We investigate the violation of the Leggett-Garg inequalities for a harmonic oscillator in various quantum states and with various choices of a projection operator for a dichotomic variable. We focus on the two-time quasiprobability distribution function with a dichotomic variable constructed with the position or momentum operator of a harmonic oscillator. Our results are the generalization of the previous work by Mawby and Halliwell [Phys. Rev. A 107, 032216 (2023)], but the new points are the following. We first obtain the explicit expression for the two-time quasiprobability distribution function for the (thermal) squeezed coherent state. Second, we find the two-time quasiprobability distribution function with the dichotomic variable and the projection operator constructed in terms of the momentum operator. Third, we demonstrate that the violation of the Leggett-Garg inequalities can be boosted by adopting the dichotomic variable and the projection operator defined by a finite range of the position/momentum, in which a larger violation appears for the ground state and the squeezed state. We give an intuitive interpretation when the violation of the Leggett-Garg inequalities appears. We also present a mathematical formula to compute the quasiprobability distribution function by using the integral representation of the Heaviside function, which is useful to generalize it to a quantum field theory.
The quantum-mechanical double harmonic oscillator is proposed as a model for the generation of Schr\"odinger-cat states. To demonstrate this the quantum-statistical properties of the double oscillator in its ground state are examined. The nonclassical nature of this state and interference effects are analyzed. The squeezing properties of the double harmonic oscillator are calculated and compared with those of the ``idealized'' cat state. The behavior of the quasiprobability distribution function is also discussed.
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 $\ensuremath{-}0.123$ for a squeezed coherent-state projector, which is equivalent to
98 of the L\"uders 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.