Phase Space Displacement, Quadrature Squeezing and Temperature Estimation from the Fock Distribution for 1-Mode Gaussian States

Abstract

We present a method to estimate the amount of displacement, squeezing and temperature of a single-mode harmonic oscillator state based on both the weighted least squares and maximum likelihood estimators applied to measured Fock state populations and fluorescence measurements. Displacement, squeezing and temperature, or equivalently the mean of the quadratures and its variances, are essential state parameters used in quantum computation and various 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 displacement, squeezing, phase and temperature. First, we simulate Fock population measurement experiments. 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), and for high squeezing (r = 2.5) we obtain fidelities greater than 99.9%. We also report confidence intervals and their coverage probabilities, mean squared error and their coverage probabilities. We also simulate fluorescence measurement experiments for different motional states of a trapped ion. Jointly fitting datasets for different relative phases helped to decrease significantly the uncertainty on the estimates. Based on the analysis of real data we proposed a correction to the model that describes the probability of measuring the ion on the spin down state.