Full monitoring of ensemble trajectories with 10 dB-sub-Heisenberg imprecision

Abstract The change of a quantum state can generally only be fully monitored through simultaneous measurements of two non-commuting observables $$\hat{X}$$ X ̂ and $$\hat{Y}$$ Y ̂ spanning a phase space. A measurement device that is coupled to the thermal environment provides at a time a pair of values that have a minimal uncertainty product set by the Heisenberg uncertainty relation, which limits the precision of the monitoring. Here, we report on an optical ensemble measurement setup that is able to monitor the time-dependent change of the quantum state’s displacement in phase space ( $$\langle \hat{X}(t)\rangle ;\langle \hat{Y}(t)\rangle$$ ⟨ X ̂ ( t ) ⟩ ; ⟨ Y ̂ ( t ) ⟩ ) with an imprecision 10 dB below the Heisenberg uncertainty limit. Our setup provides pairs of values (X(t i ); Y(t i )) from simultaneous measurements at subsequent times t i . The measurement references are not coupled to the thermal environment but are established by an entangled quantum state. Our achievement of a tenfold reduced quantum imprecision in monitoring arbitrary time-dependent displacements supports the potential of the quantum technology required for entanglement-enhanced metrology and sensing as well as measurement-based quantum computing.