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

Abstract

Macrorealism (MR) is the world-view that at all moments of time, a system is definitely in one of the observable states available to it, irrespective of past or future measurements. The Leggett-Garg (LG) inequalities were introduced to test MR and therefore assess for the presence of macroscopic quantum coherence. Since such effects could plausibly be found in various types of macroscopic oscillators, we consider the application of the LG approach to the one-dimensional quantum harmonic oscillator and 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 then 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 two and threetime LG inequalities can each be independently violated or satisfied. Similar results are found for four-time LG inequalities. We find that the violations persist (although are reduced) when the sign function defining Q is smeared to reflect experimental imprecision. We also find that LG violations diminish for higher energy eigenstates, 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, a simple example of a feature recently noted by Bose et al. [Phys. Rev. Lett. 120, 210402 (2018)]. We also show that two-time LG violations in a gaussian state are also readily found if the dichotomic variable at one of the times is taken to be the parity operator. To demonstrate the versatility of the technique we developed, we also calculate temporal correlators for the Morse potential, and present a brief analysis of the LG inequalities in this system, where we find significant violations for the first excited state.