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arXiv:cond-mat/0612576 · Physical Review B

Microwave bifurcation of a Josephson junction: Embedding-circuit requirements

Vladimir Manucharyan, Etienne Boaknin, Michael Metcalfe, Rajamani Vijay, Irfan A. Siddiqi, Michel Devoret

Published Jul 26, 2007• 89 citations
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Abstract

A Josephson tunnel junction which is rf driven near a dynamical bifurcation point can amplify quantum signals. However, the bifurcation point will exist robustly only if the electrodynamic environment of the junction meets certain criteria. We develop a general formalism for dealing with the nonlinear dynamics of a Josephson junction embedded in an arbitrary microwave circuit. We find sufficient conditions for the existence of the bifurcation regime: (a) the embedding impedance of the junction needs to present a resonance at a particular frequency ${\ensuremath{\omega}}_{R}$, with the quality factor $Q$ of the resonance and the participation ratio $p$ of the junction satisfying $Qp⪢1$, and (b) the drive frequency should be low frequency detuned away from ${\ensuremath{\omega}}_{R}$ by more than $\sqrt{3}{\ensuremath{\omega}}_{R}∕(2Q)$.

DOI
10.1103/physrevb.76.014524

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A Josephson tunnel junction which is rf driven near a dynamical bifurcation point can amplify quantum signals. However, the bifurcation point will exist robustly only if the electrodynamic environment of the junction meets certain criteria. We develop a general formalism for dealing with the nonlinear dynamics of a Josephson junction embedded in an arbitrary microwave circuit. We find sufficient conditions for the existence of the bifurcation regime: (a) the embedding impedance of the junction needs to present a resonance at a particular frequency \ensuremathωR{\ensuremath{\omega}}_{R}\ensuremathωR​, with the quality factor QQQ of the resonance and the participation ratio ppp of the junction satisfying Qp⪢1Qp⪢1Qp⪢1, and (b) the drive frequency should be low frequency detuned away from \ensuremathωR{\ensuremath{\omega}}_{R}\ensuremathωR​ by more than 3\ensuremathωR∕(2Q)\sqrt{3}{\ensuremath{\omega}}_{R}∕(2Q)3​\ensuremathωR​∕(2Q).
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).
We have investigated numerically a system consisting of a long Josephson junction embedded into a microstrip resonator. For such a configuration the Josephson vortex dynamics in the junction is driven by the oscillating currents in the resonator. We have calculated the complex impedance of the junction at the resonant frequency. The Q factor of the resonator and the change of the resonant frequency of the whole system can then be easily calculated knowing the parameters of the resonator without Josephson junction. The fluxon dynamics and the complex impedance of the junction are simulated for different values of the junction length, damping, dc bias current, and dc magnetic field. This allows us to explain the behavior of the impedance of the junction and different loss mechanisms. In particular, we predict a nonmonotonic dependence of the both real and imaginary parts of the impedance on the amplitude of the oscillating current in the resonator for some range of parameters. Our results show that such a combination of active and passive superconducting elements forms a rather interesting nonlinear physical system which might be useful for development of tunable/switchable superconducting resonators and suggests an experimental technique for detecting trapped vortices in long junctions.
A Josephson junction embedded in a dissipative circuit can be externally driven to induce nonlinear dynamics of its phase. Classically, under sufficiently strong driving and weak damping, dynamic multistability emerges, associated with dynamical bifurcations so that the often-used modeling as a Duffing oscillator, which can exhibit bistability at most, is insufficient. The present work analyzes in this regime corresponding quantum properties by mapping the problem onto a highly nonlinear quasienergy operator in a rotating frame. This allows us to identify in detail parameter regions where simplifications such as the Duffing approximation are valid, to explore classical-quantum correspondences, and to study how quantum fluctuations impact the effective junction parameters as well as the dynamics around higher-amplitude classical fixed points.
The authors present theoretical and experimental results on the effect of periodic perturbations on microwave-driven Josephson tunnel junctions close to a period-doubling bifurcation. The theory, which is quite general, describes how a periodic near-resonant perturbation changes the stability of the system. The phenomenon was investigated experimentally in small Nb-NbO/sub x/-Pb tunnel junctions, and experimental findings were found to be in agreement with the theory. The effect has importance for the performance of parametric amplifiers based on Josephson tunnel junctions.>
We analyze properties of bifurcation quantum detectors based on weakly nonlinear superconducting resonance circuits, in particular, with application to quantum readout. The developed quantitative description demonstrates strong influence of higher harmonics on their characteristics. While this effect is relevant for various circuits, including the conventional Josephson bifurcation amplifier and the parametrically driven circuit, we first focus on the period-doubling bifurcation under a force driving. This kind of bifurcation is due to nominally quadratic nonlinearity, which enables parametric down-conversion of the driving signal at nearly double resonance frequency to the basic mode. We analyze the effect of higher harmonics on the dynamics of the basic mode, inherent in a nonlinear circuit, which in our case is based on a Josephson junction with a sinusoidal current-phase relation as the origin of nonlinearity. We demonstrate that effects beyond the monochromatic approximation significantly modify the bare characteristics and evaluate their contribution. Due to high sensitivity of this circuit to small variations of parameters, it can serve as an efficient detector of the quantum state of superconducting qubits.