Exotic quantum holonomy and higher-order exceptional points in quantum kicked tops

The correspondence between exotic quantum holonomy, which occurs in families of Hermitian cycles, and exceptional points (EPs) for non-Hermitian quantum theory is examined in quantum kicked tops. Under a suitable condition, an explicit expression of the adiabatic parameter dependencies of quasienergies and stationary states, which exhibit anholonomies, is obtained. It is also shown that the quantum kicked tops with the complexified adiabatic parameter have a higher-order EP, which is broken into lower-order EPs with the application of small perturbations. The stability of exotic holonomy against such bifurcation is demonstrated.

Figure 1

Parametric evolution of quasienergies of the quantum top under a rank-1 kick (3) along the unitary cycle $C$. (a) $J=1$ case. Thick and dashed curves correspond to $\omega =2\pi /3$ and $\pi /6$, respectively. (b) $J=3/2$ case. Thick and dashed curves correspond to $\omega =\pi /2$ and $\pi /4$, respectively. As shown later in the main text, thick curves and dashed curves correspond to $(2J+1)$-EP cases (Sec. 4) and 2EP cases (Sec. 5), respectively.

Figure 2

A contour plot of $\mathrm{Re}{E}_{-1}$ in the $\Lambda$ plane [$\Lambda$ is defined in Eq. (5)]. Lighter (darker) color indicates large (smaller) value of $\mathrm{Re}{E}_{-1}$. Parameters are $J=1$ and $\omega =2\pi /(2J+1)$. There is a 3EP at $\Lambda =0$, which is indicated by the filled circle. A branch cut, which emanates from the 3EP, is drawn by a bold line. A dash-dotted curve $C^{\prime }$ is a smooth deformation of $C$. The exotic quantum holonomy induced by $C$ can be emulated by the non-Hermitian cycle $C^{\prime }$.

Figure 3

Parametric evolution of EPs of the quantum kicked top $J=1$ within the unitary cycle $C$ in the $\Lambda$ plane. $\omega$ is varied within the interval $[0,\pi ]$. There is a triple degeneracy point ${\Lambda }_0$($=1$) at $\omega =0$. Note that the system is unitary (diagonalizable) at $\Lambda ={\Lambda }_0$. As $\omega$ is increased in the interval $(0,2\pi /3)$, two 2EPs evolve along the curves that pass ${\Lambda }_1$ and ${\Lambda }_2$ at $\omega =\pi /6$. These 2EPs merge at $\omega =2\pi /3$ to form a 3EP, which locates at ${\Lambda }_3$($=0$). If $\omega$ is increased further ($2\pi /3<\omega <\pi$), there are two 2EPs, which are fragments of the 3EP ${\Lambda }_3$, in the real axis. Finally, at $\omega =\pi$, a 2EP arrives at ${\Lambda }_0$ to become a diabolic point, and the other arrives at ${\Lambda }_4$($<0$).

Figure 4

Contour plots of $\mathrm{Re}{E}_M$ in the $\Lambda$ plane with (a) $M=-1$, (b) $M=0$, and (c) $M=1$. We choose $\omega =\pi /6$, where a 3EP is split into two 2EPs indicated by filled circles. Other parameters are the same as in Fig. 2.

Figure 5

A schematic explanation of two non-Hermitian cycles $C_1$ (dashed lines) and $C_2$ (dotted lines) that emulate a unitary cycle $C$ in the $\Lambda$ plane for the quantum kicked spin $J=1$ and $\omega =\pi /6$ (see Fig. 4). $C$ start from $\Lambda =1$ and encircle EPs ${\Lambda }_1$ and ${\Lambda }_2$.

Figure 6

Contour plots of $\mathrm{Re}{E}_M$ in the $\Lambda$ plane with (a) $M=-3/2$, (b) $M=-1/2$, (c) $M=1/2$, and (d) $M=3/2$. Other parameters are $J=3/2$ and $\omega =\pi /8$. There are three 2EPs, which merge at $\omega =\pi /2$ (not shown here) to compose a 4EP. Other conventions are the same as in Fig. 2.