  Non-vanishing fluctuations of the vacuum state are a salient feature of
quantum theory. These fluctuations fundamentally limit the precision of quantum
sensors. Nowadays, several systems such as optical clocks, gravitational wave
detectors, matter-wave interferometers, magnetometers, and optomechanical
systems approach measurement sensitivities where the effect of quantum
fluctuations sets a fundamental limit, the so-called standard quantum limit
(SQL). It has been proposed that the SQL can be overcome by squeezing the
vacuum fluctuations. Realizations of this scheme have been demonstrated in
various systems. However, protocols based on squeezed vacuum crucially rely on
precise control of the relative orientation of the squeezing with respect to
the operation imprinting the measured quantity. Lack of control can lead to an
amplification of noise and reduces the sensitivity of the device. Here, we
experimentally demonstrate a novel quantum metrological paradigm based on phase
insensitive Fock states of the motional state of a trapped ion, with
applications in frequency metrology and displacement detection. The measurement
apparatus is used in two different experimental settings probing non-commuting
observables with sensitivities beyond the SQL. In both measurements, classical
preparation and detection noise are sufficiently small to preserve the quantum
gain in a full metrological protocol.
