Theory of quantum acoustomagnonics and acoustomechanics with a micromagnet

Recently, we proposed a way to engineer a flexible acoustomechanical coupling between the center-of-mass motion of an isolated micromagnet and one of its internal acoustic phonons by using a magnon as a passive mediator [C. Gonzalez-Ballestero, J. Gieseler, and O. Romero-Isart, Phys. Rev. Lett. 124, 093602 (2020)]. In our approach, the coupling is enabled by the strong magnetoelastic interaction between magnons and acoustic phonons which originates from the small particle size. Here, we substantially extend our previous work. First, we provide the full theory of the quantum acoustomagnonic interaction in small micromagnets and analytically calculate the magnon-phonon coupling rates. Second, we fully derive the acoustomechanical Hamiltonian presented in Gonzalez-Ballestero et al. Finally, we extend our previous results for the fundamental acoustic mode to higher-order modes. Specifically, we show the cooling of the center-of-mass motion with a range of internal acoustic modes. Additionally, we derive the power spectral densities of the center-of-mass motion which allow us to probe the same acoustic modes.

Figure 1

In this work, we theoretically study the interaction between the center-of-mass motion of a harmonically trapped micromagnet, the magnetization fluctuations about its fully magnetized state, namely magnons, and the quanta of elastic deformations in the magnet, i.e., acoustic phonons.

Figure 2

Lowest-order torsional (a) and spheroidal (b) acoustic mode eigenfrequencies of a YIG sphere, versus mode index $\lambda$. Different colors correspond to different values of the mode index $\nu$. The joining lines are a guide to the eye. The right vertical axis shows the eigenfrequencies for a YIG sphere of $R=500\mathrm{nm}$.

Figure 3

Magnonic eigenfrequencies of a YIG sphere for $l=1$ (red), $l=2$ (blue), and $l=3$ (orange), as a function of external static field $B_0$. The frequency ${\omega }_0$ (see main text) has been subtracted on the vertical axis for a better visualization. The shaded area shows the unstable region when $B_0<{\mu }_0{M}_S/3=246\mathrm{mT}$ for YIG.

Figure 4

Magnetoelastic couplings $g_{\alpha \beta }$ for the Kittel magnon–$S_{\nu 21}$ phonon (a), the $\left\{210\right\}$ magnon–$T_{\nu 21}$ phonon (b), the $\left\{210\right\}$ magnon–$S_{\nu 11}$ phonon (c), and the $\left\{210\right\}$ magnon–$S_{\nu 31}$ phonon pairs (d), versus acoustic mode frequency and for YIG parameters (see Table 1). The discrete acoustic modes $\nu =1,2,3,...$ are represented by solid circles. The upper axes show the external field $B_0$ required to tune the magnon in resonance with the corresponding acoustic phonon [see Eq. (31)]. All axes are normalized to be size independent. The contour plots show the normalized acoustic mode intensities corresponding to $\nu =1$, i.e., to the lowest-frequency acoustic mode in each panel.

Figure 5

(a) A weak inhomogeneous magnetic field drive ${\mathbf{H}}_d(\mathbf{r},t)$ (blue), superposed with the homogeneous bias field $H_0{\mathbf{e}}_z$ (red), results in an interaction between the center-of-mass motion of a harmonically trapped micromagnet and its internal magnetization field $\mathbf{m}(\mathbf{r},t)$ (green), i.e., its magnonic modes. (b) Some potential setups for nonmagnetic trapping of the micromagnet: quasielectrostatic levitation (top), tethering to a cantilever (middle), and deposition on a carbon nanotube (bottom).

Figure 6

(a) Energy level diagram of the proposed acoustomechanical system. (b) Magnon fraction of each hybrid magnon-phonon normal mode as a function of the adimensional parameter $g/\mathrm{\Delta }$. (c) External bias field $B_0={\mu }_0{H}_0$ required to achieve $\chi ={\chi }_0={10}^{-2}$ versus micromagnet radius $R$, for the 10 lowest energy acoustic modes, that are able to interact with the Kittel magnon.

Figure 7

Acoustomechanical figures of merit versus acoustic quality factor $Q_p$, for $Q_x={10}^8$ and ${\omega }_{tx}=2\pi \times 200\mathrm{kHz}$. (a) Coupling between center of mass and mode ${\widehat{c}}_2$ normalized to magnetic field gradient $b$ and linewidth ${\gamma }_2$ (left), and ratio of center-of-mass frequency to linewidth ${\gamma }_2$ (right). (b) Cooperativity $C=4G_{x2}^2/\left({\gamma }_x{\gamma }_m\right)$ normalized to square of the field gradient $b^2$. The solid lines for each radius correspond to the $S_{121}$ acoustic mode, whereas the lower ends of the shaded areas correspond to the $S_{10,21}$ acoustic mode. The curves for all the acoustic modes with $1\le \nu \le 10$ lie, in descending order, within the shaded area.

Figure 8

Steady-state center-of-mass occupation versus acoustic quality factor, for the $S_{121}$ acoustic phonon and three different values of field gradient $b$. The parameters are $Q_x={10}^8, {\omega }_{tx}=2\pi \times 200\mathrm{kHz}$, and $R=100\mathrm{nm}$ [panel (a)] or $R=1\mu \mathrm{m}$ [panel (b)]. The three upper (lower) curves in each panel correspond to environments at $T_e=300\mathrm{K}$ ($T_e=100\mathrm{mK}$). The dashed lines depict the approximation Eq. (52). The shaded area corresponds to ground-state cooling, ${\langle {\widehat{b}}^{†}\widehat{b}\rangle }_{\mathrm{ss}}<1$.

Figure 9

Steady-state center-of-mass occupation versus acoustic quality factor, for different acoustic $S_{\nu 21}$ modes, and for $R=100\mathrm{nm}$ [panel (a)] or $R=1\mu \mathrm{m}$ [panel (b)]. We take the same parameters as in Fig. 8, and $b={10}^3\mathrm{T}/\mathrm{m}$. The four upper (lower) curves in each panel correspond to environments at $T_e=300\mathrm{K}$ ($T_e=100\mathrm{mK}$). The shaded area corresponds to ground-state cooling, ${\langle {\widehat{b}}^{†}\widehat{b}\rangle }_{\mathrm{ss}}<1$.

Figure 10

Position power spectral density for a micromagnet with $R=100\mathrm{nm}$ [panel (a)] or $R=1\mu \mathrm{m}$ [panel (b)], for $T_e=300\mathrm{K}, Q_p={10}^7, Q_x={10}^5$, and coupling to the $S_{121}$ acoustic mode. In the main panels, different curves correspond to $b=0$ (orange), $b={10}^3\mathrm{T}/\mathrm{m}$ (blue), and $b={10}^4\mathrm{T}/\mathrm{m}$ (red). The left insets show the power spectral density near $\omega ={\omega }_{tx}$ at $b={10}^4\mathrm{T}/\mathrm{m}$, for different values of the acoustic quality factor $Q_p$. The right insets show the power spectral density in normalized units as a function of frequency $\omega$ and magnetic field gradient $b$. The dashed curves correspond to $\pm |G_{x2}|$ as a function of $b$.

Figure 11

Position power spectral density for $Q_p={10}^7, b={10}^4\mathrm{T}/\mathrm{m}$, and two different micromagnet sizes, $R=100\mathrm{nm}$ (green curves) and $R=1\mu \mathrm{m}$ (purple curves). The different panels (a) to (d) correspond to different acoustic phonons $S_{\nu 21}$ coupled to the center-of-mass motion.

Figure 12

Increase in the internal temperature of the micromagnet computed from Eq. (D11) versus homogeneous driving field $B$, at $T_e=300\mathrm{K}$ and for radius $R=100\mathrm{nm}$ [panel (a)] and $R=1\mu \mathrm{m}$ [panel (b)]. Different colors correspond to different values of the imaginary part of the permittivity, ${\epsilon }_i$. The dashed line in each panel marks the maximum field felt by the micromagnet under the inhomogeneous driving Eq. (41), on its thermal motion at a temperature $T_e$ and a field gradient $b={10}^4$ T/m [Eq. (D14)]. Unspecified parameters are taken as in the main text.

Figure 13

Same as Fig. 12 with $T_e=100\mathrm{mK}$.