Nonlinear damping, the change in damping rate with the amplitude of oscillations plays an important role in many electrical, mechanical and even biological oscillators. In novel technologies such as carbon nanotubes, graphene membranes or superconducting resonators, the origin of nonlinear damping is sometimes unclear. This presents a problem, as the damping rate is a key figure of merit in the application of these systems to extremely precise sensors or quantum computers. Through measurements of a superconducting resonator, we show that from the interplay of quantum fluctuations and the nonlinearity of a Josephson junction emerges a power-dependence in the resonator response which closely resembles nonlinear damping. The phenomenon can be understood and visualized through the flow of quasi-probability in phase space where it reveals itself as dephasing. Crucially, the effect is not restricted to superconducting circuits: we expect that quantum fluctuations or other sources of noise give rise to apparent nonlinear damping in systems with a similar conservative nonlinearity, such as nano-mechanical oscillators or even macroscopic systems.

Observing quantum phenomena in macroscopic objects, and the potential discovery of a fundamental limit in the applicability of quantum mechanics, has been a central topic of modern experimental physics. Highly coherent and heavy micromechanical oscillators controlled by superconducting circuits are a promising system for this task. Here we focus in particular on the electrostatic coupling of motion to a weakly anharmonic circuit, namely, the transmon qubit. In the case of a megahertz mechanical oscillator coupled to a gigahertz transmon, we explain the difficulties in bridging the large electromechanical frequency gap. To remedy this issue, we explore the requirements to reach phonon-number resolution in the resonant coupling of a megahertz transmon and a mechanical oscillator.

Attempting to reconcile general relativity with quantum mechanics is one of the great undertakings of contemporary physics. Here, the authors present how the incompatibility between the two theories arises in the simple thought experiment of preparing a heavy object in a quantum superposition. Following Penrose's analysis of the problem, the authors determine the requirements on physical parameters to perform experiments where both theories potentially interplay. The authors use these requirements to compare different systems, focusing on mechanical oscillators that can be coupled to superconducting circuits.

In the published paper, the Hamiltonian featured in figure 2 was incorrect. The term EJ/(12h) should be EJ/(24h), as shown in the corrected figure in this corrigendum. Note that this error was simply typographic, and was never reflected in the QuCAT software. (Figure Presented).

We present the design, measurement, and analysis of a current sensor based on a process of Josephson parametric upconversion in a superconducting microwave cavity. When a coplanar waveguide is terminated with a nanobridge-constriction Josephson junction, we observe modulation sidebands from the cavity that enable highly sensitive frequency-multiplexed output of small currents for applications such as readout of transition-edge sensor arrays. We derive an analytical model to reproduce the measurements over a wide range of bias current, detuning, and input power. When the frequency of the cavity is tuned by more than 100 MHz with a dc current, our device achieves a minimum current sensitivity of 8.9pA/Hz. Extrapolating the results of our analytical model, we predict an improved device based on our platform, capable of achieving a sensitivity down to 50fA/Hz, or even lower if one can take advantage of parametric amplification in the Josephson cavity. Taking advantage of the Josephson architecture, our approach can provide higher sensitivity than kinetic inductance designs, and potentially enables detection of currents ultimately limited by quantum noise.

Quantum circuits constructed from Josephson junctions and superconducting electronics are key to many quantum computing and quantum optics applications. Designing these circuits involves calculating the Hamiltonian describing their quantum behavior. Here we present QuCAT, or 'Quantum Circuit Analyzer Tool', an open-source framework to help in this task. This open-source Python library features an intuitive graphical or programmatical interface to create circuits, the ability to compute their Hamiltonian, and a set of complimentary functionalities such as calculating dissipation rates or visualizing current flow in the circuit.

Multimode optomechanical systems are an emerging platform for studying fundamental aspects of matter near the quantum ground state and are useful in sensitive sensing and measurement applications. We study optomechanical cooling in a system where two nearly degenerate mechanical oscillators are coupled to a single microwave cavity. Due to an optically mediated coupling the two oscillators hybridize into a bright mode with a strong optomechanical cooling rate and a dark mode nearly decoupled from the system. We find that at high coupling, sideband cooling of the dark mode is strongly suppressed. Our results are relevant to novel optomechanical systems where multiple closely spaced modes are intrinsically present.

Detecting weak radio-frequency electromagnetic fields plays a crucial role in a wide range of fields, from radio astronomy to nuclear magnetic resonance imaging. In quantum optics, the ultimate limit of a weak field is a single photon. Detecting and manipulating single photons at megahertz frequencies presents a challenge because, even at cryogenic temperatures, thermal fluctuations are appreciable. Using a gigahertz superconducting qubit, we observed the quantization of a megahertz radio-frequency resonator, cooled it to the ground state, and stabilized Fock states. Releasing the resonator from our control, we observed its rethermalization with nanosecond resolution. Extending circuit quantum electrodynamics to the megahertz regime, we have enabled the exploration of thermodynamics at the quantum scale and allowed interfacing quantum circuits with megahertz systems such as spin systems or macroscopic mechanical oscillators.

When a two-level system (TLS) is coupled to an electromagnetic resonator, its transition frequency changes in response to the quantum vacuum fluctuations of the electromagnetic field, a phenomenon known as the Lamb shift. Remarkably, by replacing the TLS by a harmonic oscillator, normal-mode splitting leads to a quantitatively similar shift, without taking quantum fluctuations into account. In a weakly anharmonic system, lying in between the harmonic oscillator and a TLS, the origins of such shifts can be unclear. An example of this is the dispersive shift of a transmon qubit in circuit quantum electrodynamics (QED). Although often referred to as a Lamb shift, the dispersive shift observed in spectroscopy in circuit QED could contain a significant contribution from normal-mode splitting that is not driven by quantum fluctuations, raising the question: how much of this shift is quantum in origin? Here we treat normal-mode splitting separately from shifts induced by quantum vacuum fluctuations in the Hamiltonian of a weakly anharmonic system, providing a framework for understanding the extent to which observed frequency shifts can be attributed to quantum fluctuations.

In this experiment, we couple a superconducting transmon qubit to a high-impedance 645Ω microwave resonator. Doing so leads to a large qubit-resonator coupling rate g, measured through a large vacuum Rabi splitting of 2g≃910 MHz. The coupling is a significant fraction of the qubit and resonator oscillation frequencies ω, placing our system close to the ultrastrong coupling regime (g=g/ω=0.071 on resonance). Combining this setup with a vacuum-gap transmon architecture shows the potential of reaching deep into the ultrastrong coupling g∼0.45 with transmon qubits.

Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines, and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number n scales as n for n up to, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a nonzero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultrastrong coupling regime of a multimode circuit QED.

We experimentally investigate dissipation in mechanical resonators made of a disordered superconducting thin film of a Molybdenum-Rhenium(MoRe) alloy. Electrostatically driving the drum with a resonant AC voltage, we detect its motion using a superconducting microwave cavity. From the temperature dependence of mechanical resonance frequencies and quality factors, we find evidence for non-resonant, mechanically active two-level systems (TLSs) limiting its quality factor at low temperature. In addition, we observe a strong suppression of mechanical dissipation at large mechanical driving amplitudes, suggesting an unconventional saturation of the non-resonant TLSs. These observations shed light on the mechanism of mechanical damping in superconducting drums and routes towards understanding dissipation in such devices.

With the introduction of superconducting circuits into the field of quantum optics, many experimental demonstrations of the quantum physics of an artificial atom coupled to a single-mode light field have been realized. Engineering such quantum systems offers the opportunity to explore extreme regimes of light-matter interaction that are inaccessible with natural systems. For instance the coupling strength g can be increased until it is comparable with the atomic or mode frequency ω_a,m and the atom can be coupled to multiple modes which has always challenged our understanding of light-matter interaction. Here, we experimentally realize a transmon qubit in the ultra-strong coupling regime, reaching coupling ratios of g/ω_m = 0.19 and we measure multi-mode interactions through a hybridization of the qubit up to the fifth mode of the resonator. This is enabled by a qubit with 88% of its capacitance formed by a vacuum-gap capacitance with the center conductor of a coplanar waveguide resonator. In addition to potential applications in quantum information technologies due to its small size, this architecture offers the potential to further explore the regime of multi-mode ultra-strong coupling.