arXiv:2009.13492 · Physical review. B./Physical review. B
In situ tunable nonlinearity and competing signal paths in coupled superconducting resonators
Michael Fischer, Qiming Chen, Christian Besson, Peter Eder, Jan Goetz, Stefan Pogorzalek, Michael Renger, Edwar Xie, Michael J. Hartmann, Kirill G. Fedorov, Achim Marx, Frank Deppe, Rudolf Groß
We have fabricated and studied a system of two tunable and coupled nonlinear superconducting resonators. The nonlinearity is introduced by galvanically coupled dc superconducting quantum interference devices. We simulate the system response by means of a circuit model, which includes an additional signal path introduced by the electromagnetic environment. Furthermore, we present two methods allowing us to experimentally determine the nonlinearity. First, we fit the measured frequency and flux dependence of the transmission data to simulations based on the equivalent circuit model. Second, we fit the power dependence of the transmission data to a model that is predicted by the nonlinear equation of motion describing the system. Our results show that we are able to tune the nonlinearity of the resonators by almost two orders of magnitude via an external coil and two on-chip antennas. The studied system represents a basic building block for larger systems, allowing for quantum simulations of bosonic many-body systems with a larger number of lattice sites.
We have fabricated and studied a system of two tunable and coupled nonlinear superconducting resonators. The nonlinearity is introduced by galvanically coupled dc superconducting quantum interference devices. We simulate the system response by means of a circuit model, which includes an additional signal path introduced by the electromagnetic environment. Furthermore, we present two methods allowing us to experimentally determine the nonlinearity. First, we fit the measured frequency and flux dependence of the transmission data to simulations based on the equivalent circuit model. Second, we fit the power dependence of the transmission data to a model that is predicted by the nonlinear equation of motion describing the system. Our results show that we are able to tune the nonlinearity of the resonators by almost two orders of magnitude via an external coil and two on-chip antennas. The studied system represents a basic building block for larger systems, allowing for quantum simulations of bosonic many-body systems with a larger number of lattice sites.
We experimentally study the behavior of a parametrically pumped nonlinear oscillator, which is based on a superconducting λ /4 resonator, and is terminated by a flux-tunable superconducting quantum interference device. We extract parameters for two devices. In particular, we study the effect of the nonlinearities in the system and compare to theory. The Duffing nonlinearity, α , is determined from the probe-power dependent frequency shift of the oscillator, and the nonlinearity, β , related to the parametric flux pumping, is determined from the pump amplitude for the onset of parametric oscillations. Both nonlinearities depend on the parameters of the device and can be tuned in situ by the applied dc flux. We also suggest how to cancel the effect of β by adding a small dc flux and a pump tone at twice the pump frequency.
We propose tunable superconducting split-ring resonators (SRRs) employing nonlinear Josephson inductance. A fraction of SRR is replaced by Nb-AlOx-Nb Josephson tunnel junctions connected in parallel and forming a superconducting quantum interference device, whose inductance is sensitive to the external dc magnetic field. Due to the lumped nature of the Josephson inductance, the SRR can be made very compact, and its resonance frequency can be tuned by applying magnetic field. We present the model, results of extensive electromagnetic-simulation and experimental data for the SRR weakly coupled to a transmission line within frequency range of 11–13 GHz.
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, 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.
We have fabricated and characterized systems consisting of superconducting quantum bits (qubits) of the transmon type, coupled to a tunable transmission line resonator. The resonator consists of a superconducting quarter wavelength coplanar waveguide (CPW) resonator that is made tunable by terminating the transmission line to ground through a superconducting quantum interference device (SQUID). Here we present measurements on two devices where we observe Rabi oscillations as we drive the qubit, and vacuum Rabi splitting both as we tune the qubits into resonance with the resonator and as we tune the resonator into resonance with the qubits. We also observe coherent interaction between two resonant qubits. In addition, we demonstrate that we can calculate the response very accurately using a multilevel Hamiltonian.
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.