Squeezed states of magnons and phonons in cavity magnomechanics

We show how to create quantum squeezed states of magnons and phonons in a cavity magnomechanical system. The magnons are embodied by a collective motion of a large number of spins in a macroscopic ferrimagnet, and couple to cavity microwave photons and phonons (vibrational modes of the ferrimagnet) via the magnetic dipole interaction and magnetostrictive interaction, respectively. The cavity is driven by a weak squeezed vacuum field generated by a flux-driven Josephson parametric amplifier, which is essential to get squeezed states of the magnons and phonons. We show that the magnons can be prepared in a squeezed state via the cavity-magnon beam-splitter interaction, and by further driving the magnon mode with a strong red-detuned microwave field, the phonons are squeezed. We show optimal parameter regimes for obtaining large squeezing of the magnons and phonons, which are robust against temperature and could be realized with experimentally reachable parameters.

Figure 1

(a) A YIG sphere is simultaneously in a uniform bias magnetic field and near the maximum magnetic field of a microwave cavity mode, which is driven by a weak squeezed vacuum field generated by a flux-driven JPA. The magnon mode is directly driven by a microwave source (not shown) to enhance the magnomechanical coupling. The bias magnetic field ($z$ direction), the drive magnetic field ($y$ direction), and the magnetic field ($x$ direction) of the cavity mode are mutually perpendicular at the site of the YIG sphere. (b) Frequencies and linewidths of the system. The magnon mode with frequency ${\omega }_m$ and linewidth ${\kappa }_m$ is driven by a microwave field at frequency ${\omega }_0$ and the mechanical motion at frequency ${\omega }_b$ scatters photons onto the two sidebands at ${\omega }_0+{\omega }_b$ (blue sideband) and ${\omega }_0-{\omega }_b$ (red sideband, not shown). If the drive squeezed field with frequency ${\omega }_s$, the cavity with frequency ${\omega }_a$ and linewidth ${\kappa }_a$, and the magnon mode are resonant, the magnon mode is squeezed, and if they are further resonant with the blue (anti-Stokes) sideband, the mechanical mode is cooled and squeezed.

Figure 2

Variance of the magnon amplitude quadrature $\langle \delta x{\left(t\right)}^2\rangle$ vs (a) detunings ${\mathrm{\Delta }}_m$ and ${\mathrm{\Delta }}_a$, and (b) squeezing parameter $r$ and phase $\theta$. In (a) $r=2$, and $\theta =0$, and in (b) ${\mathrm{\Delta }}_a={\mathrm{\Delta }}_m=0$. The blank area denotes $\langle \delta x{\left(t\right)}^2\rangle >\frac{1}{2}$, i.e., above vacuum fluctuations. See text for details of the other parameters.

Figure 3

(a) Variance of the cavity phase quadrature $\langle \delta Y{\left(t\right)}^2\rangle$ and (b) of the magnon amplitude quadrature $\langle \delta x{\left(t\right)}^2\rangle$ vs squeezing parameter $r$ and coupling $g_{ma}$. (c) $\langle \delta Y{\left(t\right)}^2\rangle$ and (d) $\langle \delta x{\left(t\right)}^2\rangle$ vs $r$ and temperature $T$ (from 10 to 500 mK). We take $T=20$ mK for (a) and (b), and $g_{ma}=4{\kappa }_a$ for (c) and (d), and ${\mathrm{\Delta }}_a={\mathrm{\Delta }}_m=0$ and $\theta =0$ for all the plots. The other parameters are as in Fig. 2.

Figure 4

Variance of the mechanical position quadrature $\langle \delta q{\left(t\right)}^2\rangle$ vs (a) detunings ${\tilde{\mathrm{\Delta }}}_m$ and ${\mathrm{\Delta }}_a$, (b) couplings $g_{ma}$ and $G_{mb}$, and (c) squeezing parameter $r$ for temperatures $T=10$ mK (solid), 100 mK (dashed), and 200 mK (dotted-dashed). We take $g_{ma}/2\pi =4.2$ MHz and $G_{mb}/2\pi =1.5$ MHz in (a) and (c), ${\tilde{\mathrm{\Delta }}}_m={\omega }_b$ and ${\mathrm{\Delta }}_a=1.1{\omega }_b$ in (b) and (c), $r=1$ in (a) and (b), and ${\mathrm{\Delta }}_s={\omega }_b$ and $\theta =0$ for all the plots. In (a) [(b)] the blank area denotes $\langle \delta q{\left(t\right)}^2\rangle >0.5$ (0.74). See text for details of the other parameters.