Floquet Recurrences in the Double Kicked Top

TL;DR

This work analyzes exact quantum recurrences in the double kicked top (DKT), a driven spin system with a second kick that breaks time-reversal symmetry. By mapping to effective kicks $k_r$ and $k_\theta$, the authors analytically prove exact periodicities of the Floquet operator at $k_r = j\pi/2$ and $k_r = j\pi/4$, with periods that depend on whether $j$ is integer or half-odd integer and are independent of $k_\theta$. They investigate the dynamical consequences using Husimi distributions, long-time-averaged von Neumann entropy, and fidelity rate functions to identify DQPTs and study integrability via level-spacing statistics near the periodic points; they find a robust, tunable transition between regular and chaotic dynamics controlled by $k_r$ and $k_\theta$. The results demonstrate that the DKT provides a scalable platform for quantum control and information processing, where exact recurrences persist despite TRS breaking and the system can display integrable behavior at Floquet resonances. The work connects exact quantum recurrences, DQPTs, and spectral statistics in a many-body spin system, with potential experimental realizations in NMR, cold-atom, and superconducting qubit platforms.

Abstract

We study exact quantum recurrences in the double kicked top (DKT), a driven spin model that extends the quantum kicked top (QKT) by introducing an additional time-reversal symmetry-breaking kick. Reformulating its dynamics in terms of effective parameters $k_r$ and $k_θ$, we analytically show exact periodicity of the Floquet operator for $k_r = jπ/2$ and $k_r = jπ/4$ with distinct periods for integer and half-odd integer $j$. These exact recurrences were found to be independent of $k_θ$. The long-time-averaged entanglement and fidelity rate function show dynamical quantum phase transition (DQPT) for $k_r = jπ/2$ at time-reversal symmetric cases $k_θ= \pm k_r$. In the other time-reversal symmetric case $k_θ= 0$, the DQPT exists only for a half-odd integer $j$. Using level statistics, a smooth transition is observed from integrable to non-integrable nature as $k_r$ is changed away from $jπ/2$. Our work demonstrates that regular and chaotic regimes can be controlled for any system size by tuning $k_r$ and $k_θ$, making the DKT a useful platform for quantum control and information processing applications.

Figures (16)

• Figure 1: Husimi function of the time-evolved initial state $|\theta_0 = 2.25,\, \phi_0 = 2.0\rangle$ over eight kicks. Here, $k_r = j\pi/2$, $k_\theta = 0$, and $j = 76$. Panels (a)–(i) correspond to $n = 0$ through $n = 8$, respectively.
• Figure 2: Husimi function of the time-evolved initial state $|\theta_0 = 2.25,\, \phi_0 = 2.0\rangle$ over eight kicks. Here, $k_r = j\pi/2$, $k_\theta = k_r$, and $j = 76$. Panels (a)–(i) correspond to $n = 0$ through $n = 8$, respectively.
• Figure 3: Long-time-averaged von Neumann entropy for the single-particle reduced density matrix $\rho_1(n)$, with total spin $j = 76$. We evolve 40,000 initial spin-coherent states $|\theta_0, \phi_0\rangle$ over $n = 1000$ steps. Here, $k_r = j\pi/2$ and (a) $k_\theta = 0$, (b) $k_\theta = 0.75 j\pi/2$, (c) $k_\theta = 0.95 j\pi/2$, (d) $k_\theta = j\pi/2$.
• Figure 4: Long-time-averaged von Neumann entropy for the initial spin-coherent states (a) $|\theta_0 = 0, \phi_0 = 0\rangle$ and (b) $|\theta_0 = \pi/2, \phi_0 = \pm \pi/2\rangle$, with $k_r = j\pi/2$ and total spin $j = 76$, evolved for $n = 1000$ time steps.
• Figure 5: Husimi function of the time-evolved initial state $|\theta_0 = 2.25,\, \phi_0 = 2.0\rangle$ over eight kicks. Here, $k_r = j\pi/2$, $k_\theta = 0$, and $j = 75.5$. Panels (a)–(i) correspond to $n = 0$ through $n = 12$, respectively.