Two-mode bosonic quantum metrology with number fluctuations
Antonella De Pasquale, Paolo Facchi, Giuseppe Florio, Vittorio Giovannetti, Koji Matsuoka, Kazuya Yuasa · Physical Review A · 2015
We search for the optimal quantum pure states of identical bosonic particles for applications in quantum metrology, in particular, in the estimation of a single parameter for the generic two-mode interferometric setup. We consider the general case in which the total number of particles is fluctuating around an average $N$ with variance $\mathrm{\ensuremath{\Delta}}{N}^{2}$. By recasting the problem in the framework of classical probability, we clarify the maximal accuracy attainable and show that it is always larger than the one reachable with a fixed number of particles (i.e., $\mathrm{\ensuremath{\Delta}}N=0)$. In particular, for larger fluctuations, the error in the estimation diminishes proportionally to $1/\mathrm{\ensuremath{\Delta}}N$, below the Heisenberg-like scaling $1/N$. We also clarify the best input state, which is a quasi-NOON state for a generic setup and, for some special cases, a two-mode Schr\"odinger-cat state with a vacuum component. In addition, we search for the best state within the class of pure Gaussian states with a given average $N$, which is revealed to be a product state (with no entanglement) with a squeezed vacuum in one mode and the vacuum in the other.