Quantum metrology of two-photon absorption

Two-photon absorption (TPA) is of fundamental importance in super-resolution imaging and spectroscopy. Its nonlinear character allows for the prospect of using quantum resources, such as entanglement, to improve measurement precision or to gain new information on, e.g., ultrafast molecular dynamics. Here, we establish the metrological properties of nonclassical squeezed light sources for precision measurements of TPA cross sections. We find that the Cramér-Rao bound does not provide a fundamental limit for the precision achievable with squeezed states in the limit of very small cross sections. Considering the most relevant measurement strategies—namely, photon-counting and quadrature measurements—we determine the quantum advantage provided by squeezed states as compared to coherent states. We find that squeezed states outperform the precision achievable by coherent states when performing quadrature measurements, which provide improved scaling of the Fisher information with respect to the mean photon number $\sim n^4$. Due to the interplay of the incoherent nature and the nonlinearity of the TPA process, unusual scaling can also be obtained with coherent states, which feature an $\sim n^3$ scaling in both quadrature and photon-counting measurements.

Figure 1

Sketch of the investigated setup: An initial quantum state of light ${\rho }_0$ (here a squeezed state) evolves under two-photon absorption (TPA) into ${\rho }_{\varepsilon }$ before being measured.

Figure 2

Classical Fisher information (CFI) vs the mean photon number for quadrature and photon number measurements with squeezed input states, giving rise to quartic $\propto$ $n^4$ and quadratic $\propto$ $n^2$ scaling, respectively, as well as coherent states, which feature cubic scaling $\propto$ $n^3$. These scalings are obtained in the limit of very weak TPA losses, i.e., $\varepsilon \to 0$.

Figure 3

(a) Negativity of the Wigner function for an initial squeezed state with squeezing parameter $r=1$ or $\langle n\rangle \simeq 1.4$ (blue, solid line) and $r=1.5$ or $\langle n\rangle \simeq 4.5$ (orange, dashed line). (b) Classical Fisher Information (CFI) vs the TPA absorbance $\varepsilon ={\gamma }_{\mathrm{TPA}}t$ for the initial squeezed state with squeezing parameter $r=1$ for measuring the squeezed $q$ quadrature (red, solid line) and the anti-squeezed $p$ quadrature (blue, dashed line).

Figure 4

(a) The dominant scaling exponent, Eq. (13), of the classical Fisher information (11) for quadrature measurements of the squeezed quadrature is plotted vs the average photon number $n$ and the absorbance $\varepsilon$ of a squeezed vacuum state. (b) The same as in panel (a) but for a coherent input state. (c) Classical Fisher information (11) for quadrature measurements is plotted vs the average photon number $n$ and the absorbance $\varepsilon$ of a squeezed vacuum state. (d) The same as in panel (c) but for a coherent input state.

Figure 5

Quantum Fisher Information (QFI) is shown vs the TPA absorbance $\varepsilon ={\gamma }_{\text{TPA}}t$. The QFI of a squeezed vacuum initial state (blue) is compared the QFI of a coherent initial state (gray).

Figure 6

(a) Logarithm of the classical Fisher information (11), ${log}_{10}{\mathcal{F}}_C({\rho }_{\varepsilon },\widehat{q})$, for quadrature measurements of the squeezed quadrature is plotted vs the photon number $n_0$ and the absorbance $\varepsilon$ of a squeezed vacuum state. (b) The same as in panel (a) but for a coherent input state.