Magnon blockade induced by parametric amplification

We propose to achieve and enhance the magnon blockade effect in an optomechanical-magnetic system based on a quantum destructive interference mechanism between three indirect transition pathways. By introducing a degenerate parametric amplifier, we analytically obtain the optimal parametric gain and phase for achieving the magnon blockade. Under the optimized parameter conditions, the driving detunings of cavity and magnon modes can be flexibly controlled to achieve the smallest second-order magnon correlation function. Moreover, the magnon blockade can exhibit fascinating features by proper driving detuning and weaker driving strength. Our scheme combines the benefits of destructive interference-induced magnon blockade and energy-level anharmonicity-induced magnon blockade, which results in a reduction of equal-time second-order magnon correlation while simultaneously avoiding time-delay second-order magnon correlation with rapid oscillation. Our work provides an alternative and experimentally feasible platform for manipulating few-magnon states and generating single-magnon sources.

Figure 1

Sketch of the hybrid quantum system. A degenerate parametric amplifier is placed in an optomechanical cavity coupled to a small YIG sphere. The cavity mode $a$ is pumped by an external weak field with amplitude $E$ and frequency ${\omega }_l$. The parametric amplification with amplitude $G$ and frequency ${\omega }_d$ on the cavity is generated by pumping the parametric amplifier. The cavity mode $a$ couples to the magnon mode $m$ and phonon mode $b$ through magnetic dipole interaction and radiation-pressure interaction, respectively.

Figure 2

(a) Time evolution of fidelity $\mathcal{F}$ for different optomechanical coupling strength $g$. (b) Second-order magnon correlation function $g_m^{\left(2\right)}\left(0\right)$ and (c) photon correlation function $g_a^{\left(2\right)}\left(0\right)$ as a function of $\kappa t$ with Hamiltonian $H_2$ (pink line) and $H_4$ (cyan line). In (b), $G$ and $\theta$ are taken as $G^{\mathrm{opt}}=|E^2({\tilde{\mathrm{\Delta }}}_a-2{\mathrm{\Delta }}_g+{\tilde{\mathrm{\Delta }}}_m)/[{\tilde{\mathrm{\Delta }}}_m({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g)-G_m^2]|$ and $\theta =\mathrm{Arg}\{E^2({\tilde{\mathrm{\Delta }}}_a-2{\mathrm{\Delta }}_g+{\tilde{\mathrm{\Delta }}}_m)/[{\tilde{\mathrm{\Delta }}}_m({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g)-G_m^2]\}$ according to Eq. (13), respectively. In (c), $G$ and $\theta$ are taken as $G_2^{\mathrm{opt}}=|E^2{\tilde{\mathrm{\Delta }}}_m^2({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g+{\tilde{\mathrm{\Delta }}}_m)/[G_m^2-{\tilde{\mathrm{\Delta }}}_m({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g)]Z|$ and $\theta =\mathrm{Arg}\{E^2{\tilde{\mathrm{\Delta }}}_m^2({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g+{\tilde{\mathrm{\Delta }}}_m)/[G_m^2-{\tilde{\mathrm{\Delta }}}_m({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g)]Z\}$ according to Eq. (14), respectively. The other parameters are ${\omega }_b=100\kappa$, ${\gamma }_b={10}^{-6}{\omega }_b$, $G_m=3\kappa$, $E=0.1\kappa$, and ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_m=0$.

Figure 3

Second-order magnon correlation function on a logarithmic scale ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ as a function of (a) $G$, (b) ($\theta$) and the driving detuning ${\mathrm{\Delta }}_c$, and (c) $G$ and $\theta$. The white dashed lines represent the optimal conditions decided by Eq. (13). ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_m=0$ is taken in (c). The other parameters are the same as given in Fig. 2.

Figure 4

(a), (b) Second-order magnon and photon correlation functions on a logarithmic scale ${\log }_{10}\left[g_a^{\left(2\right)}\left(0\right)\right]$ and ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ as a function of the driving detuning ${\mathrm{\Delta }}_c$. (c), (d) Photon-number and magnon-number distributions $P_{n,m=1,2}$ as a function of the driving detuning ${\mathrm{\Delta }}_c$. The solid and dashed lines are the exact numerical results obtained by solving Eqs. (5) and (6), and asterisks indicate the analytical solutions given by Eq. (12). (e) ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ and (f) ${\log }_{10}\left[g_a^{\left(2\right)}\left(0\right)\right]$ as a function of the driving detuning ${\mathrm{\Delta }}_c$, where blue and orange lines and black asterisks indicate that the number of Fock states is truncated to $N=3$ and $N=5$ for the two modes, respectively. $E=0.01\kappa$ is taken in all the plots. The other parameters are the same as given in Fig. 2.

Figure 5

Energy-level diagram of the system showing the zero-, one-, and two-magnon states (horizontal black lines without arrows) and the corresponding transition pathways (color lines with arrows) leading to the destructive interference for preventing the two-magnon occupation.

Figure 6

(a) Second-order magnon and photon correlation functions on a logarithmic scale ${\log }_{10}\left[g_a^{\left(2\right)}\left(0\right)\right]$ and ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ as a function of cavity-magnon coupling strength $G_m$. (b) Second-order magnon correlation function on a logarithmic scale ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ as a function of ${\mathrm{\Delta }}_c$ for different parametric amplification $G$. ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_m=\kappa$ is used in (a), and ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_m$ in (b). In (a), for magnon blockade, $G$ and $\theta$ are taken as $G^{\mathrm{opt}}=|E^2({\tilde{\mathrm{\Delta }}}_a-2{\mathrm{\Delta }}_g+{\tilde{\mathrm{\Delta }}}_m)/[{\tilde{\mathrm{\Delta }}}_m({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g)-G_m^2]|$ and $\theta =\mathrm{Arg}\{E^2({\tilde{\mathrm{\Delta }}}_a-2{\mathrm{\Delta }}_g+{\tilde{\mathrm{\Delta }}}_m)/[{\tilde{\mathrm{\Delta }}}_m({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g)-G_m^2]\}$ according to Eq. (13), respectively, and for photon blockade, $G$ and $\theta$ are taken as $G_2^{\mathrm{opt}}=|E^2{\tilde{\mathrm{\Delta }}}_m^2({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g+{\tilde{\mathrm{\Delta }}}_m)/[G_m^2-{\tilde{\mathrm{\Delta }}}_m({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g)]Z|$ and $\theta =\mathrm{Arg}\{E^2{\tilde{\mathrm{\Delta }}}_m^2({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g+{\tilde{\mathrm{\Delta }}}_m)/[G_m^2-{\tilde{\mathrm{\Delta }}}_m({\tilde{\mathrm{\Delta }}}_a-{\mathrm{\Delta }}_g)]Z\}$ according to Eq. (14), respectively. $E=0.01\kappa$ is taken in (a) and (b). The other parameters are the same as given in Fig. 2.

Figure 7

(a) Second-order magnon correlation function on a logarithmic scale ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ as a function of the driving detuning ${\mathrm{\Delta }}_c$ when ${\mathrm{\Delta }}_m/\kappa =0,1,2$, and 3 for different pairs of optimal parameters $G$ and $\theta$. (b) ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ as a function of ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_m$ for different pairs of optimal parameters $G$ and $\theta$, which result in the perfect MB occurring at ${\mathrm{\Delta }}_c/\kappa =0,1,2,3$. In (a) and (b), different pairs of optimal parameters $G$ and $\theta$ are obtained by absorbing ${\mathrm{\Delta }}_c/\kappa ={\mathrm{\Delta }}_m/\kappa =0,1,2,3$ into Eq. (13), respectively, corresponding to green, blue, pink, and red lines. (c), (d) ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ as a function of ${\mathrm{\Delta }}_c$ and ${\mathrm{\Delta }}_m$. The white dashed lines represent the optimal CMB condition $G_m^2={\mathrm{\Delta }}_c{\mathrm{\Delta }}_m$. The other parameters are the same as given in Fig. 2.

Figure 8

Evolution of time-delay second-order correlation function ${\log }_{10}\left[g_m^{\left(2\right)}\left(\tau \right)\right]$ vs time $t$ for (i) $G=0$, (ii) $G=G^{\mathrm{opt}}$, ${\mathrm{\Delta }}_c=0$, and (iii) $G=G^{\mathrm{opt}}$, ${\mathrm{\Delta }}_c/\kappa =3$. The other parameters are the same as given in Fig. 2.

Figure 9

Second-order magnon correlation function on a logarithmic scale ${\log }_{10}\left[g_m^{\left(2\right)}\left(0\right)\right]$ as a function of (a) mean photon thermal occupation ${\overline{n}}_a$ and (b) mean magnon thermal occupation ${\overline{n}}_m$ for different frequency detuning ${\mathrm{\Delta }}_c/\kappa =(0,2,3)$ under the optimal UMB condition. The other parameters are the same as given in Fig. 2.