Quantum sensing with a driven oscillator that returns to the vacuum

A weak force displaces a quantum oscillator in phase space, and the quantum Fisher information sets how precisely that displacement can be measured. The usual way to raise it is to squeeze the probe, but squeezing is fragile, has to be remade each cycle, and fades as the oscillator loses energy. We propose instead a quadratic Hamiltonian, varying in time, whose propagator traces a closed loop in the symplectic group $$\textrm{Sp}(2,\mathbb {R})$$ , the probe passes through strong parametric gain and returns to the vacuum with no squeezing left. The gain enlarges the response to the signal instead of lowering the noise, and because the probe ends where it began, cycles follow one another with nothing to prepare in between, one quadrature being read at the end by ordinary homodyne detection. One smooth function carries the whole propagator, thus a loop is designed by choosing it and reading off the drive it implies, and the Fisher information is two integrals of the signal against the propagator. We give a family of such loops in closed form and three upper bounds, the sharpest reached exactly by the signal matched to the loop. When the oscillator also loses photons to a bath at rate $$\kappa $$ , the Fisher information $$\mathcal {F}$$ over a cycle from $$t_{0}$$ to T obeys $$\mathcal {F} \le 4\textrm{e}^{\kappa (T-t_{0})}\Vert \tilde{s}\Vert ^{2}[(T-t_{0}) + \eta /\kappa ]$$ , where $$\Vert \tilde{s}\Vert ^{2}$$ is the signal energy and $$\eta $$ the noise the bath injects while the gain is on. The inequality has the shape of a second law: the first term is what a loop that merely rotates already achieves, the second is the price of the gain.