Entangled atomic ensemble and an yttrium-iron-garnet sphere in coupled microwave cavities

We present a scheme to generate distant bipartite and tripartite entanglement between an atomic ensemble and an yttrium iron garnet (YIG) sphere in coupled microwave cavities. We consider an atomic ensemble in a single-mode microwave cavity which is coupled with a second single-mode cavity having a YIG sphere. Our system, therefore, has five excitation modes, namely, cavity-1 photons, an atomic ensemble, cavity-2 photons, a magnon and a phonon mode in the YIG sphere. We show that significant bipartite entanglement exists between indirectly coupled subsystems in the cavities which is robust against temperature. Moreover, we present suitable parameters for a significant tripartite entanglement of the ensemble, magnon, and phonon modes. We also demonstrate the existence of tripartite entanglement between the magnon and phonon modes of the YIG sphere with indirectly coupled cavity photons. Interestingly, this distant tripartite entanglement is of the same order as that previously found for a single-cavity system. We show that cavity-cavity coupling strength affects both the degree and transfer of quantum entanglement between various subsystems. Therefore, an appropriate cavity-cavity coupling optimizes the distant entanglement by increasing the entanglement strength and its robustness against temperature.

Figure 1

Schematic representation of two single-mode cavities coupled to each other with coupling strength $J$ incorporating an atomic ensemble of $N$ two-level atoms characterized by intrinsic frequency ${\omega }_e$ placed in cavity 1 and a YIG sphere placed in cavity 2. An external laser field drives the cavity at frequency ${\omega }_l$ with strength ${\mathrm{\Omega }}_l$. Correspondingly, a microwave magnetic field at frequency ${\omega }_l$ with strength ${\mathrm{\Omega }}_n$ drives the magnon modes of the YIG sphere, enhancing the magnomechanical coupling. The YIG sphere is concurrently influenced by the cavity's magnetic field, the bias magnetic field, and the drive magnetic field, all orthogonal to each other at the site of the YIG sphere. The decay rates of the cavity modes $a_1$ and $a_2$, atomic ensemble $e$, magnon mode $n$, and phonon mode $d$ associated with the YIG sphere are given by ${\kappa }_a,{\gamma }_e, {\kappa }_n$, and ${\gamma }_d$, respectively.

Figure 2

Density plots of bipartite entanglements (a) $E_N^{de}$, (b) $E_N^{ne}$, (c) $E_N^{a_{1}d}$, and (d) $E_N^{a_{1}n}$ versus normalized cavity-1 detuning ${\mathrm{\Delta }}_1/{\omega }_d$ and cavity-2 detuning ${\mathrm{\Delta }}_2/{\omega }_d$. In (a) and (b) ${\mathrm{\Delta }}_e=-{\omega }_d$, whereas in (c) and (d) ${\mathrm{\Delta }}_e={\omega }_d$. In all cases, ${\tilde{\mathrm{\Delta }}}_n=0.9{\omega }_d$, and $J=0.8{\omega }_d$.

Figure 3

Density plot of $E_N^{de}$ versus normalized cavity detuning ${\mathrm{\Delta }}_a/{\omega }_d$ and (a) and (b) magnon detuning ${\tilde{\mathrm{\Delta }}}_n/{\omega }_d$ at ${\mathrm{\Delta }}_e=-{\omega }_d$ and (c) and (d) ensemble detuning ${\mathrm{\Delta }}_e/{\omega }_d$ at ${\tilde{\mathrm{\Delta }}}_n=0.9{\omega }_d$. In (a) and (c) ${\mathrm{\Delta }}_a/{\omega }_d={\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$. However, ${\mathrm{\Delta }}_a/{\omega }_d=-{\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$ in (b) and (d). The cavity-cavity coupling strength is taken to be $J={\omega }_d$.

Figure 4

Density plot of $E_N^{ne}$ versus normalized cavity detuning ${\mathrm{\Delta }}_a/{\omega }_d$ and (a) and (b) magnon detuning ${\tilde{\mathrm{\Delta }}}_n/{\omega }_d$ at ${\mathrm{\Delta }}_e=-{\omega }_d$ and (c) and (d) ensemble detuning ${\mathrm{\Delta }}_e/{\omega }_d$ at ${\tilde{\mathrm{\Delta }}}_n=0.9{\omega }_d$. In (a) and (c) ${\mathrm{\Delta }}_a/{\omega }_d={\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$. However, ${\mathrm{\Delta }}_a/{\omega }_d=-{\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$ in (b) and (d). The cavity-cavity coupling strength is taken to be $J=0.8{\omega }_d$.

Figure 5

Density plot of $E_N^{a_{1}d}$ and $E_N^{a_{1}n}$ versus normalized cavity detuning ${\mathrm{\Delta }}_a/{\omega }_d$ and ensemble detuning ${\mathrm{\Delta }}_e/{\omega }_d$ at $J=0.8{\omega }_d$ and ${\tilde{\mathrm{\Delta }}}_n=0.9{\omega }_d$. In (a) and (c) ${\mathrm{\Delta }}_a/{\omega }_d={\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$. However, ${\mathrm{\Delta }}_a/{\omega }_d=-{\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$ in (b) and (d).

Figure 6

Density plot of $E_N^{a_{1}d}$ and $E_N^{a_{1}n}$ versus normalized cavity detuning ${\mathrm{\Delta }}_a/{\omega }_d$ and coupling strength $J/{\omega }_d$ at ${\tilde{\mathrm{\Delta }}}_n=0.9{\omega }_d$ and ${\mathrm{\Delta }}_e={\omega }_d$. In (a) and (c) ${\mathrm{\Delta }}_a/{\omega }_d={\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$. However, ${\mathrm{\Delta }}_a/{\omega }_d=-{\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$ in (b) and (d).

Figure 7

Line plots illustrating the effect of the cavity-cavity coupling rate $J$ against cavity detuning ${\mathrm{\Delta }}_a/{\omega }_d=-{\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$ on bipartite entanglement of the cavity-2 photon-magnon ($a_{2}n$), phonon-ensemble ($de$), magnon-ensemble ($ne$), magnon-phonon ($nd$), cavity-2 photon-phonon ($a_{2}d$), cavity-1 photon-phonon ($a_{1}d$), and cavity-1 photon-magnon ($a_{1}n$) modes varied in regular intervals from $J=0.4{\omega }_d$ to $J=1.4{\omega }_d$ in (a)–(f) at ${\tilde{\mathrm{\Delta }}}_n=0.9{\omega }_d$ and ${\mathrm{\Delta }}_e=-{\omega }_d$.

Figure 8

Line plots for $E_N^{de}, E_N^{ne}, E_N^{a_{1}d}$, and $E_N^{a_{1}n}$ as a function of temperature considered at the optimized values of cavity detunings ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, effective magnon detuning ${\tilde{\mathrm{\Delta }}}_n$, ensemble detuning ${\mathrm{\Delta }}_e$, and cavity-cavity coupling rate $J$ shown in Table 2.

Figure 9

Density plots of (a) $E_N^{de}$, (b) $E_N^{ne}$ (c) $E_N^{a_{1}d}$, and (d) $E_N^{a_{1}n}$ as a function of temperature $T$ and normalized coupling rate $J/{\omega }_d$. Other parameters are optimized as shown in Table 2.

Figure 10

Tripartite entanglement of the cavity-1 photon-magnon-phonon ($a_{1}dn$) and ensemble-magnon-phonon ($nde$) modes as a function of ${\mathrm{\Delta }}_a/{\omega }_d=-{\mathrm{\Delta }}_1/{\omega }_d={\mathrm{\Delta }}_2/{\omega }_d$ at ${\mathrm{\Delta }}_e=2{\omega }_d$ and ${\tilde{\mathrm{\Delta }}}_n=0.25{\omega }_d$ for $a_{1}dn$ and ${\mathrm{\Delta }}_e=-0.5{\omega }_d$ and ${\tilde{\mathrm{\Delta }}}_n=0.65{\omega }_d$ for $nde$ tripartite subsystems. The cavity-cavity coupling strength is $J={\omega }_d$.

Figure 11

Line plot of $E_N$ against the cavity-cavity coupling rate $J$. The rest of the conditions and parameters are the same as in Fig. 7 with ${\mathrm{\Delta }}_a=-{\omega }_d$.

Figure 12

Density plots illustrating the effect of the cavity-cavity coupling rate $J$ and cavity detuning ${\mathrm{\Delta }}_a$ on bipartite entanglement of (a) the cavity-1 photon-phonon ($a_{1}d$) and (b) cavity-1 photon-magnon ($a_{1}n$) modes. The rest of the conditions and parameters are the same as in Fig. 7.