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We show that in some cases the coherent state can have a larger violation ofthe Leggett-Garg inequality (LGI) than the cat state by numerical calculations.To achieve this result, we consider the LGI of the cavity mode weakly coupledto a zero-temperature environment as a practical instance of the physicalsystem. We assume that the bosonic mode undergoes dissipation because of aninteraction with the environment but is not affected by dephasing. Solving themaster equation exactly, we derive an explicit form of the violation of theinequality for both systems prepared initially in the coherent state$|\alpha\rangle$ and the cat state $(|\alpha\rangle+|-\alpha\rangle)$. For theevaluation of the inequality, we choose the displaced parity operatorscharacterized by a complex number $\beta$. We look for the optimum parameter$\beta$ that lets the upper bound of the inequality be maximum numerically.Contrary to our expectations, the coherent state occasionally exhibits quantumquality more strongly than the cat state for the upper bound of the violationof the LGI in a specific range of three equally spaced measurement times(spacing $\tau$). Moreover, as we let $\tau$ approach zero, the optimizedparameter $\beta$ diverges and the LGI reveals intense singularity.

the european physical journal. d, atomic, molecular and optical physics/the european physical journal. d, atomic, molecular, optical and plasma physics

Violations of the Leggett-Garg inequality for coherent and cat states