Tuning entanglement by squeezing magnons in anisotropic magnets

We theoretically study the entanglement between two arbitrary spins in a magnetic material where magnons naturally form a general squeezed coherent state in the presence of an applied magnetic field and axial anisotropies. Employing concurrence as a measure of entanglement, we demonstrate that spins are generally entangled in thermodynamic equilibrium, with the amount of entanglement controlled by the external fields and anisotropies. As a result, the magnetic medium can serve as a resource to store and process quantum information. We furthermore show that the entanglement can jump discontinuously when decreasing the transverse magnetic field. This tunable entanglement can be potentially used as an efficient switch in quantum-information processing tasks.

Figure 1

(a) A general coherent state $|\alpha \rangle =D(\alpha \left)\right|0\rangle$, where $D\left(\alpha \right)\equiv exp\alpha a^{†}-{\alpha }^{*}a$ is a displacement operator with $\alpha =\left|\alpha \right|e^{i\theta }$, is the minimum uncertainty state $\left(\mathrm{\Delta }{S}^x=\mathrm{\Delta }{S}^y=1/2\right)$. All spins deviate from the $\mathbf{B}\propto \widehat{\mathbf{z}}$ direction coherently. (b) The general squeezed vacuum state $S\left(\zeta \right)|0\rangle$, where $S\left(\zeta \right)\equiv e^{\left[{\zeta }^{*}{a}^2-\zeta {\left(a^{†}\right)}^2\right]/2}$ is the squeezing operator with a squeezing parameter $\zeta =r{e}^{i\varphi }$, is also a minimum uncertainty state with uncertainties being squeezed (blue ellipse) compared with the vacuum uncertainties (green disk). The direction of the squeezing (the orientation of the semiminor axis of the ellipse with respect to the $S^x$ axis) is $\varphi /2$. The length of the semiminor axis is $e^{-r}/2$, and the length of the semimajor axis is $e^r/2$. The average direction of the spin in such that the state is along $\mathbf{B}$. (c) For the general squeezed coherent state $D\left(\alpha \right)S\left(\zeta \right)|0\rangle$, the degree of the deviation from $\widehat{\mathbf{z}}$ is determined by the parameter $\alpha$, and the degree of squeezing of the uncertainty is determined by the squeezing parameter $\zeta$.

Figure 2

(a) Concurrence $\mathcal{C}$ as a function of the number of squeezed magnons $N_s={sinh}^{2}r$ for different numbers of coherent magnons $N_c={\left|\alpha \right|}^2$. $\left|\alpha \right|=0$ corresponds to the squeezed vacuum state [see Eq. (20)], where the maximal concurrence is $1/N_0$. As we increase the number of coherent magnons, the physics is dominated by the coherent part when the number of squeezed magnons is small. As a result, the concurrence is zero. However, the presence of the coherent part can increase the upper bound of the concurrence once the number of squeezed magnons is above some critical value ${sinh}^2{r}_c$, which depends on the amount of coherent magnons we put into the system. (b) Critical values ${\left(N_s\right)}^{1/4}=\sqrt{sinh{r}_c}$ as a function of $\left|\alpha \right|$, which can be fitted using a linear relation. When the number of coherent magnons is larger than $\sqrt{2N_0{N}_s}$, the coherent part dominates the physics, and the concurrence is zero. Otherwise, we have a finite concurrence. In the numerical study, we set $N_0={10}^8$.

Figure 3

(a) Concurrence as a function of the distance $\gamma$ between two spins in dimensions $d=1,d=2$, and $d=3$. The overall value of the concurrence will decrease as we increase the dimension of our system. In a given dimension, the concurrence is finite but will decrease as we increase the distance between these two spins within a critical distance ${\gamma }_c$, beyond which the concurrence vanishes. We have set $\lambda =10,\eta =0.5$, and $\widehat{\mathbf{R}}=\widehat{\mathbf{x}}$ in higher dimension. (b) Critical distance ${\gamma }_c$ as a function of the correlation length $\lambda$, which can be fitted well with a linear relation. We have set $\eta =0.5$.

Figure 4

Concurrence as a function of one angle (setting the other angle to zero). For the pink dashed curve, we plot the concurrence as a function of the angle $\varphi$ and set $\theta =0$. For the black solid curve, we plot the concurrence as a function of the angle $\theta$ and set $\varphi =0$. We find that concurrence is periodic with period $2\pi$ in $\varphi$ and $\pi$ in $\theta$. In both cases, we set $N_s={sinh}^{2}4,N_c=9\times {10}^4$, and $N_0={10}^8$.

Figure 5

Spins A and B are placed above a ferromagnetic insulator subjected to magnetic fields $\mathbf{h},\mathbf{B}$ and anisotropies, which realizes the Hamiltonian that we discussed in Sec. 2. We turn on the coupling $\tilde{J}\left(t\right)$ between spins and the insulator adiabatically so that these spins behave like a part of the insulator, and thus, the concurrence ${\mathcal{C}}_{AB}$ grows correspondingly. This entanglement remains even after the coupling $\tilde{J}$ is turned off as long as this turnoff process is rapid enough.