shifts if the mode photon population changes. If the drive power was increased above a threshold, self-oscillations were observed in this device. A fabrication process for DJJAAs using overlap Josephson junctions was carried out iteratively. Several issues in the fabrication process were resolved. In the fabricated DJJAAs it was possible to identify and drive resonant modes for gain G > 20 dB. The resonant modes in a fabricated device did not form dimers due to offset in the Josephson inductance from its Josephson junctions compared to the design. Future efforts can improve the quality of fabricated overlap Josephson junctions and on studying nonlinear effects that were observed in the device.

In this project, we have done a calibrated measurement on a previously designed and fabricated current pumped Josephson Parametric amplifier. We have installed a microwave switch into our crygenic fridge tobe able to get a calibrated response measurement using Short-Open-Load calibration, which we used withmeasurements of the gain and flux tunability of the device. This showed that our previous measurements onthe gain deviated significantly, with about 10dB. We also did a thermal calibration on the amplifier by heatingthe plate it was attached to, and found that the input noise approximates the limit of a half quantum of noise.

Superconducting microwave resonators based on coplanar waveguides (CPWs), which allow for on-chip implementation, have a wide variety of uses, from the coupling of qubits to the detection of photons from interstellar clouds. With the integration of a bias circuitry, the already versatile resonator will become even more so. However, applying a DC voltage or current bias to the resonator without significantly degrading its quality factor Q is no trivial task. One of the first superconducting microwave resonators with the ability to allow for the application of a DC bias, made use of symmetry points. In order to not rely on those symmetry points, our group came up with a different design that used a shunted capacitor instead. However, this design needs a more extensive and in particular more complicated fabrication process. As the groups focus is now completely away from the original design, we go back in this thesis, to honestly evaluate such a resonator by applying a bias to its center. We present a design with the cavity length l to be a full wavelength (l = λ) terminated on both ends to ground, fed in by a capacitively coupled AC-feedline at λ/4 and a DC-bias line at its center (λ/2). After the first fabrication trial we were unable to successfully judge the performance of the microwave resonator as it suffered from bad internal loss, leading to a low internal Q ~ 300 at 4 K. The only thing that this showed us, is that its design is perhaps not as straightforward. On the other hand, interesting results specific to the design were obtained from QUCS simulations (using a combination of lumped and distributed elements). These simulations showed that the quality of the resonator is highly sensitive to the position of its galvanically connected bias line, with respect to the center of the cavity (voltage node). Where Q_int drops down to 10 % of its maximum for < 0.1 mm (= l/320) away from the center. Furthermore, we saw that asymmetry in the ports causes the maximum Q_int (voltage node) to be off-center by 9 μm, and that adding an extra feedline in symmetry to the existing one puts it back on-center. Demonstrating the voltage node susceptibility to asymmetries of the cavity mode. Despite the unpromising signs the results show, suggestions have been put forward about better isolation of the bias line that have the potential to still make the design attractive. While these isolations still baring the traits of a faster and less complex fabrication.

Quantum-limited parametric amplifiers have become increasingly interesting and relevant with the progressing field of quantum computing. However, currently it is still challenging to fabricate these complex devices. In this thesis, we developed cross type Josephson junctions (JJs), lumped element LC resonators, lumped element Josephson parametric amplifiers (JPAs) and Josephson traveling wave parametric amplifiers (JTWPAs). We found and resolved several fabrication issues however we suspect that both the JPA and JTWPA suffered from microshorts in the circuits during the cryogenic measurements. Nevertheless, we managed to measure the resonance frequency of the JPA and have noticed the flux focusing effect when we flux tune it. We preformed room temperature measurements on over 300 JJs and calculated the relative standard deviation (RSD) for various JJ sizes. We found the spread of 30 fabricated resonators and a rough estimate of their Q factors. The JTWPAs developed are non-degenerate four wave mixing amplifiers that use the resonant phase matching (RPM) technique to reduce the phase mismatch problem. The transmission through the measured JTWPA showed 50 dB loss probably caused by presumably a short from the transmission line to ground. However, when the JTWPA was driven with a pump we observed 20 dB increase in transmission in some frequency bands.

The goal of the research described in this thesis was to investigate the basis of dissipation dilution as well as make it comprehensible. In the written literature we could find that the kinetic energy of an oscillating beam is stored into two types of potential energy~\cite{Kippenberg, Kotthaus}. A dissipative bending component and a conservative component due to elongation.

After this the different types of damping were introduced. Of which structural damping was most important, it is experimentally found to be approximately constant for many materials over a large band of frequencies. The loss tangent and quality factor for this type of damping are both constant. The physical origin of this behaviour isn't really understood. But there have been ideas hinted that it is due to surface imperfections~\cite{Kippenberg}.

To simulate dissipation dilution a spring system has been developed. In this system part of the energy is stored in torsion springs and another part in elongation springs. From this model it is observed that the effective spring constant of the total system depends on the initial strain. At low amounts of strain the spring constant is similar to that of torsion springs while at higher strains it becomes more like the elongation spring model. The quality factor of the beam is found to increase linearly with the strain.

This thesis analyses a lumped element circuit proposed for an analogue quantum simulation of opto-mechanics. The circuit consists of two resonators, a simple LC-resonator, and a similar resonator in which the inductor is replaced by a SQUID as a flux tuneable inductance. The interaction between the two resonators is established by mutual inductance between the inductor in the LC-resonator and the SQUID loop part of the other resonator. This makes the resonance frequency of the SQUID-resonator a function of the flux in the LC-resonator. It is shown that mutual and self inductances in the SQUID loop give rise to two transcendental constraints, for the magnetic flux in the SQUID loop, and the generalised flux across the SQUID. By an approximate solution for the constraints and under the assumption that the loop inductance is small compared to the SQUID inductance, we derive an approximate description for the circuit dynamics. We demonstrate that the circuit Hamiltonian contains the asymmetric opto-mechanical interaction, but in addition also a self-Kerr non-linearity in the analogue optical cavity, as well as a weak cross-Kerr interaction between the two resonators.

In this thesis, a we have designed and fabricated a Josephson Parametric Amplifier (JPA) using a new double-angle evaporation method without a Dolan bridge. We have found and resolved several issues in the fabrication procedure, but it requires further tuning before being fully functional. We have also simulated the behaviour of a general parametric amplifier with an additional Duffing non-linear term, and found that this term appears to limit the oscillation amplitude. We have attempted to characterize Josephson junctions fabricated with the new double-angle evaporation procedure, but without much success. Using a different fabrication method, a JPA was made and successfully characterized. The maximum measured gain is 16 dB, with a bandwidth of 1 MHz. The noise temperature is comparable to the cryostat temperature of 250 mK, but it was not characterized accurately.

A multilayer graphene oscillator is coupled with a superconducting microwave cavity. A nonlinear term in the restoring force is determined and the behavior of the oscillator is investigated.