Mean square flux noise in SQUIDs and qubits: numerical calculations

Abstract

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 2 from independently fluctuating spins on the surface of thin-film loops of arbitrary geometry. By reciprocity, 2 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 2 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 2 ∝ 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 2 is significant-even dominant-in narrow-linewidth loops.

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

“…Actual devices will show some degree of asymmetry for the current densities at the wire edges. For example, the SQUID is a closed SC wire loop, so to satisfy the Meissner effect it produces higher current density at the inside wire edge, 32 thus minimizing the value of magnetic field inside the wire. Modifying Eq.…”

“…31 More recently, extensive numerical studies 25,32 showed that these calculations greatly underestimated the value of the flux produced by spins located at the wire surfaces, because they did not take account of the singular nature of the spin's dipole field. Indeed, analytic expressions for the flux noise power that take into account the spin's singular behaviour are still absent from the literature.…”

“…Previous calculations of flux noise due to localized spin 15 were based on modelling the spin as a square loop of side 0.1 µm, the minimum feature size allowed by the finite element software FastHenry. 31 More recently, extensive numerical studies 25,32 showed that these calculations greatly underestimated the value of the flux produced by spins located at the wire surfaces, because they did not take account of the singular nature of the spin's dipole field. Indeed, analytic expressions for the flux noise power that take into account the spin's singular behaviour are still absent from the literature.…”

“…We shall show below that our expressions lead to much larger values than previous ones, 6 but are in close agreement with the most recent numerical studies. 32 The article is organized as follows: Section II introduces our flux-vector model, and develops a general theory on how flux noise depends on flux vector and the 7,25 and expected location of the spin species causing flux noise. The superconducting wire is made of niobium, the insulator of partially oxidized aluminum (likely amorphous), and the substrate of silicon-dioxide.…”

Magnetic field calculations in the vicinity of superconductive circuit structures

“…InductEx and the modified FastHenry are freeware modules [15], [18] executed from a console or command prompt in MS Windows, Linux, Mac OS X and Mac Power PC. The magnetic field calculation code that solves equations (3) to (5) at arbitrary coordinates was developed as part of a calculation of mean square flux noise in SQUIDs [17] and is open source [19] Matlab code. Mapping of InductEx output files to the vectors required by the magnetic field calculator is done with the InductEx add-on IDensity [18].…”