  Non-Hermitian two-site ``dimers'' serve as minimal models in which to explore
the interplay of gain and loss in dynamical systems. In this paper, we
experimentally and theoretically investigate the dynamics of non-Hermitian
dimer models with non-reciprocal hoppings between the two sites. We investigate
two types of non-Hermitian couplings; one is when asymmetric hoppings are
externally introduced, and the other is when the non-reciprocal hoppings depend
on the population imbalance between the two sites, thus introducing the
non-Hermiticity in a dynamical manner. We engineer the models in our synthetic
mechanical set-up comprised of two classical harmonic oscillators coupled by
measurement-based feedback. For fixed non-reciprocal hoppings, we observe that,
when the strength of these hoppings is increased, there is an expected
transition from a $\mathcal{PT}$-symmetric regime, where oscillations in the
population are stable and bounded, to a $\mathcal{PT}$-broken regime, where the
oscillations are unstable and the population grows/decays exponentially.
However, when the non-Hermiticity is dynamically introduced, we also find a
third intermediate regime in which these two behaviors coexist, meaning that we
can tune from stable to unstable population dynamics by simply changing the
initial phase difference between the two sites. As we explain, this behavior
can be understood by theoretically exploring the emergent fixed points of a
related dimer model in which the non-reciprocal hoppings depends on the
normalized population imbalance. Our study opens the way for the future
exploration of non-Hermitian dynamics and exotic lattice models in synthetic
mechanical networks.
