  We find a different mechanism for suppression of decay in an open
one-dimensional lattice system, which originates from a dark Floquet state, a
sink state to which the system is asymptotically driven, whose overall
probability is determined only by the parameters of the periodic driving field.
The zero-quasienergy of dark Floquet state has been shown to be not a real
zero, but a vanishingly small negative imaginary number which will cause
undesirable physical effect in long-time evolution of quantum states, which is
extremely different from the conservative counterpart. Another important
finding is that the value of the system's effective decay, determined by the
size of the non-zero imaginary part of the dark-Floquet-state-related
quasienergy, depends not on how many localized lossy sites there are but on
which one of the lossy sites is nearest to the driven site. Thus, for specially
designed local dissipation, by controlling the driving parameters, it is
possible for us to drive the system to a dark Floquet state with a much lower
level of overall probability loss as compared to the undriven case and with
good stability over enough longer evolution time. These results are applicable
to the multisite lattice system with an odd number of sites and may be
significant for long-time control of decay in a vast family of multistate
physical systems with localized dissipation.
