Figure 1

The ground state parameters $\langle S_x^2\rangle$ and $\langle S_y^2\rangle$ (blue dash-dot line) and $\overline{s}=\langle S_z\rangle$ (red solid line) as the function of $\kappa$. In the lower part of the figure the graphic presentation of the variables $\langle S_x^2\rangle$ and $\langle S_y^2\rangle$ showing the shape of the cross section of the quadrupolar ellipsoid in the $xy$ plane for four values of $\kappa$; from left to right $\kappa =0$, 0.25, 0.5, and 0.75.Reuse & Permissions

Figure 2

Graphic presentation of the variables $s$ and $\gamma$ and their evolution. The thick red arrow represents the mean value of the spin. The quadrupolar variables are shown by the quadrupolar ellipsoid, see the text, drawn in blue. (a) The ground state and (b) the standard transverse dynamics, i.e., the spin precession. (c)–(e) Transient values of the variables for longitudinal oscillations. (c) and (e) Correspond to the longest and shortest length of the spin, and at the moment depicted in (d) the spin length equals to its equilibrium value, but the quadrupolar ellipsoid is turned on the angle $\gamma$ with respect to the $x$ axis. (c)–(e) The shape of the unperturbed ellipsoid is shown in light gray.Reuse & Permissions

Figure 3

Phase plane representations for dissipation-free nonlinear longitudinal spin oscillations for different values of the parameter $\kappa$; $\kappa =0.6$, 0.5, and 0.2 for (a), (b), and (c), respectively. The green and red ellipses present the minima and maxima, respectively; the standard saddle points are depicted by red rectangles, while the standard separatrix trajectories are drawn by red lines. The singular trajectories with $s=\pm 1$ and the separatrix trajectories entering the nonstandard saddle points on these lines $s=\pm 1$ are shown by blue lines in (a) and (c). For the critical value $\kappa =0.5$, all the separatrix trajectories and the singular trajectories with $s=\pm 1$ organize a common net; and on the corresponding frame (b) all of them are presented by red lines.Reuse & Permissions

Figure 4

Phase plane representation of damped longitudinal spin oscillations for $\kappa =0.6$. Here and in Fig. 6, the dashed lines (obtained analytically before) represent the phase trajectories without dissipation, while the full lines are trajectories for dissipation constant $\lambda =0.2$, found numerically. The separatrix lines are drawn by red curves.Reuse & Permissions

Figure 5

Time evolution of the spin $s$ and quadrupolar variables ${\sigma }_{1,2}$, see Eq. (6), describing the switching for the magnet with $\kappa =0.6$. The data for two characteristic initial conditions, corresponding, to relatively small (or large) initial deviation of $s$ and relatively large and (or small) initial deviation of $\gamma$ are present by full and dash-dot lines, respectively. For concrete calculations, the values $s\left(0\right)=0.5$, $\gamma \left(0\right)=0.1\pi$ and $s\left(0\right)=0.1$, $\gamma \left(0\right)=0.03\pi$ are used in these cases. The dash lines demonstrate the switching for the equilibrium initial value of spin $s\left(0\right)=\sqrt{1-{\kappa }^2}=0.8$ and large enough $\gamma \left(0\right)=0.19\pi$. The values of spin $s$ (red lines) and ${\sigma }_1=\langle S_y^2-S_x^2\rangle$ (blue lines) are going to their finite equilibrium values $-\sqrt{1-{\kappa }^2}$ and $\kappa =0.6$, whereas ${\sigma }_2\to 0$ (green lines).Reuse & Permissions

Figure 6

Phase plane for the damped spin evolutions for low anisotropy $\kappa =0.2$. The central frame shows the full diagram; left and right panels demonstrate the details of the behavior near the equilibrium values $s=-\overline{s}$ and $s=\overline{s}$, respectively. On this frame, the regions colored by green and yellow correspond to different basins of attraction with initial values leading to the equilibrium states with $s=\overline{s}$ and $s=-\overline{s}$, respectively.Reuse & Permissions

Figure 7

The same as in Fig. 5, but for small anisotropy $\kappa =0.2$. Here the full and dash-dot lines show the behavior for minimal deviation $s\left(0\right)=0.55$, with the large value of $\gamma \left(0\right)=0.4\pi$ and the small values $s\left(0\right)=0.2$, $\gamma \left(0\right)=0.1\pi$, respectively. Note here much larger amplitude of the sign-alternating oscillations of ${\sigma }_1$ within the initial stage of evolution, that corresponds with the change of $\gamma$ on $\pi$, see Fig. 6.Reuse & Permissions