Leggett-Garg tests for macrorealism in the quantum harmonic oscillator and more general bound systems

TL;DR

This paper applies Leggett-Garg inequalities to quantum harmonic oscillators and other bound systems, demonstrating conditions under which quantum coherence can be experimentally tested and observed, including in classical-like states.

Contribution

It introduces a simple method to compute LG correlations for bound systems and explores LG violations across various states and potentials, including harmonic and Morse.

Findings

LG violations occur in low-energy states and persist under experimental imprecision.

Violations diminish in higher energy eigenstates, indicating classicalization.

LG violations can be observed with different coarse-grained definitions of the dichotomic variable.

Abstract

The Leggett-Garg (LG) inequalities were introduced to test for the possible presence of macroscopic quantum coherence. Since such effects may be found in various types of macroscopic oscillators, we consider the application of the LG approach to the one-dimensional quantum harmonic oscillator and also to more general bound systems, using a single dichotomic variable $Q$ given by the sign of the oscillator position. We present a simple method to calculate the temporal correlation functions appearing in the LG inequalities for any bound system for which the eigenspectrum is (exactly or numerically) known. We apply this result to the quantum harmonic oscillator for a variety of experimentally accessible states, namely energy eigenstates, and superpositions thereof. For the subspace of states spanned by only the ground state and first excited state, we readily find substantial regions of parameter space in which the LG inequalities at two, three and four times can each be independently violated or satisfied. We also find that the violations persist, although are reduced, when the sign function defining $Q$ is smeared to reflect experimental imprecision. For higher energy eigenstates, we find that LG violations diminish, showing the expected classicalization. With a $Q$ defined using a more general type of position coarse graining, we find two-time LG violations even in the ground state. We also show that two-time LG violations in a gaussian state are readily found if the dichotomic variable at one of the times is taken to be the parity operator. To demonstrate the versatility of the approach, we repeat much of the LG analysis for the Morse potential, finding qualitatively similar physical results.