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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.Comment: 17 pages, 8 figure