Quantum parametric amplification of phonon-mediated magnon-spin interaction

Hybrid quantum systems (HQSs) combine different physical components with complementary functionalities, thus providing multitasking capabilities that individual components cannot offer Kurizki et al. (2015). In particular, benefiting from the unprecedented tunability and compatibility, HQSs based on magnonics, or say hybrid magnonics, promote the development of novel quantum technologies, such as microwave-to-optical quantum transducers Hisatomi et al. (2016); Zhu et al. (2020a); Chai et al. (2022) and quantum detection and sensing Cao and Yan (2019); Colombano et al. (2020); Wolski et al. (2020); Crescini et al. (2020); Lu et al. (2021). However, a current challenge of functionalizing hybrid magnonics is how to coherently couple the magnons to quantum systems that either weakly couple to magnons or have large energy difference from magnon modes. Specifically, the former is mainly caused by the inadequate spatial matching, thus making the coupling strengths lower than intrinsic losses and decoherence of both systems. For the latter, the coherent process of quantum state transfer can hardly take place if the energy cannot be conserved, even in the presence of strong interactions. A typical case is to coherently couple magnons to external mechanical oscillators (corresponding to mechanical phonons aside from acoustic phonons due to the deformation of the magnetic system itself). This is generally difficult to achieve due to their dissimilar excitation energies or frequency mismatch Zhang et al. (2016b); Gonzalez-Ballestero et al. (2020a, b). This cannot enable the coherent information transfer between magnons and mechanical phonons, and thus may limit the development of practical applications based on hybrid magnon-phonon systems Holanda et al. (2018); Bozhko et al. (2020), such as high-precision metrology of small magnetic fields and mechanical displacements. In addition, recent studies have found that magnons can strongly interact with solid-state spins such as nitrogen-vacancy centers and silicon-vacancy centers in diamond Neuman et al. (2020); Wang et al. (2021); Skogvoll et al. (2021); Fukami et al. (2021); Xiong et al. (2022). However, reaching the single magnon-spin strong-coupling regime (SCR) often requires the spins to be positioned very close to the magnet surface, which inevitably causes difficulties of manipulating single spins.

In this work, we propose an experimentally feasible method for strongly coupling magnons to mechanical phonons and to single solid-state spin in a hybrid tripartite system. Our proposal is based on a hybrid device where a micromechanical cantilever, pumped by a detuned parametric drive, interacts simultaneously with a nanomagnet and a single nitrogen-vacancy (NV) center via magnetic field gradients. We show that the magnon mode can be coupled, in a coherent and tunable way, to the phonon mode by adjusting the mechanical parametric drive (MPD). With experimentally accessible parameters, the magnon-mechanical coupling can be exponentially enhanced, thus driving the initially weak-coupling system into the SCR and even the ultrastrong-coupling regime (UCR). The coupling enhancement can facilitate the extension of coherence time of the quantum system where the mechanical mode acts as a long-lived quantum memory. We also extend our protocol by coupling an NV spin to the mechanical phonon. In the dispersive-coupling regime, we realize coherent quantum state transfer between the nanomagnet and the NV center. Remarkably, the mechanical noise amplified by the MPD can hardly corrupt the coherent and strong-coupling dynamics of the proposed system without using additional engineered-reservoir techniques that are generally required for suppressing the amplified noise Lü et al. (2015); Qin et al. (2019, 2018); Leroux et al. (2018); Chen et al. (2021); Tang et al. (2022); Li et al. (2020b); Wang et al. (2022), which could simplify the experimental requirement and implementation.

Related forms of using the quantum parametric amplification (QPA) to significantly enhance and controllably manipulate the interaction between quantum objects have been studied in various coupled systems ranging from optomechanics Lü et al. (2015); Lemonde et al. (2016); Qin et al. (2019); Zhao et al. (2020); Liu et al. (2023), cavity or circuit QED Qin et al. (2018); Leroux et al. (2018); Wang et al. (2019); Zhu et al. (2020b); Groszkowski et al. (2020); Wang et al. (2020a, b); Qin et al. (2021); Chen et al. (2021); Tang et al. (2022); Villiers et al. , trapped ions Ge et al. (2019); Burd et al. (2021); Affolter et al. (2023) and hybrid spin-mechanical systems Li et al. (2020b); Wang et al. (2022); Pan et al. (2023). Our work further extends and determines the capabilities of exploiting the QPA to enhance the magnon-phonon and the phonon-mediated magnon-spin interactions in a hybrid bipartite system. Notice that a recent study has shown that the QPA can be used to enhance the direct tripartite coupling among spins, magnons and phonons Hei et al. (2023), where markedly different physical model and coupling mechanism were considered in a different hybrid setup. Our findings in achieving the effective enhancement of the bipartite interaction between the magnon and an external phonon provide a versatile tool for various intriguing applications, such as the improvement of magnon-assisted ground-state cooling of the mechanical motion Gonzalez-Ballestero et al. (2020a); Asjad et al. (2023), and the preparation of highly entangled cat states between the magnon and the solid spin in the weak-coupling regime Sharma et al. (2021); Sun et al. (2021). As for the enhancement of the bipartite interaction between the magnon and the NV spin, our strategy strengthens the ability for processing complicated quantum information tasks where both the NV spin and the magnons can act as collaborate quantum memories. Our scheme may also find inspiring applications to design quantum transducers, quantum memories and high-precision measurements.

2   Quantum dynamics of hybrid magnon-phonon-spin system

We consider a hybrid tripartite system, as schematically illustrated in Fig. 1(a), where a spherical nanomagnet of radius $R$ is attached on top of a singly-clamped micromechanical cantilever of dimensions ($\ell,w,t$) Vinante et al. (2011); Burgess et al. (2013); Vinante and Falferi (2013); den Haan et al. (2015), and a single NV center is placed under a magnetic tip fixed at the bottom of the cantilever Kolkowitz et al. (2012). The nanomagnet interacts with an external magnetic field, which has a homogeneous component $\textbf{B}_{z}=B_{z}\textbf{e}_{z}$, and a linear gradient $\textbf{B}_{0}(x)=-b_{0}x\textbf{e}_{x}$, with $b_{0}$ the magnetic field gradient. The homogeneous field $\textbf{B}_{z}$ is introduced to align the magnetization of the nanomagnet to the $z$ direction, while the linear gradient $\textbf{B}_{0}(x)$ is used to couple the nanomagnet to the mechanical motion of the cantilever (see below). The motion of the magnetic tip (assumed to be only along the $x$ direction) produces a strong magnetic gradient $\textbf{B}_{1}(x)=-b_{1}x\textbf{e}_{x}$, which causes Zeeman shifts of the NV center, giving rise to the magnetic coupling between the NV spin and the mechanical motion of the cantilever (see below). Note here that the magnetic gradient $\textbf{B}_{0}(x)$ has no effect on the spin-mechanical coupling because it does not change the local magnetic field sensed by the motionless NV spin. Additionally, we assume that the cantilever is electrically pumped by a periodic drive that modulates the mechanical spring constant in time Rugar and Grütter (1991); Szorkovszky et al. (2011), thereby amplifying the zero-point fluctuations of the mechanical motion. To be specific, a time-varying voltage originating from a tunable oscillating pump under the cantilever is applied to form a general capacitor between the two electrodes, one of which is coated on the lower surface of the cantilever and the other is placed on top of the pump, as illustrated in Fig. 1(a).

2.1   Free Hamiltonians for phonon, spin and magnon

We begin with the Hamiltonian of the mechanical cantilever. By modeling the mechanical motion of the cantilever as a harmonic oscillator Rabl et al. (2009, 2010) and following the standard quantization procedure Lemonde et al. (2016); Li et al. (2020b); Wang et al. (2022), the Hamiltonian of the cantilever oscillator is obtained as

where $\hat{b}$ ($\hat{b}^{\dagger}$) is the annihilation (creation) operator of the motional quanta, i.e., phonon, with frequency $\omega_{x}$, and $\omega_{p}$, $\Omega_{p}$ are the pump frequency and amplitude, respectively. Equation (1) shows that a two-phonon (i.e., parametric) drive of the mechanical mode can be achieved by applying the time-dependent pump to the mechanical oscillator. This detuned parametric drive, as detailed below, modifies the eigenstates of the mechanical Hamiltonian to be squeezed Fock states, whose amplified fluctuations in the anti-squeezed quadrature yield larger interactions with both the magnon and the NV spin [see Fig. 1(b)]. In addition, working with the typical mechanical zero-point fluctuations of hundreds of femtometers, the proper distances between the two electrode plates, e.g., tens of nanometers, and the accessible parameters with respect to the electric drive module, the nonlinear amplitude $\Omega_{p}$ of up to GHz could be realized Li et al. (2020b).

For the single NV center, its electronic ground state is a $S=1$ spin triplet with basis states $\left|m_{s}\right\rangle$, where $m_{s}=0,\pm 1$. The zero-field splitting between the degenerate sublevels $\left|m_{s}=\pm 1\right\rangle$ and $\left|m_{s}=0\right\rangle$ is $D=2\pi\times 2.87$ GHz Childress et al. (2006); Kolkowitz et al. (2012); Taylor et al. (2008). We assume that the NV symmetry axis is along the $z$ direction. The magnetic field $\textbf{B}_{z}$ lifts the degeneracy of the states $|m_{s}=\pm 1\rangle$, causing a Zeeman splitting $\delta=2g_{e}\mu_{B}B_{z}/\hbar$, where $g_{e}\simeq 2$ is the Land$\acute{\mathrm{e}}$ factor and $\mu_{B}$ the Bohr magneton. Taking the state $|0\rangle$ to be the energy reference, the free Hamiltonian of the NV center has the form

where $\delta_{j}\equiv D\pm\delta/2$. By choosing a proper magnetic strength $B_{z}$ such that $\left|\delta_{+1}-\omega_{x}\right|\gg\left|\delta_{-1}-\omega_{x}\right|$, the contribution $\propto$ $\left(\left|+1\rangle\langle 0\right|+\left|0\rangle\langle+1\right|\right)(\hat{b}^{\dagger}+\hat{b})$ oscillates rapidly and thus can be excluded from the dynamics. The Hamiltonian (2) can then be rewritten as

where $\omega_{\mathrm{NV}}\equiv\delta_{-1}$ and $\hat{\sigma}_{z}=|-1\rangle\langle-1|-|0\rangle\langle 0|$.

Next, we turn our attention to the spherical nanomagnet that supports the magnetization-wave or spin-wave (magnon) modes Zare Rameshti et al. (2022). Under the dipolar, isotropic, and magnetostatic approximations Gonzalez-Ballestero et al. (2020b), one can obtain the magnetization eigenmodes, known as the magnetostatic dipolar spin waves, or Walker modes for spherical magnets Walker (1957, 1958); Röschmann and Dötsch (1977). The eigenfrequencies of the Walker modes, $\omega_{\beta}\equiv\omega_{lmn}$, depend on the external homogeneous field $B_{z}\equiv\mu_{0}H_{z}$, where $l=1,2,3,\dots$, $m\in[-l,l]$, $n=0,1,2,\dots,n_{\mathrm{max}}(l)$, and $\mu_{0}$ is the vacuum permeability. These Walker modes can be further quantized Mills (2006), resulting in the free magnon Hamiltonian

with the bosonic ladder operators $[\hat{s}_{\beta},\hat{s}_{\beta^{\prime}}^{\dagger}]=\delta_{\beta\beta^{\prime}}$. Correspondingly, the spin-wave magnetization operator in the Schr$\mathrm{\ddot{o}}$dinger picture is given by

where $\tilde{\mathbf{m}}_{\beta}(\mathbf{r})$ is the mode function of the spin wave, and $\mathcal{M}_{0\beta}\equiv\sqrt{\hbar|\gamma|M_{S}/\tilde{\Lambda}_{\beta}}$ is the zero-point magnetization, with $|\gamma|=g_{e}\mu_{B}/\hbar$ the absolute value of the gyromagnetic ratio, $M_{S}$ the saturation magnetization and $\tilde{\Lambda}_{\beta}\equiv 2\operatorname{Im}\int dV\tilde{m}_{x}^{*}(\mathbf{r})\tilde{m}_{y}(\mathbf{r})$ the normalization constant.

For the proposed hybrid system in Fig. 1(a), we desire to provide a mechanism where the spin-phonon coupling is determined only by the magnetic gradient $\textbf{B}_{1}(x)$ originating from the vibration of magnetic tip, while the inhomogeneous external field $\textbf{B}_{0}(x)$ is responsible only for the magnon-phonon coupling. To this end, we first select a magnetic tip with a size of $\sim$100 nm, and the magnetic substance is selected as Dysprosium (Dy) due to a high saturation magnetization of up to 3.7 T Mamin et al. (2012), which is larger than that of yttrium iron garnet (YIG) with $\mu_{0}M_{S}\approx 0.74$ T. This allows us to obtain a strong magnetic gradient exceeding $10^{7}$ T/m at short distances for the NV center, as demonstrated in the simulation below. Moreover, we consider a specific arrangement of magnets such that the magnetic tip is positioned, along the $z$ axis, at a distance of hundreds of nanometers away from the nanomagnet. This can avoid the interplay of the magnetic fields generated by the magnetic tip and the nanomagnet due to an adequate spatial separation. To verify this, we numerically simulate the relevant magnetic fields and compute the value of magnetic gradient using the finite-element analysis method, and the results are shown in Figs. 1(c)-1(e). Figure 1(c) is an example of the simulated magnetic field lines and strength distribution for a Dy tip with underside diameter and height of 100 nm, and a YIG sphere with radius of 100 nm, which are separated by 300 nm in the direction of $z$-axis. One can see in this configuration that the field distribution around the Dy tip can hardly be affected by the YIG sphere and vice versa. Moreover, Figs. 1(d) and 1(e) show values of the magnetic strength $|B_{1}(x)|$ and the gradient $b_{1}$ along the $x$ direction for $z=0$. Here, the effect of the YIG sphere on the Dy tip is evaluated by comparing the values of $B_{1}(x)$ and $b_{1}$ after removing the YIG sphere (labeled with “Air” in the plots). We find that both the magnetic strength and the magnetic gradient remain unchanged in the presence or absence of the YIG sphere, revealing the negligible influence of the YIG sphere on the magnetic tip in the present configuration.

The oscillations of the mechanical cantilever as well as the Dy tip produce a strong magnetic gradient that causes the Zeeman shifts of the NV center Li et al. (2016); Sánchez Muñoz et al. (2018); Zhou et al. (2022), thereby coupling the spin and mechanical degrees of freedom. The magnetic interaction Hamiltonian can be written as

where $\hat{\textbf{S}}$ denotes the spin operator and $\textbf{r}_{0}$ the position of the NV center. In our setup, the bending motion of the cantilever (along the $x$ direction) modulates the local magnetic field sensed by the NV center, giving rise to the spin-mechanical coupling with the interaction Hamiltonian

After using again $\hat{x}=x_{\mathrm{zpf}}(\hat{b}^{\dagger}+\hat{b})$, this interaction Hamiltonian becomes

where $\lambda\equiv-\mu_{B}g_{e}b_{1}x_{\mathrm{zpf}}/\hbar$ denotes the spin-mechanical coupling rate and the spin operators $\hat{\sigma}_{+}=\hat{\sigma}_{-}^{\dagger}=|-1\rangle\langle 0|$. This spin-mechanical coupling depends on the magnetic gradient $b_{1}$, which can be adjusted by positioning the NV spin at the suitable distances from the Dy tip. For instance, a large gradient $b_{1}\approx 4.5\times 10^{7}$ T/m can be obtained for a distance of about 10 nm between the Dy tip and the NV spin, as shown in Fig. 1(e).

In addition to the coherent dynamics described by Eq. (17), we take into account the couplings of the mechanical oscillator and the nanomagnet to the Markovian baths, which cause the mechanics to be damped at a rate $\gamma_{x}$ and the magnon at a rate $\gamma_{s}$. In the presence of the MPD, the noise coming from the mechanical bath is also amplified. In the weak mechanical dissipation limit $\gamma_{x}\ll\Delta_{s}$, the mechanical bath with a thermal occupation $\bar{n}_{x}$ corresponds to a bath for the $\hat{b}_{s}$ mode with a thermal occupation $\bar{n}_{x}^{s}=\bar{n}_{x}\cosh(2r_{p})+\sinh^{2}(r_{p})$ (see Appendix B for details). That is, the $\hat{b}_{s}$ mode can be considered as being coupled, with an unchanged vacuum damping rate $\gamma_{x}$, to a thermal bath but with thermal occupation $\bar{n}_{x}^{s}$. The system dynamics under dissipation can then be described by the Lindblad master equation

Here, $\hat{H}_{\mathrm{mm}}^{\mathrm{S}}$ is given by Eq. (16), $\hat{\rho}$ is the density operator, $\mathcal{L}_{x}[\hat{\rho}]=\gamma_{x}[\left(\bar{n}_{x}^{s}+1\right)L_{\hat{b}_{s}}\hat{\rho}+\bar{n}_{x}^{s}L_{\hat{b}_{s}^{\dagger}}\hat{\rho}]$ and $\mathcal{L}_{s}[\hat{\rho}]=\gamma_{s}[\left(\bar{n}_{s}+1\right)L_{\hat{s}}\hat{\rho}+\bar{n}_{s}L_{\hat{s}^{\dagger}}\hat{\rho}]$, where $L_{\hat{o}}\hat{\rho}\equiv\hat{o}\hat{\rho}\hat{o}^{\dagger}-[\hat{o}^{\dagger}\hat{o},\hat{\rho}]/2$. Moreover, $\bar{n}_{j}=\left[\exp\left(\hbar\omega_{j}/k_{B}T_{j}\right)-1\right]^{-1}$ ($j=x,s$), with $k_{B}$ the Boltzmann constant and $T_{j}$ the temperature of the thermal bath of each degree of freedom. We assume that all modes couple with the thermal reservoirs at a common temperature $T$. It is also convenient to define the mechanical quality ($Q$) factor as $Q_{x}=\omega_{x}/\gamma_{x}$. The experimental values of $Q_{x}$ exceeding $10^{8}$ have been demonstrated recently in nanomechanical resonators Norte et al. (2016); Tsaturyan et al. (2017); Ghadimi et al. (2018). Regarding the magnon linewidth, we consider $\gamma_{s}=2\pi\times 1$ MHz for the YIG material Zhang et al. (2014); Tabuchi et al. (2015); Shen et al. (2022).

Next, we explore the physical regimes where our proposed magnon-mechanical system can be achieved using realistic parameters. We assume that the dimensions of the micromechanical cantilever are $(\ell,w,t)=(4,0.1,0.02)$ $\mu$m. Then, the frequency and the effective mass of this mechanical oscillator are estimated, respectively, as $\omega_{x}\approx 3.516\times\left(t/\ell^{2}\right)\sqrt{E/12\rho}\approx 2\pi\times 3.8$ MHz and $M=\rho\ell wt/4\approx 7\times 10^{-18}$ kg, with the Young’s modulus $E\approx 1.22\times 10^{12}\mathrm{~{}Pa}$ and the mass density $\rho\approx 3.52\times 10^{3}\mathrm{~{}kg/m^{3}}$ for diamond. For the YIG nanomagnet, the sizes in the range of 10 $\leq R\leq$ 100 nm are considered hereafter, in accordance with the approximations for deriving magnon eigenmodes Gonzalez-Ballestero et al. (2020b). The magnon frequency $\omega_{K}=|\gamma|B_{z}$ can be tuned by the external magnetic field $B_{z}$, and we apply a field $B_{z}\approx 0.084$ T such that the spin Hamiltonian (3) is valid. This leads to $\omega_{K}\approx 2\pi\times 2.35$ GHz. In addition, we consider a large field gradient $b_{0}\approx 10^{7}$ T/m for the nanomagnet Mamin et al. (2007, 2012); Poggio and Degen (2010). Once the parameters $b_{0}$, $R$ and $x_{\mathrm{zpf}}$ are given, the magnon-mechanical coupling can be determined via $g=b_{0}x_{\mathrm{zpf}}\mathcal{M}_{K}V/2\hbar$. We list the main system parameters used for our numerical calculations in Table 1 (see Appendix C).

In Fig. 2(a), we plot the ratio $g_{r}/\gamma_{s}$ versus $r_{p}$ for two values of the nanomagnet radius $R$. Because we always have $\gamma_{x}\ll\gamma_{s}$ for typical values of $Q_{x}$, the ratio $g_{r}/\gamma_{s}>1$ can thus be seen as an indication of the system entering the SCR, whereas $g_{r}/\gamma_{s}\leq 1$ corresponds to the weak-coupling regime (WCR). One can see from Fig. 2(a) that $g_{r}/\gamma_{s}$ increases gradually as a function of $r_{p}$ due to the effect of the MPD. For a relatively small $R$ of 10 nm, the current system cannot reach the SCR even for $r_{p}=2.5$, whereas for a larger $R$ of 100 nm, the system can be enhanced into the SCR as long as $r_{p}>0.93$. Remarkably, one can obtain $g_{r}/|\Delta_{s}|>0.1$ for $r_{p}\geq 2.42$, indicating that the system enters the UCR. Figure 2(b) shows the cooperativity $C_{r}=4g_{r}^{2}/[\gamma_{x}\gamma_{s}(1+\bar{n}_{x}^{s})]$ versus the mechanical $Q$ factor $Q_{x}$ ranging from $10^{5}$ to $10^{8}$ for different $r_{p}$, considering $R=100$ nm and a cryogenic temperature $T=10$ mK Tsaturyan et al. (2017). The system resides in the high-cooperativity ($C>1$) regime even at a moderate $Q_{x}\sim 10^{5}$, and $C_{r}$ increases significantly with increasing $Q_{x}$. In contrast, increasing $r_{p}$ leads to the decrease in $C_{r}$, due to the amplified mechanical noise responsible for the rapidly accumulated thermal phonons $\bar{n}_{x}^{s}$ with increasing the parametric drive. Note that this is generally a prominent problem for realizing coherent energy transfer in previous schemes, however it can be circumvented in our proposal as verified in the following.

By numerically solving the master equation (18), the time evolution of the occupation probabilities of the mean phonon and magnon numbers can be evaluated. Figure 2(d) shows $\langle\hat{b}_{s}^{\dagger}\hat{b}_{s}\rangle$ (solid curve) and $\langle\hat{s}^{\dagger}\hat{s}\rangle$ (dashed curve) versus the evolution time for $r_{p}=2.5$ and $Q_{x}=10^{8}$. The hollow marks are obtained by solving the master equation (18) with the effective Hamiltonian (17), and are in good agreement with the exact numerical results (solid and dashed curves). The typical vacuum Rabi oscillations occur for multiple periods, revealing the SCR that the system resides. Notably, we also find that the coherent dynamics can hardly be interrupted in the presence of the amplified mechanical noise, and this is shown to be valid at least for $Q_{x}\geq 10^{5}$. For larger mechanical damping, e.g., $Q_{x}=10^{3}$, the coherent dynamics of the system are corrupted drastically by the amplified thermal noise, with the state transfer between the mechanical phonon and the magnon being completely decoupled, as depicted in Fig. 2(e) (see the top two curves). To be more specific, the enhancement of the mechanical noise induced by the phonon squeezing can be effectively evaluated via $\gamma_{x}^{\prime}/\gamma_{x}=\cosh(2r_{p})$, where $\gamma_{x}^{\prime}$ is the effective damping rate of the $\hat{b}_{s}$ mode. The enhancement of the magnon-mechanical coupling is $g_{r}/g=\cosh(r_{p})$. For $Q_{x}\geq 10^{5}$ or $\gamma_{x}\leq 2\pi\times 38$ Hz, the enhanced damping rate $\gamma_{x}^{\prime}\leq 2\pi\times 2.8$ kHz, which is at least three orders of magnitude smaller than the parametrically-enhanced coupling $g_{r}\approx 2\pi\times 4.23$ MHz. This allows the observation of the coherent state transfer in the system. In contrast, for relatively low $Q_{x}=10^{3}$ or $\gamma_{x}=2\pi\times 3.8$ kHz, the damping rate is effectively enhanced by the phonon squeezing to $\gamma_{x}^{\prime}=2\pi\times 0.28$ MHz. In this case, $g_{r}$ is just about 15 times larger than $\gamma_{x}^{\prime}$, and therefore, the amplified noise corrupts the coherent state transfer in the system. This may lead to a certain limitation of the proposed scheme in case of the mechanical resonator possessing a extremely-low $Q$. To eliminate completely the amplified noise from the mechanical bath, a possible strategy is to inject an orthogonally squeezed vacuum, such that the bath of the $\hat{b}_{s}$ mode remains in vacuum (see Appendix B for more details). The coherent dynamics of the system can then be recovered, as verified in Fig. 2(e) (see the bottom two curves). Remarkably, the cooperativity $C_{r}$ after introducing the auxiliary reservoir is renewed as $C_{r}^{\prime}\equiv 4g_{r}^{2}/[\gamma_{x}\gamma_{s}(1+\bar{n}_{x})]$, which can be parametrically-enhanced by increasing $r_{p}$, as shown Fig. 2(c).

3.2   Quantum dynamics with enhanced magnon-spin coupling

Now let us integrate the NV spin into the abovementioned magnon-mechanical system. The Hamiltonian describing the hybrid tripartite system, in the rotating frame with $\omega_{p}$, is given by

with $\Delta_{\mathrm{NV}}=\omega_{\mathrm{NV}}-\omega_{p}$. We use the squeezing transformation $\hat{U}_{s}$ again to diagonalize the mechanics-only parts in $\hat{H}_{\mathrm{Hybrid}}$, and the resulting Hamiltonian reads

where $\lambda_{r}=\lambda\cosh(r_{p})$ and $\lambda_{c}=\lambda\sinh(r_{p})$ are the enhanced spin-mechanical couplings by the MPD. In the dispersive regime with properly large detunings (see Appendix D), the squeezed phonon mode $\hat{b}_{s}$ is only virtually excited, which allows us to adiabatically eliminate the mechanical degree of freedom from the dynamics. We further consider that the magnon frequency is very close to the NV spin frequency, namely $\omega_{K}\approx\omega_{\mathrm{NV}}$, which can be achieved by introducing an additional static magnetic field (with the opposite direction to $\textbf{B}_{z}$) for the nanomagnet (see relevant realistic parameters below). Furthermore, the magnetic gradient $b_{1}$ is adjusted such that the bare spin-mechanical coupling $\lambda$ is nearly equal to the bare magnon-mechanical coupling $g$. Given these settings, the effective Hamiltonian describing the phonon-mediated magnon-spin interactions can then be written as (see Appendix D)

where $\Delta_{k}=g_{r}^{2}/(\Delta_{K}-\Delta_{s})$, $\Delta_{n}=\lambda_{r}^{2}/(\Delta_{\mathrm{NV}}-\Delta_{s})$ are the effective detunings, and $g_{ms}=g_{r}\lambda_{r}/[\Delta_{K(\mathrm{NV})}-\Delta_{s}]$ is the effective magnon-spin coupling rate. The Hamiltonian (21) indicates that we have achieved an effective two-mode system where the magnon is resonantly coupled to the NV spin with an enhanced coupling rate.

Under a realistic consideration, the dissipative dynamics of the tripartite system is governed by the master equation

where $\hat{H}_{\mathrm{Hybrid}}^{\mathrm{S}}$ is given in Eq. (20), and $\mathcal{L}_{z}[\hat{\rho}]=\gamma_{z}L_{\hat{\sigma}_{z}}\hat{\rho}$ with $\gamma_{z}$ the single spin dephasing rate of the NV center. Experimentally, the NV dephasing time $T_{2}=1/\gamma_{z}$ can readily reach several hundreds of kilohertz at both room temperature and cryogenic temperature Ishikawa et al. (2012); Mamin et al. (2013); Bar-Gill et al. (2013); Ovartchaiyapong et al. (2014), and hereafter we consider $\gamma_{z}\approx 2\pi\times 1$ kHz. As for other experimental parameters, we recall the scenario considered in Sec. 3.1, where relevant experimental parameters are $\omega_{x}\approx 2\pi\times 3.8$ MHz, $Q_{x}=10^{8}$, and $g\approx 2\pi\times 0.69$ MHz for $R=100$ nm. For the NV spin, we apply a static magnetic field $B_{z}\approx 0.068$ T, which gives $\omega_{\mathrm{NV}}\approx 2\pi\times 0.96$ GHz. Furthermore, an additional local field $B_{z}^{\prime}\approx 0.034$ T is introduced to adjust the frequency of the nanomagnet, such that $\omega_{K}=|\gamma|(B_{z}-B_{z}^{\prime})\approx\omega_{\mathrm{NV}}$ Zhang et al. (2014). Regarding the bare coupling between the NV spin and the mechanical oscillator, we consider $b_{1}\approx 4.56\times 10^{7}$ T/m, resulting $\lambda\approx 2\pi\times 0.69$ MHz, namely $\lambda\approx g$. Then, the system dynamics can be evaluated by exactly solving the master equation (22).

The dynamics of the hybrid system is examined for a proper MPD and the suitably large detunings, for example, $r_{p}=1.54$, $\Delta_{s}=-55g_{r}$ and $\Delta_{K}=-45g_{r}$. The numerical results are shown in Fig. 3(a). Here, we assume that the NV spin is initially in the excited state, while the nanomagnet and the mechanical oscillator are in their vacuum states. The evident vacuum Rabi oscillations between the nanomagnet and the NV spin with excited-state population $\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle$ can be observed clearly, whereas the mechanical oscillator is nearly unexcited (namely, virtually excited) with $\langle\hat{b}_{s}^{\dagger}\hat{b}_{s}\rangle\approx 0$ during the evolution. This implies that the quantum state initially stored in the NV spin can be transferred to the nanomagnet through the mechanical oscillator serving as a quantum bus, and vice versa. In our simulations, the decay rate of the magnon mode is set to be $\gamma_{s}=2\pi\times 0.1$ MHz. Experimentally, a narrow magnon linewidth $<2\pi\times 0.6$ MHz has been demonstrated in a YIG sphere Zhang et al. (2016b), which could be more smaller when considering an ultrapure YIG Rameshti et al. (2022). For the relatively strong magnon dissipation, the state transfer becomes more inefficient with increasing $\gamma_{s}$, while it exhibits strong robustness against the mechanical dissipation for $Q_{x}$ as low as $10^{4}$, as shown in Figs. 3(b) and 3(c).

4  Conclusion

In conclusion, we have presented an experimentally feasible method to realize the strong magnon-mechanical and phonon-mediated magnon-spin couplings in a hybrid tripartite system. We have shown that the magnon mode supported by the nanomagnet can be coupled, in a strong and tunable way, to the phonon mode of the micromechanical cantilever by introducing the parametric amplification of the mechanical motion. With experimentally accessible parameters, this magnon-mechanical coupling can readily reach the strong-coupling regime, thus removing the obstacle to achieving the strong coupling between the two quantum systems with a large energy difference. By further coupling an NV spin to the mechanical motion and working in the dispersive regime, we have engineered an efficient strategy for quantum state transfer between the nanomagnet and the NV center. This provides a versatile platform for processing complicated quantum information tasks where the NV spin and the magnon can both act as collaborate quantum memories. Our proposal could also find other potential applications such as quantum control and measurement of magnon excitations, and the design of quantum transducers and sensors, etc.

Appendix B  Master equation and squeezing-induced noise

1  Derivation of the master equation

Considering the dissipation of the mechanical mode to a Markovian bath of thermal occupation $\bar{n}_{x}$ and damping rate $\gamma_{x}$, as well as the magnon mode to a bath of thermal occupation $\bar{n}_{s}$ and damping rate $\gamma_{s}$, the dynamics of the hybrid magnon-mechanical system, in the absence of the MPD, is described by the following master equation

where $\hat{H}_{\mathrm{mm}}^{\prime}$ is given by Eq. (15) with $\Omega_{p}=\omega_{p}=0$, $\hat{\rho}$ is the density operator, and $L_{\hat{o}}\hat{\rho}\equiv\hat{o}\hat{\rho}\hat{o}^{\dagger}-[\hat{o}^{\dagger}\hat{o},\hat{\rho}]/2$.

After introducing the MPD, the mechanical phonons can be squeezed, resulting in a squeezed-phonon mode $\hat{b}_{s}=\hat{b}\cosh(r_{p})+\hat{b}^{\dagger}\sinh(r_{p})$. In terms of the mode $\hat{b}_{s}$, the master equation (25) can be rewritten as

where $\hat{H}_{\mathrm{mm}}^{\mathrm{S}}$ is given by Eq. (16), $\bar{n}_{x}^{s}=\bar{n}_{x}\cosh(2r_{p})+\sinh^{2}(r_{p})$, $M=\cosh(r_{p})\sinh(r_{p})$, and $\mathscr{L}_{\hat{o}}\hat{\rho}\equiv\hat{o}\hat{\rho}\hat{o}-[\hat{o}\hat{o},\hat{\rho}]/2$. Equation (26) shows that when the mechanical mode $\hat{b}$ is coupled to a Markovian bath of thermal occupancy $\bar{n}_{x}$, the $\hat{b}_{s}$ mode is driven by a squeezed reservoir of thermal population $\bar{n}_{x}^{s}$.

In the limit of a weak mechanical dissipation, i.e., $\gamma_{x}\sinh(2r_{p})\ll\Delta_{s}$, the two terms (associated with $\mathcal{L}_{\hat{b}_{s}}$ and $\mathcal{L}_{\hat{b}_{s}^{\dagger}}$) describing the two-phonon correlation in Eq. (26) are high-frequency oscillating terms and can thus be neglected. Therefore, the master equation (26) is reduced to

which is exactly the master equation (18) in the main text. From Eq. (27), we find that the squeezed-phonon mode $\hat{b}_{s}$ can be considered as being coupled, with an unchanged vacuum damping rate $\gamma_{x}$, to a thermal bath but with an amplified thermal occupation $\bar{n}_{x}^{s}$. Apparently, the thermal noise with thermal occupation $\bar{n}_{x}^{s}$ increases drastically with increasing the MPD. In general, this amplified thermal noise could corrupt the coherent dynamics of the system even with an exponentially enhanced coupling, thus limiting the application of the system in the quantum regime. However, it can hardly affect the coherent state transfer in our system at least for $Q\geq 10^{5}$, as the amplified thermal noise remains far less than the enhanced coupling strength even for a strong MPD (e.g., $r_{p}=2.5$). We note that although the intrinsic mechanical damping can be extremely low in state-of-the-art experiments, e.g., $Q_{x}\geq 10^{8}$ in nanomechanical resonators Norte et al. (2016); Tsaturyan et al. (2017); Ghadimi et al. (2018), our scheme presents a clear advantage in relaxing the high-$Q$ requirement of the mechanical resonator in realistic experiments.

2  Elimination of the squeezing-induced noise

To eliminate the amplified noise from the mechanical bath, a possible strategy is to use the squeezed-vacuum-reservoir technique, which has been widely used in previous schemes Lü et al. (2015); Qin et al. (2019, 2018); Leroux et al. (2018); Chen et al. (2021); Tang et al. (2022); Li et al. (2020b); Wang et al. (2022). The technique was first proposed in an optomechanical system to eliminate the noise of the squeezed cavity mode Lü et al. (2015). It employs an auxiliary, broadband squeezed-vacuum field to drive the cavity, which is phase matched with the parametric drive that squeezes the cavity mode. This ensures that the squeezed cavity mode is equivalently coupled to a vacuum bath, thereby allowing one to describe the system dynamics with a simplified master equation in the standard Lindblad form.

To demonstrate more explicitly how to eliminate the amplified mechanical noise in our scheme using the above auxiliary-reservoir technique, we now derive the master equation for the hybrid magnon-mechanical system. We first assume that the mechanical mode is coupled to a squeezed thermal reservoir with a squeezing parameter $r_{e}$ and a reference phase $\theta_{e}$. The dynamics of the system, in the absence of the MPD, is described by the master equation