Quantum entanglement generation on magnons assisted with microwave cavities coupled to a superconducting qubit

We consider a hybrid system, see Fig.1, in which two microwave cavities cross each other, two yttrium iron garnet (YIG) spheres are coupled to the microwave cavities, respectively. A superconducting (SC) qubit, represented by black spot in the Fig.1, is placed at the center of the crossing in order to interact with the two microwave cavities simultaneously. The YIG spheres are placed at the antinode of two microwave magnetic fields, respectively, and a static magnetic field is locally biased in each YIG sphere. In our model, the SC qubit is a two-level system with ground state $|g\rangle\!_{q}$ and excited state $|e\rangle\!_{q}$.

The magnetostatic modes in YIG can be excited when the magnetic component of the microwave cavity field is perpendicular to the biased magnetic field. We only consider the Kittel mode Kittel in the hybrid system, namely, the magnon modes can be excited in YIG. The frequency of the magnon is in the gigahertz range. Thus the magnon generally interacts with the microwave photon via the magnetic dipole interaction. The frequency of the magnon is given by $\omega_{m}=\gamma H$, where $H$ is the biased magnetic field and $\gamma/2\pi=28$ GHz/T is the gyromagnetic ratio.

In recent years, some experiments have already realized the strong and ultrastrong magnon-magnon coupling mm1 ; mm2 ; mm3 as well as the magnon-qubit interaction MSC ; PMS , which means that in the hybrid system shown in Fig.1 the magnon is both coupled to the SC qubit and another magnon. However, we mainly consider that the magnons which frequencies are tuned by the locally biased static magnetic fields can be resonant with the cavities. In the meantime, the two cavities modes interact indirectly in the far-detuning regime for exchanging photons. The entanglement of two nonlocal magnons can be constructed by using two cavities and the SC qubit. Given that there are magnon-magnon and magnon-qubit interactions, the magnon can be detuned with the qubit and another magnon in order to neglect their interactions. In the rotating wave approximation the Hamiltonian of the hybrid system is ($\hbar=1$ hereafter) Hm

Here, $H_{0}$ is the free Hamiltonian of the two cavities, two magnons and the SC qubit. $H_{\mathrm{int}}$ is the interaction Hamiltonian among the cavities, magnons and SC qubit. $\omega_{m_{1}}$ and $\omega_{m_{2}}$ are the frequencies of the two magnons, which are tunable under biased magnetic fields, respectively. $\omega_{a_{1}}$ and $\omega_{a_{2}}$ are the frequencies of two cavities, and $\omega_{q}$ is the state transition frequency between $|g\rangle\!_{q}\leftrightarrow|e\rangle\!_{q}$ of the SC qubit. In the Kittel mode, the collective spins in YIGs can be expressed by the boson operators. $m_{1}$ ($m_{2}$) and $m_{1}^{\dagger}$ ($m_{2}^{\dagger}$) are the annihilation and creation operators of magnon mode 1 (2). $a_{1}$ ($a_{2}$) and $a_{1}^{\dagger}$ ($a_{2}^{\dagger}$) denote the annihilation and creation operators of cavity mode 1 (2), respectively. They satisfy commutation relations $[O,O^{\dagger}]=1$ for $O=a_{1},a_{2},m_{1},m_{2}$. $\sigma_{z}=|e\rangle_{q}\langle e|-|g\rangle_{q}\langle g|$. $\sigma=|g\rangle_{q}\langle e|$ and $\sigma^{+}=|e\rangle_{q}\langle g|$ are the lowing and raising operators of the SC qubit. $\lambda_{q_{1}}$ ($\lambda_{q_{2}}$) is the coupling strength between the SC qubit and the cavity mode 1 (2). $\lambda_{m_{1}}$ ($\lambda_{m_{2}}$) is the coupling between the magnon mode 1 (2) and the cavity mode 1 (2).

As mentioned above, the two microwave cavities are identical ones with the same frequency $\omega_{a_{1}}=\omega_{a_{2}}=\omega_{a}$. Meanwhile, one can assume that $\lambda_{q_{1}}=\lambda_{q_{2}}=\lambda_{q}$. In the interaction picture with respect to $e^{-\mathrm{i}H_{0}t}$, the Hamiltonian is expressed as

where $\delta_{1}=\omega_{m_{1}}-\omega_{a}$, $\delta_{2}=\omega_{m_{2}}-\omega_{a}$, $\Delta_{1}=\omega_{q}-\omega_{a}$ and $\Delta_{2}=\omega_{q}-\omega_{a}$. The SC qubit is coupled to the two cavities simultaneously. Owing to $\Delta_{1}=\Delta_{2}=\Delta_{0}\neq 0$ and $\Delta_{0}\gg\lambda_{q}$, the two identical microwave cavities indirectly interact with each other in the far-detuning regime. Therefore, the effective Hamiltonian of the subsystem composed of the two microwave cavities and the SC qubit in the far-detuning regime is given by EH

where $\widetilde{\lambda}_{q}=\lambda_{q}^{2}/\Delta_{0}$.

We now give the protocol of quantum entanglement generation on two nonlocal magnons. Generally, the magnon can be excited by a drive magnetic field. For convenience the state of magnon 1 is prepared as $|1\rangle\!_{m\!_{1}}$ via the magnetic field. The initial state of the hybrid system is $|\varphi\rangle_{0}=|1\rangle\!_{m\!_{1}}|0\rangle\!_{m\!_{2}}|0\rangle\!_{a\!_{1}}|0\rangle\!_{a\!_{2}}|g\rangle\!_{q}$, in which the two cavities are all in the vacuum state, magnon 2 is in the state $|0\rangle\!_{m\!_{2}}$, and the SC qubit is in state $|g\rangle\!_{q}$ which is unaltered all the time due to the indirect interaction between the two cavities.

step 1: The frequency of magnon 1 is tuned to be $\omega_{m_{1}}\!=\!\omega_{a_{1}}$ so that the cavity 1 could be resonated with it. Therefore, the magnon 1 and cavity 1 are in a superposed state after time $T_{1}=\pi/4\lambda_{m_{1}}$. The local evolution is $|1\rangle\!_{m\!_{1}}|0\rangle\!_{a\!_{1}}\rightarrow\frac{1}{\sqrt{2}}(|1\rangle\!_{m\!_{1}}|0\rangle\!_{a\!_{1}}-\mathrm{i}|0\rangle\!_{m\!_{1}}|1\rangle\!_{a\!_{1}})$, which means that the states of SC qubit, magnon 2 and cavity 2 are unchanged due to decoupling between the SC and two cavities, and the magnon 2 is far-detuned with cavity 2. The state evolves to

step 2: The magnons are tuned to far detune with respective cavities. From Eq. (3), the evolution of subsystem composed of two microwave cavities and SC qubit is given by

under the condition $\Delta_{0}\gg\lambda_{q}$.

After time $T_{2}=\pi/2\widetilde{\lambda}_{q}$, the evolution between two cavities is $|1\rangle\!_{a\!_{1}}|0\rangle\!_{a\!_{2}}\rightarrow-|0\rangle\!_{a\!_{1}}|1\rangle\!_{a\!_{2}}$, which indicates that the photon can be indirectly transmitted between the two cavities, with the state of SC qubit unchanged. Therefore, the state after this step changes to

step 3: The frequency of magnon 2 is tuned with $\omega_{m_{2}}=\omega_{a_{2}}$ to resonate with the cavity 2. In the meantime the cavities are decoupled to the SC qubit and the magnon 1 is far detuned with the cavity 1. After time $T_{3}=\pi/2\lambda_{m_{2}}$, the local evolution $|0\rangle\!_{m\!_{2}}|1\rangle\!_{a\!_{2}}\rightarrow-\mathrm{i}|1\rangle\!_{m\!_{2}}|0\rangle\!_{a\!_{2}}$ is attained. The final state is

which is just the single-excitation Bell state on two nonlocal magnons.

In the whole process, we mainly consider the interactions between the magnons and the cavities, and between the cavities and the SC qubit. However, the SC qubit can be coupled to the magnons. In terms of Ref.MSC , the interactions between the magnons and the SC qubit are described as $H_{qm,1}=\lambda_{qm,1}(\sigma^{+}m_{1}+H.c.)$ and $H_{qm,2}=\lambda_{qm,2}(\sigma^{+}m_{2}+H.c.)$ where $\lambda_{qm,1}=\lambda_{q}\lambda_{m_{1}}/\Delta_{0}$ and $\lambda_{qm,2}=\lambda_{q}\lambda_{m_{2}}/\Delta_{0}$, while the conditions $\omega_{q}=\omega_{m_{1}}$ and $\omega_{q}=\omega_{m_{2}}$ are attained. In the meantime, the two magnons are interacts each other by using the SC qubit. Generally, the frequencies of two magnon modes are tuned by the locally biased magnetic fields. Therefore, the magnon can be detuned with the SC qubit and another magnon in order to neglect the interactions between the magnons and the SC qubit.

We here simulate SIM the fidelity of the Bell state on two nonlocal magnons by considering the dissipations of all constituents of the hybrid system. The realistic evolution of the hybrid system composed of magnons, microwave cavities and SC qubit is governed by the master equation

Here, $\rho$ is the density operator of the hybrid system, $\kappa_{m_{1}}$ and $\kappa_{m_{2}}$ are the dissipation rates of magnon 1 and 2, $\kappa_{a_{1}}$ and $\kappa_{a_{2}}$ denote the dissipation rates for the two microwave cavities 1 and 2, $\gamma_{q}$ is the dissipation rate of the SC qubit, $D[X]\rho\!\!=\!\!(2X\rho X^{\dagger}-X^{\dagger}X\rho-\rho X^{\dagger}X)/2$ for $X=m_{1},m_{2},a_{1},a_{2},\sigma$. The fidelity of the entangled state of two nonlocal magnons is defined by $F=_{3}\!\!\langle\varphi|\rho|\varphi\rangle_{3}$.

The related parameters are chosen as $\omega_{q}/2\pi=7.92$ GHz, $\omega_{a}/2\pi=6.98$ GHz, $\lambda_{q}/2\pi=83.2$ MHz, $\lambda_{m_{1}}/2\pi=15.3$ MHz, $\lambda_{m_{2}}/2\pi=15.3$ MHz PMS , $\kappa_{m_{1}}/2\pi$=$\kappa_{m_{2}}/2\pi=\kappa_{m}/2\pi=1.06$ MHz, $\kappa_{a_{1}}/2\pi=\kappa_{a_{2}}/2\pi=\kappa_{a}/2\pi=1.35$ MHz SU , $\gamma_{q}/2\pi=1.2$ MHz MSC . The fidelity of the entanglement between two nonlocal magnons can reach 92.9$\%$.

The influences of the imperfect relationship among parameters is discussed next. The Fig.2(a) shows the fidelity influenced by the coupling strength between the microwave cavities and the SC qubit. Since $\widetilde{\lambda}_{q}=\lambda_{q}^{2}/\Delta_{0}$ in Eq.(3), the fidelity is similar to parabola. In Fig.2(b)-(d), we give the fidelity varied by the dissipations of cavities, magnons, and SC qubit. As a result of the virtual photon, the fidelity is almost unaffected by the SC qubit, shown in Fig.2(d).

Similar to the protocol of entangled state generation for two nonlocal magnons in two microwave cavities, we consider the protocol for entanglement of three nonlocal magnons. As shown in Fig.3, similar to the hybrid system composed of two magnons coupled to the respective microwave cavities and a SC qubit in Fig.1, there are three magnons in three YIGs coupled to respective microwave cavities and a SC qubit placed at the center of the three identical cavities ($\omega_{a_{1}}\!=\!\omega_{a_{2}}\!=\!\omega_{a_{3}}\!=\!\omega_{a}$). Each magnon is in biased static magnetic field and is located at the antinode of the microwave magnetic field.

In the interaction picture, the Hamiltonian of the hybrid system depicted in Fig.3 is

where $\lambda_{m_{3}}$ is the coupling strength between magnon 3 and microwave cavity 3, $a_{3}$ and $m^{\dagger}_{3}$ are annihilation operator of the cavity 3 and creation operator of the magnon 3, respectively. $\lambda_{q}$ is the coupling between the SC qubit and three cavities, $\delta_{3}=\omega_{m_{3}}-\omega_{a}$. The frequency $\omega_{m_{3}}$ can be tuned by the biased magnetic field in microwave cavity 3. $\Delta_{3}=\omega_{q}-\omega_{a}=\Delta_{0}$.

At the beginning we have the initial state $|\psi\rangle_{0}^{(3)}\!\!\!=\!\!|\psi\rangle\!_{m}^{(3)}\otimes|\psi\rangle\!_{a}^{(3)}\otimes|g\rangle\!_{q}$ with $|\psi\rangle\!_{m}^{(3)}\!\!\!=\!\!|1\rangle\!_{m\!_{1}}|0\rangle\!_{m\!_{2}}|0\rangle\!_{m\!_{3}}\!=\!|100\rangle\!_{m}$ and $|\psi\rangle\!_{a}^{(3)}\!\!\!=\!\!|0\rangle\!_{a\!_{1}}|0\rangle\!_{a\!_{2}}|0\rangle\!_{a\!_{3}}\!=\!|000\rangle\!_{a}$. The single-excitation is set in the magnon 1. The magnon 1 is resonant with the cavity 1 by tuning the frequency of magnon 1, and the SC qubit is decoupled to the cavities. After time $T_{1}^{(3)}=\pi/2\lambda_{m_{1}}$, the local evolution $|1\rangle\!_{m\!_{1}}|0\rangle\!_{a\!_{1}}\rightarrow-\mathrm{i}|0\rangle\!_{m\!_{1}}|1\rangle\!_{a\!_{1}}$ is attained. The state is evolved to

The SC qubit is coupled to the three identical microwave cavities at the same time in far-detuning regime $\Delta_{0}\gg\lambda_{q}$. Therefore, the effective Hamiltonian of the subsystem composed of the SC qubit and the three identical cavities is of the form EH

The magnons are then all detuned with the cavities. The local evolution $e^{-\mathrm{i}H_{\mathrm{eff}}^{(3)}t}|100\rangle\!_{a}|g\rangle$ of the subsystem is given by

where $C^{(3)}_{1,t}=\frac{e^{\mathrm{i}3\widetilde{\lambda}_{q}t}+2}{3}$ and $C^{(3)}_{2,t}=C^{(3)}_{3,t}=\frac{e^{\mathrm{i}3\widetilde{\lambda}_{q}t}-1}{3}$. It is easy to derive that

Fig.4 shows the probability related to the states $|100\rangle\!_{a}|000\rangle\!_{m}|g\rangle\!_{q}$, $|010\rangle\!_{a}|000\rangle\!_{m}|g\rangle\!_{q}$ and $|001\rangle\!_{a}|000\rangle\!_{m}|g\rangle\!_{q}$. In particular, one has $|C^{(3)}_{1,t}|^{2}=|C^{(3)}_{2,t}|^{2}=|C^{(3)}_{3,t}|^{2}=\frac{1}{3}$, with

at time $T_{2}^{(3)}=2\pi/9\widetilde{\lambda}_{q}$. Correspondingly, the state evolves to

Finally, the magnons can be resonated with the respective cavities under the condition $\{\delta_{1},\delta_{2},\delta_{3}\}=0$. The local evolution and the time are $|0\rangle\!_{m\!_{k}}|1\rangle\!_{a\!_{k}}\rightarrow-\mathrm{i}|1\rangle\!_{m\!_{k}}|0\rangle\!_{a\!_{k}}$ and $T_{3k}^{(3)}=\pi/2\lambda_{m_{k}}$ ($k=1,2,3$), respectively. Thus the final state is

In the whole process, the state of the SC qubit is kept unchanged.

The entanglement fidelity of three nonlocal magnons is given here by taking into account the dissipations of hybrid system. Firstly, the master equation which governs the realistic evolution of the hybrid system composed of three magnons, three microwave cavities and a SC qubit can be expressed as

where $\rho^{(3)}$ is the density operator of realistic evolution of the hybrid system, $\kappa_{m_{3}}$ is the dissipation rate of magnon 3 with $\kappa_{m_{3}}/2\pi=\kappa_{m}/2\pi=1.06$ MHz SU , $\kappa_{a_{3}}$ denotes the dissipation rate for the microwave cavities 3 with $\kappa_{a_{3}}/2\pi=\kappa_{a}/2\pi=1.35$ MHz SU , $D[X]\rho^{(3)}\!\!=\!\!(2X\rho^{(3)}X^{\dagger}-X^{\dagger}X\rho^{(3)}-\rho^{(3)}X^{\dagger}X)/2$ for any $X=m_{1},m_{2},m_{3},a_{1},a_{2},a_{3},\sigma$.

The entanglement fidelity for three nonlocal magnons is defined by $F^{(3)}=_{3}^{(3)}\!\!\langle\psi|\rho^{(3)}|\psi\rangle_{3}^{(3)}$, which can reach 84.9$\%$. The fidelity with respect to the parameters is shown in Fig.5.