Entanglement of solitons in the Frenkel-Kontorova model

We investigate entanglement of solitons in the continuum limit of the nonlinear Frenkel-Kontorova chain. We find that the entanglement of solitons manifests particlelike behavior as they are characterized by localization of entanglement. The von Neumann entropy of solitons mixes critical with noncritical behaviors. Inside the core of the soliton the logarithmic increase of the entropy is faster than the universal increase of a critical field, whereas outside the core the entropy decreases and saturates the constant value of the corresponding massive noncritical field. In addition, two solitons manifest long-range entanglement that decreases with the separation of the solitons more slowly than the universal decrease of the critical field. Interestingly, in the noncritical regime of the Frenkel-Kontorova model, entanglement can even increase with the separation of the solitons. We show that most of the entanglement of the so-called internal modes of the solitons is saturated by local degrees of freedom inside the core, and therefore we suggest using the internal modes as carriers of quantum information.

Figure 1

(Color online) Schematic structure of the FK chain for $N=10$. The blue circles are the vacuum equilibrium locations, where the black squares represent the soliton solution.

Figure 2

(Color online) Single-soliton ${\varphi }_{\mathrm{SG}}$ and double-soliton ${\varphi }_{2\mathrm{SG}}$ solutions of the finite SG equation sampled at $N=1000$ points. The length of the region is given by $2L=16{\lambda }_J$ where ${\lambda }_J$ is the Josephson length. For the single-soliton solution the stability region for the magnetic field is $H∊[0.0019,1.0052)$ and for the double soliton $H∊[0.1023,1.0622)$. See details in the Appendix.

Figure 3

(Color online) Normal modes in the background of the single-soliton solution, shown in Fig. 2. The four first normal modes and the tenth normal mode are shown. The first normal mode is an internal mode.

Figure 4

(Color online) Normal modes in the background of the double-soliton solution shown in Fig. 2. The four first normal modes and the tenth normal mode are shown. The first two normal modes are internal modes.

Figure 5

(Color online) Log-log scale of ${\xi }_n$, where $0⩽n⩽500$. The plots are given for the vacuum sector and the soliton sector in the critical $(g={10}^{10})$, noncritical $(g={10}^4)$, and weak-coupling $(g=5)$ regimes. In the soliton sector the center of the soliton is located at $n=0$.

Figure 6

(Color online) Localization of entanglement in the soliton sector of the FK model. In the noncritical regime $(g={10}^4)$ the size of the core is $\sqrt{g}=100$. The critical limit is given for $g={10}^8$. Localization of entanglement is seen in the noncritical regime, where maximal entropy is obtained for $l=\sqrt{g}$.

Figure 7

(Color online) Localization of entanglement in the soliton sector of the FK model, logarithmic scale. The increase of entanglement inside the soliton is faster than the corresponding one in the vacuum sector. Both entropies coincide outside the soliton. The entanglement of the soliton sector has a maximal value and is therefore localized. Note that the logarithmic increase inside the soliton is even faster than the corresponding one in the critical vacuum sector.

Figure 8

(Color online) Localization of entanglement in the soliton sector for $g=200,1000,25000$. Entanglement localization “measures” the size of the soliton, as maximal entanglement is achieved for $l=\sqrt{g}$ in all three cases.

Figure 9

(Color online) The prefactor $\alpha \left(g\right)$ of $E_S=\alpha \ln l$ for $l⪡\sqrt{g}$, logarithmic scale. Interestingly, for any value of $g$ in the strong-coupling regime $(g>g_{\mathrm{con}})$, ${\alpha }^{\mathrm{sol}}\left(g\right)>{\alpha }^{\mathrm{vac}}(g\to \infty )=1∕3$. In the soliton sector ${\alpha }_{\max }^{\mathrm{sol}}\left(g\right)\sim 4∕9$, and is obtained for $g=N$.

Figure 10

(Color online) Maximal entropy as a function of $g$. ${\omega }_1$ is minimal for $g=200$, exactly where the von Neumann entropy reaches its maximal value.

Figure 11

(Color online) Long-range entanglement between two sliding blocks in the presence of two solitons with $g=3000$, $l=l_{\mathrm{sol}}=56$, and $d=d_{\mathrm{sol}}=281$. LN is maximized when the blocks coincide with the solitons’ cores (the average of the centers is located in the center of the chain, $n=0$).

Figure 12

(Color online) First and second participation functions of the double-soliton and vacuum sectors, shown in logarithmic scale, where the chain is divided into two equal-sized blocks. In contrast to the localized behavior of the participation functions in the vacuum sector, the participation functions in the solitons sector peak at the core of the soliton.

Figure 13

(Color online) Long-range entanglement as a function of the solitons separation. The centers of the two blocks coincide with the solitons, $d=d_{\mathrm{sol}}$ and $l=l_{\mathrm{sol}}$. The plots are given for $g=1600,{10}^4$. $\ln E_{\mathrm{LN}}(g=1000)$ of the vacuum sector is plotted for reference, where $\beta \left(g\right)=5.57$. $l\sim \sqrt{g}=40,96$, respectively. In the case $g={10}^4$, the entanglement of solitons decreases slower than its critical vacuum sector correspondence for small values of $d∕l$. In the case $g=1600$, the entanglement oscillates and is not a decreasing monotonic function of the separation for large values of $d∕l$.

Figure 14

(Color online) Particlelike entanglement of the solitons: Dependence of $\ln E_{\mathrm{LN}}$ on the ratio between the internal modes’ eigenfrequencies for $g={10}^4$, where $20\ln {\omega }_1$ and $20\ln {\omega }_2$ are also plotted. The three left-hand vertical arrows are examples of drastic increase of the gap, which correspondingly changes the entanglement. Notice that as the gap is closed $(d∕l>6)$, the entanglement decreases significantly. The entanglement of the toy model, $E\left(\alpha \right)$, is shown for reference.

Figure 15

(Color online) The exponential coefficient $\beta$ in the asymptotic behavior $E_{\mathrm{LN}}(d⪢l)\sim e^{-\beta d∕l}$ as a function of $g$. In the vacuum sector $\beta$ decreases monotonically to the universal value 2.7 (not seen in graph). The lower bound $\beta >2$ is satisfied in the soliton sector for approximately $g>2000$. For smaller values of $g$ the exponential approximation breaks down.

Figure 16

(Color online) Topological nature of ${\omega }_1$. $\ln E_{\mathrm{LN}}$ versus $d$ for $g=731$ (deep in the noncritical regime). The computed logarithmic negativity is not a monotonically decreasing function of the separation between the solitons. For reference we plot $2\ln \left({\omega }_1\right)-20$. Note that maximal entanglement is achieved for minimal $w_1$, which characterizes the regime in which $g$ is small.

Figure 17

(Color online) ${\omega }_2∕{\omega }_1$ and $\ln {\omega }_1$ as a function of $\ln E_{\mathrm{LN}}$ for three coupling strengths: $g=29500,7900,1000$.

Figure 18

(Color online) $E_S$ as a function of $l$ for small values of $g$.

Figure 19

(Color online) Entangling a soliton and an external oscillator $Q$ by two-mode squeezing with $r=0.99$. Shown $E_S\left(A\right)-E_S\left(B\right)⩽E_D\left(A\right)$ as a function of $l$ for $g=1000$. $E_S\left(Q\right)$ is shown for reference. Note that the distillable entanglement saturates the inserted entanglement for quite small values of $l$.

Figure 20

(Color online) Saturation of the hashing inequality—schematic presentation. At $t=0$ we perform two-mode squeezing of $\chi$ and $Q$ so that $\chi$ decay significantly outside the squeezing block $W$ and we measure the entanglement between $A$ and $Q$ such that $A⪢W$. In a simplified mode-wise decomposition of $A$ and $B\cup Q$ we treat only two modes: $\mid 1_A,1_B⟩$, in which the participation function is localized at the boundary and $\mid 2_A,2_B⟩$, in which the participation function is localized inside $W$.

Figure 21

(Color online) Double-soliton solution. ${\eta }_{\pm }=\frac{1}{\sqrt{2}}({\eta }_1\pm {\eta }_2)$ for $g=500$ and $g={10}^4$, where ${\omega }_1(g=500)=0.0011$, ${\omega }_2(g=500)=0.0012$, ${\omega }_1(g=10000)=0.0832$, and ${\omega }_2(g=10000)=0.3075$. Note that for almost degenerate modes the transformed collective modes ${\eta }_{\pm }$ are well separated, while for nondegenerate modes the collective modes overlap and the squeezing is not optimal.

Figure 22

(Color online) Double-soliton solution. Two-mode squeezing of ${\eta }_{\pm }$ with $r=2$. The graph shows $E_S\left(A_1\right)-E_S(A_1\cup A_2)⩽E_D(A_1,A_2)$ as a function of $l$ for $g=500$, $k=0.9415$, and $d\left(l\right)=331-l$ (the separation between the blocks decreases as their size increases). $E(\pm )$ is plotted for reference. Note that the distillable entanglement actually exceeds the inserted entanglement by squeezing, as for large values of $l$, $d\left(l\right)$ decreases to very small numbers, for which the vacuum entanglement has a non-negligible contribution.