Dynamic Dissipative Cooling of a Magnomechanical Resonator in The Strong Magnomechanical Coupling Regime

Abstract

A cavity magnomechanical system with a yttrium iron garnet (YIG) sphere is proposed to realize the ground-state cooling of the magnomechanical resonator in the strong coupling regime. We theoretically investigate the cooling dynamics of the system and find that the magnomechanical resonator can be effectively cooled by dynamic dissipative modulation of the magnon mode within the range of experimentally feasible parameters. Moreover, we show that the cooling process is significantly accelerated and its cooling limit can be reduced remarkably by employing the periodic pulse dissipation. The scheme provides a new perspective for the research of ground-state cooling of magnomechanical resonators, which is helpful for the quantum manipulation of macroscopic mechanical devices.

Appendix

Equation (16) corresponds to dynamics differential equation of the second-order moments.

$$\begin{array}{c}\frac{d}{dt}\langle \delta {m}^{\dagger}\delta m\rangle =-i(G\langle \delta {m}^{\dagger}\delta b\rangle -{G}^{*}{\langle \delta {m}^{\dagger}\delta b\rangle }^{*}+G{\langle \delta m\delta b\rangle }^{*}-{G}^{*}\langle \delta m\delta b\rangle )-{\kappa }_{eff}\langle \delta {m}^{\dagger}\delta m\rangle ,\\ \frac{d}{dt}\langle \delta {b}^{\dagger}\delta b\rangle =-i(-G\langle \delta {m}^{\dagger}\delta b\rangle +{G}^{*}{\langle \delta {m}^{\dagger}\delta b\rangle }^{*}+G{\langle \delta m\delta b\rangle }^{*}-{G}^{*}\langle \delta m\delta b\rangle )-{\gamma }_{b}\langle \delta {b}^{\dagger}\delta b\rangle +{\gamma }_{b}{n}_{th},\\ \frac{d}{dt}\langle \delta {m}^{\dagger}\delta b\rangle =[-i({\Delta }_{eff}+{\omega }_{b})-\frac{{\kappa }_{eff}+{\gamma }_{b}}{2}]\langle \delta {m}^{\dagger}\delta b\rangle -i({G}^{*}\langle \delta {m}^{\dagger}\delta m\rangle -{G}^{*}\langle \delta {b}^{\dagger}\delta b\rangle +G{\langle \delta {m}^{2}\rangle }^{*}-{G}^{*}{\langle \delta {b}^{2}\rangle }^{*}),\\ \frac{d}{dt}\langle \delta m\delta b\rangle =[i({\Delta }_{eff}-{\omega }_{b})-\frac{{\kappa }_{eff}+{\gamma }_{b}}{2}]\langle \delta m\delta b\rangle -i(G\langle \delta {m}^{\dagger}\delta m\rangle +G\langle \delta {b}^{\dagger}\delta b\rangle +G+{G}^{*}\langle \delta {m}^{2}\rangle +G\langle \delta {b}^{2}\rangle ),\\ \frac{d}{dt}\langle \delta {m}^{2}\rangle =(2i{\Delta }_{eff}-{\kappa }_{eff})\langle \delta {m}^{2}\rangle -2iG(\langle \delta m\delta b\rangle +{\langle \delta {m}^{\dagger}\delta b\rangle }^{*}),\\ \frac{d}{dt}\langle \delta {b}^{2}\rangle =(-2i{\omega }_{b}-{\gamma }_{b})\langle \delta {b}^{2}\rangle -2i({G}^{*}\langle \delta m\delta b\rangle +G\langle \delta {m}^{\dagger}\delta b\rangle ).\end{array}$$

(16)

When the stability conditions are satisfied, the system will evolve to the steady-state. Then, let all derivative terms on the left side of the above system of covariance differential equations be zero, and the steady-state cooling limit is obtained by solving it. We focus on the resolved sideband regime ${\kappa }_{eff}<{\omega }_{b}$, and set ${\Delta }_{eff}=-{\omega }_{b}$. In addition, the cooperativity is $C=4{\left|G\right|}^{2}/\left(\kappa \gamma \right)\gg 1$. The steady-state average phonon number is given by

$${\overline{N} }_{std}\simeq \frac{4{|G|}^{2}+{\kappa }_{eff}^{2}}{4{|G|}^{2}({\kappa }_{eff}+{\gamma }_{b})}{\gamma }_{b}{n}_{th}+\frac{4{\omega }_{b}^{2}({\kappa }_{eff}^{2}+8{|G|}^{2})+{\kappa }_{eff}^{2}({\kappa }_{eff}^{2}-8{|G|}^{2})}{16{\omega }_{b}^{2}(4{\omega }_{b}^{2}+{\kappa }_{eff}^{2}-16{|G|}^{2})}.$$

(17)

In the case of the resolved sideband, it can be simplified to

$${\overline{N} }_{std}\simeq \frac{{\gamma }_{b}{n}_{th}(4{|G|}^{2}+{\kappa }_{eff}^{2})}{4{|G|}^{2}({\kappa }_{eff}+{\gamma }_{b})}+\frac{{\kappa }_{eff}^{2}+8{|G|}^{2}}{16{\omega }_{b}^{2}({\omega }_{b}^{2}-4{|G|}^{2})},$$

(18)

By adjusting the intensity of the external drive magnetic field, the magnomechanical coupling strength can be achieved to the strong coupling regime $\left(G>{\kappa }_{eff}\right)$. The steady-state average phonon number can be further simplified as

$${\overline{N} }_{std}\simeq \frac{{\gamma }_{b}{n}_{th}}{{\kappa }_{eff}+{\gamma }_{b}}+\frac{{|G|}^{2}}{2({\omega }_{b}^{2}-4{|G|}^{2})}.$$

(19)