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Non-local electrodynamics of superconducting wires: implications for flux noise and inductance

P. Senarath Yapa

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

The simplest model for superconductor electrodynamics are the London equations, which treats the impact of electromagnetic fields on the current density as a localized phenomenon. However, the charge carriers of superconductivity are quantum mechanical objects, and their wavefunctions are delocalized within the superconductor, leading to non-local effects. The Pippard equation is the generalization of London electrodynamics which incorporates this intrinsic non-locality through the introduction of a new superconducting characteristic length, \xi_0, called the Pippard coherence length. When building nano-scale superconducting devices, the inclusion of the coherence length into electrodynamics calculations becomes paramount. In this thesis, we provide numerical calculations of various electrodynamic quantities of interest in the non-local regime, and discuss their implications for building superconducting devices. We place special emphasis on Superconducting QUantum Inteference Devices (SQUIDs), and their usage as flux quantum bits (qubits) in quantum computation. One of the main limitations of these flux qubits is the presence of intrinsic flux noise, which leads to decoherence of the qubits. Although the origin of this flux noise is not known, there is evidence that it is related to spin impurities within the superconducting material. We present calculations which show that the flux noise in the non-local regime is signi cantly different from the local case. We also demonstrate that non-local electrodynamics greatly affect the self-inductance of the qubit.

Related papers

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

Flux noise in superconducting circuits: Electron versus nuclear spins

S. LaForest, Rogério de Sousa

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

•Jan 15, 2015•0 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

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

Noise theory of dc nano-SQUIDs based on Dayem nanobridges

C. Granata, Antonio Vettoliere, M. Russo +1 more

In the recent years, nanoscale superconducting quantum interference devices (nano-SQUIDs) played a fundamental role in the study of small spin systems. Nano-SQUIDs typically employ nano-Dayem bridges having dimensions (length $L$ and/or width $W$) greater than coherence length $\ensuremath{\xi}$ of the superconducting film. They exhibit characteristics different from those of standard SQUIDs because the current-phase relationship (CPR) is nonsinusoidal, rendering most of the theoretical predictions based on the standard SQUID theory usually unreliable. Here, we present a noise theory of dc nano-SQUIDs based on Dayem nanobridges. We have computed the main characteristics of this quantum device including current-voltage and voltage-magnetic flux characteristics, magnetic flux-to-voltage transfer factor, and spectral densities of voltage and magnetic flux noise for $L$/$\ensuremath{\xi}$ ratios ranging from 1 (sinusoidal limit) to 3.5 (hysteretic limit). The CPRs have been computed by using the theory of Josephson weak links based on the Ginzburg--Landau equation. The results show a dependence of the magnetic flux noise spectral density on ($L$/$\ensuremath{\xi}$)${}^{4/3}$ involving a degradation of about a factor of five between the two extreme cases and are consistent with experimentally measured magnetic flux noises reported in the literature. These results provide useful information for both device physics and their applications.

Physical Review B•Dec 27, 2011•34 citations
The simplest model for superconductor electrodynamics are the London equations, which treats the impact of electromagnetic fields on the current density as a localized phenomenon. However, the charge carriers of superconductivity are quantum mechanical objects, and their wavefunctions are delocalized within the superconductor, leading to non-local effects. The Pippard equation is the generalization of London electrodynamics which incorporates this intrinsic non-locality through the introduction of a new superconducting characteristic length, \xi_0, called the Pippard coherence length. When building nano-scale superconducting devices, the inclusion of the coherence length into electrodynamics calculations becomes paramount. In this thesis, we provide numerical calculations of various electrodynamic quantities of interest in the non-local regime, and discuss their implications for building superconducting devices. We place special emphasis on Superconducting QUantum Inteference Devices (SQUIDs), and their usage as flux quantum bits (qubits) in quantum computation. One of the main limitations of these flux qubits is the presence of intrinsic flux noise, which leads to decoherence of the qubits. Although the origin of this flux noise is not known, there is evidence that it is related to spin impurities within the superconducting material. We present calculations which show that the flux noise in the non-local regime is signi cantly different from the local case. We also demonstrate that non-local electrodynamics greatly affect the self-inductance of the qubit.
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
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) 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.
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.
In the recent years, nanoscale superconducting quantum interference devices (nano-SQUIDs) played a fundamental role in the study of small spin systems. Nano-SQUIDs typically employ nano-Dayem bridges having dimensions (length LLL and/or width WWW) greater than coherence length $\ensuremath{\xi}$ of the superconducting film. They exhibit characteristics different from those of standard SQUIDs because the current-phase relationship (CPR) is nonsinusoidal, rendering most of the theoretical predictions based on the standard SQUID theory usually unreliable. Here, we present a noise theory of dc nano-SQUIDs based on Dayem nanobridges. We have computed the main characteristics of this quantum device including current-voltage and voltage-magnetic flux characteristics, magnetic flux-to-voltage transfer factor, and spectral densities of voltage and magnetic flux noise for LLL/$\ensuremath{\xi}$ ratios ranging from 1 (sinusoidal limit) to 3.5 (hysteretic limit). The CPRs have been computed by using the theory of Josephson weak links based on the Ginzburg--Landau equation. The results show a dependence of the magnetic flux noise spectral density on (LLL/$\ensuremath{\xi}$)4/3{}^{4/3}4/3 involving a degradation of about a factor of five between the two extreme cases and are consistent with experimentally measured magnetic flux noises reported in the literature. These results provide useful information for both device physics and their applications.