Quantum Fisher Information Measure in a Strongly Confined Harmonic Paul Trap Lattice System

In this work, we examine how the informational and structural properties of a single ion respond to controlled changes of the effective potential in a Paul trap modified by an optical lattice. We consider the ground state of the system where confinement is strongest. And by treating the trap frequency $ω$ and lattice $κ$ as independent tunning parameters, we show that Fisher information, Shannon entropy, and Fisher-Shannon complexity track the curvature of the effective potential $ω_{\mathrm{eff}}=ω^2,\sqrt{1-κ}$. The $ω$ and $κ$ sweeps confirm that curvature and not the choice of control parameter determines the behaviour of the system. This gives the trapped-ion platform a clear advantage that the curvature can be engineered without altering the harmonic characteristics of the system. The interplay between $ω$ and $κ$ thus provides a practical route for precision quantum control and offers Information-theoretic framework for experiments that probe confinement, quantization scale, and information flow in engineered ion traps.

💡 Research Summary

This paper presents a theoretical investigation into the informational properties of a single ion confined in a hybrid potential created by superimposing an optical lattice onto a standard harmonic Paul trap. The central objective is to understand how measures from information theory—specifically, the Fisher information, Shannon entropy, and Fisher-Shannon complexity—respond to controlled variations of the trapping potential’s curvature.

The system is modeled by a one-dimensional potential V(x) combining a harmonic term (from the Paul trap with frequency ω) and a periodic sinusoidal term (from the optical lattice with strength parameter κ). Under the condition of strong confinement, where the ion’s motion is restricted to a region much smaller than the lattice period, the potential is approximated to second order around its minimum. This yields an effective harmonic potential characterized by a single parameter: the effective curvature ω_eff = ω√(1-κ). This ω_eff becomes the key physical quantity governing the system’s dynamics. The ground-state wavefunctions in position and momentum space are derived analytically from this effective harmonic oscillator model.

The core analysis involves calculating exact analytical expressions for the position- and momentum-space Fisher information (I_x, I_p) and Shannon entropies (S_x, S_p) for the ground state. The Fisher information quantifies the local sharpness or gradient content of the probability distribution, while the Shannon entropy measures its global spreading. The product of exponential Shannon entropy and Fisher information defines the Fisher-Shannon complexity (P_x, P_p), a statistical measure that captures the interplay between order and disorder.

The principal finding is that all these information-theoretic measures are solely governed by the effective curvature ω_eff, not by the individual control parameters ω or κ. Sweeping ω (trap frequency) or κ (lattice depth) independently, as long as they result in the same ω_eff, produces identical informational properties. Specifically, increasing ω_eff (tightening the confinement) leads to: an increase in I_x (position distribution becomes sharper), a decrease in S_x (position distribution becomes more localized), a decrease in I_p, an increase in S_p, and a significant increase in the momentum-space complexity P_p. The opposite trends occur when ω_eff is decreased (softening the confinement).

The study demonstrates that the trapped-ion platform offers a unique advantage: the fundamental curvature of the confining potential, which dictates the system’s informational structure, can be engineered without altering its harmonic character. This provides experimentalists with multiple, equivalent control knobs (ω and κ) to tailor quantum states. The work establishes an information-theoretic framework that connects fundamental physical parameters to measurable informational quantities, offering new tools for precision quantum control, probing confinement effects, and studying information flow in engineered quantum systems like ion-based quantum heat engines or nanofriction simulators.