  We demonstrate that two-time correlation functions, which are generalizations
of out-of-time-ordered correlators (OTOCs), can show 'false-flags' of chaos by
exhibiting behaviour predicted by random matrix theory even in a system with
classically regular dynamics. In particular, we analyze a system of bosons
trapped in a double-well potential and probed by a quantum dot which is coupled
to the bosons dispersively. This is an integrable system (considered both as
separate parts and in total). Despite the continuous time evolution generated
by the actual Hamiltonian, we find that the n-fold two-time correlation
function for the probe describes an effective stroboscopic or Floquet dynamics
whereby the bosons appear to be alternately driven by two different
non-commuting Hamiltonians in a manner reminiscent of the Trotterized time
evolution that occurs in digital quantum simulation. The classical limit of
this effective dynamics can have a nonzero Lyapunov exponent, while the
effective level statistics and return probability show traditional signatures
of chaotic behaviour. In line with several other recent studies, this work
highlights the fact that the behavior of OTOCs and their generalizations must
be interpreted with some care.
