On the coupling of magnetic moments to superconducting quantum interference devices

We investigate the coupling factor $\phi_\mu$ that quantifies the magnetic flux $\Phi$ per magnetic moment $\mu$ of a point-like magnetic dipole that couples to a superconducting quantum interference device (SQUID). Representing the dipole by a current-carrying loop, the reciprocity of mutual inductances of SQUID and loop provides a way of calculating $\phi_\mu(\vec{r}, \vec{e}_\mu)$ vs.~position $\vec{r}$ and orientation $\vec{e}_\mu$ of the dipole anywhere in space from the magnetic field $B(\vec{r})$ produced by a supercurrent circulating in the SQUID loop. We use numerical simulations based on London and Ginzburg-Landau theory to calculate $\phi_\mu$ from the supercurrent density distributions in various SQUID geometries. We treat the far-field regime ($r\gtrsim a=$ inner size of the SQUID loop) with the dipole placed on the symmetry axis of circular or square shaped loops. We compare expressions for $\phi_\mu$ from filamentary loop models with simulation results for loops with finite width $w$ (outer size $A>a$), thickness $d$ and London penetration depth $\lambda_L$ and show that for thin ($d\ll a$) and narrow ($w < a$) loops the introduction of an effective loop size $a_{\rm eff}$ in the filamentary loop-model expressions results in agreement with simulations. For a dipole placed in the center of the loop, simulations provide an expression $\phi_\mu(a,A,d,\lambda_L)$ 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. Here, we compare simulations with an analytical expression derived for a homogeneous current density distribution, which yields excellent agreement for $\lambda_L>w,d$. Moreover, we analyze $\phi_\mu$ provided by the introduction of a constriction in the SQUID arm below the magnetic dipole.