  Open physical systems with balanced loss and gain, described by non-Hermitian
parity-time ($\mathcal{PT}$) reflection symmetric Hamiltonians, exhibit a
transition which could engenders modes that exponentially decay or grow with
time and thus spontaneously breaks the $\mathcal{PT}$-symmetry. Such
$\mathcal{PT}$-symmetry breaking transitions have attracted many interests
because of their extraordinary behaviors and functionalities absent in closed
systems. Here we report on the observation of $\mathcal{PT}$-symmetry breaking
transitions by engineering time-periodic dissipation and coupling, which are
realized through state-dependent atom loss in an optical dipole trap of
ultracold $^6$Li atoms. Comparing with a single transition appearing for static
dissipation, the time-periodic counterpart undergoes $\mathcal{PT}$-symmetry
breaking and restoring transitions at vanishingly small dissipation strength in
both single and multiphoton transition domains, revealing rich phase structures
associated to a Floquet open system. The results enable ultracold atoms to be a
versatile tool for studying $\mathcal{PT}$-symmetric quantum systems.
