Quantum Zeno Dynamics of Two Interacting Particles

TL;DR

This work investigates confinement of interacting quantum particles via Quantum Zeno Dynamics by numerically simulating two protons in a 1D region under frequent position measurements. It develops a practical framework using finite-difference methods to solve the two-body problem and Crank-Nicolson time evolution, introducing a leakage measure and Zeno time to predict required measurement frequencies. The authors provide open-source code (2IonQZD) that scales to multi-particle systems and demonstrate QZD confinement across different region sizes, highlighting the trade-offs between measurement rate and leakage. The study advances understanding of measurement-induced confinement in infinite-dimensional Hilbert spaces and offers a computational platform for exploring Zeno-based control in quantum information and precision ion-trapping contexts.

Abstract

According to quantum Zeno dynamics (QZD), the evolution of a quantum system can be restricted to a subspace of its Hilbert space by frequent measurements. A crucial question in QZD of a particle's position is: how short the time interval between successive measurements should be, in order to confine the particle in its initial spatial region? To address this question, we consider a toy model with two ions initially known to be, for simplicity, in a one-dimensional spatial region. By simulating the evolution of this two-body quantum system, we estimate the measurement frequency needed to keep the ions within their initial confined region at a desired confidence level. Two key parameters we employ in our calculations are the Zeno time and the leakage probability of the quantum system. The measurement frequencies are calculated and compared when ions are located initially at different spatial regions. For our simulation, we introduce the Python code {\tt 2IonQZD}.

Figures (6)

• Figure 1: The ground state probability density associated with the wavefunction solution to the Schrödinger equation for the two-proton system with electrostatic interaction in pico meter scale.
• Figure 2: The probability density against the ions' positions is shown for nine time steps $10^{-18}$ s. The wavefunction is mostly collapsed after $10^{-17}$ s.
• Figure 3: The probability density versus the protons' positions is shown for nine time steps, $10^{-21}$ s. The ions are observed with high probability inside the confinement region.
• Figure 4: Shown are the leakage probability for various proton confinement sizes, from $d=10^{-12}$ m, to $d=10^{-7}$ m, as a function of time.
• Figure 5: The leakage probability vs. time is shown for different confinement sizes $d$.