  In quantum metrology, entanglement represents a valuable resource that can be
used to overcome the Standard Quantum Limit (SQL) that bounds the precision of
sensors that operate with independent particles. Measurements beyond the SQL
are typically enabled by relatively simple entangled states (squeezed states
with Gaussian probability distributions), where quantum noise is redistributed
between different quadratures. However, due to both fundamental limitations and
the finite measurement resolution achieved in practice, sensors based on
squeezed states typically operate far from the true fundamental limit of
quantum metrology, the Heisenberg Limit. Here, by implementing an effective
time-reversal protocol through a controlled sign change in an optically
engineered many-body Hamiltonian, we demonstrate atomic-sensor performance with
non-Gaussian states beyond the limitations of spin squeezing, and without the
requirement of extreme measurement resolution. Using a system of 350 neutral
$^{171}$Yb atoms, this signal amplification through time-reversed interaction
(SATIN) protocol achieves the largest sensitivity improvement beyond the SQL
($11.8 \pm 0.5$~dB) demonstrated in any interferometer to date. Furthermore, we
demonstrate a precision improving in proportion to the particle number
(Heisenberg scaling), at fixed distance of 12.6~dB from the Heisenberg Limit.
These results pave the way for quantum metrology using complex entangled
states, with potential broad impact in science and technology. Potential
applications include searches for dark matter and for physics beyond the
standard model, tests of the fundamental laws of physics, timekeeping, and
geodesy.
