Quantum chaos on the separatrix of the periodically perturbed Harper model

TL;DR

This work investigates quantum chaos arising at the separatrix of a sinusoidally perturbed Harper model by formulating a finite-dimensional Floquet system on a torus and quantizing it with a discrete phase-space representation. The authors show a strong quantum–classical correspondence: Floquet eigenstates possess Husimi distributions that resemble classical orbits, and the energy dispersion σ_{h0} of the unperturbed operator tracks ergodicity in both classical and quantum pictures. They derive a quantum analogue of the classical separatrix width using the first term of the Magnus expansion in the interaction representation, connecting the width ΔH_Q to averaged perturbations evaluated near separatrix energies. Numerical results corroborate that chaotic regions host ergodic quantum subspaces, with off-diagonal perturbation matrix elements peaking near separatrices and with quasi-energy statistics transitioning toward GOE-like behavior in ergodic subspaces. The work provides a framework for quantum control and Floquet engineering in finite-dimensional systems and deepens understanding of quantum chaos on compact phase spaces.

Abstract

We explore the relation between a classical periodic Hamiltonian system and an associated discrete quantum system on a torus in phase space. The model is a sinusoidally perturbed Harper model and is similar to the sinusoidally perturbed pendulum. Separatrices connecting hyperbolic fixed points in the unperturbed classical system become chaotic under sinusoidal perturbation. We numerically compute eigenstates of the Floquet propagator for the associated quantum system. For each propagator eigenstate we compute a Husimi distribution in phase space and an energy and energy dispersion from the expectation value of the unperturbed Hamiltonian operator. The Husimi distribution of each Floquet eigenstate resembles a classical orbit with a similar energy and similar energy dispersion. Chaotic orbits in the mixed classical system are related to Floquet eigenstates that appear ergodic. For a mixed regular and chaotic system, the energy dispersion can separate the Floquet eigenstates into ergodic and integrable subspaces. The width of a chaotic region in the classical system is estimated by integrating the perturbation along a separatrix orbit. We derive a related expression for the associated quantum system from the averaged perturbation in the interaction representation evaluated at states with energy close to the separatrix.

Figures (9)

• Figure 1: a) We show a surface of section for regular (not chaotic) classical system, the Harper model with Hamiltonian in equation \ref{['eqn:Hclassical']}), and with parameters of the model printed on top of the figure. Points are plotted once every perturbation period of $2\pi$. Each orbit is plotted with the same color points, and we choose different colors for different orbits. This particular system has $\mu = \mu' = 0$ and lacks sinusoidal time-dependent perturbations, but the figures are made in the same way as subsequent figures that have periodic perturbations. b) The related quantum model (with the same parameters $a, \epsilon, \mu, \mu'$ as the classical one) has Hamiltonian operator in equation \ref{['eqn:Hquantum']} and Floquet propagator given in equation \ref{['eqn:U_T']}. We show Husimi distributions for the eigenstates of the unitary propagator for the discrete quantized model with $N=100$. Eigenstates of the propagator are arranged in order of their expectation value $\mu_{h0,j} = \langle \hat{h}_0 \rangle$. c) The top panel shows with solid brown dots the expectation value of the unperturbed Hamiltonian $\hat{h}_0$ for each eigenstate of the Floquet propagator. The mean values of the unperturbed Hamiltonian $\mu_{h0,j} = \langle H_0 \rangle$ are also computed for classically integrated orbits and these are shown as blue squares. The x-axis shows the state index $j$ denoted a state for the quantum system, but the classical values are plotted over the same range in order of increasing mean value $\langle H_0 \rangle$, averaged from points in an orbit. The bottom panel shows (with brown dots) the standard deviations $\sigma_{h0,j}$ (equation \ref{['eqn:sigh0']}) of the eigenvectors of the propagator. The standard deviation of the unperturbed Hamiltonian function for the same classical orbits as plotted in the top panel are plotted as blue squares.
• Figure 2: a) Similar to Figure \ref{['fig:vanil']}a except for a classical system that has chaotic regions. The parameters $a,\epsilon,\mu, \mu'$ of the Hamiltonian are printed on the top left. b) We show the Hussimi distributions of the eigenstates of the associated quantum model with $N=100$. The Hussimi distributions of some eigenstates of the Floquet propagator fill the chaotic regions in phase space seen in the associated classical system. The quantum system has $N=100$ states. c) Similar to Figure \ref{['fig:vanil']}c. We show energy means and dispersions from both classical orbits and eigenstates. Chaotic eigenstates have large standard deviations $\sigma_{h0,j}$ (defined in equation \ref{['eqn:sigh0']}). Chaotic orbits also have large energy standard deviations.
• Figure 3: Similar to Figure \ref{['fig:Q9b']} except for a classical system that has larger chaotic regions and has an asymmetric perturbation; $\mu \ne \mu'$. This system also has $N=100$ states.
• Figure 4: Similar to Figure \ref{['fig:Q9b']}b except that the number of states $N=255$ instead of 100. The Hamiltonian has the same parameters $a,\epsilon, \mu, \mu'$ as the system shown in Figure \ref{['fig:Q9b']}. The Husimi distributions are similar with larger $N$, though there may be additional localized states located with the chaotic region.
• Figure 5: A sequence of Husimi distributions as a function of increasing dimension $N$. We show Husimi functions constructed from eigenstates of the Floquet propagator $\hat{U}_T$ that have energy near the separatrix for a Hamiltonian model with parameters printed on the top left side of the plot. Each panel shows a model that has the same parameters but has a different dimension $N$. The dimensions are printed on the bottom left of each panel and are powers of 2. This figure illustrates that in a region that is chaotic in the associated classical model, Husimi functions of the eigenstates appear increasingly diffuse and evenly distributed at larger $N$. This suggests that they belong to a subspace that is ergodic in the sense described by Shnirelman_1974 (also see Kurlberg_2001).