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arXiv:quant-ph/0506188 · Europhysics Letters (EPL)

Scattering-induced dynamical entanglement and the quantum-classical correspondence

M. Lombardi, A. Matzkin

Published Apr 11, 2006• 15 citations
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Abstract

The generation of entanglement produced by a local potential interaction in a bipartite system is investigated. The degree of entanglement is contrasted with the underlying classical dynamics for a Rydberg molecule (a charged particle colliding on a kicked top). Entanglement is seen to depend on the structure of classical phase-space rather than on the global dynamical regime. As a consequence, regular classical dynamics can in certain circumstances be associated with higher entanglement generation than chaotic dynamics. In addition, quantum effects also come into play: for example, partial revivals, which are expected to persist in the semiclassical limit, affect the long-time behaviour of the reduced linear entropy. These results suggest that entanglement may not be a pertinent universal signature of chaos.

DOI
10.1209/epl/i2006-10028-6

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The generation of entanglement produced by a local potential interaction in a bipartite system is investigated. The degree of entanglement is contrasted with the underlying classical dynamics for a Rydberg molecule (a charged particle colliding on a kicked top). Entanglement is seen to depend on the structure of classical phase-space rather than on the global dynamical regime. As a consequence, regular classical dynamics can in certain circumstances be associated with higher entanglement generation than chaotic dynamics. In addition, quantum effects also come into play: for example, partial revivals, which are expected to persist in the semiclassical limit, affect the long-time behaviour of the reduced linear entropy. These results suggest that entanglement may not be a pertinent universal signature of chaos.
PACS. 03.67.Mn – Entanglement production, characterization, and manipulation. PACS. 05.45.Mt – Quantum chaos; semiclassical methods. PACS. 34.60.+z – Scattering in highly excited states (e.g., Rydberg states). Abstract. – The generation of entanglement produced by a local potential interaction in a bipartite system is investigated. The degree of entanglement is contrasted with the underlying classical dynamics for a Rydberg molecule (a charged particle colliding on a kicked top). En-tanglement is seen to depend on the structure of classical phase-space rather than on the global dynamical regime. As a consequence, regular classical dynamics can in certain circumstances be associated with higher entanglement generation than chaotic dynamics. In addition, quan-tum effects also come into play: for example, partial revivals, which are expected to persist in the semiclassical limit, affect the long-time behaviour of the reduced linear entropy. These results suggest that entanglement may not be a pertinent universal signature of chaos. Entanglement, i.e. the nonseparability intrinsic to composite systems, is one of the most peculiar features of the quantum world. In its most popular form, found for example in the original EPR proposal [1], the entanglement is of geometrical nature. However any usual
A Rydberg molecule is composed of an outer electron that collides on the residual ionic core. Typical states of Rydberg molecules display entanglement between the outer electron and the core. In this work, we quantify the average entanglement of molecular eigenstates and further investigate the time evolution of entanglement production from initially unentangled states. The results are contrasted with the underlying classical dynamics, obtained from the semiclassical limit of the core-electron collision. Our findings indicate that entanglement is not simply correlated with the degree of classical chaos, but rather depends on the specific phase-space features that give rise to inelastic scattering. Hence mixed phase-space or even regular classical dynamics can be associated with high entanglement generation.
The relationship between classically chaotic dynamics and the entanglement properties of the corresponding quantum system is examined in the semiclassical limit. Numerical results are computed for a modified kicked top, keeping the classical dynamics constant while investigating the entanglement for several versions of the corresponding quantum system characterized by a different value of the effective Planck constant ℏ eff. Our findings indicate that as ℏ eff → 0, the apparent signatures of classical chaos in the entanglement properties, such as characteristic oscillations in the time-dependence of the linear entropy, can also be obtained in the regular regime. These results suggest that entanglement is not a universal marker of chaotic dynamics of the corresponding classical system.
We explore the dynamics of entanglement in classically chaotic systems by considering a multiqubit system that behaves collectively as a spin system obeying the dynamics of the quantum kicked top. In the classical limit, the kicked top exhibits both regular and chaotic dynamics depending on the strength of the chaoticity parameter kappa in the Hamiltonian. We show that the entanglement of the multiqubit system, considered for both the bipartite and the pairwise entanglement, yields a signature of quantum chaos. Whereas bipartite entanglement is enhanced in the chaotic region, pairwise entanglement is suppressed. Furthermore, we define a time-averaged entangling power and show that this entangling power changes markedly as kappa moves the system from being predominantly regular to being predominantly chaotic, thus sharply identifying the edge of chaos. When this entangling power is averaged over all states, it yields a signature of global chaos. The qualitative behavior of this global entangling power is similar to that of the classical Lyapunov exponent.
It is widely recognized that entanglement generation and dynamical chaos are intimately related in semiclassical models via the process of decoherence. In this paper, we propose a unifying framework which directly connects the bipartite and multipartite entanglement growth to the quantifiers of classical and quantum chaos. In the semiclassical regime, the dynamics of the von Neumann entanglement entropy, the spin squeezing, the quantum Fisher information, and the out-of-time-order square commutator are governed by the divergence of nearby phase-space trajectories via the local Lyapunov spectrum, as suggested by previous conjectures in the literature. General analytical predictions are confirmed by detailed numerical calculations for two paradigmatic models, relevant in atomic and optical experiments, which exhibit a regular-to-chaotic transition: the quantum kicked top and the Dicke model.
We study the dynamical generation of entanglement as a signature of chaos in a system of periodically kicked coupled tops, where chaos and entanglement arise from the same physical mechanism. The long-time-averaged entanglement as a function of the position of an initially localized wave packet very closely correlates with the classical phase space surface of section--it is nearly uniform in the chaotic sea, and reproduces the detailed structure of the regular islands. The uniform value in the chaotic sea is explained by the random state conjecture. As classically chaotic dynamics take localized distributions in phase space to random distributions, quantized versions take localized coherent states to pseudorandom states in Hilbert space. Such random states are highly entangled, with an average value near that of the maximally entangled state. For a map with global chaos, we derive that value based on analytic results for the entropy of random states. For a mixed phase space, we use the Percival conjecture to identify a "chaotic subspace" of the Hilbert space. The typical entanglement, averaged over the unitarily invariant Haar measure in this subspace, agrees with the long-time-averaged entanglement for initial states in the chaotic sea. In all cases the dynamically generated entanglement is that of a random complex vector, even though the system is time-reversal invariant, and the Floquet operator is a member of the circular orthogonal ensemble.