  According to the Heisenberg uncertainty principle, the energy or frequency
uncertainty of a measurement can be at the best inversely proportional to the
observation time ($T$). The observation time in an experiment using a quantum
mechanical probe is ultimately limited by the coherence time of the probe.
Therefore the inverse proportionality of the statistical uncertainty of a
frequency measurement to the observation time is also limited up to the
coherence time of the probe, provided the systematic uncertainties are well
below the statistical uncertainties. With a single laser-cooled barium ion as a
quantum probe, we show that the uncertainty in the frequency measurement for a
general time-dependent Hamiltonian scales as $1/T^{1.75\pm 0.03}$ as opposed to
$1/T$, given by the Heisenberg limit for time-independent Hamiltonian. These
measurements, based on controlled feedback Hamiltonian and implemented on a
laser cooled single ion, allowed precise measurement of noise frequency in the
kHz range. Moreover, based on the observed sensitivity of a single ion
experiment presented here, we propose the use of a similar protocol with
enhanced sensitivity as a tool to directly verify the existence of certain
types of light mass axion-like dark matter particles where no direct
measurement protocol exists.
