Manifestation of the Coupling Phase in Microwave Cavity Magnonics

Alan Gardin, Jeremy Bourhill, Vincent Vlaminck, Christian Person, Christophe Fumeaux, Vincent Castel, and Giuseppe C. Tettamanzi

Phys. Rev. Applied 19, 054069 – Published 22 May 2023

Abstract

The interaction between microwave photons and magnons is well understood and originates from the Zeeman coupling between spins and a magnetic field. Interestingly, the magnon-photon interaction is accompanied by a phase factor, which can usually be neglected. However, under the rotating wave approximation, if two magnon modes simultaneously couple with two cavity resonances, this phase cannot be ignored as it changes the physics of the system. We consider two such systems, each differing by the sign of one of the magnon-photon coupling strengths. This simple difference, originating from the various coupling phases in the system, is shown to preserve, or destroy, two potential applications of hybrid photon-magnon systems, namely dark-mode memories and cavity-mediated coupling. The observable consequences of the coupling phase in this system is akin to the manifestation of a discrete Pancharatnam-Berry phase, which may be useful for quantum information processing and the creation of nonreciprocal devices using proper cavity engineering.

Received 11 January 2023
Revised 1 February 2023
Accepted 14 April 2023

DOI:https://doi.org/10.1103/PhysRevApplied.19.054069

Physics Subject Headings (PhySH)

Geometric & topological phases Magnons Photonics Polaritons Quantum information with hybrid systems

Ferrimagnets Optical microcavities

Microwave techniques

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Alan Gardin ^1,2,3,*, Jeremy Bourhill^2,4, Vincent Vlaminck ^2,3, Christian Person^2,3, Christophe Fumeaux ⁵, Vincent Castel^2,3, and Giuseppe C. Tettamanzi ^1,6

¹School of Physics, The University of Adelaide, Adelaide, SA 5005, Australia
²IMT Atlantique, Technopole Brest-Iroise, CS 83818, Brest Cedex 3 29238, France
³Lab-STICC (UMR 6285), CNRS, Technopole Brest-Iroise, CS 83818, Brest Cedex 3 29238, France
⁴ARC Centre of Excellence for Engineered Quantum Systems and ARC Centre of Excellence for Dark Matter Particle Physics, Department of Physics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
⁵School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia
⁶School of Chemical Engineering and Advanced Materials, The University of Adelaide, Adelaide, SA 5005, Australia

^*alan.gardin@adelaide.edu.au

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Vol. 19, Iss. 5 — May 2023

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Images

Figure 1
Illustration of the appearance of a physical phase $θ$ when two magnon modes $m_{0}, m_{1}$ (blue circles) both couple to two cavity modes $c_{0}, c_{1}$ (red circles). The black lines between each circle represent a coupling described by Eq. (7) (the Hermitian conjugate term is omitted for readability). The coupling phase measures the difference in orientation of the yellow arrows coming out of each circle. By moving to an appropriate reference frame using a unitary transformation, we can align the yellow arrows of two interacting modes, hence removing the coupling phase. We successively (from left to right) rotate the cavity mode $c_{0}$ , the cavity mode $c_{1}$ , and finally the magnon mode $m_{1}$ to align the yellow arrows with that of $m_{0}$ . After successive rotation of the operators, all yellow arrows point in the same direction, and yet a physical phase $θ = φ_{11} - φ_{01} - (φ_{10} - φ_{00})$ remains. Note that the free Hamiltonian is unaffected by the successive rotation of the modes, see Appendix pp2.
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Figure 2
Numerical spectra of $H_{θ}$ for $θ = 0$ and $θ = π$ as a function of the average magnon frequency $ω_{m} / 2 π$ , for two values of the magnon detuning $δ_{m}$ . Parameters are $ω_{c, 0} = 4 GHz$ , $ω_{c, 1} = 6 GHz$ , and $g_{0} = g_{1} = 0.03 ω_{c}$ . The dashed line corresponds to $ω = ω_{m}$ . The legend in (a) is common to (b),(c) and (d).
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Figure 3
Numerical spectra of $H_{θ}$ for $θ = 0$ (solid lines), and $θ = π$ (dots) as a function of the magnon detuning $δ_{m}$ at $ω_{m} = ω_{c}$ . The dashed lines are at $ω_{c}^{'} \pm | G_{π} |$ while the dotted lines are at $ω_{c}^{'} \pm | G_{0} |$ , with $G_{θ}$ evaluated using Eq. (20) for $δ_{m} = 0$ . We set $g_{0} = g_{1} = 0.01 ω_{c}$ in (a), $g_{0} = 0.01 ω_{c, 0}$ and $g_{1} = 0.01 ω_{c, 1}$ in (b). The definition of $ω_{c}^{'}$ is per the main text. Other parameters and legend are as in Fig. 2.
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Figure 4
Schematic of the interactions present in $H_{θ}$ in the rotated magnon basis [see Eqs. (22) and (23)] for (a) arbitrary $θ$ , (b) $θ = 0$ , (c) $θ = π$ . The black lines are labeled by the magnitude of the couplings, and the dotted black line represents the possibility of vanishing $δ_{m}$ . Note that $M_{0} = M_{2 π}$ .
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Figure 5
Numerical calculation of the microwave transmission through the system obtained using an input-output theory for $H_{θ}$ at different magnon detuning $δ_{m}$ . See the main text for the parameters. In (a) we see that the spectral line associated to $ω_{1}$ (see Fig. 2) does not lead to a maximum of transmission, contrary to (b)–(d). This shows that the eigenmode associated with $ω_{1}$ is a dark mode.
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Figure 6
Example diagrams showing coupling phase removal by unitary transformations. The conventions used are that of Fig. 1. (a) Rotation of the cavity mode $c$ using $U_{c} = e^{i φ c^{†} c}$ . The magnon mode $m$ is unaffected. (b) Rotation of the magnon mode $m$ using $U_{m} = e^{- i φ m^{†} m}$ . (c) Rotation of both magnon modes using $U = e^{- i φ_{j} m_{j}^{†} m_{j}}$ for $j \in {0, 1}$ . (d) Rotation of the cavity mode $c$ using $U = e^{i φ_{0} c^{†} c}$ and the magnon mode $m_{1}$ using $U = e^{- i (φ_{1} - φ_{0}) m_{1}^{†} m_{1}}$ .
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