Quadrature squeezing and temperature estimation from the Fock distribution

We present a method to estimate the amount of squeezing and temperature of a single-mode Gaussian harmonic oscillator state based on the weighted least-squares estimator applied to measured Fock state populations. Squeezing and temperature, or equivalently the quadrature variances, are essential parameters states used in various quantum communication and sensing protocols. They are often measured with homodyne-style detection, which requires a phase reference such as a local oscillator. Our method allows estimation without a phase reference, by using for example a photon-number-resolving detector. To evaluate the performance of our estimator, we simulated experiments with different values of squeezing and temperature. From 10,000 Fock measurement events, we produced estimates for states whose fidelities to the true state are greater than 99.99% for small squeezing (r<1.0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r<1.0$$\end{document}), and for high squeezing (r=2.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=2.5$$\end{document}), we obtain fidelities greater than 99.9%. We also report confidence intervals and their coverage probabilities.