  We study a system of $N$ noninteracting particles on a line in the presence
of a harmonic trap $U(x)=\mu \bigl[x-z(t)\bigr]^2/2$, where the trap center
$z(t)$ undergoes a bounded stochastic modulation. We show that this stochastic
modulation drives the system into a nonequilibrium stationary state, where the
joint distribution of the positions of the particles is not factorizable. This
indicates strong correlations between the positions of the particles that are
not inbuilt, but rather get generated by the dynamics itself. Moreover, we show
that the stationary joint distribution can be fully characterized and has a
special conditionally independent and identically distributed (CIID) structure.
This special structure allows us to compute several observables analytically
even in such a strongly correlated system, for an arbitrary bounded drive
$z(t)$. These observables include the average density profile, the correlations
between particle positions, the order and gap statistics, as well as the full
counting statistics. We then apply our general results to two specific examples
where (i) $z(t)$ represents a dichotomous telegraphic noise, and (ii) $z(t)$
represents an Ornstein-Uhlenbeck process. Our analytical predictions are
verified in numerical simulations, finding excellent agreement.
