Tunable cooperativity in coupled spin-cavity systems

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Letter

Tunable cooperativity in coupled spin-cavity systems

Lukas Liensberger, Franz X. Haslbeck, Andreas Bauer, Helmuth Berger, Rudolf Gross, Hans Huebl, Christian Pfleiderer, and Mathias Weiler

Phys. Rev. B 104, L100415 – Published 22 September 2021

Abstract

We experimentally study the tunability of the cooperativity in coupled spin-cavity systems by changing the magnetic state of the spin system via an external control parameter. As a model system, we use the skyrmion host material ${Cu}_{2} {OSeO}_{3}$ coupled to a microwave cavity resonator. We measure a dispersive coupling between the resonator and magnon modes in different magnetic phases of the material and model our results by using the input-output formalism. Our results show a strong tunability of the normalized coupling rate by magnetic field, allowing us to change the magnon-photon cooperativity from 1 to 60 at the phase boundaries of the skyrmion lattice state.

Received 23 February 2021
Revised 1 September 2021
Accepted 2 September 2021

DOI:https://doi.org/10.1103/PhysRevB.104.L100415

Physics Subject Headings (PhySH)

Magnetic phase transitions Magnetization dynamics Magnons Spin dynamics Spin texture

Helimagnets Magnetic insulators Noncollinear magnets

Ferromagnetic resonance Microwave techniques

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Lukas Liensberger^1,2,*, Franz X. Haslbeck ^3,4,†, Andreas Bauer^3,5, Helmuth Berger⁶, Rudolf Gross ^1,2,7, Hans Huebl ^1,2,7, Christian Pfleiderer^3,5,7, and Mathias Weiler ^1,2,8,‡

¹Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
²Physik-Department, Technische Universität München, 85748 Garching, Germany
³Lehrstuhl für Topologie korrelierter Systeme, Physik-Department, Technische Universität München, 85748 Garching, Germany
⁴Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany
⁵Zentrum für QuantumEngineering (ZQE), Technische Universität München, D-85748 Garching, Germany
⁶Institut de Physique de la Matière Complexe, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
⁷Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
⁸Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany

^*Lukas.Liensberger@wmi.badw.de
^†Present address: Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany.
^‡weiler@physik.uni-kl.de

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Issue

Vol. 104, Iss. 10 — 1 September 2021

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Images

Figure 1
Microwave dynamics in ${Cu}_{2} {OSeO}_{3}$ . (a) Magnetic phase diagram extracted from broadband magnetic resonance measurements. The following magnetic phases are distinguished: field-polarized (fp), conical (c), helical (h) phase, skyrmion lattice phase (S), paramagnetic (pm) phase. Note that the crossover between the fp and the pm regime above $T_{c}$ is not resolved. (b) Schematic magnetic field dependence of the uncoupled, uniform resonance frequencies of the magnon modes $f_{mag}$ in a chiral magnet. The resonance frequency of the resonator $f_{res}$ is lower than all magnon frequencies. Normalized and absolute frequencies are shown on the left and right ordinate, respectively. (c) Illustration of the input-output formalism and the parameters of Eq. (1). The following parameters are established: magnon resonance frequency $f_{mag}$ , magnon loss rate $κ_{mag}$ , cavity resonance frequency $f_{res}$ , cavity internal $κ_{int}$ and external loss rate $κ_{ext}$ , and effective coupling rate $g_{eff}$ . (d) Frequency shift of the resonator mode in a coupled magnon-photon system calculated using the input-output formalism with a constant coupling rate $g_{eff} / (2 π) = 120 MHz$ in all magnetic phases. The hybridized magnon-cavity modes are denoted by tildes.
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Figure 2
Experimental setup. (a) Schematic of the loop-gap resonator and feedline design. (b) Side view of the resonator and the ${Cu}_{2} {OSeO}_{3}$ crystal mounted on a copper rod and placed inside the cavity. Note the different configurations of static external magnetic field $H_{0}$ relative to the oscillating magnetic field $h_{ac}$ .
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Figure 3
Magnon-photon coupling in a skyrmion host material. (a) Illustration of the two dominant skyrmion resonance modes. The blue and red color correspond to the local magnetization pointing parallel and antiparallel to $H_{0}$ , respectively, and the gray color to noncollinear to $H_{0}$ . The oscillating driving field $h_{ac}$ is either orthogonal (ccw mode) or parallel (breathing mode) to the external magnetic field $H_{0}$ . (b) Measured reflection parameter $| S_{11} |$ as a function of the magnetic field $H_{0}$ and the applied microwave frequency $f$ . The vertical dashed lines indicate phase boundaries (cf. Fig. S6 [34]). The magnetic fields are normalized to the critical field $H_{c 2}$ . (c) Simulated spectrum calculated with Eq. (1). (d) Coupling rate $g_{eff}$ extracted from fitting Eq. (1) to frequency traces at fixed magnetic field and magnon loss rate $κ_{mag}$ from broadband magnetic resonance (BMR) measurements (cf. Fig. S8 [34]) used to calculate the spectrum shown in (c). The fit error bars are smaller than the symbol size. The rather broad phase transition $S \leftrightarrow c$ due to the irregular shape of the single crystal is indicated by the orange shading around the dashed gray lines.
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Figure 4
Cooperativity calculated by using Eq. (2) and the parameters shown in Fig. 3. (a) Orthogonal excitation (OE). (b) Parallel excitation (PE), for which the cooperativity can be tuned from 1 to 60. The fit error bars are smaller than the symbol size.
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