  The canonical ensemble plays a crucial role in statistical mechanics in and
out of equilibrium. For example, the standard derivation of the fluctuation
theorem relies on the assumption that the initial state of the heat bath is the
canonical ensemble. On the other hand, the recent progress in the foundation of
statistical mechanics has revealed that a thermal equilibrium state is not
necessarily described by the canonical ensemble but can be a quantum pure state
or even a single energy eigenstate, as formulated by the eigenstate
thermalization hypothesis (ETH). Then, a question raised is how these two
pictures, the canonical ensemble and a single energy eigenstate as a thermal
equilibrium state, are compatible in the fluctuation theorem. In this paper, we
theoretically and numerically show that the fluctuation theorem holds in both
of the long and short-time regimes, even when the initial state of the bath is
a single energy eigenstate of a many-body system. Our proof of the fluctuation
theorem in the long-time regime is based on the ETH, while it was previously
shown in the short-time regime on the basis of the Lieb-Robinson bound and the
ETH [Phys. Rev. Lett. 119, 100601 (2017)]. The proofs for these time regimes
are theoretically independent and complementary, implying the fluctuation
theorem in the entire time domain. We also perform a systematic numerical
simulation of hard-core bosons by exact diagonalization and verify the
fluctuation theorem in both of the time regimes by focusing on the finite-size
scaling. Our results contribute to the understanding of the mechanism that the
fluctuation theorem emerges from unitary dynamics of quantum many-body systems,
and can be tested by experiments with, e.g., ultracold atoms.
