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Japanese Journal of Applied Physics

dc SQUID Microwave-Induced Characteristics and Magnetic Field Response, Derived Using the Stewart-McCumber Model

Seiichi Naitô

Published Apr 1, 1985• 3 citations
Publisher

Abstract

This paper describes the I-V characteristics (as a function of applied magnetic field) and magnetic field response of a dc SQUID which incorporates two weak-link Josephson junctions in a superconducting ring. These characteristics are derived by digital computer simulation, using the Stewart-McCumber model. The effects of asymmetry (dissimilarity between the junctions) on SQUID critical current I c, normal resistance R and capacitance C are also simulated. With microwave excitation applied, the Josephson oscillations of the dc SQUID are found to be “frequency-locked” to the applied microwave signal even when critical currents I c of the two junctions differed, Subharmonic voltage steps are obtained for a symmetric dc SQUID in a magnetic field even when the current-phase relationship is purely sinusoidal.

DOI
10.1143/jjap.24.449

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Journal of Applied Physics•Nov 1, 1981•29 citations
This paper describes the I-V characteristics (as a function of applied magnetic field) and magnetic field response of a dc SQUID which incorporates two weak-link Josephson junctions in a superconducting ring. These characteristics are derived by digital computer simulation, using the Stewart-McCumber model. The effects of asymmetry (dissimilarity between the junctions) on SQUID critical current I c, normal resistance R and capacitance C are also simulated. With microwave excitation applied, the Josephson oscillations of the dc SQUID are found to be “frequency-locked” to the applied microwave signal even when critical currents I c of the two junctions differed, Subharmonic voltage steps are obtained for a symmetric dc SQUID in a magnetic field even when the current-phase relationship is purely sinusoidal.
The dynamics and noise of a dc SQUID (the Superconducting Quantum Interference Device), with the McCumber parameter /spl beta//sub c/=2/spl pi/R/sup 2/l/sub c/C//spl Phi//sub 0/ close to the unity (where I/sub c/, R, C is the critical current, the shunt resistance and the capacitance of the Josephson junctions comprising the SQUID, respectively, /spl Phi//sub 0/=2.07 10/sup -15/ Wb is the magnetic flux quantum) integrated with a planar spiral input coil, have been experimentally studied. The length of the spiral input coil was chosen so as to match its /spl lambda//4 microwave resonance frequency to the plasma resonance frequency of the SQUID. The input coil resonance is found to enhance the overall quality factor Q of the Josephson oscillations in the SQUID and, as a result, to increase the dynamic resistance R/sub d/ and the gradient of the flux-to-voltage characteristics, /spl delta/V//spl delta//spl Phi/, without hysteresis. A dc SQUID with a loop inductance L=30.5 pH, /spl beta//sub c/=0.72, and a 6 turn input coil demonstrated a non-distorted quasi-sinusoidal flux-to-voltage transfer function with an exceptionally large modulation depth of 140 /spl mu/V peak-to-peak. The spectral density of the magnetic flux noise was as low as 3.5/spl times/10/sup -7/ /spl Phi//sub 0//Hz/sup 1/2/ in the double stage configuration, measured at a temperature 4.2 K using direct read out electronics. In combination with an intermediary transformer, the current resolution of the SQUID is as low as 1.25 pA/Hz/sup 1/2/ with an input coil inductance of 100 nH.
I discuss experimental and theoretical results on an LC filtered dc SQUID phase qubit. This qubit is an asymmetric aluminum dc SQUID, with junction critical currents 1.5 and 26.8 μA, on a sapphire substrate. The layout differs from earlier designs by incorporating a superconducting ground plane and weakly coupled coplanar waveguide microwave drive line to control microwave-qubit coupling. I begin with a discussion of quantizing lumped element circuit models. I use nodal analysis to construct a 2d model for the dc SQUID phase qubit that goes beyond a single junction approximation. I then discuss an extension of this ``normal modes'' SQUID model to include the on-chip LC filter with design frequency ∼ 180 MHz. I show that the filter plus SQUID model yields an effective Jaynes-Cummings Hamiltonian for the filter-SQUID system with coupling g / 2 π ∼ 32 MHz. I present the qubit design, including a noise model predicting a lifetime T1 = 1.2 μs for the qubit based on the design parameters. I characterized the qubit with measurements of the current-flux characteristic, spectroscopy, and Rabi oscillations. I measured T1 = 230 ns, close to the value 320 ns given by the noise model using the measured parameters. Rabi oscillations show a pure dephasing time Tφ = 1100 ns. The spectroscopic and Rabi data suggest two-level qubit dynamics are inadequate for describing the system. I show that the effective Jaynes-Cummings model reproduces some of the unusual features.
The LC\mathrm{LC}LC (inductance-capacitance) resonance phenomenon in the dc SQUID (superconducting quantum interference device) creates a current peak in the $I\ensuremath{-}V$ characteristics at a voltage determined by the resonance frequency. These $I\ensuremath{-}V$ characteristics have been analyzed with the resistively shunted Josephson-junction model in the low-damping and small-oscillation regime. The resonance $I\ensuremath{-}V$ characteristics contain sufficient information to determine all circuit parameters for the dc SQUID's. Specifically, we can apply this analysis to symmetric, tunnel-junction dc SQUID's to deduce the pair-quasiparticle interference conductance ($cos\ensuremath{\theta}$ term).
Relaxation Oscillation SQUIDs (ROSs) based on 4x4 micrometers 2 Nb/AlOx Josephson tunnel junctions have been fabricated and characterized. A ROS consists of a hysteretic dc SQUID shunted by an inductor L and a resistor R in series to induce the relaxation oscillations. The values of L range from 20 nH up to about 300 nH, whereas the time constant L/R are between 8 and 45 ns. Frequency-flux characteristics have been recorded with the help of a spectrum analyzer, directly connected to the ROS. The relaxation frequencies range from 5 to 180 MHz. The experimental characteristics can be explained very well with a simple model describing the oscillation cycle. The effect of the self-induced magnetic field due to a magnetic coupling between the dc SQUID and the shunt circuit has been studied in detail. The sensitivity of ROSs and DROSs (Double Relaxation Oscillation SQUIDs) improves with increasing relaxation frequency. In (D)ROSs based on unshunted, hysteretic tunnel junction dc SQUIDs, the maximum relaxation frequency and the sensitivity are limited by LC resonances due to the SQUID capacitance and the shunt inductance. It is shown that the relaxation frequency can be increased up to frequencies of the order of 1 GHz if an extra resistor is integrated to damp these resonances. The optimum value of the damping resistor can be obtained from the ROS parameters. For stable operation of a (D)ROS, the shunt resistance should not be too large. The optimum value of this resistance can be calculated from the effective McCumber parameter and the bias current. Theoretically, the sensitivity of a ROS with a SQUID capacitance of 1 pF, a SQUID inductance of 20 pH and a relaxation frequency of 1 GHz equals about 5h, where h is Planck's constant. In a DROS with voltage readout based on similar SQUIDs, the theoretical sensitivity at a relaxation frequency of 1 GHz is 17h, with an estimated flux-to-voltage transfer coefficient of 5 mV/(phi) 0.
The dynamics of two inductively-coupled Josephson junctions, driven by a current source and by an external flux were studied. Approximate analytical arguments and numerical simulations were used. The various characteristics times of the DC-SQUID were systematically obtained as functions of the parameters of the junctions and the superconducting loop. We have studied the effects of the external magnetic field and quasiparticle conductance on the structure of the I-V characteristics. The transition times between the various branches of the I-V characteristics were calculated using approximate analytical arguments as well as numerical simulations. The results, both for the I-V characteristics and the transition times, were compared with existing experiments.