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Journal of Modern Optics

New photon-counting formula in thermo field dynamics

Gang Ren, Jian-ming Du, Hai-jun Yu, Hong-Yi Fan

Published Nov 13, 2016• 2 citations
Publisher

Abstract

We show that the probability distribution pm,T of registering m photoelectrons can be explained in Thermo Field Dynamics theory as a matrix element of thermo entangling operator e1-ξa†a~† between ρ=∑mm,m~ρ and m,m~=mm~ where m~ is a fictitious Fock state. Based on this, the new integration formula for calculating pm,T, which involves both Laguerre polynomial and the characteristic function χη of ρ, is obtained. This provides us with convenience for evaluating photoelectric counting probability for various optical fields.

DOI
10.1080/09500340.2016.1257748

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We show that the probability distribution pm,T of registering m photoelectrons can be explained in Thermo Field Dynamics theory as a matrix element of thermo entangling operator e1-ξa†a~† between ρ=∑mm,mρ and m,m=mm~ where m~ is a fictitious Fock state. Based on this, the new integration formula for calculating pm,T, which involves both Laguerre polynomial and the characteristic function χη of ρ, is obtained. This provides us with convenience for evaluating photoelectric counting probability for various optical fields.
We find that in optical fields' diffusion process, the l-photocounting probability formula [Formula: see text] at time t = 0 should be generalized to a new formula [Formula: see text] at time t, where ρ(0) is the initial density operator of the field, Ll is the Laguerre polynomial, : : denotes normal ordering, and ξ is the quantum efficiency of the photocounting detector. This new formula brings much convenience for different ρ0, because for deriving [Formula: see text] there is no need to derive the corresponding ρ(t) in the diffusion channel.
A general formula is derived relating the generating function of the N-fold photoelectron counting distribution to the characteristic functional of the optical field. For a stationary Gaussian radiation field the generating function is expressed as an infinite product involving eigenvalues of a set of integral equations. The results of Bédard, Dialetis, and Cantrell are shown to hold as special cases. Addition of photoelectric counts from correlated Gaussian light is analyzed and the appropriate generating function is given.
We express the exact probability density distribution function as the product of a gamma distribution and a series of associated Laguerre polynomials, with the expansion coefficients determined by moments of the integrated intensity. Orthogonal polynomials with respect to the exact probability distribution function are then expanded in similar fashion. These polynomials are then used to construct an expansion of the joint probability distribution function in the second-order photoelectron statistics. Since the polynomials are identical with the corresponding Laguerre polynomials when the exact probability distribution function is the gamma distribution, the new polynomials are generalized versions of the associated Laguerre polynomials. The joint photoelectron statistics may be studied with these new polynomials.
A theory of electromagnetic field measurement by means of photoionization is developed and applied to photoelectron counting. A probability theory involving multitime joint probability functions for a sequence of photoionizations is formulated. A general quantum-theory definition is proposed for the nonexclusive probability function which occurs in the probability theory. Approximations are then introduced to derive expressions for this probability function which involve correlation functions of the photoionization detector and the electromagnetic field-plus-source. The general theory is used to derive quantities of interest in photo-ionization counting experiments. Expressions are derived for (1) the probability Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) that nnn photo-ionizations are observed in the time interval ttt to t+Tt+Tt+T, and (2) quantities related to Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T), such as its generating function and various moments. Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) is found to be a compound Poisson distribution determined by the density operator of the field when the latter is expressed in Glauber's PPP representation. Using this result, the character of Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) is examined for several specific density operators. These correspond to a coherent state, various fields with the mode phases distributed independently of the mode amplitudes, and a "spread-out" coherent state.