On the coupling of magnetic moments to superconducting quantum interference devices

Abstract

Abstract We investigate the coupling factor φ µ that quantifies the magnetic flux Φ per magnetic moment µ of a point-like magnetic dipole that couples to a superconducting quantum interference device (SQUID). Representing the dipole by a tiny current-carrying (Amperian) loop, the reciprocity of mutual inductances of SQUID and Amperian loop provides an elegant way of calculating ϕ μ ( r , e ˆ μ ) vs. position r and orientation e ˆ μ of the dipole anywhere in space from the magnetic field B J ( r ) produced by a supercurrent circulating in the SQUID loop. We use numerical simulations based on London and Ginzburg–Landau theory to calculate φ µ from the supercurrent density distributions in various superconducting loop geometries. We treat the far-field regime ( r ≳ a = inner size of the SQUID loop) with the dipole placed on (oriented along) the symmetry axis of circular or square shaped loops. We compare expressions for φ µ from simple filamentary loop models with simulation results for loops with finite width w (outer size A > a), thickness d and London penetration depth λ L and show that for thin ( d ≪ a ) and narrow (w < a) loops the introduction of an effective loop size a eff in the filamentary loop-model expressions results in good agreement with simulations. For a dipole placed right in the center of the loop, simulations provide an expression ϕ μ ( a , A , d , λ L ) that covers a wide parameter range. In the near-field regime (dipole centered at small distance z above one SQUID arm) only coupling to a single strip representing the SQUID arm has to be considered. For this case, we compare simulations with an analytical expression derived for a homogeneous current density distribution, which yields excellent agreement for λ L > w , d . Moreover, we analyze the improvement of φ µ provided by the introduction of a narrow constriction in the SQUID arm below the magnetic dipole.