Magnon photon coupling for magnetization antiparallel to the magnetic field

Current studies of cavity magnon polaritons are focused on ferromagnetic magnons for which the frequency increases with a static magnetic field. In this paper, we propose a ferromagnetic system with magnon frequency decreasing with a static magnetic field. It is achieved by antiparallel alignment between the magnetization and static magnetic field. The magnetization precession is stabilized by a large anisotropic field along the direction of magnetization. The analysis of the Polder tensor shows that the magnon modes in parallel and antiparallel alignments are analogous to those in an antiferromagnet. The strong coupling between a magnon and photon for antiparallel alignment results in an anticrossing gap in the transmission spectrum. Based on the Tavis-Cummings Hamiltonian and Bloch sphere representation, we show that the photon absorption decreases (increases) the spin angular momentum in antiparallel (parallel) alignment. The coupled Hamiltonians of harmonic oscillators are derived and have the same form for both parallel and antiparallel cases. The method developed and results presented are expected to be helpful to realize low-frequency magnon photon coupling that is similar to those in an antiferromagnet.

Figure 1

Magnetization precession in (a) parallel and (b) antiparallel alignment between $\vec{M}$ and ${\vec{H}}_0$. In (b), a large anisotropic field ($H_A>H_0$) is applied to stabilize the magnetization.

Figure 2

$|S_{21}|$ spectrum of a ferromagnet-embedded cavity for (a) parallel alignment and (b) antiparallel alignment calculated with the method given in Sec. 2. In each figure, the horizontal blue lines represent the odd mode (12.8 GHz) and even mode (14.3 GHz) of cavity resonance. The tilted lines represent the magnon mode and cross with cavity resonance to form level repulsion in (a) and (b). In the calculation of (b), we set a large anisotropic field of $H_A=0.5$ T. (c) $|S_{21}|$ spectrum of an antiferromagnet-embedded cavity calculated with the method given in Ref. [37].

Figure 3

Magnetization $\vec{M}$ and spin angular momentum $\vec{S}$ in the Bloch sphere representation. The static magnetic field ${\vec{H}}_0$ aligns along the $+\vec{z}$ direction while the effective magnetic field follows the magnetization. (a) $\vec{M}\parallel {\vec{H}}_0$ and $S^z=-S$ in the ground state; (b) $\vec{M}\parallel (-{\vec{H}}_0)$ and $S^z=+S$ in the ground state. (c) and (d) are the corresponding $|S_{21}|$ spectra of (a) and (b). The parameters used are ${\omega }_c=5$ GHz, $g=0.1$ GHz, and ${\kappa }_c=T_m=0.01$ GHz.