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Physical Review Letters

Magnetic Flux Noise in dc SQUIDs: Temperature and Geometry Dependence

Steven M. Anton, Jeffrey Birenbaum, Sean R. O’Kelley, Vladimir Bolkhovsky, Danielle Braje, George Fitch, M. Neeley, Gene C Hilton, Hsiao-Mei Cho, K. D. Irwin, F. C. Wellstood, William D. Oliver, Alexander Shnirman, John L. Clarke

Published Apr 5, 2013• 110 citations
Publisher

Abstract

The spectral density S(Φ)(f) = A(2)/(f/1 Hz)(α) of magnetic flux noise in ten dc superconducting quantum interference devices (SQUIDs) with systematically varied geometries shows that α increases as the temperature is lowered; in so doing, each spectrum pivots about a nearly constant frequency. The mean-square flux noise, inferred by integrating the power spectra, grows rapidly with temperature and at a given temperature is approximately independent of the outer dimension of a given SQUID. These results are incompatible with a model based on the random reversal of independent, surface spins.

DOI
10.1103/physrevlett.110.147002

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Superconductor Science and Technology•Jun 6, 2013•5 citations
The spectral density S(Φ)(f) = A(2)/(f/1 Hz)(α) of magnetic flux noise in ten dc superconducting quantum interference devices (SQUIDs) with systematically varied geometries shows that α increases as the temperature is lowered; in so doing, each spectrum pivots about a nearly constant frequency. The mean-square flux noise, inferred by integrating the power spectra, grows rapidly with temperature and at a given temperature is approximately independent of the outer dimension of a given SQUID. These results are incompatible with a model based on the random reversal of independent, surface spins.
At temperatures below about 1 K, a series of dc superconducting quantum interference devices (SQUID’s) exhibited an apparent flux noise with a spectral density scaling as 1/f α, where 0.58<α<0.80. Typically, the magnitude of the noise increased as the temperature was lowered below 1 K, tending to flatten out at low temperatures with a value of 7±3 μΦ0Hz−1/2 at 1 Hz that was nearly independent of the parameters and materials of the SQUID’s. Although a large number of hypothetical sources of the noise have been eliminated, the origin remains unidentified.
For over three decades, the presence of magnetic flux noise with a power spectral density scaling roughly as SPhi(f)propto1/falphaS_Phi(f) propto 1/f^alphaSP​hi(f)propto1/falpha, where alphalesssim1alpha lesssim 1alphalesssim1, has been known to limit the low-frequency performance of dc superconducting quantum interference devices (SQUIDs). In recent years, experiments indicate that this same noise persists to frequencies up to 1~GHz and is a dominant source of dephasing in flux-sensitive superconducting quantum bits (qubits). Thus, the reduction of flux noise presents a major hurdle towards the successful realization of scalable quantum computers that are based on flux-based qubits. In this thesis, we present experimental measurements, theoretical analyses, and numerical simulations that support a more detailed understanding of both the microscopic and macroscopic properties of flux noise.Our experimental work begins with flux noise measurements of a large number of SQUIDs in the temperature range from 0.1~K to 4~K. We report on measurements of ten SQUIDs with systematically varied geometries and show that alphaalphaalpha increases as the temperature is lowered; in so doing, each spectrum pivots about a nearly constant frequency. The mean square flux noise, inferred by integrating the power spectra, grows rapidly with temperature and at a given temperature is approximately independent of the outer dimension of a given SQUID washer. We show that these results are incompatible with a model based on the random reversal of independent, spins that are located at the surface of the SQUID washer.In the course of our flux noise measurements, we became aware of a spurious contribution to low-frequency critical current noise in Josephson junctions---normally attributed to charge trapping in the barrier---arising from temperature instabilities inherent in cryogenic systems. These temperature fluctuations modify the critical current via its temperature dependence. By computing cross-correlations between measured temperature and critical current noise in Al-AlOx-Al junctions, we show that, despite excellent temperature stability, temperature fluctuations induce observable critical current fluctuations. Particularly, because 1/f1/f1/f critical current noise has decreased with improved fabrication techniques in recent years, it is important to understand and eliminate this additional noise source.Next, we introduce a numerical method of calculating the mean square flux noise PhisqPhisqPhisq from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity, PhisqPhisqPhisq is proportional to langleBbf(rbf)2ranglelangleBbf(rbf)^2ranglelangleBbf(rbf)2rangle, where Bbf(rbf)Bbf(rbf)Bbf(rbf) is the magnetic field generated by a circulating current around the loop and rbfrbfrbf varies over the loop surface. By discretizing the loop nonuniformly, we efficiently and accurately compute the current distribution and resulting magnetic field, which may vary rapidly across the loop. We use this method to compute PhisqPhisqPhisq in a number of scenarios in which we systematically vary physical parameters of the loop. We compare our simulations to an earlier analytic result predicting that PhisqproptoR/WPhisq propto R/WPhisqproptoR/W in the limit where the loop radius RRR is much greater than the linewidth WWW. We further show that the previously neglected contribution of edge spins to PhisqPhisqPhisq is significant---even dominant---in narrow-linewidth loops.To calculate theoretical dephasing rates in qubits, we consider flux noise with a spectral density SPhi(f)=A2/(f/1 hboxHz)alphaS_Phi(f) = A^2/(f/1~hbox{Hz})^alphaSP​hi(f)=A2/(f/1 hboxHz)alpha, where AAA is of the order of 1~muPhi0,hboxHz−1/2muPhi_0 , hbox{Hz}^{-1/2}muPhi0​,hboxHz−1/2 and 0.6leqalphaleq1.20.6 leq alpha leq 1.20.6leqalphaleq1.2; Phi0Phi_{0}Phi0​ is the flux quantum. For a qubit with an energy level splitting linearly coupled to the applied flux, our calculations of the dependence of the pure dephasing time tauphitau_phitaup​hi of Ramsey and echo pulse sequences on alphaalphaalpha for fixed AAA show that tauphitau_phitaup​hi decreases rapidly as alphaalphaalpha is reduced. We find that tauphitau_phitaup​hi is relatively insensitive to the noise bandwidth, f1leqfleqf2f_1 leq f leq f_2f1​leqfleqf2​, for all alphaalphaalpha provided the ultraviolet cutoff frequency f2>1/tauphif_2 > 1/tau_phif2​>1/taup​hi. We calculate the ratio tauphi,E/tauphi,Rtau_{phi,E} / tau_{phi,R}tauphi,E​/tauphi,R​ of the echo (EEE) and Ramsey (RRR) sequences, and the dependence of the decay function on alphaalphaalpha and f2f_2f2​. We investigate the case in which SPhi(f0)S_Phi(f_0)SP​hi(f0​) is fixed at the ``pivot frequency'' f0neq1f_0 neq 1f0​neq1~Hz while alphaalphaalpha is varied, and find that the choice of f0f_0f0​ can greatly influence the sensitivity of tauphi,Etau_{phi,E}tauphi,E​ and tauphi,Rtau_{phi,R}tauphi,R​ to the value of alphaalphaalpha.Finally, we conclude with a brief review of our principal results and conclusions. We also comment on promising avenues of future research.
We examine the behavior of low frequency flux noise measured in ten Nb-NbOx-Pbln dc SQUIDs in the temperature range 1 K to 20 mK. As previously reported, the flux noise power spectrum SΦ(f) typically scales with frequency as A/fα. Remark ably, the excess noise power A at 1 Hz increases as the temperature is decreased below about 0.7 K, saturating to a value around (10 μΦα)2/Hz below about 0.2 K. Here we report on the dependence of both the magnitude A and the slope a on the size and linewidth of the SQUID loops. In particular, at the lowest temperatures we find that a declines to values as low as 0.5 in the smallest devices, while a is in the range 0.9-1 for the largest devices.
The spectral density SΦ(f ) of the low-frequency 1/f noise of high transition temperature dc superconducting quantum interference devices (SQUIDs) with narrow linewidths was independent of B0, the magnetic field in which they were cooled, up to a threshold value, about 33 μT in the best case. Above this threshold, which is associated with the entry of flux vortices into the film, the noise increased rapidly. By contrast, for large square washer SQUIDs, SΦ(f ) scaled linearly with B0. Estimates indicate that the 1/f flux noise produced by the pickup loop of a directly coupled magnetometer is negligible.
Abstract Noise in superconducting quantum interference devices (dc SQUIDs) with different design, based on Nb/AlO x /Nb technology, has been systematically measured at temperature T = 4.2 K. We demonstrate that for all our devices the power spectral density of the white noise (above a frequency of about 1 kHz) is determined by the shunt resistors, reaching about 5 times Planck’s constant. The low-frequency noise level was measured for washer- and multi-loop-type SQUIDs with systematically varied effective area, SQUID loop perimeter, and inductance. We show that at 1 Hz the noise spectral density is approximately 40 times higher than the white noise level and its scaling with washer width and effective area is negligible. These results are incompatible with several models of the flicker noise source. Possible origins of this noise are discussed.
The performance of SQUIDs and superconducting qubits based on magnetic flux is degraded by the presence of magnetic flux noise with a spectral density scaling approximately inversely with frequency. It is generally accepted that the noise arises from the random reversal of spins on the surface of the superconductors. We introduce a numerical method of calculating the mean square flux noise from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity, is proportional to 〈 B ( r ) 2 〉, where B ( r ) is the magnetic field generated by a circulating current around the loop and r varies over the loop surface. By discretizing the loop nonuniformly, we efficiently and accurately compute the current distribution and resulting magnetic field, which may vary rapidly across the loop. We use this method to compute in a number of scenarios in which we systematically vary physical parameters of the loop. We compare our simulations to an earlier analytic result predicting that in the limit where the loop radius R is much greater than the linewidth W . We further show that the previously neglected contribution of edge spins to is significant—even dominant—in narrow-linewidth loops.