A superconducting quantum interference device (SQUID) inserted in a superconducting waveguide resonator imposes current and voltage boundary conditions that makes it suitable as a tuning element for the resonator modes. If such a SQUID element is subject to a periodically varying magnetic flux, the resonator modes acquire frequency sidebands. In this work we calculate the multifrequency eigenmodes of resonators coupled to periodically driven SQUIDs and we use the Lagrange formalism to propose a theory for their quantization. The elementary excitations of a multifrequency mode can couple resonantly to physical systems with different transition frequencies and this makes the resonator an efficient quantum bus for state transfer and coherent quantum operations in hybrid quantum systems. As an example of the application of our multifrequency modes, we determine their coupling to transmon qubits with different frequencies and we present a bichromatic scheme for entanglement and gate operations.
A superconducting quantum interference device (SQUID) inserted in a superconducting waveguide resonator imposes current and voltage boundary conditions that makes it suitable as a tuning element for the resonator modes. If such a SQUID element is subject to a periodically varying magnetic flux, the resonator modes acquire frequency sidebands. In this work we calculate the multifrequency eigenmodes of resonators coupled to periodically driven SQUIDs and we use the Lagrange formalism to propose a theory for their quantization. The elementary excitations of a multifrequency mode can couple resonantly to physical systems with different transition frequencies and this makes the resonator an efficient quantum bus for state transfer and coherent quantum operations in hybrid quantum systems. As an example of the application of our multifrequency modes, we determine their coupling to transmon qubits with different frequencies and we present a bichromatic scheme for entanglement and gate operations.
Hybrid quantum systems play an essential role in efforts towards the advancements of quantum information technology and quantum sensing by bridging distinct technological platforms. In particular, hybrid systems comprising superconducting microwave circuits are being pursued intensely in light of existing achievements in superconducting qubits and precise quantum control in microwave circuits. Their realization relies on the design of superconducting resonators capable of coupling to an oscillatory mode of a second quantum system, facilitating the probing and manipulation of one via the other. In order to accommodate the coupled system, its particular constraints and requirements must be respected, which may be quite extreme from the perspective of established superconducting circuit design and operation. Overcoming this challenge has proved to be a substantial obstacle for the attainment of strongly coupled hybrid systems, despite spectacular advances in superconducting microwave circuits individually. In this work, we examine two approaches tackling this issue for different hybrid quantum systems, focusing on the optimization and characterization of the superconducting circuits. The first device aims at a superconductor-atom hybrid system through the coupling to Rydberg-Rydberg transitions in ultra-cold Rb-87 atoms trapped near the superconducting chip. Using simulations in conjunction with theoretical considerations, we demonstrate a comprehensive geometry parameter optimization procedure for maximum coupling rate. Presenting an experimental implementation of the resulting circuit, we validate its suitability for use under the intended operation conditions, including the option of tuning the electronic transition frequencies using a dc voltage. Secondly, we investigate a device designed for applications in large magnetic in-plane fields used to mediate coupling to integrated micromechanical oscillators; more specifically it is a flux-tunable SQUID resonator comprising Josephson nano-constrictions as nonlinear inductive components. Performing a thorough characterization of its frequency and Kerr anharmonicity tuning behavior at fields up to several hundred mT, we discover a pronounced asymmetry in its flux response at large fields. We show this asymmetry to be caused by the nano-constrictions turning into Josephson diodes, develop an intuitive macroscopic model for its field-induced emergence and reconstruct the diode current-phase relation from the flux tuning data. Several figures of merit informing the suitability of the SQUID resonators for optomechanical systems as well as other applications are improved by the diode effect, highlighting the potential of Josephson diodes in superconducting microwave circuits. Our results provide a promising stepping stone for the advancement of hybrid systems involving superconducting microwave circuits, putting new and exciting operation and coupling regimes within reach.
In this thesis I present different types of manipulations of microwave fields using tunable superconducting resonators. A resonator is made tunable by adding one or more superconducting quantum interference devices, SQUIDs. The SQUID consists of a superconducting loop with two Josephson junctions and acts as a tunable nonlinear inductor. Modulation of the SQUID nonlinearity can be performed to induce different types of non-trivial oscillator dynamics. The first project I present is on subharmonic oscillations. Here, the SQUID in a tunable resonator is driven with an external signal at an integer multiple of the frequency w. When w is placed slightly below the first resonator mode, I show generation of radiation at w, which is known as frequency down-conversion. In my measurements, subharmonic oscillations have been detected from period doubling up to period quintupling. For the specific case of period tripling, theory is developed and I show good agreement between theory and experiments. The second project of the thesis is on a doubly tunable resonator. Here, I show creation of a superconducting resonator with two independently tunable boundary conditions. The idea with this system is to operate the resonator in a breathing or a translational mode and compare the two. For static dc magnetic flux the performance is very good. As a second step, I perform fast modulation of both SQUIDs to generate radiation. Even though some of the measurement results are promising, I also show some contradicting observations. These indicate that the actual modulation mechanism of the SQUID is not pumping of the magnetic flux in the SQUID loop as intended, but rather direct current driving of the SQUID. However, the system remains a promising idea for microwave manipulation and creation of interesting non-classical states. As a possible application of the doubly tunable resonator, I present a theory proposal on how to use it for measurements on relativistic effects. By using the tunability of the resonator boundary conditions, the resonator can simulate a space rocket and time dilation could be measured. However, due to the crosstalk problems in the doubly tunable resonator this experiment was never realized. Finally, I finish the thesis by presenting a design for a tunable microwave coupling. Papers E and F show a high on-off ratio for the coupler and demonstrate how it can be used for storage of microwave signals. Furthermore, the tunable coupler can be used to shape a microwave signal by tuning the strength of the coupling.
Qubit-resonator coupled systems have been intensively studied because of their potential usefulness in quantum information processing. We investigated a superconducting flux qubit coupled to a resonator mode (SQUID plasma mode) associated with a dc SQUID used to read out the qubit state. Higher-order red and blue sideband transitions with | M | = 1, 2, and 3, where M represents the change in the quantum number of the resonator state, were clearly observed away from the symmetry point. We calculated the transition matrix elements between the dressed states and examined their dependence on both the coupling strength and the flux bias for the qubit. The observation of the sidebands up to | M | = 3 is consistent with the calculation results.
Controllable coupling between the odd and even fundamental modes of an asymmetric half-wavelength superconducting coplanar waveguide resonator is demonstrated. The resonant frequency of the even mode Ωe could be tuned by an external magnetic field, while the resonant frequency of the odd mode Ωo is field independent. To realize the tunability of Ωe, the central conductor of the NbN-based resonator was galvanically coupled to an array of Al-based rf-SQUIDs (radio frequency superconducting quantum interferometer device). These rf-SQUIDs are placed in only one resonator gap, ensuring its strong asymmetry. By adjusting the appropriate external magnetic flux Φ, equal frequencies of both modes were obtained. At this resonant point Ωo(Φ)=Ωe(Φ) an avoided level crossing of the eigenfrequencies was observed, demonstrating the coupling between the odd and even fundamental modes.
We propose a scheme for implementing quantum information transfer based on frequency modes of microwave photons in a superconducting circuit. In our proposal, quantum information can be encoded on frequency modes of microwave photons, which act as a qubit in the resonator. Operations for the qubit, which is a process involving parametric frequency conversion, can be implemented by adjusting biased-dc superconducting quantum interference (SQUID). The coupling between two resonators can be controlled by tuning the frequency of the LC circuit inserted by a dc SQUID with two Josephson-junctions (2JJ-SQUID). Compared with previous ones, our work can avoid dephasing and decoherence resulting from atom decay. In addition, the resonator which includes multiple photons in two frequency modes can play a role of an identical atomic ensemble, which could lead to photon blockade.
In this thesis, measurements on superconducting electrical circuits that exhibit quantum mechanical effects are presented. Nonlinear superconducting circuits can be regarded as artificial atoms with energy spectra that can be engineered. The artificial atoms can be used as quantum bits for quantum computation. By coupling the artificial atoms to the electromagnetic field inside a transmission line resonator, quantum optical effects can be studied in these electrical circuits.The possibility of using a frequency tunable transmission line resonators for coupling of superconducting quantum bits is investigated. Tunable superconducting resonators consisting of coplanar waveguide structures, terminated to ground in one end via a superconducting quantum interference device (SQUID), are characterized. By applying a magnetic field to the SQUID loop, the boundary condition of the resonator changes leading to a change in resonance frequency.A change of the resonance frequency by more than 250 linewidths is demonstrated. It is also demonstrated that the resonator can be tune by several hundred MHz on a time scale that is much shorter than the photon lifetime of the resonator.Circuits containing both one and two transmon qubits coupled to the resonator are characterized. Quantum mechanical interactions between the resonator and the qubits, in form of vacuum Rabi splitting, are observed. Coherent control of a single qubit coupled to the resonator is demonstrated by performing Rabi oscillations of the qubit. In the two qubit case, coherent interaction between the qubits is observed in terms of a nonlinear scaling of the coupling strength between the resonator and the collective two qubit system.Measurements on a strongly driven single Cooper-pair box, using a quantum capacitance readout, are also presented. A rich structure in the response is well explained in terms of longitudinal dressed states of the single-Cooper pair box.When measuring the response of a single-electron box, a gate dependent excess dissipation is observed. The dissipation is understood by considering the internal dynamics of the single electron box. The excess dissipation is modeled as an effective resistance named the Sisyphus resistance.