Large violation of Leggett-Garg inequalities with coherent-state projectors for a harmonic oscillator and chiral scalar field

We investigate violations of Leggett-Garg inequalities (LGIs) for a harmonic oscillator and a $(1+1)$-dimensional chiral scalar field with coherent-state projectors, which is equivalent to a heterodyne-type measurement scheme. For the harmonic oscillator, we found that the vacuum and thermal states violated the LGIs by evaluating the two-time quasiprobability distribution function. In particular, we demonstrate that the value of the two-time quasiprobability reaches $-0.123$ for a squeezed coherent-state projector, which is equivalent to $98\%$ of the Lüders bound corresponding to the maximal violation of the LGIs. We also find a violation of the LGIs for the local mode of a quantum chiral scalar field by constructing a coherent-state projector similar to the harmonic-oscillator case. In contrast with the harmonic oscillator, the periodicity in the time direction of the quasiprobability disappears, which is related to the existence of quantum entanglement between the local mode and its complementary degrees of freedom.