Floquet Recurrences in the Double Kicked Top

Technical Summary: Floquet Recurrences in the Double Kicked Top

Problem Statement
The paper investigates exact quantum recurrences in the Double Kicked Top (DKT), a driven spin model that extends the standard Quantum Kicked Top (QKT) by introducing a second instantaneous kick per driving period. This additional kick breaks time-reversal symmetry. While previous studies established that the QKT exhibits exact state-independent recurrences at specific chaoticity parameters (rational multiples of $j\pi$), it remained unclear whether the DKT exhibits similar exact recurrences and how these depend on the time-reversal symmetry-breaking parameter, $k_\theta$. Furthermore, the distinction between local dynamical changes and global structural changes induced by broken time-reversal symmetry was not fully understood.

Methodology
The authors employ a combination of analytical derivations and numerical simulations to analyze the periodicity of the DKT Floquet operator ($U$).

• Model Reformulation: The DKT dynamics are reformulated using effective kick parameters $k_r = (k + k')/2$ and $k_\theta = (k - k')/2$, where $k_r$ governs the QKT-equivalent dynamics and $k_\theta$ controls time-reversal symmetry breaking.
• Analytical Proofs: The authors derive the periodicity of the Floquet operator for specific values of $k_r$ ($j\pi/2$ and $j\pi/4$) by distinguishing between integer and half-odd-integer total spin $j$. They utilize Pauli matrix representations, operator identities, and the Cayley-Hamilton theorem to simplify powers of $U$ and demonstrate independence from $k_\theta$.
• Numerical Analysis:
  • Husimi Distributions: Used to visualize the phase-space evolution of spin-coherent states.
  • Entanglement Dynamics: Long-time-averaged von Neumann entropy of the single-qubit reduced density matrix is computed to quantify entanglement.
  • Fidelity and DQPTs: The fidelity decay and its associated rate function (analogous to free energy density) are calculated to detect Dynamical Quantum Phase Transitions (DQPTs).
  • Level Statistics: Spacing-ratio statistics (SRF) of quasi-energies are analyzed to distinguish between integrable (Poisson) and non-integrable (Gaussian Orthogonal Ensemble) regimes.

Key Contributions and Results

1. Exact Periodicity Independent of Symmetry Breaking:
   The primary analytical result is that the Floquet operator becomes exactly periodic at specific values of the effective kick strength $k_r$, regardless of the symmetry-breaking parameter $k_\theta$.

   • Case $k_r = j\pi/2$:
     • For integer $j$, the period is 8 ($U^8 = I$).
     • For half-odd-integer $j$, the period is 12 ($U^{12} = \pm I$).
   • Case $k_r = j\pi/4$:
     • For integer $j$, the period is 48 ($U^{48} = I$).
     • For half-odd-integer $j$, the Floquet operator does not exhibit periodicity. The authors provide an analytical proof showing that the operator acting on a specific subspace of unpaired qubits yields eigenvalues that are not roots of unity, preventing periodicity.

2. Entanglement and Time-Reversal Symmetry:
   While the periodicity of the operator is independent of $k_\theta$, the entanglement dynamics are not.

   • Perturbing $k_\theta$ away from zero has minimal effect on entanglement for most initial states near $k_\theta = 0$.
   • However, near the time-reversal symmetric cases $k_\theta = \pm k_r$, breaking the symmetry induces significant changes, including the emergence of "vortex-like" structures in the von Neumann entropy landscape around trivial fixed points.
   • Specific initial states (e.g., $|\theta_0 = 0, \phi_0 = 0\rangle$) show sharp transitions in entanglement only when $k_\theta = k_r$, whereas others ($|\theta_0 = \pi/2, \phi_0 = -\pi/2\rangle$) show vortex formation just before reaching $k_\theta = k_r$.

3. Dynamical Quantum Phase Transitions (DQPTs):
   Analysis of the fidelity rate function reveals:

   • Pronounced peaks (signatures of DQPTs) occur at $k_\theta = \pm k_r$ for both integer and half-odd-integer $j$.
   • For the case $k_\theta = 0$, sharp changes in the rate function are observed only for half-odd-integer $j$, indicating a qualitative distinction between the two time-reversal symmetric cases ($k_\theta = 0$ vs. $k_\theta = \pm k_r$).

4. Integrability and Chaos Transition:
   Near the exact recurrence point $k_r = j\pi/2$, the system exhibits signatures of quantum integrability (high degeneracy, Poissonian level statistics). As $k_r$ is perturbed away from this value, the system transitions smoothly to non-integrable (chaotic) behavior characterized by GOE statistics.

   • A unique bifurcation phenomenon is observed in the long-time-averaged von Neumann entropy near $k_r = j\pi/2$, where low-entropy regions merge. The authors note this is a purely quantum effect with no classical analogue, demonstrating how quantum effects can suppress underlying classical chaos.

Significance and Claims
The paper claims to generalize previous findings on quantum recurrences in the QKT by providing explicit analytical derivations for the DKT. The work demonstrates that:

• Regular and chaotic regimes in the DKT can be controlled for any system size by tuning the effective parameters $k_r$ and $k_\theta$.
• Exact quantum recurrences are robust against time-reversal symmetry breaking ($k_\theta$), even though the resulting entanglement dynamics are sensitive to it.
• The DKT serves as a useful platform for studying the interplay between time-reversal symmetry, quantum integrability, and chaos.
• The system is realizable in existing experimental platforms, including NMR, cold atoms, and superconducting qubits, by extending standard QKT protocols with a second kick or nonlinear twist.

The authors conclude that the DKT offers a versatile framework for quantum control and information processing applications, specifically through the ability to tune system size and parameters to achieve desired entanglement dynamics and recurrence behaviors.