Fisher Information Measures under Lattice Combined Paul Trap

TL;DR

This paper investigates how optical lattice modifications in a Paul trap affect information measures like Fisher information and Shannon entropy, revealing invariance properties and the effects of non-harmonic corrections.

Contribution

It demonstrates the invariance of Fisher-Shannon complexity under effective frequency control and analyzes the impact of lattice corrections on motional states.

Findings

Fisher information and Shannon entropy track the effective frequency of the potential.

Fisher-Shannon complexity remains invariant under optical modulation of the lattice.

Non-harmonic corrections break the invariance, especially in excited states.

Abstract

We examine how the informational properties of a confined single ion response in a Paul trap modified by optical-lattice. We focus on the ground and first excited motional states and show that Fisher information, Shannon entropy, and Fisher-Shannon complexity track the effective frequency $\omega_{\mathrm{eff}}=\omega\sqrt{1-\kappa}$ of the potential. We show that the Fisher information and Shannon entropy reflect an effective frequency-driven redistribution of information between conjugate spaces. Our results show that the Fisher-Shannon complexity measure remains invariant under effective frequency control. The invariance demonstrates that optical modulation of $\kappa$ rescales localization, without altering the harmonic structure of the motional states. These results establish a controlled information-theoretic baseline for lattice-assisted Paul traps. Beyond the harmonic limit, retaining the quartic lattice correction introduces non-Gaussian wavefunction features through state-dependent mixing of higher eigenstates, which breaks the mutual compensation between Fisher information and Shannon entropy that sustains the invariant. The departure of $P'$ from its harmonic reference value intensifies with $\kappa$ and is stronger for the excited state, which confirms that the Fisher-Shannon complexity invariance is a distinctive property of the small-oscillation harmonic regime.