Macroscopically distinguishable superposition in infinitely many degrees of freedom

TL;DR

This paper analyzes macroscopically distinguishable superpositions in an infinite array of bosonic sites by contrasting local and nonlocal dynamics. Using nonlocal coherent states (NCS) built on a prime-indexed Fock space, it shows that evolution under a globally quadratic Hamiltonian $\hat{N}^2$ drives NCS into nonlocal cat states, whereas locally quadratic dynamics remain separable across sites. The work formalizes the evolution through Q-functions and extends the framework to generalized bosons, identifying Möbius states as a distinct route to cat-state creation. These results illuminate the conditions under which coherent-state structure can be fused with nonlocal cat-state phenomena and point to potential physical realizations in engineered quantum systems with implications for physics and materials science.

Abstract

We investigate the concept of macroscopically distinguishable superpositions within an infinite array of boson sites. Our approach is rigorous within the frame of Hilbert space theory. In this context, it is natural to differentiate between states -- and corresponding dynamics -- that involve only finitely many degrees of freedom, referred to as local, and those that are inherently nonlocal. Previous studies have shown that such systems can support nonlocal coherent states (NCS). In this work, we demonstrate that NCS can dynamically evolve into nonlocal cat states under the influence of a nonlocal Hamiltonian -- specifically, the square of the total number operator. Crucially, the resulting dynamics cannot be decomposed into local factors. Furthermore, we explore broader mathematical implications of these phenomena within the framework of generalized bosons. Our findings highlight that the concepts of coherent states and nonlocal cat states are not inherently bound together; rather, their fusion is a distinctive feature of standard bosons. Finally, we propose that if the generalized boson framework can be physically realized in engineered quantum systems, the phenomena described here may hold significant relevance for both physics and materials science.

Figures (2)

• Figure 1: An example of $\psi_{p^{-s}z_p} (x_p)$ given by (\ref{['psip']}). Here: $p = 2, s = 1+1i, z_p = 3+8i, \theta_p = \pi/3$. The Gaussian $|\psi_{p^{-s}z_p}|^2$ is centered around $x_p \simeq 5.2178$, and has variance $1/\sqrt{2}$.
• Figure 2: The interference pattern for $P(x)$ based on $\psi$ as in (\ref{['psip']}), $|\alpha| = 5$ and $\eta = .99$. The NCS case driven by the globally quadratic Hamiltonian is completely analogous and results in the same pattern detected via a local observation at any single site $p$ in the infinite array, where $\alpha = p^{-s} z_p$.