Resonant fragility and nonresonant robustness of Floquet eigenstates in kicked spin systems

Abstract

In classical systems, the Kolmogorov-Arnold-Moser (KAM) theorem establishes that resonant tori of integrable Hamiltonians are destroyed by any nonintegrable perturbation, whereas nonresonant tori are only deformed up to a finite value of the perturbation parameter. In this contribution, we identify a quantum analog of this differentiated sensitivity for one-degree-of-freedom spin Hamiltonians subject to periodic instantaneous kicks. After detecting quantum signatures of resonances in the participation ratio and in the quasiprobability phase-space distribution of Floquet eigenstates of the perturbed Hamiltonian, we show that eigenstates of the unperturbed Hamiltonian exhibit greater sensitivity against the perturbation when they satisfy a resonant condition. The sensitivity is quantified through the fidelity between perturbed and unperturbed eigenstates. This differentiated sensitivity becomes increasingly pronounced as the system size grows. Our findings are supported by numerical results and insights from analytical calculations based on unitary perturbation theory. Although our analysis focuses on kicked models, the mechanism could be extended to more general periodic drivings, providing a preliminary step toward a quantum counterpart of the classical breaking of resonant tori.

Figures (3)

• Figure 1: (a) Classical time period (orange curve) and quantum periods, $T_{k,1}=2\pi/(E_{k+1}-E_k)$ (blue dots), as a function of energy of $h_0$ (classical) and scaled energies, $\bar{E}_k/J = (E_{k+1} + E_k)/(2J)$ (quantum). (b) Logarithm of the participation ratio of the Floquet eigenstates, $\ket{f_k}$, with respect to the $\hat{H}_0$ eigenbasis plotted as a function of $V_k=\langle f_k|\hat{H}_0| f_k\rangle/J$. The inset is a zoom of the $3{:}4$ resonance, showing that third neighbours states are involved in this resonance. (c) Maximum perturbation strength $\epsilon_{k,\text{max}}$, Eq.\ref{['eq: max epsilon']}, as a function of the index $k$ of the Floquet eigenstates $|f_k\rangle$ associated, respectively, to the states $|E_k\rangle$. Dashed lines with labels $m{:}n$ in the panels indicate some rational multiples of the kick period $m\tau/n$ and the energies associated, $E_{m:n}$. Parameters used: $J=500$ and $\tau=8$ for (a-c), and $\epsilon=10^{-3}$ for (b). Data for (c) available at Raw_data_doi
• Figure 2: First row: stroboscopic Poincaré sections of the classical LMG Hamiltonian (a) without and (b) with a small perturbation; panel (c) is a zoom of the low-energy region of (b). Second row: (d) Husimi function of a representative LMG eigenstate in that energy region, and (e)-(h) of Floquet eigenstates close to 1:1 resonance. Parameters used: $\tau=8$ and $\epsilon=10^{-3}$, and for second row $J=500$.
• Figure 3: $\epsilon_{k,\text{max}}$ as a function of $J$ in log-log scale for (a) non-resonant states, (b) states close to resonance and (c) exact resonant states. Numerical fits to the numerical data are indicated in each panel. (d) Time period of the kick over its closest quantum period, $\tau/T_{k,1}$, as a function of $J$ in linear-linear scale. The exact resonant condition $1{:}1$ is $\tau/T_{k,1}=1$ (see Sec. IV in SM for similar plots for other resonances). (e) Scaling of kick matrix elements for states involved in resonances $1\!:\!1$ ($l=k+1$), $2\!:\!3$ ($l=k+2$) and $3\!:\!4$ ($l=k+3$). In panels (a), (b) and (d) the kick period is $\tau=8$. In panel (c) $\tau$ is adjusted to fulfill exactly the quantum resonant conditions with $\tau\approx 8$. Non-resonant states in (a) are obtained by selecting $\hat{H}_0$ eigenstates closest to the energies $E/J=0.4$ and $E/J=-0.85$, where, according to Fig.\ref{['fig:1']}, no resonances are present. Data for (a-c) available at Raw_data_doi.