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arXiv:cond-mat/0702445

Dispersive microwave bifurcation of a superconducting resonator cavity incorporating a Josephson junction

Etienne Boaknin, Vladimir Manucharyan, Simon Fissette, Michael Metcalfe, Luigi Frunzio, Rajamani Vijay, Irfan A. Siddiqi, Andreas J. Wallraff, Robert J. Schoelkopf, Michel Devoret

Published Feb 19, 2007• 7 citations
arXivHTMLPDF

Abstract

We have observed the dynamical bistability of a microwave superconducting Fabry-Perot cavity incorporating a non-linear element in the form of Josephson tunnel junction. The effect, which is the analog of optical bistability, manifests itself in the transmission and reflection characteristics of the cavity and is governed by a competition between the wave amplitude dependence of the resonant frequency and the finite residence time of the field energy inside the cavity. This finite residence time is solely due to extraction of radiation from the cavity by the measurement process. The tight quantitative agreement with a simple model based on the Duffing oscillator equation shows that the nonlinearity, and hence the bifurcation phenomenon, is solely dispersive.

DOI
10.48550/arxiv.cond-mat/0702445

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We have observed the dynamical bistability of a microwave superconducting Fabry-Perot cavity incorporating a non-linear element in the form of Josephson tunnel junction. The effect, which is the analog of optical bistability, manifests itself in the transmission and reflection characteristics of the cavity and is governed by a competition between the wave amplitude dependence of the resonant frequency and the finite residence time of the field energy inside the cavity. This finite residence time is solely due to extraction of radiation from the cavity by the measurement process. The tight quantitative agreement with a simple model based on the Duffing oscillator equation shows that the nonlinearity, and hence the bifurcation phenomenon, is solely dispersive.
We report microwave transmission measurements of superconducting Fabry-Perot resonators, having a superconducting nanowire placed at a supercurrent antinode. As the plasma oscillation is excited, the supercurrent is forced to flow through the nanowire. The microwave transmission of the resonator-nanowire device shows a nonlinear resonance behavior, significantly dependent on the amplitude of the supercurrent oscillation. We show that such amplitude-dependent response is due to the nonlinearity of the current-phase relationship of the nanowire. The results are explained within a nonlinear oscillator model of the Duffing oscillator, in which the nanowire acts as a purely inductive element, in the limit of low temperatures and low amplitudes. The low-quality factor sample exhibits a ``crater'' at the resonance peak at higher driving power, which is due to dissipation. We observe a hysteretic bifurcation behavior of the transmission response to frequency sweep in a sample with a higher quality factor. The Duffing model is used to explain the Duffing bistability diagram. We also propose a concept of a nanowire-based qubit that relies on the current dependence of the kinetic inductance of a superconducting nanowire.
We performed a novel phase-sensitive microwave reflection experiment which directly probes the dynamics of the Josephson plasma resonance in both the linear and the nonlinear regime. When the junction was driven below the plasma frequency into the nonlinear regime, we observed for the first time the transition between two different dynamical states predicted for nonlinear systems. In our experiment, this transition appears as an abrupt change in the reflected signal phase at a critical excitation power. This controlled dynamical switching can form the basis of a sensitive amplifier, in particular, for the readout of superconducting qubits.
We review the theory, fabrication, and implementation of the Josephson bifurcation amplifier (JBA). At the core of the JBA is a nonlinear oscillator based on a reactively shunted Josephson junction. A weak input signal to the amplifier couples to the junction critical current I(0) and results in a dispersive shift in the resonator plasma frequency omega(p). This shift is enhanced by biasing the junction with a sufficiently strong microwave current I(rf) to access the nonlinear regime where omega(p) varies with I(rf). For a drive frequency omega(d) such that Omega=2Q(1-omega(d)/omega(p))>3, the oscillator enters the bistable regime where two nondissipative dynamical states O(L) and O(H), which differ in amplitude and phase, can exist. The sharp I(0) dependent transition from O(L) to O(H) forms the basis for a sensitive digital threshold amplifier. In the vicinity of the bistable regime (Omega<3), analog amplification of continuous signals is also possible. We present experimental data characterizing amplifier performance and discuss two specific applications--the readout of superconducting qubits (digital mode) and dispersive microwave magnetometry (analog mode).
Recent experiments have realized a familiar model of quantum optics, dispersive optical bistability, in the context of microwave coplanar resonators with internal Josephson junctions that are used to achieve large effective Kerr nonlinearities with very low dissipation. A Josephson junction qubit constitutes an effective atom that may be coupled to such a nonlinear resonator to provide a very useful probe that is sensitive to the quantum noise in the nonlinear cavity. In particular, I will explain how sidebands on the qubit excitation spectrum are sensitive to the effective temperature, oscillation frequency and damping rate of the intracavity field fluctuations.
Recent experiments have realized a familiar model of quantum optics, dispersive optical bistability, in the context of microwave coplanar resonators with internal Josephson junctions that are used to achieve large effective Kerr nonlinearities with very low dissipation. A Josephson junction qubit constitutes an effective atom that may be coupled to such a nonlinear resonator to provide a very useful probe that is sensitive to the quantum noise in the nonlinear cavity. In particular, I will explain how sidebands on the qubit excitation spectrum are sensitive to the effective temperature, oscillation frequency and damping rate of the intracavity field fluctuations.
The Josephson non-linearity provides a route to high sensitivity measurements through the bifurcation of a resonant mode into two metastable dynamical states. We probe the bifurcation regime of a superconducting microwave co-planar resonator incorporating a controlled non-linearity made of an array of SQUIDs. The switching probability between metastable states may be determined by repeated measurements under nominally identical conditions. We show that measurements of this probability and the inter-mode coupling introduced by the SQUIDs may be used for spectroscopy of other resonant modes. We also find the width of the switching-curve is within a factor two of quantitative agreement with Dykman's model, but that the switching probability is affected by a low frequency noise that has its source on-chip.