Precision measurements with photon-subtracted or photon-added Gaussian states

Photon-subtracted and photon-added Gaussian states are amongst the simplest non-Gaussian states that are experimentally available. It is generally believed that they are some of the best candidates to enhance sensitivity in parameter extraction. We derive here the quantum Cramér-Rao bound for such states and find that for large photon numbers photon subtraction or addition only leads to a small correction of the quantum Fisher information (QFI). On the other hand, a divergence of the QFI appears for very small squeezing in the limit of vanishing photon number in the case of photon subtraction, implying an arbitrarily precise measurement with almost no light. However, at least for the standard and experimentally established preparation scheme, the decreasing success probability of the preparation in that limit exactly cancels the divergence, leading to finite sensitivity per square root of Hz, when the duration of the preparation is taken into account.

Figure 1

Wigner function of photon-subtracted state for ${\Gamma }_{\theta }=1.1$ and ${\overline{x}}_{\theta }=1/20$.

Figure 2

Fisher information $I(\widehat{\rho }\left(\theta \right))$ as function of $\overline{x}\equiv {\overline{x}}_{\theta }$ and $\Gamma \equiv {\Gamma }_{\theta }$ for ${\Gamma }_{\theta }^{\prime }={\overline{p}}_{\theta }=0$ in units of its value $2\pi$ for a coherent state (${\Gamma }_{\theta }=1$) with the same $\overline{x}$. The plot is cut at $I(\widehat{\rho }\left(\theta \right))=200.$

Figure 3

Success probability ($p_1\equiv P_1^{\mathrm{out}}$) for preparation of the single-photon-subtracted squeezed coherent state for ${\overline{x}}_{\theta }=0$ as a function of squeeze parameter $r$ and beam splitter mixing angle $\delta$.