  The Heisenberg uncertainty principle sets a lower bound on the sensitivity of
continuous optical measurements of force. This bound, the standard quantum
limit, can only be reached when a mechanical oscillator subjected to the force
is unperturbed by its environment, and when measurement imprecision from photon
shot-noise is balanced against disturbance from measurement backaction. We
apply an external force to the center-of-mass motion of an ultracold atom cloud
in a high-finesse optical cavity. The optomechanically transduced response
clearly demonstrates the trade-off between measurement imprecision and
back-action noise. We achieve a sensitivity that is consistent with theoretical
predictions for the quantum limit given the atoms' slight residual thermal
disturbance and the photodetection quantum efficiency, and is a factor of 4
above the absolute standard quantum limit.
