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Journal of Applied Physics

Thermal magnetic noise in rf SQUIDS coupled to ferromagnetic cores

Stefano Vitale, Giovanni Andrea Prodi, MASSIMO CERDONIO

Published Mar 1, 1989• 29 citations
Publisher

Abstract

We present a simple model for the thermal magnetic noise in a rf SQUID coupled to a ferromagnetic core via a superconducting dc flux transformer. We show that, under conditions that are almost always fulfilled in practice, the coupling to the core can be described in terms of a SQUID frequency-dependent complex inductance Ls (ω). This inductance reduces to an ordinary, frequency-independent real value above a roll-off frequency ωc . To this inductance, a Nyquist flux noise generator is associated, with spectral density S′2φi(ω) =−4kBT Im[Ls(ω)]/ ω, where kB is the Boltzmann constant, T the temperature, and Im[Ls (ω)] the imaginary part of Ls (ω). If ωc ≪ωrf , with ωrf the frequency of the rf pump, then the response of the SQUID to low-frequency signals is not modified by the coupling to the core, while the above Nyquist noise acts as an additive input noise. We experimentally demonstrate the model for a SQUID coupled to a toroidal core.

DOI
10.1063/1.342862

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For over three decades, the presence of magnetic flux noise with a power spectral density scaling roughly as $S_Phi(f) propto 1/f^alpha$, where $alpha lesssim 1$, 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 $alpha$ 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 $1/f$ 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 $Phisq$ from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity, $Phisq$ is proportional to $langleBbf(rbf)^2rangle$, where $Bbf(rbf)$ is the magnetic field generated by a circulating current around the loop and $rbf$ 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 $Phisq$ 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 $Phisq propto R/W$ 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 $Phisq$ is significant---even dominant---in narrow-linewidth loops.To calculate theoretical dephasing rates in qubits, we consider flux noise with a spectral density $S_Phi(f) = A^2/(f/1~hbox{Hz})^alpha$, where $A$ is of the order of 1~$muPhi_0 , hbox{Hz}^{-1/2}$ and $0.6 leq alpha leq 1.2$; $Phi_{0}$ 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 $tau_phi$ of Ramsey and echo pulse sequences on $alpha$ for fixed $A$ show that $tau_phi$ decreases rapidly as $alpha$ is reduced. We find that $tau_phi$ is relatively insensitive to the noise bandwidth, $f_1 leq f leq f_2$, for all $alpha$ provided the ultraviolet cutoff frequency $f_2 > 1/tau_phi$. We calculate the ratio $tau_{phi,E} / tau_{phi,R}$ of the echo ($E$) and Ramsey ($R$) sequences, and the dependence of the decay function on $alpha$ and $f_2$. We investigate the case in which $S_Phi(f_0)$ is fixed at the ``pivot frequency'' $f_0 neq 1$~Hz while $alpha$ is varied, and find that the choice of $f_0$ can greatly influence the sensitivity of $tau_{phi,E}$ and $tau_{phi,R}$ to the value of $alpha$.Finally, we conclude with a brief review of our principal results and conclusions. We also comment on promising avenues of future research.

•Jan 1, 2013•0 citations
We present a simple model for the thermal magnetic noise in a rf SQUID coupled to a ferromagnetic core via a superconducting dc flux transformer. We show that, under conditions that are almost always fulfilled in practice, the coupling to the core can be described in terms of a SQUID frequency-dependent complex inductance Ls (ω). This inductance reduces to an ordinary, frequency-independent real value above a roll-off frequency ωc . To this inductance, a Nyquist flux noise generator is associated, with spectral density S′2φi(ω) =−4kBT Im[Ls(ω)]/ ω, where kB is the Boltzmann constant, T the temperature, and Im[Ls (ω)] the imaginary part of Ls (ω). If ωc ≪ωrf , with ωrf the frequency of the rf pump, then the response of the SQUID to low-frequency signals is not modified by the coupling to the core, while the above Nyquist noise acts as an additive input noise. We experimentally demonstrate the model for a SQUID coupled to a toroidal core.
We have performed a study of the low-frequency noise in high-temperature radio frequency (rf) superconducting quantum interference devices (SQUIDs) cooled in static magnetic fields. At 77 K, the noise of different thin-film YB2C3O7−x washer rf SQUIDs was measured in static magnetic fields up to 55 μT. The SQUID inductances ranged from 150 to 300 pH. The measured low-frequency flux noise of the SQUIDs SΦ1/2 at 1 Hz was between 70 and 350 μΦ0/√Hz, and in terms of magnetic field noise SB1/2 between 120 and 1100 fT/√Hz. The low-frequency noise of all devices at 1 Hz showed no significant increase for all applied static fields.
We have fabricated RF and DC SQUIDS with step-edge Josephson junctions (SEJ) and YBa 2 Cu 3 O 7 thin films. The low-frequency noise of these SQUIDS has been investigated as a function of temperature, bias current and magnetic field. Typically, the energy resolution in the white noise region of the DC SQUIDS with an inductance of 45 pH was 3*10 -30 J Hz -1 at 77.5 K. Depending on the bias current, the low-frequency spectrum could be of either 1/f type or Lorentzian type. The Lorentzian spectrum was generated by random telegraph signals (RTSS). An Arrhenius plot was obtained from the temperature dependence of the switching time of these RTSS. We have determined two barrier lengths of 0.23 eV and 1 eV of the two-level fluctuator (TLF) generating this RTS noise. From the magnetic-field dependence of the low-frequency noise we have found that the noise is periodic with the flux corresponding to the area of the SEJ. The results were discussed in terms of a model treating the SEJ as a multi-junction interferometer which forms a two-level fluctuator when the junction is placed inside a SQUID loop. The noise is generated by thermally activated switching of the SQUID between the two different states of the SQUID.
Applications of Superconducting Quantum Interference Devices (SQUIDs) usually require high sensitivity at relatively low frequencies, often down to 1 Hz or lower. Excess noise, typically with a spectral density scaling inversely as the frequency, can substantially reduce the sensitivity of SQUIDs at low frequencies. We have studied 1/f noise in niobium rf-SQUIDs in some detail. To reduce any noise contribution of the readout electronics, a cryogenic preamplifier was used. When one measures the signal voltage of the SQUID directly, a pronounced 1/f noise was observed in all samples, and was nearly independent of the bias frequency used. The crossover between 1/f and white noise moved towards higher frequencies as the bias frequency was increased, because of a lower white noise contribution. The 1/f noise scaled approximately as the inductance of the SQUID. When operated in a flux locked loop, however, no 1/f noise could be observed above 0.5 Hz. Operating a rf SQUID in a flux locked loop can thus substantially reduce 1/f noise in rf SQUIDs.
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.
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.
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.