Quantum Acoustomechanics with a Micromagnet

We show theoretically how to strongly couple the center-of-mass motion of a micromagnet in a harmonic potential to one of its acoustic phononic modes. The coupling is induced by a combination of an oscillating magnetic field gradient and a static homogeneous magnetic field. The former parametrically couples the center-of-mass motion to a magnonic mode while the latter tunes the magnonic mode in resonance with a given acoustic phononic mode. The magnetic fields can be adjusted to either cool the center-of-mass motion to the ground state or to enter into the strong quantum coupling regime. The center of mass can thus be used to probe and manipulate an acoustic mode, thereby opening new possibilities for out-of-equilibrium quantum mesoscopic physics. Our results hold for experimentally feasible parameters and apply to levitated micromagnets as well as micromagnets deposited on a clamped nanomechanical oscillator.

Figure 1

(a),(b) Schematic illustration of our proposal. (c) Magnetoelastic coupling strength between the Kittel magnon and the first 30 acoustic spheroidal modes with angular (azimuthal) mode number 2 (1) versus acoustic mode frequency. The upper scale indicates the $B_0$ needed to tune magnon and phonon in resonance. All axes are normalized to be independent on the micromagnet radius $R$. Parameters correspond to yttrium iron garnet (YIG): ${\rho }_m=5170\mathrm{kg}/{\mathrm{m}}^3$, $M_S=5.87\times {10}^5\mathrm{A}/\mathrm{m}$, $|\gamma |=1.76\times {10}^{11}{\mathrm{T}}^{-1}{\mathrm{s}}^{-1}$ [18, 19].

Figure 2

Relevant acoustomechanical parameters for $Q_x={10}^8$ versus acoustic quality factor $Q_p$. The dependence with the field gradient $b_g$ has been factored out explicitly, so the figures are $b_g$ independent. (a) Coupling (left) and mechanical frequency (right) normalized to linewidth of mode ${\widehat{c}}_2$. (b) Single-phonon cooperativity $C\equiv 4G_{x2}^2/{\gamma }_2{\gamma }_x$. In both panels we fix $\chi ={10}^{-2}$ by applying an external static field $B_0\approx \{5.4,0.45,0.018\}\mathrm{T}$ for $R=\{10,{10}^2,{10}^3\}\mathrm{nm}$, respectively.

Figure 3

Steady-state center-of-mass occupation ($Q_x={10}^8$) versus acoustic quality factor, for $R=100\mathrm{nm}$ (a) and $R=1\mu \mathrm{m}$ (b) and three values of the magnetic gradient $b_g$. Solid and dashed lines indicate results at room ($T_{e,j}=300\mathrm{K}$) and cryogenic temperatures ($T_{e,j}=100\mathrm{mK}$), respectively. The shaded area indicates the ground-state cooling region $⟨{\widehat{b}}^{†}\widehat{b}{⟩}_{\mathrm{ss}}<1$. The right-hand axes indicate the steady-state center-of-mass temperature $k_B{T}_x\approx \hslash {\omega }_x⟨{\widehat{b}}^{†}\widehat{b}{⟩}_{\mathrm{ss}}$ (${\omega }_x=2\pi \times 200\mathrm{kHz}$).

Figure 4

Power spectral density $S_{xx}(\omega )$, for $Q_p={10}^6$, $Q_x={10}^5$, $R=100\mathrm{nm}$, and $T_{e,j}=300\mathrm{K}$, at three different values of the field gradient $b_g$. Left inset: Peak splitting at $b_g={10}^4\mathrm{T}/\mathrm{m}$, for different acoustic quality factors $Q_p$. Right inset: Normalized power spectral density as a function of $b_g$ and detuning $\omega -{\omega }_x$. The dashed lines indicate the function $\pm G_{x2}(b_g)$. Strong coupling is reached at $b_g\approx {10}^4\mathrm{T}/\mathrm{m}$.