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The Measurement on High-order Photon Number and its Variances

Yuzhen Li

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

A method of measuring expectation values of quadrature moments of a multimode field is introduced through two-levels probe presented by Enrique Solano et al.Similarly,the measurements of expectation values of high-order photon numbers and variances are also discussed.It is pointed out that high-order photon numbers(n~k) and variances are expanded in terms of normal order of creation(a~+) and annihilation(a) operator of night field,and we have derived the relation between high-order photon numbers and variances and probe observables.The results show that information of high-order photon numbers and variances only contain the first or the second derivative,at τ=0,of the measured probe population.Note that knowing derivative at τ=0 only requires knowing the function in a vicinity and the influence of decoherence is minimized.

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A method of measuring expectation values of quadrature moments of a multimode field is introduced through two-levels probe presented by Enrique Solano et al.Similarly,the measurements of expectation values of high-order photon numbers and variances are also discussed.It is pointed out that high-order photon numbers(nk) and variances are expanded in terms of normal order of creation(a+) and annihilation(a) operator of night field,and we have derived the relation between high-order photon numbers and variances and probe observables.The results show that information of high-order photon numbers and variances only contain the first or the second derivative,at τ=0,of the measured probe population.Note that knowing derivative at τ=0 only requires knowing the function in a vicinity and the influence of decoherence is minimized.
We present a method of measuring expectation values of quadrature moments of a multimode field through two-level probe "homodyning". Our approach is based on an integral transform formalism of measurable probe observables, where analytically derived kernels unravel efficiently the required field information at zero interaction time, minimizing decoherence effects. The proposed scheme is suitable for fields that, while inaccessible to a direct measurement, enjoy one and two-photon Jaynes-Cummings interactions with a two-level probe, like spin, phonon, or cavity fields. Available data from previous experiments are used to confirm our predictions.
Two measurable characteristics of microwave one-mode photon states are discussed: a rotated quadrature distribution (tomogram) and normally and antinormally ordered moments of photon creation and annihilation operators. Extraction of these characteristics from an amplified microwave signal is presented. Relations between the tomogram and the moments are found and can be used as a cross-check of experiments. Formalism of the ordered moments is developed. The state purity and generalized uncertainty relations are considered in terms of moments. Unitary and nonunitary time evolution of moments is obtained in the form of a system of linear differential equations in contrast to partial differential equations for quasidistributions. Time evolution is specified for the cases of a harmonic oscillator and a damped harmonic oscillator which describe noiseless and decoherence processes, respectively.
Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.
High-order photon correlations of light fields are important for characterizing the quantum nature. Since Hanbury Brown and Twiss conducted the pioneering experiments in the 1950s, the HBT effect has inspired extensive research on high-order photon correlation in quantum optics, quantum information, and quantum imaging. The Single-photon counting module is one of the most widely used single-photon detectors. Due to its high detection efficiency and low dark counts in the visible and near-infrared region, it is reasonably chosen for basic research on quantum mechanics. Many researches have demonstrated that the maximum value of second-order photon correlation g(2)(τ) at zero delay (τ = 0) can be used to distinguish different light fields. Therefore, the HBT scheme containing two single photon detectors have been widely used in many advanced studies, such as space interference, ghost imaging, single photon detection with high efficiency, etc. However, higher-order photon correlations g(n) (n > 2) can reveal more measurable characteristics of light fields, such as information about the non-Gaussian scattering process, the skewness and kurtosis of photon number distribution, etc. When the extended HBT scheme is used to measure higher-order photon correlations, the experimental conditions including quantum efficiency and background noise greatly affect the photon correlation measurement. The influences of the counting rate and resolution time of the detection system on the measurements are also very important and cannot be ignored. Therefore, the comprehensive considering of various influence factors is necessary for accurately measuring the high-order photon correlations and also a challenge. In this paper, we present a method based on double Hanbury Brown-Twiss scheme for the accurate measuring of high-order photon correlations g(n) (n > 2). The system consists of four single photon counting modules and is used to detect and analyze the joint distribution probability of temporal photon correlation. Considering the effects of the background noise and overall efficiency, theoretically, we analyze the correlations of the third- and fourth-order photon with the incident light intensity, squeezing parameter and photon number respectively for thermal state, coherent state, squeezed vacuum state, and Fock state. Meanwhile, experimentally we study the influences of resolution time and counting rate on correlations of the coherent state and thermal state with third- and fourth-order photon. On condition that the resolution time is 210 ns and the counting rate is 80 kc/s, the correlations of third and fourth-order photon with the thermal state at zero time delay are accurately measured, and the relative statistical deviations of the measured vales from the theoretical values are 0.3% and 0.8%, respectively. In addition, the third- and fourth-order photon correlations of the thermal state at different delay times are also observed. It is demonstrated that the high-order photon correlations of light fields are measured accurately by comprehensively analyzing various influencing factors. This technique provides a promising and useful tool to investigate quantum correlated imaging and quantum coherence of light fields.
A new way to expand an arbitrary boson-field operator into ordered products of creation and annihilation operators is introduced: alternate ordering. A continuous order parameter s characterizes the order and takes the value s=+1 for normal-alternate order, s=0 for symmetric-alternate order, and s=-1 for antinormal-alternate order. All of the alternate moments of a number of well-known states of light are calculated. The expansion of density matrices and operators in terms of alternate products is briefly discussed as well as the relation with the concepts of phase and higher-order phase dependence. Finally, the relevance of the new ordering scheme in the study of photon statistics is outlined and illustrated in the derivation of moment equations for phase-sensitive light amplifiers and absorbers.