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Modern Physics Letters B

Evolution law of photocounting probability in a diffusion channel

Hong-Yi Fan, Fang Jia, Pengfei Zhang, Rui He

Published Jul 9, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217984914501450

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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.
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.
We examine how a Laguerre-polynomial-weighted chaotic photon field (LPWCPF), whose density operator is [Formula: see text], evolves in an amplitude-damping channel. By using a newly derived generating function of two-variable Hermite polynomials we obtain the evolution law of LPWCPF, which turns out to be a new LPWCPF with a new parameter, depending on 1 − e−2κt, where κ represents decay rate. The technique of integration (summation) within an ordered product of operators is used in our discussions.
We propose a new photon field in quantum field theory, named Laguerre-polynomial-weighted chaotic field. This field will emerge when an initial photon-added chaotic field, which is represented by its density operator [Formula: see text], dissipates in an amplitude damping channel described by time evolution equation [Formula: see text], where κ is a damping coefficient, that is, the initial field will evolve into [Formula: see text] with [Formula: see text], where Ls is the s-order Laguerre-polynomial, and : : denotes normal ordering. We employ the method of summation within an ordered product of operators to obtain our result.