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We investigate non-reciprocal scattering within the modes of a microwave frequency comb. Adjusting the pump frequencies, amplitudes, and phases of a Josephson parametric oscillator, we control constructive interference for the $m \longrightarrow \ell$ scattering processes, while concurrently achieving destructive interference for the inverse process $\ell \longrightarrow m$. We outline the methodology for realizing non-reciprocity in the context of two-mode isolation and a three-mode circulation, which we extend to multiple modes. We find good agreement between the experiments and a linearized theoretical model. Nonreciprocal scattering expands the toolset for parametric control, with the potential to engineer novel quantum correlations., Comment: 9 pages, 5 figures

We describe an experiment demonstrating the generation of three independent square-ladder continuous-variable cluster states with up to 94 qumodes of a microwave frequency comb. This entanglement structure at a large scale is realized by injecting vacuum fluctuations into a Josephson Parametric Amplifier pumped by three coherent signals around twice its resonance frequency, each having a particular well-defined phase relation. We reach up to 1.4 dB of squeezing of the nullifier which verifies the cluster state on the square ladder graph. Our results are consistent with a more familiar measure of two-mode squeezing, where we find up to 5.42 dB for one pump, and up to 1 dB for three pumps., Comment: 5 pages, 4 figures (main article) and 7 pages, 6 figures (suppl. material)

We study force-gradient sensing with a coherently driven mechanical resonator and phase-sensitive detection of motion through the two-tone backaction evading measurement of cavity optomechanics. The response of the optomechanical system, solved by numerical integration of the classical equations of motion, shows an extended region which is monotonic to changes in force gradient. We use Floquet theory to model the fluctuations, which rise only slightly above that of the usual backaction evading measurement in the presence of the mechanical drive. The monotonic response and minimal backaction are advantageous for applications such as atomic force microscopy., Comment: 13 pages, 3 figures

Two-level system (TLS) loss is typically limiting the coherence of superconducting quantum circuits. The loss induced by TLS defects is nonlinear, resulting in quality factors with a strong dependence on the circulating microwave power. We observe frequency mixing due to this nonlinearity by applying a two-tone drive to a coplanar waveguide resonator and measuring the intermodulation products using a multifrequency lock-in technique. This intermodulation spectroscopy method provides an efficient approach to characterizing TLS loss in superconducting circuits. Using harmonic balance reconstruction, we recover the nonlinear parameters of the device-TLS interaction, which are in good agreement with the standard tunnelling model for TLSs., Comment: 10 pages, 11 figures

Control over the coupling between multiple modes of a frequency comb is an important step toward measurement-based quantum computation with a continuous-variable system. We demonstrate the creation of square-ladder correlation graphs in a microwave comb with 95 modes. The graphs are engineered through precise control of the relative phase of three pumps applied to a Josephson parametric oscillator. Experimental measurement of the mode scattering matrix is in good agreement with theoretical predictions based on a linearized equation of motion of the parametric oscillator. The digital methods used to create and measure the correlations are easily scaled to more modes and more pumps, with the potential to tailor a specific correlation graph topology., Comment: 9 pages, 7 figures

Electromechanical transduction gain of 21 dB is realized in a micro-cantilever resonant force sensor operated in the unresolved-sideband regime. Strain-dependent kinetic inductance weakly couples cantilever motion to a superconducting nonlinear resonant circuit. A single pump generates motional sidebands and parametrically amplifies them via four-wave mixing. We study the gain and added noise, and we analyze potential benefits of this integrated amplification process in the context force sensitivity., Comment: 6 pages, 4 figures, supplementary attached

We study the response of several microwave resonators made from superconducting NbTiN thin-film meandering nanowires with large kinetic inductance, having different circuit topology and coupling to the transmission line. Reflection measurements reveal the parameters of the circuit and analysis of their temperature dependence in the range 1.7-6 K extract the superconducting energy gap and critical temperature. The lumped-element LC resonator, valid in our frequency range of interest, allows us to predict the quasiparticle contribution to internal loss, independent of circuit topology and characteristic impedance. Our analysis shows that the internal quality factor is limited not by thermal-equilibrium quasiparticles, but an additional temperature-dependent source of internal microwave loss., Comment: 10 pages, 6 figures

We describe a transducer for low-temperature atomic force microscopy based on electromechanical coupling due to a strain-dependent kinetic inductance of a superconducting nanowire. The force sensor is a bending triangular plate (cantilever) whose deflection is measured via a shift in the resonant frequency of a high-Q superconducting microwave resonator at 4.5 GHz. We present design simulations including mechanical finite-element modeling of surface strain and electromagnetic simulations of meandering nanowires with large kinetic inductance. We discuss a lumped-element model of the force sensor and describe the role of an additional shunt inductance for tuning the coupling to the transmission line used to measure the microwave resonance. A detailed description of our fabrication is presented, including information about the process parameters used for each layer. We also discuss the fabrication of sharp tips on the cantilever using focused electron beam-induced deposition of platinum. Finally, we present measurements that characterize the spread of mechanical resonant frequency, the temperature dependence of the microwave resonance, and the sensor's operation as an electromechanical transducer of force., Comment: 15 pages, 10 figures, 1 table

We implement an iSWAP gate with two transmon qubits using a flux-tunable coupler. Precise control of the relative phase of the qubit-control pulses and the parametric-coupler drive is achieved with a multi-channel instrument called Presto using direct digital synthesis (DDS), a promising technique for scaling up quantum systems. We describe the process of tuning and benchmarking the iSWAP gate, where the relative phase of the pulses is controlled via software. We perform the iSWAP gate in 290 ns, validate it with quantum-state tomography, and measure 2\% error with interleaved randomized benchmarking.

We use the principles of cavity optomechanics to design a resonant mechanical force sensor for atomic force microscopy. The sensor is based on a type of electromechanical coupling, dual to traditional capacitive coupling, whereby the motion of a cantilever induces surface strain that causes a change in the kinetic inductance of a superconducting nanowire. The cavity is realized by a compact microwave-plasma mode with an equivalent $LC$ circuit involving the kinetic inductance of the nanowire. The device is fully coplanar and we show how to transform the cavity impedance for optimal coupling to the transmission line and the following amplifier. For the device presented here, we estimate the bare kinetic inductive mechano-electric coupling (KIMEC) rate $g_0 / 2 \pi$ in the range 3-10 Hz. We demonstrate phase-sensitive detection of cantilever motion using a multifrequency pumping and measurement scheme., Comment: 10 pages, 4 figures

We describe a digital microwave platform called Presto, designed for measurement and control of multiple quantum bits (qubits) and based on the third-generation radio-frequency system on a chip. Presto uses direct digital synthesis to create signals up to 9 GHz on 16 synchronous output ports, while synchronously analyzing response on 16 input ports. Presto has 16 DC-bias outputs, 4 inputs and 4 outputs for digital triggers or markers, and two continuous-wave outputs for synthesizing frequencies up to 15 GHz. Scaling to a large number of qubits is enabled through deterministic synchronization of multiple Presto units. A Python application programming interface configures a firmware for synthesis and analysis of pulses, coordinated by an event sequencer. The analysis integrates template matching (matched filtering) and low-latency (184 - 254 ns) feedback to enable a wide range of multi-qubit experiments. We demonstrate Presto's capabilities with experiments on a sample consisting of two superconducting qubits connected via a flux-tunable coupler. We show single-shot readout and active reset of a single qubit; randomized benchmarking of single-qubit gates showing 99.972% fidelity, limited by the coherence time of the qubit; and calibration of a two-qubit iSWAP gate., Comment: v3: peer reviewed, accepted manuscript; v2: correct theoretical gate fidelity in Sec. III C

Significant progress has been made with multipartite entanglement of discrete qubits, but continuous variable systems may provide a more scalable path toward entanglement of large ensembles. We demonstrate multipartite entanglement in a microwave frequency comb generated by a Josephson parametric amplifier subject to a bichromatic pump. We find 64 correlated modes in the transmission line using a multifrequency digital signal processing platform. Full inseparability is verified in a subset of seven modes. Our method can be expanded to generate even more entangled modes in the near future., Comment: Updated: 18 pages and 11 figures, including supplementary material. Some textual changes to the main text, details about data analysis moved to the supplementary material. Figures in the main text are updated with better axis labels. Supplementary materials have a new section added and more supporting figures. Updated the labeling on most old figures

Hosting non-classical states of light in three-dimensional microwave cavities has emerged as a promising paradigm for continuous-variable quantum information processing. Here we experimentally demonstrate high-fidelity generation of a range of Wigner-negative states useful for quantum computation, such as Schr\"{o}dinger-cat states, binomial states, Gottesman-Kitaev-Preskill (GKP) states, as well as cubic phase states. The latter states have been long sought after in quantum optics and were never achieved experimentally before. To do so, we use a sequence of interleaved selective number-dependent arbitrary phase (SNAP) gates and displacements. We optimize the state preparation in two steps. First we use a gradient-descent algorithm to optimize the parameters of the SNAP and displacement gates. Then we optimize the envelope of the pulses implementing the SNAP gates. Our results show that this way of creating highly non-classical states in a harmonic oscillator is robust to fluctuations of the system parameters such as the qubit frequency and the dispersive shift.

Exploiting multiple modes in a quantum acoustic device could enable applications in quantum information in a hardware-efficient setup, including quantum simulation in a synthetic dimension and continuous-variable quantum computing with cluster states.We develop a multimode surface acoustic wave (SAW) resonator with a superconducting quantum interference device (SQUID) integrated in one of the Bragg reflectors. The interaction with the SQUID-shunted mirror gives rise to coupling between the more than 20 accessible resonator modes. We exploit this coupling to demonstrate two-mode squeezing of SAW phonons, as well as four-mode multipartite entanglement. Our results open avenues for continuous-variable quantum computing in a compact hybrid quantum system.

We present an alternative approach to pump-probe spectroscopy for measuring fast charge dynamics with an atomic force microscope (AFM). Our approach is based on coherent multifrequency lock-in measurement of the intermodulation between a mechanical drive and an optical or electrical excitation. In response to the excitation, the charge dynamics of the sample is reconstructed by fitting a theoretical model to the measured frequency spectrum of the electrostatic force near resonance of the AFM cantilever. We discuss the time resolution, which in theory is limited only by the measurement time, but in practice is of order one nanosecond for standard cantilevers and imaging speeds. We verify the method with simulations and demonstrate it with a control experiment, achieving a time resolution of $20~\mathrm{ns}$ in ambient conditions, limited by thermal noise.

We describe a phase-coherent multifrequency lock-in measurement technique that uses the inverse Fourier transform to reconstruct the nonlinear current-voltage characteristic (IVC) of a nanoscale junction. The method provides for a separation of the galvanic and displacement currents in the junction, and easy cancellation of the parasitic displacement current from the measurement leads. These two features allow us to overcome traditional limitations imposed by the low conductance of the junction and high capacitance of the leads, thus providing an increase in measurement speed of several orders of magnitude. We demonstrate the method in the context of conductive atomic force microscopy, acquiring IVCs at every pixel while scanning at standard imaging speed.

Non-invasive thermal noise calibration of both torsional and flexural eigenmodes is performed on numerous cantilevers of 10 different types. We show that for all tipless and short-tipped cantilevers, the ratio of torsional to flexural mode stiffness is given by a constant, times the ratio of their resonant frequencies. By determining this constant we enable a calibration of the torsional eigenmode, starting from a calibration of the flexural eigenmode. Our results are well motivated from beam theory and we verify them with finite element simulation., Comment: 6 pages, 3 figures

The interaction between a rapidly oscillating atomic force microscope tip and a soft material surface is described using both elastic and viscous forces with a moving surface model. We derive the simplest form of this model, motivating it as a way to capture the impact dynamics of the tip and sample with an interaction consisting of two components: interfacial or surface force, and bulk or volumetric force. Analytic solutions to the piece-wise linear model identify characteristic time constants, providing a physical explanation of the hysteresis observed in the measured dynamic force quadrature curves. Numerical simulation is used to fit the model to experimental data and excellent agreement is found with a variety of different samples. The model parameters form a dimensionless impact-rheology factor, giving a quantitative physical number to characterize a viscoelastic surface that does not depend on the tip shape or cantilever frequency., Comment: 13 pages, 7 figures

Background forces are linear long-range interactions of the cantilever body with its surroundings that must be compensated for in order to reveal tip-surface force, the quantity of interest for determining material properties in atomic force microscopy. We provide a mathematical derivation of a method to compensate for background forces, apply it to experimental data, and discuss how to include background forces in simulation. Our method, based on linear-response theory in the frequency domain, provides a general way of measuring and compensating for any background force and it can be readily applied to different force reconstruction methods in dynamic AFM., Comment: Author prepared version of a manuscript published at Physical Review Applied. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI

Atomic force microscope (AFM) users often calibrate the spring constants of cantilevers using functionality built into individual instruments. This is performed without reference to a global standard, which hinders robust comparison of force measurements reported by different laboratories. In this article, we describe a virtual instrument (an internet-based initiative) whereby users from all laboratories can instantly and quantitatively compare their calibration measurements to those of others - standardising AFM force measurements - and simultaneously enabling non-invasive calibration of AFM cantilevers of any geometry. This global calibration initiative requires no additional instrumentation or data processing on the part of the user. It utilises a single website where users upload currently available data. A proof-of-principle demonstration of this initiative is presented using measured data from five independent laboratories across three countries, which also allows for an assessment of current calibration., Comment: 34 pages, 14 figures

Friction is a complicated phenomenon involving nonlinear dynamics at different length and time scales[1, 2]. The microscopic origin of friction is poorly understood, due in part to a lack of methods for measuring the force on a nanometer-scale asperity sliding at velocity of the order of cm/s.[3, 4] Despite enormous advance in experimental techniques[5], this combination of small length scale and high velocity remained illusive. Here we present a technique for rapidly measuring the frictional forces on a single asperity (an AFM tip) over a velocity range from zero to several cm/s. At each image pixel we obtain the velocity dependence of both conservative and dissipative forces, revealing the transition from stick-slip to a smooth sliding friction[1, 6]. We explain measurements on graphite using a modified Prandtl-Tomlinson model that takes into account the damped elastic deformation of the asperity. With its greatly improved force sensitivity and very small sliding amplitude, our method enables rapid and detailed surface mapping of the full velocity-dependence of frictional forces with less than 10~nm spatial resolution., Comment: 7 pages, 4 figures

We demonstrate an alternative to Kelvin Probe Force Microscopy for imaging surface potential. The open-loop, single-pass technique applies a low-frequency AC voltage to the atomic force microscopy tip while driving the cantilever near its resonance frequency. Frequency mixing due to the nonlinear capacitance gives intermodulation products of the two drive frequencies near the cantilever resonance, where they are measured with high signal to noise ratio. Analysis of this intermodulation response allows for quantitative reconstruction of the contact potential difference. We derive the theory of the method, validate it with numerical simulation and a control experiment, and we demonstrate its utility for fast imaging of the surface photo-voltage on an organic photo-voltaic material., Comment: 4 pages, 3 figures, peer-reviewed, preprint

Conventional dynamic atomic force microscopy (AFM) can be extended to bimodal and multimodal AFM in which the cantilever is simultaneously excited at two ore more resonance frequencies. Such excitation schemes result in one additional amplitude and phase images for each driven resonance, and potentially convey more information about the surface under investigation. Here we present a theoretical basis for using this information to approximate the parameters of a tip-surface interaction model. The theory is verified by simulations with added noise corresponding to room-temperature measurements.

We present a theoretical framework for the dynamic calibration of the higher eigenmode parameters (stiffness and optical lever responsivity) of a cantilever. The method is based on the tip-surface force reconstruction technique and does not require any prior knowledge of the eigenmode shape or the particular form of the tip-surface interaction. The calibration method proposed requires a single-point force measurement using a multimodal drive and its accuracy is independent of the unknown physical amplitude of a higher eigenmode., Comment: 4 pages, 4 figures

We have studied quantum phase-slip (QPS) phenomena in long one-dimensional Josephson junction series arrays with tunable Josephson coupling. These chains were fabricated with as many as 2888 junctions, where one sample had a tunable weak link in the middle. Measurements were made of the zero-bias resistance, $R_0$, as well as current-voltage characteristics (IVC). The finite $R_0$ is explained by QPS and shows an exponential dependence on $\sqrt{E_J/E_C}$ with a distinct change in the exponent at $R_0=R_Q=h/4e^2$. When $R_0 > R_Q$ the IVC clearly shows a remnant of the Coulomb blockade, which evolves to a zero-current state with a sharp critical voltage as $E_J$ is tuned to a smaller value. The zero-current state below the critical voltage is due to coherent QPS and we show that these are enhanced at the central weak link. Above the critical voltage a negative differential resistance is observed which nearly restores the zero-current state.

We propose a theoretical framework for reconstructing tip-surface interactions using the intermodulation technique when more than one eigenmode is required to describe the cantilever motion. Two particular cases of bimodal motion are studied numerically: one bending and one torsional mode, and two bending modes. We demonstrate the possibility of accurate reconstruction of a two-dimensional conservative force field for the former case, while dissipative forces are studied for the latter., Comment: 11 pages, 10 figures

We studied current-voltage characteristics of long one dimensional Josephson junction chains with Josephson energy much larger than charging energy, $E_J \gg E_C$. In this regime, typical IV curves of the samples consist of a supercurrent branch at low bias voltages followed by a voltage-independent chain current branch, $I_{Chain}$ at high bias. Our experiments showed that $I_{Chain}$ is not only voltage-independent but it is also practically temperature-independent up to $T_C$. We have successfully model the transport properties in these chains using a capacitively shunted junction model with nonlinear damping.

We demonstrate quantitative force imaging of long-range magnetic forces simultaneously with near-surface van-der-Waals and contact-mechanics forces using intermodulation atomic force microscopy. Magnetic forces at the 200 pN level are separated from near-surface forces at the 30 nN level. Imaging of these forces is performed in both the contact and non-contact regimes of near-surface interactions.

We perform simulations and experiments on an oscillating atomic force microscope cantilever approaching a surface, where the intermodulation response of the cantilever driven with two pure harmonic tones is investigated. In the simulations, the tip and surface interact with a conservative nonlinear force, and the parameter space of approach distance and surface stiffness is explored. Approach experiments are carried out on three surfaces with widely varying stiffness. Qualitative similarities between simulations and experiment can be seen, but quantitative comparison is difficult due to the overly idealized tip-surface force model used in the simulation., Comment: 8 pages, 5 figures

We present polynomial force reconstruction from experimental intermodulation atomic force microscopy (ImAFM) data. We study the tip-surface force during a slow surface approach and compare the results with amplitude-dependence force spectroscopy. Based on polynomial force reconstruction we generate high-resolution surface property maps of polymer blend samples. The polynomial method is described as a special example of a more general approximataive force reconstruction, where the aim is to determine model parameters which best approximate the measured force spectrum. This approximative approach is not limited to spectral data and we demonstrate how is can adapted to a force quadrature picture., Comment: 10 pages, 5 figures, 1 mp4 movie as ancillary file

Intermodulation atomic force microscopy (ImAFM) is a mode of dynamic atomic force microscopy that probes the nonlinear tip-surface force by measurement of the mixing of multiple tones in a frequency comb. A high $Q$ cantilever resonance and a suitable drive comb will result in tip motion described by a narrow-band frequency comb. We show by a separation of time scales, that such motion is equivalent to rapid oscillations at the cantilever resonance with a slow amplitude and phase or frequency modulation. With this time domain perspective we analyze single oscillation cycles in ImAFM to extract the Fourier components of the tip-surface force that are in-phase with tip motion ($F_I$) and quadrature to the motion ($F_Q$). Traditionally, these force components have been considered as a function of the static probe height only. Here we show that $F_I$ and $F_Q$ actually depend on both static probe height and oscillation amplitude. We demonstrate on simulated data how to reconstruct the amplitude dependence of $F_I$ and $F_Q$ from a single ImAFM measurement. Furthermore, we introduce ImAFM approach measurements with which we reconstruct the full amplitude and probe height dependence of the force components $F_I$ and $F_Q$, providing deeper insight into the tip-surface interaction. We demonstrate the capabilities of ImAFM approach measurements on a polystyrene polymer surface., Comment: 12 pages, 7 figures

In atomic force microscopy (AFM) tip-surface interactions are usually considered as functions of the tip position only, so-called force curves. However, tip-surface interactions often depend on the tip velocity and the past tip trajectory. Here, we introduce a compact and general description of these interactions appropriate to dynamic AFM where the measurement of force is restricted to a narrow frequency band. We represent the tip-surface interaction in terms of a force disk in the phase space of position and velocity. Determination of the amplitude dependence of tip-surface forces at a fixed static probe height allows for a comprehensive treatment of conservative and dissipative interactions. We illuminate the fundamental limitations of force reconstruction with narrow band dynamic AFM and we show how the amplitude dependence of the Fourier component of the force at the tip oscillation frequency, gives qualitative insight into the detailed nature of the tip-surface interaction. With minimal assumptions this amplitude dependence force spectroscopy allows for a quantitative reconstruction of the effective conservative tip-surface force as well as a position-dependent damping factor. We demonstrate this reconstruction on simulated intermodulation AFM data.

Knowledge of surface forces is the key to understanding a large number of processes in fields ranging from physics to material science and biology. The most common method to study surfaces is dynamic atomic force microscopy (AFM). Dynamic AFM has been enormously successful in imaging surface topography, even to atomic resolution, but the force between the AFM tip and the surface remains unknown during imaging. Here, we present a new approach that combines high accuracy force measurements and high resolution scanning. The method, called amplitude-dependence force spectroscopy (ADFS) is based on the amplitude-dependence of the cantilever's response near resonance and allows for separate determination of both conservative and dissipative tip-surface interactions. We use ADFS to quantitatively study and map the nano-mechanical interaction between the AFM tip and heterogeneous polymer surfaces. ADFS is compatible with commercial atomic force microscopes and we anticipate its wide-spread use in taking AFM toward quantitative microscopy.

Various methods of force measurement with the Atomic Force Microscope (AFM) are compared for their ability to accurately determine the tip-surface force from analysis of the nonlinear cantilever motion. It is explained how intermodulation, or the frequency mixing of multiple drive tones by the nonlinear tip-surface force, can be used to concentrate the nonlinear motion in a narrow band of frequency near the cantilevers fundamental resonance, where accuracy and sensitivity of force measurement are greatest. Two different methods for reconstructing tip-surface forces from intermodulation spectra are explained. The reconstruction of both conservative and dissipative tip-surface interactions from intermodulation spectra are demonstrated on simulated data., Comment: 25 pages (preprint, double space) 7 figures

Nonlinear systems can be probed by driving them with two or more pure tones while measuring the intermodulation products of the drive tones in the response. We describe a digital lock-in analyzer which is designed explicitly for this purpose. The analyzer is implemented on a field-programmable gate array, providing speed in analysis, real-time feedback and stability in operation. The use of the analyzer is demonstrated for Intermodulation Atomic Force Microscopy. A generalization of the intermodulation spectral technique to arbitrary drive waveforms is discussed.

Intermodulation atomic force microscopy (IMAFM) is a dynamic mode of atomic force microscopy (AFM) with two-tone excitation. The oscillating AFM cantilever in close proximity to a surface experiences the nonlinear tip-sample force which mixes the drive tones and generates new frequency components in the cantilever response known as intermodulation products (IMPs). We present a procedure for extracting the phase at each IMP and demonstrate phase images made by recording this phase while scanning. Amplitude and phase images at intermodulation frequencies exhibit enhanced topographic and material contrast., Comment: 6 pages, 6 pages

Nonlinear kinetic inductance in a high Q superconducting coplanar waveguide microresonator can cause a bifurcation of the resonance curve. Near the critical pumping power and frequency for bifurcation, large parametric gain is observed for signals in the frequency band near resonance. We show experimental results on signal and intermodulation gain which are well described by a theory of the parametric amplification based on a Kerr nonlinearity. Phase dependent gain, or signal squeezing, is verified with a homodyne detection scheme., Comment: Submitted to Physica Scripta, topical issue: Nobel Symposium on Quantum Bits, 2009. 10 pages, 5 figures. Version 2 contains a few new sentences about the current-phase relation of weak links

Experimental investigations of the nonlinear properties of superconducting niobium coplanar waveguide resonators are reported. The nonlinearity due to a current dependent kinetic inductance of the center conductor is strong enough to realize bifurcation of the nonlinear oscillator. When driven with two frequencies near the threshold for bifurcation, parametric amplification with a gain of +22.4 dB is observed., Comment: 4 pages, 5 figures, 1 table. In version 2: Added appendix with model description and fits to measurements. Minor corrections and rephrasing

We investigate by theory and experiment, the Josephson junction switching current detector in an environment with frequency dependent damping. Analysis of the circuit's phase space show that a favorable topology for switching can be obtained with overdamped dynamics at high frequencies. A pulse-and-hold method is described, where a fast switch pulse brings the circuit close to an unstable point in the phase space when biased at the hold level. Experiments are performed on Cooper pair transistors and Quantronium circuits, which are overdamped at high frequencies with an on-chip RC shunt. For 20 us switch pulses the switching process is well described by thermal equilibrium escape, based on a generalization of Kramers formula to the case of frequency dependent damping. A capacitor bias method is used to create very rapid, 25 ns switch pulses, where it is observed that the switching process is not governed by thermal equilibrium noise., Comment: 11 pages, 8 figures, added references in chapter I, changed text in chapter II, III, V and VII

We have measured the current-voltage (I-V) characteristics of small-capacitance single Josephson junctions at low temperatures (T=0.02-0.6 K), where the strength of the coupling between the single junction and the electromagnetic environment was controlled with one-dimensional arrays of dc SQUIDs. The single-junction I-V curve is sensitive to the impedance of the environment, which can be tuned IN SITU. We have observed Coulomb blockade of Cooper-pair tunneling and even a region of negative differential resistance, when the zero-bias resistance R_0' of the SQUID arrays is much higher than the quantum resistance R_K = h/e^2 = 26 kohm. The negative differential resistance is evidence of coherent single-Cooper-pair tunneling within the theory of current-biased single Josephson junctions. Based on the theory, we have calculated the I-V curves numerically in order to compare with the experimental ones at R_0' >> R_K. The numerical calculation agrees with the experiments qualitatively. We also discuss the R_0' dependence of the single-Josephson-junction I-V curve in terms of the superconductor-insulator transition driven by changing the coupling to the environment., Comment: 11 pages with 14 embedded figures, RevTeX4, final version

We review experiments on small-capacitance Josephson junctions. When the Josephson junction is fabricated in the configuration of dc superconducting quantum interference device (SQUID), the Josephson coupling can be tuned IN SITU with an external magnetic field. The electrical transport properties of one-dimensional arrays of small-capacitance dc SQUIDs have been investigated in order to study the superconductor-insulator transition. The arrays have also been used to bias a single Josephson junction, and a clear Coulomb blockade of Cooper-pair tunneling has been observed in the single junction., Comment: 22 pages with 12 embedded figures, LaTeX2e, in "Studies of High Temperature Superconductors (Advances in Research and Applications) Vol. 43, BSCCO Tapes and More on Josephson Structures and Superconducting Electronics," edited by A. Narlikar (Nova Science Publishers, New York, 2002, ISBN1-590-33343-8) chap. 6

We have measured the current-voltage characteristics of small-capacitance single Josephson junctions at low temperatures (T < 0.04 K), where the strength of the coupling between the single junction and the electromagnetic environment was controlled with one-dimensional arrays of dc SQUIDs. We have clearly observed Coulomb blockade of Cooper-pair tunneling and even a region of negative differential resistance, when the zero-bias resistance of the SQUID arrays is much higher than the quantum resistance h/e^2 = 26 kohm. The negative differential resistance is evidence of coherent single-Cooper-pair tunneling in the single Josephson junction., Comment: RevTeX, 4 pages with 6 embedded figures