On the coupling of magnetic moments to superconducting quantum interference devices
2024-01-04

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>φ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">r</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mover> <mml:mi mathvariant="bold-italic">e</mml:mi> <mml:mo stretchy="false">ˆ</mml:mo> </mml:mover> <mml:mi>μ</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> vs. position <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="bold-italic">r</mml:mi> </mml:math> and orientation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mover> <mml:mi mathvariant="bold-italic">e</mml:mi> <mml:mo stretchy="false">ˆ</mml:mo> </mml:mover> <mml:mi>μ</mml:mi> </mml:msub> </mml:math> of the dipole anywhere in space from the magnetic field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi mathvariant="bold-italic">B</mml:mi> <mml:mi>J</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="bold-italic">r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> 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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≳</mml:mo> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> </mml:mrow> </mml:math> 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 &gt; a ), thickness d and London penetration depth λ L and show that for thin ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo>≪</mml:mo> <mml:mi>a</mml:mi> </mml:math> ) and narrow ( w &lt; a ) loops the introduction of an effective loop size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow> <mml:mtext>eff</mml:mtext> </mml:mrow> </mml:msub> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>φ</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mtext>L</mml:mtext> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mtext>L</mml:mtext> </mml:mrow> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:math> . Moreover, we analyze the improvement of φ μ provided by the introduction of a narrow constriction in the SQUID arm below the magnetic dipole.