Magnon-atom interaction via dispersive cavities: Magnon entanglement

In general, a magnon does not interact directly with an atom. Here we present a scheme to generate coherent coupling between two magnons and a superconducting artificial atom via exchange of virtual photons in dispersive cavities. By probing into the induced magnon-atom interactions, we identify the two-magnon process, which causes almost perfect two-mode squeezing and entanglement of the two magnons for specific conditions. The quantum entanglement of the magnon modes in two massive ferrimagnets survives at the steady state and represents genuinely macroscopic quantum state. Our work offers an interesting alternative in designing experiments for generating the coherent coupling between the magnons and the atom. It finds broad applications in the investigation of macroscopic quantum effects and the preparation of quantum devices.

Figure 1

(a) Proposed model for creating the coherent coupling between two magnons and the superconducting artificial atom in crossing dispersive microwave cavities. The superconducting artificial atom locates at the intersection of the cavities, and the two YIG spheres are placed, respectively, in two different cavities between the intersection and the end mirrors. The cavity electric fields are along $z$-axis direction and coupled to the atom, the cavity magnetic fields are, respectively, along $x,y$ directions and coupled, respectively, to the two magnons. (b) The sketch for the atom energy levels and the interaction of the atom with the cavity fields and the dressing fields through two electric dipole transitions $|0\rangle \leftrightarrow |1,2\rangle$ in $V$ configuration. In (a) and (b), by the annihilation operators $a_{1,2}$ we describe the cavity fields of the circular frequencies ${\omega }_{a_{1,2}}$, and by the annihilation operators $m_{1,2}$ we denote the magnon modes of the circular frequencies ${\omega }_{m_{1,2}}$; ${\mathrm{\Omega }}_{1,2}$ is the the Rabi frequencies for the interaction of the atom with the two external classical fields of the circular frequencies ${\omega }_{1,2}$.

Figure 2

Induced atom-magnon interaction through the dispersive cavities as an intermediate in an appropriate rotating frame. (a) Simultaneous interactions of the two cavity fields $a_{1,2}$ with each of the atomic dressed spin-flips ${\sigma }_{\tilde{1}\tilde{0}},{\sigma }_{\tilde{2}\tilde{0}}$ and the interaction of $a_l$ with the magnon $m_l$ ($l=1,2$). (b) The induced atom-magnon interaction, in which each dressed atomic transition is simultaneously accompanied with annihilation of one magnon mode and creation of the other.

Figure 3

Diagrammatic sketch for the induced atom-magnon interaction. In terms of the individual magnon modes $(m_1,m_2)$, the magnon interaction with two dressed atomic spins (${\sigma }_{\tilde{1}\tilde{0}},{\sigma }_{\tilde{2}\tilde{0}}$ as spin-down operators) is in a quadrilateral configuration, where there appear, alternately, the parameter amplifier interactions (linked circles, yielding two-mode squeeze states) and the beam-splitter interactions (solid lines, transferring quantum states). Under appropriate conditions, generated two-mode squeezing of $m_l,{\sigma }_{\tilde{l}\tilde{0}}$ ($l=1,2$) is transferred to $m_{1,2}$. Alternatively, in terms of the Bogoliubov modes $b_{1,2}$ as in Eq. (15), the atom-magnon interaction appears independently between $b_1$ and ${\sigma }_{\tilde{2}\tilde{0}}$ and between $b_2$ and ${\sigma }_{\tilde{1}\tilde{0}}$, as shown by Eq. (16).

Figure 4

Squeezing parameter $r$ (the potentially at most reachable squeezing, the red solid line, the left vertical axis) and the net dissipation or amplification rate $|A-B|$ (in the unit of Hz, the blue dashed line, the right vertical axis) versus the normalized detuning $\mathrm{\Delta }/\mathrm{\Omega }$. Note that for the sake of clarity, we have widened the positive detuning regime relative to the negative detuning regime. The other parameters are $\gamma /2\pi =1$ MHZ and $g=20\gamma$.

Figure 5

The variances $\langle {\left(\delta x_{m_{\pm }}\right)}^2\rangle$ for five different regimes as given in Table 1. The simplified expressions in Eq. (23) (red dashed lines) are in an excellent agreement with the complete expressions in Eq. (35) (blue solid lines); (b, c, and e) correspond to the dissipative regimes, while (a and d) stand for the amplifying regimes. The parameters are chosen the same as in Fig. 4 and ${\kappa }_m=0.01\gamma$.

Figure 6

Density plot of two mode magnon variances $\langle {\left(\delta x_{m_{+}}\right)}^2\rangle$ (a, b) and $\langle {\left(\delta x_{m_{-}}\right)}^2\rangle$ (c–e) versus the normalized detuning $\mathrm{\Delta }/\mathrm{\Omega }$ and the ratio of the vacuum damping rates ${\kappa }_m/\gamma$. Five different regimes are plotted separately because of their remarkably different squeezing degrees. The other parameters are the same as in Fig. 4.