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arXiv:2302.12316 · Applied Physics Letters

Model for 1/f flux noise in superconducting aluminum devices: Impact of external magnetic fields

José Alberto Nava Aquino, Rogério de Sousa

Published May 29, 2023• 2 citations
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Abstract

Superconducting quantum interference devices and related circuits made of aluminum are known to display 1/ω flux noise, where ω is frequency. A recent experiment showed that the application of an external magnetic field in the 10–100 G range changed the noise to a single Lorentzian peaked at ω = 0. Here, it is shown that a model based on independent impurity spin flips with coexisting cross and direct mechanisms of spin relaxation may explain these experiments. The model shows that application of an external magnetic field can be used to reduce the impact of flux noise in qubits.

DOI
10.1063/5.0147500

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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
Superconducting quantum interference devices and related circuits made of aluminum are known to display 1/ω flux noise, where ω is frequency. A recent experiment showed that the application of an external magnetic field in the 10–100 G range changed the noise to a single Lorentzian peaked at ω = 0. Here, it is shown that a model based on independent impurity spin flips with coexisting cross and direct mechanisms of spin relaxation may explain these experiments. The model shows that application of an external magnetic field can be used to reduce the impact of flux noise in qubits.
The microscopic description of 1/f magnetic flux noise in superconducting circuits has remained an open question for several decades despite extensive experimental and theoretical investigation. Recent progress in superconducting devices for quantum information has highlighted the need to mitigate sources of qubit decoherence, driving a renewed interest in understanding the underlying noise mechanism(s). Though a consensus has emerged attributing flux noise to surface spins, their identity and interaction mechanisms remain unclear, prompting further study. Here, we apply weak in-plane magnetic fields to a capacitively shunted flux qubit (where the Zeeman splitting of surface spins lies below the device temperature) and study the flux-noise-limited qubit dephasing, revealing previously unexplored trends that may shed light on the dynamics behind the emergent 1/f noise. Notably, we observe an enhancement (suppression) of the spin-echo (Ramsey) pure-dephasing time in fields up to B=100 G. With direct noise spectroscopy, we further observe a transition from a 1/f to approximately Lorentzian frequency dependence below 10 Hz and a reduction of the noise above 1 MHz with increasing magnetic field. We suggest that these trends are qualitatively consistent with an increase of spin cluster sizes with magnetic field. These results should help to inform a complete microscopic theory of 1/f flux noise in superconducting circuits.
Recent experiments implicate spins on the surface of metals as the source of flux noise in superconducting quantum interference devices and indicate that these spins are able to relax without conserving total magnetization. We present a model of 1/f flux noise in which electron spins on the surface of metals can relax via hyperfine interactions. Our results indicate that flux noise would be significantly reduced in superconducting materials where the most abundant isotopes do not have nuclear moments, such as zinc and lead.
Superconducting quantum interference devices (SQUIDs) and other superconducting circuits are limited by intrinsic flux noise with spectral density $1/{f}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}<1$ whose origin is believed to be due to spin impurities. Here, we present a theory of flux noise that takes into account the vectorial nature of the coupling of spins to superconducting wires. We present explicit numerical calculations of the flux-noise power (spectral density integrated over all frequencies) for electron impurities and lattice nuclear spins under several different assumptions. The noise power is shown to be dominated by surface electron spins near the wire edges, with bulk lattice nuclear spins contributing $\ensuremath{\sim}5%$ of the noise power in aluminum and niobium wires. We consider the role of electron spin phase transitions, showing that the spin-spin correlation length (describing, e.g., the average size of ferromagnetic spin clusters) greatly impacts the scaling of flux noise with wire geometry. Remarkably, the flux-noise power is exactly equal to zero when the spins are polarized along the flux-vector direction, forming what we call a poloidal state. Flux noise is nonzero for other spin textures, but gets reduced in the presence of correlated ferromagnetic fluctuations between the top and bottom wire surfaces, where the flux vectors are antiparallel. This demonstrates that engineering spin textures and/or intersurface correlation provides a method to reduce flux noise in superconducting devices.
Superconducting Quantum Interference Devices (SQUIDs) and other superconducting circuits are limited by intrinsic flux noise with spectral density 1/fα1/f^{\alpha}1/fα with α<1\alpha<1α<1 whose origin is believed to be due to spin impurities. Here we present a theory of flux noise that takes into account the vectorial nature of the coupling of spins to superconducting wires. We present explicit numerical calculations of the flux noise power (spectral density integrated over all frequencies) for electron impurities and lattice nuclear spins under several different assumptions. The noise power is shown to be dominated by surface electron spins near the wire edges, with bulk lattice nuclear spins contributing ∼5\sim 5∼5% of the noise power in aluminum and niobium wires. We consider the role of electron spin phase transitions, showing that the spin-spin correlation length (describing e.g. the average size of ferromagnetic spin clusters) greatly impacts the scaling of flux noise with wire geometry. Remarkably, flux noise power is exactly equal to zero when the spins are polarized along the flux vector direction, forming what we call a poloidal state. Flux noise is non-zero for other spin textures, but gets reduced in the presence of correlated ferromagnetic fluctuations between the top and bottom wire surfaces, where the flux vectors are antiparallel. This demonstrates that engineering spin textures and/or inter-surface correlation provides a method to reduce flux noise in superconducting devices.
Superconducting quantum devices, from qubits and magnetometers to dark matter detectors, are influenced by magnetic flux noise originating from paramagnetic surface defects and impurities. These spin systems can feature complex dynamics, including a Berezinskii-Kosterlitz-Thouless transition, that depend on the lattice, interactions, external fields, and disorder. However, the disorder included in typical models is not materials-specific, diminishing the ability to accurately capture measured flux noise phenomena. We present a first principles-based simulation of a spin lattice consisting of paramagnetic O2_22​ molecules on an Al2_22​O3_33​ surface, a likely flux noise source in superconducting qubits, to elucidate opportunities to mitigate flux noise. We simulate an ensemble of surface adsorbates with disordered orientations and calculate the orientation-dependent exchange couplings using density functional theory. Thus, our spin simulation has no free parameters or assumed functional form of the disorder, and captures correlation in the defect landscape that would appear in real systems. We calculate a range of exchange interactions between electron pairs, with the smallest values, 0.016 meV and -0.023 meV, being in the range required to act as a two-level system and couple to GHz resonators. We calculate the flux noise frequency, temperature, and applied external magnetic field dependence, as well as the susceptibility-flux noise cross-correlation. Calculated trends agree with experiment, demonstrating that a surface harboring paramagnetic adsorbates arranged with materials-specific disorder and interactions captures the various properties of magnetic flux noise observed in superconducting circuits. In addition, we find that an external electric field can tune the spin-spin interaction strength and reduce magnetic flux noise.
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.