We study the largest Lyapunov exponents λ and dynamical complexity for an open quantum driven double-well oscillator, mapping its dependence on coupling to the environment Γ as well as effective Planck’s constant β2. We show that in general λ increases with effective Hilbert space size (as β decreases, or the system becomes larger and closer to the classical limit). However, if the classical limit is regular, there is always a quantum system with λ greater than the classical λ, with several examples where the quantum system is chaotic even though the classical system is regular. While the quantum chaotic attractors are generally of the same family as the classical attractors, we also find quantum attractors with no classical counterpart. Contrary to the standard wisdom, the correspondence limit can thus be the most difficult to achieve for certain classically chaotic systems. These phenomena occur in experimentally accessible regimes.

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Abstract:The driven double-well Duffing oscillator is a well-studied system that manifests a wide variety of dynamics, from periodic behavior to chaos, and describes a diverse array of physical systems. It has been shown to be relevant in understanding chaos in the classical to quantum transition. Here we explore the complexity of its dynamics in the classical and semi-classical regimes, using the technique of ordinal pattern analysis. This is of particular relevance to potential experiments in the semi-classical regime. We unveil different dynamical regimes within the chaotic range, which cannot be detected with more traditional statistical tools. These regimes are characterized by different hierarchies and probabilities of the ordinal patterns. Correlation between the Lyapunov exponent and the permutation entropy is revealed that leads us to interpret dips in the Lyapunov exponent as transitions in the dynamics of the system.

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We study the number of irreducible factors (over Q) of iterates of polynomials of the form fr(x) = x 2 + r for r ∈ Q. When the number of such factors is bounded independent of n, we call fr(x) eventually stable (over Q). Previous work of Hamblen, Jones, and Madhu [7] shows that fr is eventually stable unless r has the form 1/c for some non-zero integer c, in which case existing methods break down. We study this family, and prove that several conditions on c of various flavors imply that all iterates of f 1/c are irreducible. We give an algorithm that checks the eventual stability of f 1/c in time O(log c), and applies to most c-values. We also study the two infinite families of c-values for which either the first iterate of f 1/c is reducible, or the first iterate is irreducible but the second iterate is reducible. We find all c-values for which the fourth iterate of f 1/c has at least four irreducible factors, and all c-values such that f 1/c is irreducible but its third iterate has at least three irreducible factors. This last result requires finding all integral points on a genus-2 hyperelliptic curve for which the method of Chabauty and Coleman does not apply; we apply the more recent variant known as elliptic Chabauty. Finally, we use all these results to completely determine the number of irreducible factors of any iterate of f 1/c , for all c with absolute value at most 10 9 .