  In the present paper, we investigate the Leggett-Garg inequality of the
cavity mode weakly coupled to a zero-temperature environment. We assume that
the boson system undergoes dissipation because of an interaction with the
environment but is not affected by dephasing. Solving the master equation
exactly, we derive an explicit form of violation of the inequality for both
cases where systems are prepared initially in the coherent state
$|\alpha\rangle$ and the cat state $(|\alpha\rangle+|-\alpha\rangle)$. We
choose the displaced parity operators for observing the cavity mode at three
equally spaced measurement times (spacing $\tau$) in the derivation of the
inequality. These operators are characterized by a complex number $\beta$. We
look for the optimum parameter $\beta$ that lets the upper bound of the
inequality be maximum by numerical calculations. We compare the maximized upper
bounds for both the initial states, the coherent and cat states. Contrary to
our expectations, the coherent state occasionally exhibits quantum quality more
strongly than the cat state for the upper bound of the Leggett-Garg inequality
if its time difference $\tau$ is in a specified range. Moreover, as we let the
time difference $\tau$ approach zero, the optimized parameter $\beta$ diverges,
that is $|\beta|\to\infty$, and the Leggett-Garg inequality reveals intense
singularity.
