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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 64 , Issue 1 : 210311(2021) https://doi.org/10.1007/s11433-020-1587-5

Enhanced entanglement and asymmetric EPR steering between magnons

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  • ReceivedApr 26, 2020
  • AcceptedJun 1, 2020
  • PublishedAug 20, 2020
PACS numbers

Abstract


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11975026, 61475007, and 61704071), the National Key Research and Development Program of China (Grant Nos. 2018YFB1107200, and 2016YFA0301302), the Key Research and Development Program of Guangzhou Province (Grant No. 2018B030329001), and the Beijing Natural Science Foundation (Grant No. Z190005).


Supplement

The supporting information is available online at phys.scichina.com and link.springer.com The supporting materials are published as submitted, without typesetting or editing. The responsibility for scientific accuracy and content remains entirely with the authors.


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  • Figure 1

    (Color online) Schematic illustration of a two-sublattice ferrimagnet coupled with a circularly polarized electromagnetic wave. The red and blue arrows represent spins on sublattices $A$ and $B$, respectively. $H$ pointing along the $z$ axis is the external static field.

  • Figure 2

    (Color online) (a) Variation of the magnon-magnon-entanglement measure $E_{N}$ with $H/H_{\rm~sp}$ for the magnons coupled to the photon mode $c$ and for different spin amplitudes, $S_B/S_A$. The green long dashed horizontal line indicates the degree of entanglement between magnons in the absence of mode $c$ for $S_B/S_A=1.6$. The inset shows the variation of the maximum value of the magnon-magnon entanglement of ratio $S_B/S_A$ when optimizing the external field. The blue solid and red dashed traces correspond to the cases in which the magnons are respectively coupled to and decoupled from the photon mode. (b) Variation of the steering parameters $\mathcal{G}^{a\rightarrow~b}$ (solid lines) and $\mathcal{G}^{b~\rightarrow~a}$ (dashed lines) with $H/H_{\rm~sp}$ in the presence of the photon mode $c$ for $S_B/S_A=1.3$ and $S_B/S_A=1$. The other parameters are the photon frequency $\omega_c/H_{\rm~sp}=~0.85$, $\omega_{a,\mathrm{an}}=0.0163\omega_{b,\mathrm{ex}},~\kappa_a=\kappa_b=0.001\omega_{b,\mathrm{ex}},~\kappa_c=0.003\omega_{b,\mathrm{ex}},~g_{ac}=0.01\omega_{b,\mathrm{ex}}$; thus, $~\omega_{b,\mathrm{an}}=(S_B/S_A)\omega_{a,\mathrm{an}},~g_{bc}=g_{ac}\sqrt{S_B/S_A}$, and $g_{ab}=\sqrt{S_B/S_A}\omega_{b,\mathrm{ex}}$. We set $\omega_{b,\mathrm{ex}}=1$ throughout this paper, so all parameters are measured relative to this quantity.

  • Figure 3

    (Color online) (a) Variation of $\mathcal{G}^{a\rightarrow~b}$ and $\mathcal{G}^{b~\rightarrow~a}$ with the ratio $\kappa_{b}/\kappa_{a}$ when the magnon modes are decoupled from the photon mode $c$ and for $S_B/S_A=1$. (b) Illustration of the variation of the one-way-steering parameter $\mathcal{G}^{b~\rightarrow~a}$ with $\ln{(S_B/S_A)}$ for $\kappa_b/\kappa_a=0.8$. The other parameters are: $\omega_{a,\mathrm{an}}=0.0163\omega_{b,\mathrm{ex}},~\kappa_a=0.001\omega_{b,\mathrm{ex}},~\omega_{b,\mathrm{an}}=(S_B/S_A)\omega_{a,\mathrm{an}}$, and $g_{ab}=\sqrt{S_B/S_A}\omega_{b,\mathrm{ex}}$.

  • Figure 4

    (Color online) (a), (b) Variation of the frequencies of the Bogoliubov modes $\omega_{\alpha,~\beta}$, the photon mode $\omega_c$, and the eigenmodes of the coupled system $\omega_{1,2,3}$ with $H/H_{\rm~sp}$ for two different values of the ratio $S_B/S_A$: (a) $S_B/S_A=1.3$ and (b) $S_B/S_A=1.0$. Note that $\omega_\beta$ and $\omega_c$ superpose to form $\omega_{2}$ and $\omega_3$, while $\omega_\alpha$ is left unchanged and forms $\omega_1$. (c), (d) The populations of modes $\alpha,\beta$, and $c$ plotted as functions of $H/H_{\rm~sp}$ for two different values of the ratio $S_B/S_A$: (c) $S_B/S_A=1.3$ and (d) $S_B/S_A=1$. The populations refer to the mean numbers of magnons and photons, i.e., $\langle~\alpha^\dagger~\alpha\rangle,~\langle~\beta^\dagger~\beta\rangle,~\langle~c^\dagger~c\rangle$, which are dimensionless quantities. Other parameters are the same as in Figure 2.

  • Figure 5

    (Color online) (a) The squeezing parameter $r$ and the coupling strength $g_{\beta~c}$ plotted as functions of the spin amplitudes $S_B/S_A$ for the optimized external field, $H$. (b) The populations of the modes $\alpha,~\beta$, and $c$ when the magnon modes are coupled to the photon mode. Black stars represent the population of mode $\alpha$ $(\langle~\alpha^\dagger~\alpha\rangle_{\rm~one})$ and red triangles represent population of mode $\beta$ $(\langle~\beta^\dagger~\beta\rangle_{\rm~one})$ when the magnon modes are decoupled from the photon mode. The parameters are the same as in Figure 2.

  • Figure 6

    (Color online) (a) Degree of the first-order coherence $\gamma_{bc}^{(1)}$ between the magnon mode $b$ and the photon mode $c$, plotted as a function of $g_{ac}$ with damping rates $\kappa_a=\kappa_b=0.001\omega_{b,\mathrm{ex}}$, and $\kappa_c=0.003\omega_{b,\mathrm{ex}}$. (b) Same as (a), but with $\kappa_a=\kappa_b=\kappa_c=0.001\omega_{b,\mathrm{ex}}$. (c) Variation of the populations of the modes $\alpha,~\beta$ and $c$ with $g_{ac}$. (d) Variation of the magnon-magnon-entanglement measure $E_{N}$ (blue solid) and the steering parameters $\mathcal{G}^{a\rightarrow~b}$ (green dash-dotted) and $\mathcal{G}^{b\rightarrow~a}$ (black dotted) with the coupling strength $g_{ac}$ when the magnon modes are coupled to the photon mode $c$. The horizontal red dashed line indicates the magnon-magnon entanglement when the magnons are decoupled from the mode $c$. The other parameters are $\omega_{a,\mathrm{an}}=\omega_{b,\mathrm{an}}=\omega_{\mathrm{an}}=0.0163\omega_{b,\mathrm{ex}},~\omega_c/H_{\rm~sp}=0.85,~H/H_{\rm~sp}=0.15,~g_{bc}=0.01\omega_{b,\mathrm{ex}},~g_{ab}=\omega_{b,\mathrm{ex}}$.

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