Violation of the two-time Leggett-Garg inequalities for a harmonic oscillator

We investigate the violation of the Leggett-Garg inequalities for a harmonic oscillator in various quantum states. We focus on the two-time quasi-probability distribution function with a dichotomic variable constructed with the position operator of a harmonic oscillator. First, we developed a new formula to compute the two-time quasi-probability distribution function, whose validity is demonstrated in comparison with the formula developed in the recent paper by Mawby and Halliwell[Phys.Rev.A, 107 032216 (2023)]. Second, we demonstrated the variety of the violation of the two-time Leggett-Garg inequalities assuming various quantum states of a harmonic oscillator including the squeezed coherent state and the thermal squeezed coherent state. Third, we demonstrated that a certain type of extension of the dichotomic variable and the corresponding projection operator can boost violation of the Leggett-Garg inequalities for the ground state and the squeezed state. We also discuss when the Leggett-Garg inequalities are violated in an intuitive manner.