  A bound state between a quantum emitter (QE) and surface plasmon polaritons
(SPPs) can be formed, where the QE is partially stabilized in its excited
state. We put forward a general approach for calculating the energy level shift
at a negative frequency $\omega$, which is just the negative of the nonresonant
part for the energy level shift at positive frequency $-\omega$. We also
propose an efficient formalism for obtaining the long-time value of the
excited-state population without calculating the eigenfrequency of the bound
state or performing a time evolution of the system, in which the probability
amplitude for the excited state in the steady limit is equal to one minus the
integral of the evolution spectrum over the positive frequency range. With the
above two quantities obtained, we show that the non-Markovian decay dynamics in
the presence of a bound state can be obtained by the method based on the
Green's function expression for the evolution operator. A general criterion for
identifying the existence of a bound state is presented. These are numerically
demonstrated for a QE located around a nanosphere and in a gap plasmonic
nanocavity. These findings are instructive in the fields of coherent
light-matter interactions.
