Cavity-magnon-polariton spectroscopy of strongly hybridized electro-nuclear spin excitations in LiHoF4

Abstract We first present a formalism that incorporates the input-output formalism and the linear response theory to employ cavity-magnon-polariton coupling as a spectroscopic tool for investigating strongly hybridized electro-nuclear spin excitations. A microscopic relation between the generalized susceptibility and the scattering parameter $$|S_{11}|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>11</mml:mn> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> </mml:mrow> </mml:math> in strongly hybridized cavity-magnon-polariton systems has been derived without resorting to semi-classical approximations. The formalism is then applied to both analyze and simulate a specific systems comprising a model quantum Ising magnet ( $$\hbox {LiHoF}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>LiHoF</mml:mtext> <mml:mn>4</mml:mn> </mml:msub> </mml:math> ) and a high-finesse 3D re-entrant cavity resonator. Quantitative information on the electro-nuclear spin states in $$\hbox {LiHoF}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>LiHoF</mml:mtext> <mml:mn>4</mml:mn> </mml:msub> </mml:math> is extracted, and the experimental observations across a broad parameter range were numerically reproduced, including an external magnetic field traversing a quantum critical point. The method potentially opens a new avenue not only for further studies on the quantum phase transition in $$\hbox {LiHoF}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>LiHoF</mml:mtext> <mml:mn>4</mml:mn> </mml:msub> </mml:math> but also for a wide range of complex magnetic systems.