Enhancement of magnon-magnon entanglement inside a cavity

Magnon-photon coupling inside a cavity has been experimentally realized and has attracted significant attention for its potential docking with quantum information science. Whether this coupling implies the steady entanglement of photons and magnons is crucial for its usage in quantum information but is still an open question. Here we study the entanglement properties among magnons and photons in an antiferromagnet-light system and find that the entanglement between a magnon and a photon is nearly zero, while the magnon-magnon entanglement is very strong and can be even further enhanced through the coupling with the cavity photons. The maximum enhancement occurs when the antiferromagnet is resonant with the light. The essential physics can be well understood within the picture of cavity-induced cooling of the magnon-magnon state near its joint vacuum with stronger entanglement. Our findings can be used to cool magnetic magnons toward their ground state and may also be significant to extend the cavity spintronics to quantum manipulation. Furthermore, the hybrid antiferromagnet-light system provides a natural platform to manipulate the deep strong correlations of continuous modes with a generic stable condition and easy tunability.

Figure 1

Scheme of a two-sublattice ferrimagnet and its coupling with an electromagnetic wave. The magnetic moments on the two sublattices (red and blue arrows) pointing along $\pm z$ directions represent the classical ground state of the system. The bottom panel indicates the existence of magnon-magnon coupling.

Figure 2

(a) The enhanced entanglement $E_N$ between two magnons $a$ and $b$ in the presence of light mode $c$ as a function of external field $H/H_{\mathrm{sp}}$ for photon frequency ${\omega }_c/H_{\mathrm{sp}}=0.80$ (black line), 0.85 (red line), 0.90 (blue line). The horizontal dashed line at $E_N=0.6851$ is the magnon-magnon entanglement without light, while the dot-dashed line at $E_N=0$ indicates no entanglement between $a-c$ and $b-c$. (b) The dispersion relations for an antiferromagnet with ${\omega }_c/H_{\mathrm{sp}}=0.85$ and $g_{ac}=g_{bc}=0.01H_{\mathrm{ex}}$. Obviously, the optical magnon band ${\omega }_{\alpha }$ is left unchanged, while the acoustic band ${\omega }_{\beta }$ is hybridized with the photon mode ${\omega }_c$ to form modes ${\omega }_{2,3}$. (c) The population of modes $c$ and $\alpha ,\beta$. The red dashed line represents the occupation of $\alpha /\beta$ mode without light. (d)–(f) The same as (a)–(c), but with $g_{ac}=0.01H_{\mathrm{ex}}$ and $g_{bc}=0$. Other parameters are $g_{ab}=H_{\mathrm{ex}},H_{\mathrm{an}}=0.0163H_{\mathrm{ex}},{\gamma }_{\mathrm{m}}=0.001H_{\mathrm{ex}}$, and ${\gamma }_c=0.003H_{\mathrm{ex}}$. For simplicity in the following all the parameters are expressed in units of $H_{\mathrm{ex}}$.

Figure 3

The entanglement as a function of magnon-photon coupling strength $g_{\mathrm{mp}}$ when the system is (a) in the strong-coupling regime $g_{ab}/H_{\mathrm{ex}}=0.01$ and (b) in the deep strong-coupling regime $g_{ab}/H_{\mathrm{ex}}=1$. The red dashed lines represent the magnon-magnon entanglement in the absence of the photon mode. ${\gamma }_{\mathrm{m}}=0.001H_{\mathrm{ex}},{\gamma }_c=0.003H_{\mathrm{ex}}$. (c) The steady magnon-magnon entanglement distribution in the $g_{ab}-g_{\mathrm{mp}}$ phase plane. The right side marks the deep strong-coupling regime, where considerable enhancement is observed. Other parameters are $H/H_{\mathrm{sp}}=0.15,{\omega }_c/H_{\mathrm{sp}}=0.85,{\gamma }_{\mathrm{m}}=0.001H_{\mathrm{ex}},{\gamma }_c=0.003H_{\mathrm{ex}}$.

Figure 4

(a) The entanglement of $a$ and $b$ as a function of the dissipation rate of magnons at the resonant condition. $g_{ab}=H_{\mathrm{ex}},H_{\mathrm{an}}=0.0163H_{\mathrm{ex}},{\gamma }_c/H_{\mathrm{ex}}=0.003$. (b) The entanglement of $a$ and $b$ as a function of the dissipation rate of photons. $H/H_{\mathrm{sp}}=0.15,{\omega }_c/H_{\mathrm{sp}}=0.85,{\gamma }_{\mathrm{m}}/H_{\mathrm{ex}}=0.001$.

Figure 5

Particle number occupation as a function of time. The left and right insets show the initial and final particle distributions in Fock space, respectively. The initial state is a tensor product of coherent photons and zero-occupied magnons. $g_{\beta c}=0.02{\omega }_c,\rho =0.01{\omega }_c$.

Figure 6

(a) The stability regime of the magnon-photon system with $g_{ab}=H_{\mathrm{ex}},H_{\mathrm{an}}=0.0163H_{\mathrm{ex}},H/H_{\mathrm{sp}}=0.15,{\omega }_c/H_{\mathrm{sp}}=0.85,{\gamma }_{\mathrm{m}}/H_{\mathrm{ex}}=0.001$. (b) The stability regime when the self-energy terms are disregarded, i.e., ${\omega }_a={\omega }_b={\omega }_c=0$.

Figure 7

(a) The entanglement of $a$ and $b$ as a function of external field for (a) DPPH, (b) ${\mathrm{MnF}}_2$, (c) ${\mathrm{NaNiO}}_2$, and (d) $\mathrm{NiO}$. The dashed lines represent the positions of the resonance frequency.