Entanglement storage by classical fixed points in the two-axis countertwisting model

We analyze a scheme for storage of entanglement quantified by the quantum Fisher information in the two-axis countertwisting model. A characteristic feature of the two-axis countertwisting Hamiltonian is the existence of the four stable center and two unstable saddle fixed points in the mean-field phase portrait. The entangled state is generated dynamically from an initial spin-coherent state located around an unstable saddle fixed point. At an optimal moment of time the state is shifted to a position around the stable center fixed points by a single rotation, where its dynamics and properties are approximately frozen. We also discuss evolution with noise. In some cases the effect of noise turns out to be relatively weak, which is explained by parity conservation.

Figure 1

Mean-field phase portraits for (a) ${\tilde{\gamma }}_x={\tilde{\gamma }}_y={\tilde{\gamma }}_z=0$; (b) ${\tilde{\gamma }}_x=0.4,{\tilde{\gamma }}_y={\tilde{\gamma }}_z=0$; (c) ${\tilde{\gamma }}_x=1,{\tilde{\gamma }}_y={\tilde{\gamma }}_z=0$; (d) ${\tilde{\gamma }}_y=0.4,{\tilde{\gamma }}_x={\tilde{\gamma }}_z=0$; (e) ${\tilde{\gamma }}_z=0.4,{\tilde{\gamma }}_x={\tilde{\gamma }}_y=0$. Red disks visualize the initial spin-coherent state.

Figure 2

Time evolution of the QFI for $\vec{\gamma }=\vec{0}$ (solid red line) and for $\vec{\gamma }\ne \vec{0}$ (gray lines), for $N=50$ atoms and $M=2\times {10}^3$ trajectories. The noise term is nonzero along the (a) $Z$, (b) $Y$, or (c) $X$ axis, with ${\sigma }_j=\chi$ (dotted lines), ${\sigma }_j=5\chi$ (dot-dashed lines), ${\sigma }_j=15\chi$ (dashed lines), ${\sigma }_j=50\chi$ (dot-dot-dashed lines). Insets in panels (c) show the QFI at $\chi t=ln\left(2\pi N\right)/N$ as a function of ${\sigma }_x$ (left) and ${\gamma }_x$ (right).

Figure 3

Illustration of the storage scheme by classical stable fixed points. The initial state for the evolution is the spin-coherent state located around the unstable fixed point. At the optimal moment of time $\tau =ln\left(2\pi N\right)/2N\chi$ (the first maximum of the QFI), the position of the state is shifted by the single rotation $\widehat{U}=e^{i{\widehat{S}}_x\pi /4}$ to regions around the two stable fixed points. Thereafter, the dynamics is limited to a narrow region of the phase space around the two antipodal stable fixed points.

Figure 4

(a) Evolution of the QFI: red dashed line by the TACT Hamiltonian, black solid line by the TACT Hamiltonian with the rotation $\widehat{U}=e^{i{\widehat{S}}_x\pi /4}$ at $t=\tau =ln\left(2\pi N\right)/2N\chi$. Both for the noiseless system and $N=50$. (b) QFI with noise and the rotation at $\tau$ in time for ${\sigma }_x=\chi /10,{\sigma }_{y,z}=0$ (blue dashed line), ${\sigma }_y=\chi /10,{\sigma }_{x,z}=0$ (red solid line), ${\sigma }_z=\chi /10,{\sigma }_{x,y}=0$ (green dotted line) and $N=50,M=2\times {10}^3$.

Figure 5

The Wigner function with noise and the rotation at $\tau$ for (a) ${\sigma }_x=\chi /10,{\sigma }_{y,z}=0$ and (b) ${\sigma }_y=\chi /10,{\sigma }_{x,z}=0$, both at $t=30ln\left(2\pi N\right)/N\chi$ and $N=50,M={10}^4$. The QFI has the meaning of the speed of change of statistics due to rotation on the Bloch sphere. A small rotation of the state results in small shifts of the distribution of quasiprobability; here the $\mathrm{SU}\left(2\right)$ Wigner function [27]. Thus the appearance of small thin structures in the Wigner function leads to a high value of the QFI, as rotating the state perpendicularly to them would strongly affect the state and the quasiprobability distribution. The high narrow fringes, and also the high value of the QFI, remain even in the case with noise along the $X$ axis, but they are mostly smeared out by noise along $Y$ or $Z$ axis.

Figure 6

(a) The half-decay time ${\tau }_{1/2}$ of the QFI for ${\gamma }_y\ne 0$ and ${\gamma }_{x,z}=0$ (open symbols), and ${\gamma }_z\ne 0$ and ${\gamma }_{x,y}=0$ (filled symbols). Dashed line is the best fit ${\tau }_{1/2}=0.977{\left(N{\sigma }_{y,z}\right)}^{-1.096}$ to the numerical data. (b) Averaged values of the QFI after the rotation and noise along the $X$ axis. In insets: solid lines are a few of the largest eigenvalues of the system density matrix while the dashed line is the fidelity function (23) in time. Parameters are the same as in Fig. 4.

Figure 7

Comparison between the QFI (solid lines) and the Fisher Information $F(\widehat{\mathrm{\Pi }},\vec{n})$ associated with specific choice of the interferometric direction and measurements of the parity operator (dot-dashed lines fitting closely to the solid lines). From top to bottom: ${\sigma }_x=\chi ,{\sigma }_x=10\chi$, and ${\sigma }_x=15\chi$.