Leggett-Garg violations for continuous variable systems with gaussian states

TL;DR

This paper investigates violations of Leggett-Garg inequalities in continuous variable quantum systems, specifically Gaussian states in a harmonic oscillator, revealing regimes of macrorealism violation through analytic correlator expressions and phase space analysis.

Contribution

It provides the first analytic expressions for LG correlators in Gaussian states and explores LG violations across various quantum states and measurement regimes.

Findings

Significant LG violations occur in Gaussian states under certain parameters.

Analytic formulas for two-time correlators are derived.

Phase space analysis elucidates the physical origin of violations.

Abstract

Macrorealism (MR) is the world view that certain quantities may take definite values at all times irrespective of past or future measurements and may be experimentally falsified via the Leggett-Garg (LG) inequalities. We put this world view to the test for systems described by a continuous variable $x$ by seeking LG violations for measurements of a dichotomic variable $Q = \textrm{sign}(x)$, in the case of gaussian initial states in a quantum harmonic oscillator. Extending our earlier analysis [C. Mawby and J. J. Halliwell, Phys. Rev. A 105, 022221 (2022)] we find analytic expressions for the temporal correlators. An exploration of parameter space reveals significant regimes in which the two-time LG inequalities are violated, and likewise at three and four times. To obtain a physical picture of the LG violations, we exploit the continuous nature of the underlying position variable and analyse the relevant quantum-mechanical currents, Bohm trajectories, and Wigner function. Further, we extend the analysis LG tests using coherent state projectors, thermal coherent states, and squeezed states.