Theory of quantum entanglement and structure of the two-mode squeezed antiferromagnetic magnon vacuum

Recently, investigations of the quantum properties of an antiferromagnet in the spin wave approximation have identified the eigenstates as two-mode squeezed sublattice-magnon states. The uniform magnon states were shown to display a massive sublattice entanglement. Here we extend this idea and study the squeezing properties of all sublattice Fock states throughout the magnetic Brillouin zone. We derive the full statistics of the sublattice magnon number with wave number $\vec{k}$ in the ground state and show that sublattice magnons occur in pairs with opposite wave vectors, hence, resulting in entanglement of both modes. To quantify the degree of entanglement we apply the Duan-Giedke-Cirac-Zoller inequality and show that it can be violated for all modes. The degree of entanglement decreases towards the corners of the Brillouin zone. We relate the entanglement to measurable correlations of components of the Néel and the magnetization vectors, thus allowing one to experimentally test the quantum nature of the squeezed vacuum. The distinct $k$-space structure of the entanglement shows that the squeezed vacuum has a nonuniform shape that is revealed through the $\vec{k}$-dependent correlators for the magnetization and the Néel vectors.

Figure 1

Energy dispersion for a two-dimensional square lattice antiferromagnet, with $\left|J\right|/K={10}^4$.

Figure 2

Both figures were made for $\left|J\right|/K={10}^4$. (a) Probability $P_{\vec{k}}\left(n\right)$ to find $n$ magnons in sublattice $A/B$ with wave vector $\vec{k}=\vec{0}$ (orange) and $\vec{k}=\frac{\pi }{200}\vec{1}$ (purple) in the squeezed vacuum state (dashed) and the probability $Q_{\vec{k}}\left(n\right)$ to find $n$ magnons in sublattice $A/B$ and $n+1$ in $B/A$ in the squeezed magnon state (solid). Here $\vec{1}={(1,1)}^T$. (b) Probability $p_{\vec{k}}$ to have at least one magnon in each sublattice. We see a sharp peak at $\vec{k}=\vec{0}$, with $p_{\vec{0}}\approx 0.6$, implying that the probability to find any number of magnons different from 0 for the other wave vectors is very small.

Figure 3

Values for $c$, depending on $p_{\vec{k}}$, which violate the DGCZ inequality (purple area). The orange (solid) [purple (dashed)] line satisfies the equality and corresponds to solution $c_1$ ($c_2$).

Figure 4

Correlators of the $x$ components of the magnetization (left) and Néel vector (right) in the $k$ space for $\left|J\right|/K={10}^4$. The $k_x$ and $k_y$ values are limited to $k_0=0.076$ in picture (b) to keep the focus on points with $C_{\vec{k}}^{N,x}>1$. For the value of $k_0$ see Appendix pp2.

Figure 5

Double logarithmic plot of $\langle \tilde{n}\rangle$ (a) and ${\langle {\left(\mathrm{\Delta }\widehat{n}\right)}^2\rangle }_{\text{sq}}/N_{\text{mod}}$ (b) depending on $\left|J\right|/K$. The number of total modes $N_{\text{mod}}=N^2$, with $N$ being the number of sites in one dimension, varies from graph to graph. The vertical, dotted line in (a) shows the value for $\left|J\right|/K$, at which the $\vec{k}=\vec{0}$ contribution to the total magnon number is as big as the total contribution from all other modes. This indicates the start of the linear behavior for the magnon number in the double logarithmic plot.

Figure 6

Ratio of energy and $\left|J\right|$ for different values of $\left|J\right|/K$ and $J=-1$ showing the lowering of the energy gap for smaller $K$ and the approach to a linear dispersion. Dashed black line gives the linear dispersion with ${\varepsilon }_{\vec{k}}/\left|J\right|=2\sqrt{2}\left|\vec{k}\right|$.