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Flux noise in superconducting circuits: Electron versus nuclear spins

S. LaForest, Rogério de Sousa

Published Jan 15, 2015• 0 citations
OpenAlex

Abstract

Superconducting Quantum Interference Devices (SQUIDs) and other superconducting circuits are limited by intrinsic flux noise with spectral density $1/f^{\alpha}$ with $\alpha<1$ 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 $\sim 5$% 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, 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.

Related papers

Flux-vector model of spin noise in superconducting circuits: Electron versus nuclear spins and role of phase transition

S. LaForest, Rogério de Sousa

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.

arXiv:1501.03776•Physical Review B•Aug 3, 2015•18 citations

Flux noise in disordered spin systems

José Alberto Nava Aquino, Rogério de Sousa

Impurity spins randomly distributed at the surfaces and interfaces of superconducting wires are known to cause flux noise in superconducting quantum interference devices (SQUIDs), providing a dominant mechanism for decoherence in flux-tunable superconducting qubits. While flux noise is well characterized experimentally, the microscopic model underlying spin dynamics remains a great puzzle. The main problem is that first-principles theories based on an integration of the quantum Heisenberg equations of motion for interacting spins are too computationally expensive to capture spin diffusion over large length scales, hindering comparisons between microscopic models and experimental data. In contrast, third-principles approaches lump spin dynamics into a single phenomenological spin-diffusion operator $D{\ensuremath{\nabla}}^{2}$ that is not able to describe the quantum noise regime and connect to microscopic models and different disorder scenarios such as spin clusters. Here we propose an intermediate ``second-principles'' method to describe general spin dissipation and flux noise in the quantum regime. It leads to the interpretation that flux noise arises from the density of paramagnon excitations at the edge of the superconducting wire, with paramagnon-paramagnon interactions leading to spin diffusion, and interactions between paramagnons and other degrees of freedom such as phonons, electrons, and two-level systems leading to spin energy relaxation. At high frequency $\ensuremath{\omega}$, we obtain an upper bound for flux noise, showing that the (super)Ohmic noise observed in experiments is not originating from interacting spin impurities. We apply the method to Heisenberg models in two-dimensional square lattices with a random distribution of vacancies, with nearest-neighbor spins coupled by a constant exchange. Explicit numerical calculations of flux noise show that it follows the observed power law $A/{\ensuremath{\omega}}^{\ensuremath{\alpha}}$, with amplitude $A$ and exponent $\ensuremath{\alpha}$ depending on temperature and inhomogeneities such as spatial confinement and disorder. These results are compared to experiments in niobium and aluminum devices. The method establishes a connection between flux noise experiments and microscopic Hamiltonians with the goal of identifying relevant microscopic mechanisms and guiding strategies for reducing flux noise.

arXiv:2207.10033•Physical review. B./Physical review. B•Oct 14, 2022•4 citations

Flux Noise due to Spins in SQUIDs

S. LaForest

Superconducting Quantum Interference Devices (SQUIDs) are currently being used as flux qubits and read-out detectors in a variety of solid-state quantum computer architectures. The main limitation of SQUID qubits is that they have a coherence time of the order of 10 us, due to the presence of intrinsic flux noise that is not yet fully understood. The origin of flux noise is currently believed to be related to spin impurities present in the materials and interfaces that form the device. Here we present a novel numerical method that enables calculations of the flux produced by spin impurities even when they are located quite close to the SQUID wire. We show that the SQUID will be particularly sensitive to spins located at its wire edges, generating flux shifts of up to 4 nano flux quanta, much higher than previous calculations based on the software package FastHenry. This shows that spin impurities in a particular region along the wire's surface play a much more important role in producing flux noise than other spin impurities located elsewhere in the device.

•Jan 1, 2013•0 citations

Magnetic flux noise in SQUIDs and qubits

Steven M. Anton

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

Theory of charge and flux noise in superconducting wires

José Alberto Nava Aquino

Superconducting qubits are at the forefront of efforts to develop scalable quantum computers due to their potential to perform complex computations beyond the capabilities of classical systems. However, maintaining the quantum coherence of these qubits remains a significant challenge, primarily due to various noise sources such as flux noise, dielectric loss, and quasiparticle poisoning. This dissertation presents a detailed theoretical investigation into two noise mechanisms affecting superconducting qubits: flux noise from spin impurities and charge/flux noise from non-equilibrium superconducting quasiparticle distributions. The first part of the research focuses on developing a general theoretical framework to calculate flux noise arising from spin impurities. This framework accounts for spin diffusion and spin-lattice relaxation, incorporating a discrete diffusion model to handle confinement effects and inhomogeneities. Analytical and numerical results show that the spin relaxation model aligns with experimental observations in aluminum devices, while the spin diffusion model better matches experiments in niobium devices. The second part of the thesis proposes a theory addressing charge and flux noise due to non-equilibrium superconducting quasiparticle distributions within superconducting wires. This theory highlights the significant impact of ohmic loss generated by these quasiparticles, revealing their contribution to charge noise at intermediate frequencies and a nearly white flux noise background. Comparative analysis with experimental data provides some validation for the theoretical models and gives insights into the temperature-dependent behavior of flux noise and the distinctive noise characteristics in aluminum and niobium devices. The findings highlight the necessity of addressing wire-resident quasiparticles

•Jan 1, 2024•0 citations

Mean square flux noise in SQUIDs and qubits: numerical calculations

Steven M. Anton, I A B Sognnaes, Jeffrey Birenbaum +3 more

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.

Superconductor Science and Technology•Jun 6, 2013•5 citations
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 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.
Impurity spins randomly distributed at the surfaces and interfaces of superconducting wires are known to cause flux noise in superconducting quantum interference devices (SQUIDs), providing a dominant mechanism for decoherence in flux-tunable superconducting qubits. While flux noise is well characterized experimentally, the microscopic model underlying spin dynamics remains a great puzzle. The main problem is that first-principles theories based on an integration of the quantum Heisenberg equations of motion for interacting spins are too computationally expensive to capture spin diffusion over large length scales, hindering comparisons between microscopic models and experimental data. In contrast, third-principles approaches lump spin dynamics into a single phenomenological spin-diffusion operator $D{\ensuremath{\nabla}}^{2}$ that is not able to describe the quantum noise regime and connect to microscopic models and different disorder scenarios such as spin clusters. Here we propose an intermediate ``second-principles'' method to describe general spin dissipation and flux noise in the quantum regime. It leads to the interpretation that flux noise arises from the density of paramagnon excitations at the edge of the superconducting wire, with paramagnon-paramagnon interactions leading to spin diffusion, and interactions between paramagnons and other degrees of freedom such as phonons, electrons, and two-level systems leading to spin energy relaxation. At high frequency $\ensuremath{\omega}$, we obtain an upper bound for flux noise, showing that the (super)Ohmic noise observed in experiments is not originating from interacting spin impurities. We apply the method to Heisenberg models in two-dimensional square lattices with a random distribution of vacancies, with nearest-neighbor spins coupled by a constant exchange. Explicit numerical calculations of flux noise show that it follows the observed power law $A/{\ensuremath{\omega}}^{\ensuremath{\alpha}}$, with amplitude AAA and exponent $\ensuremath{\alpha}$ depending on temperature and inhomogeneities such as spatial confinement and disorder. These results are compared to experiments in niobium and aluminum devices. The method establishes a connection between flux noise experiments and microscopic Hamiltonians with the goal of identifying relevant microscopic mechanisms and guiding strategies for reducing flux noise.
Superconducting Quantum Interference Devices (SQUIDs) are currently being used as flux qubits and read-out detectors in a variety of solid-state quantum computer architectures. The main limitation of SQUID qubits is that they have a coherence time of the order of 10 us, due to the presence of intrinsic flux noise that is not yet fully understood. The origin of flux noise is currently believed to be related to spin impurities present in the materials and interfaces that form the device. Here we present a novel numerical method that enables calculations of the flux produced by spin impurities even when they are located quite close to the SQUID wire. We show that the SQUID will be particularly sensitive to spins located at its wire edges, generating flux shifts of up to 4 nano flux quanta, much higher than previous calculations based on the software package FastHenry. This shows that spin impurities in a particular region along the wire's surface play a much more important role in producing flux noise than other spin impurities located elsewhere in the device.
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.
Superconducting qubits are at the forefront of efforts to develop scalable quantum computers due to their potential to perform complex computations beyond the capabilities of classical systems. However, maintaining the quantum coherence of these qubits remains a significant challenge, primarily due to various noise sources such as flux noise, dielectric loss, and quasiparticle poisoning. This dissertation presents a detailed theoretical investigation into two noise mechanisms affecting superconducting qubits: flux noise from spin impurities and charge/flux noise from non-equilibrium superconducting quasiparticle distributions. The first part of the research focuses on developing a general theoretical framework to calculate flux noise arising from spin impurities. This framework accounts for spin diffusion and spin-lattice relaxation, incorporating a discrete diffusion model to handle confinement effects and inhomogeneities. Analytical and numerical results show that the spin relaxation model aligns with experimental observations in aluminum devices, while the spin diffusion model better matches experiments in niobium devices. The second part of the thesis proposes a theory addressing charge and flux noise due to non-equilibrium superconducting quasiparticle distributions within superconducting wires. This theory highlights the significant impact of ohmic loss generated by these quasiparticles, revealing their contribution to charge noise at intermediate frequencies and a nearly white flux noise background. Comparative analysis with experimental data provides some validation for the theoretical models and gives insights into the temperature-dependent behavior of flux noise and the distinctive noise characteristics in aluminum and niobium devices. The findings highlight the necessity of addressing wire-resident quasiparticles
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.