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arXiv:2107.07971

Theory of high-power excitation spectra of rf-SQUID

Olesia Dmytruk, Ramiro H. Rodriguez, Çağlar Girit, Marco Schirò

Published Jul 16, 2021• 0 citations
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Abstract

We discuss the theory of linear and non-linear spectroscopy of an rf-SQUID coupled to a Josephson spectrometer. Recent experimental measurements on this system have shown a strongly non-linear absorption lineshape, whose current peak maximum undergoes a forward-backward bending transition depending on the value of the rf-SQUID phase. We show that this transition can be qualitatively understood by mapping the dynamics of the driven rf-SQUID onto a generalized Duffing oscillator, with tunable drive and non-linearity, undergoing a bifurcation. Finally we show that in order to quantitatively reproduce the experimental data reported in arXiv:2106.02632, it is crucial to include the feedback from the load-line, leading to an additional source of non-linearity.

DOI
10.1103/physrevb.104.214508

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We discuss the theory of linear and non-linear spectroscopy of an rf-SQUID coupled to a Josephson spectrometer. Recent experimental measurements on this system have shown a strongly non-linear absorption lineshape, whose current peak maximum undergoes a forward-backward bending transition depending on the value of the rf-SQUID phase. We show that this transition can be qualitatively understood by mapping the dynamics of the driven rf-SQUID onto a generalized Duffing oscillator, with tunable drive and non-linearity, undergoing a bifurcation. Finally we show that in order to quantitatively reproduce the experimental data reported in arXiv:2106.02632, it is crucial to include the feedback from the load-line, leading to an additional source of non-linearity.
We discuss the theory of linear and nonlinear spectroscopy of an rf superconducting quantum interference device (rf-SQUID) coupled to a Josephson spectrometer. Recent experimental measurements on this system have shown a strongly nonlinear absorption line shape, whose current peak maximum undergoes a forward-backward bending transition depending on the value of the rf-SQUID phase. We show that this transition can be qualitatively understood by mapping the dynamics of the driven rf-SQUID onto a generalized Duffing oscillator, with tunable drive and nonlinearity, undergoing a bifurcation. Finally we show that in order to quantitatively reproduce the experimental data reported by Griesmar et al. [Phys. Rev. Research 3, 043078 (2021)], it is crucial to include the feedback from the load line, leading to an additional source of nonlinearity.
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.
Using a superconducting ring containing a Nb point contact inductively coupled to a resonant circuit fed with an rf current source, (typical rf SQUID configuration) we have measured the tank voltage as a function of frequency in the regimeLI_{c}< \Phi_{0}/2\pi. ForLI_{c} \ll \Phi_{0}/2\pi, the curves of tank voltage versus drive current for various values of external flux\Phi_{x}through the ring agree with the calculations of Hansma. For higher critical currents, there is an asymmetry in these curves for frequencies above and below ω0, the resonant frequency of the circuit. Calculations using an expanded theory including the effects of a conductanceG_{o}(1+\alpha \cos \phi)and of screening flux give good agreement with the data. In principle α can be determined from these measurements, however the\cos \phiterm and the screening flux produce qualitatively similar effects. A quantitative determination of α requires a precise knowledge of Go/Ic. Using values of Goand Icfrom independent impedance measurements, the data yield a negative value of α in agreement with another experiment.
The response of nonlinear metamaterials and superconducting electronics to two-tone excitation is critical for understanding their use as low-noise amplifiers and tunable filters. A new setting for such studies is that of metamaterials made of radio frequency superconducting quantum interference devices (rf-SQUIDs). The two-tone response of self-resonant rf-SQUID meta-atoms and metamaterials is studied here via intermodulation (IM) measurement over a broad range of tone frequencies and tone powers. A sharp onset followed by a surprising strongly suppressed IM region near the resonance is observed. Using a two time scale analysis technique, we present an analytical theory that successfully explains our experimental observations. The theory predicts that the IM can be manipulated with tone power, center frequency, frequency difference between the two tones, and temperature. This quantitative understanding potentially allows for the design of rf-SQUID metamaterials with either very low or very high IM response.
We realize tunable coupling between two superconducting transmission line resonators. The coupling is mediated by a non-hysteretic rf SQUID acting as a flux-tunable mutual inductance between the resonators. We present a spectroscopic characterization of the device. In particular, we observe couplings g/2πg/2\pig/2π ranging between −320 MHz and 37 MHz. In the case of g≃0g \simeq 0g≃0 , the microwave power cross transmission between the two resonators is reduced by almost four orders of magnitude as compared to the case where the coupling is switched on.
The mechanisms of the decay of metastable current states in the superconducting loop of an rf SQUID in the hysteresis state are studied. Clamped point contacts are used as the Josephson weak link. The first derivative of the radio-frequency IVC dVT/dIRF, which gives the dependence of the decay probability density on the amplitude of the oscillations of the magnetic flux in the ring, was measured experimentally. Studies of the temperature dependence of the width of the probability distribution in the interval from 4.2 to 0.5 K show that for most high-resistance Nb–Nb, Nb–Ag–Nb contacts the quantum mechanisms of decay become dominant at a temperature of the order of 2 K. The experimentally measured parameters of the decay probability distribution in the quantum limit are compared with the theoretical values calculated based on the theory of macroscopic quantum tunneling in the limit of strong and weak dissipation. The experimental magnitude of the width of the probability density as well as the characteristic quantum temperature exceed the theoretical values, which can be explained by the deviation of the current-phase dependence of the contact from a sinusoidal dependence. In addition, some contacts apparently correspond to the case of an intermediate magnitude of dissipation. When the frequency of the rf oscillations changed from 30 to 6 MHz, the width of the distribution remained unchanged, which follows from the theory of quantum tunneling decay of the metastable current state in the loop in the limit of strong damping. At low temperatures (T ∼ 50 K) and for a sufficiently small damping coefficient, anomalous peaks were observed in the probability density with an amplitude of the rf oscillations significantly smaller than the critical magnitude of the magnetic flux in the loop.