Quantum Fisher information in many-photon states from shift current shot noise

Abstract

Quantum Fisher information (QFI) sets the ultimate precision of optical phase measurements and reveals multiphoton entanglement, but it is not accessible with conventional photodetection. We theoretically predict that a photodetector utilizing the shot noise of the quantum-geometric shift current of exciton polaritons can directly measure the QFI of nonclassical light. By solving the Lindblad equation, we obtain the time-dependent nonlinear photocurrent for an arbitrary initial photon state. It turns out that, regardless of the quantum state of the incident light, the integrated current depends only on the mean photon number. In stark contrast, the shot noise retains the quantum information: its Fano factor is proportional to the photon number variance and therefore encodes the QFI. Numerical calculations confirm these relations for illumination with optical Schrödinger cat and squeezed vacuum states. Quantum correlations in nonclassical light, usually hidden from direct detection, become observable in the form of shift current shot noise

Figures (2)

• Figure 1: Exciton polariton shift current and photon statistics. (A) Upper and lower polariton dispersion branches (solid lines). Dashed lines indicate the bare exciton and photon dispersions ($\omega_{\rm ex} = 1$ sets the energy unit, and $\omega_{\rm ph}=1.1$ at $k=0$). Parameters are $g_1 =0.25i$, a real $g_2$ that sets the current scale, and the damping constant $\Gamma=0.2$. (B),(C) Time evolution of the total particle number and the shift current for an initial single-photon state. Solid and dashed lines show numerically computed and analytical results, which coincide. The time scale is set by $T={2\pi}/{\Delta}$, where $\Delta$ ($\approx0.5$ for these parameters) is the polariton gap. (D) Time-integrated current as a function of $\Gamma$, calculated for an arbitrary initial photon state and normalized by the mean photon number $\langle n_{\rm ph} \rangle_0$. The dashed line shows the analytic single-photon response -- the shift charge $q=|g_2/g_1|$. Solid lines show numerical values that converge with increasing total simulation time $T_{\rm end}$. (E)-(G) Wigner functions in optical phase space $\alpha = X + i P$ for the coherent, Schrödinger cat, and squeezed vacuum states. (H) Fano factor, normalized by $q$, as a function of the initial mean photon number for these states. Solid lines show analytic solutions; markers show numerical data. (I) Corresponding quantum Fisher information density, $f_{\rm Q} = F_{\rm Q}/\left(4\langle n_{\rm ph} \rangle_0\right)$.
• Figure 2: Time dependencies of current-current corelation functions. (a) Analytical result normalized with $F_{\rm Q}$. (b), (c) real and imaginary part of the numerical counterpart. Here $\Gamma=0.1$ and the other model parameters are the same as those in Fig. \ref{['fig:Fano']}. The initial state is the optical cat state with $\alpha =0.5$.