Mathematical Optics

12.5.2  LINEAR ATTENUATION PROCESS

The time-evolution generator $\hat{\mathcal{K}}$ provides the linear attenuation process [16] when the inequality ϕ₋₊ > ϕ₊₋ ≥ 0 is satisfied. In this case, it is convenient to introduce parameters κ and $\overline{n}$ by

$\text{κ}={\text{ϕ}}_{-+}-{\text{ϕ}}_{+-},\overline{n}={\text{ϕ}}_{+-}/({\text{ϕ}}_{-+}-{\text{ϕ}}_{+-}),$

(12.225)

where κ > 0 and $\overline{n}\ge 0$ are satisfied. We first consider the normal-order diagonalization of the time-evolution generator $\hat{\mathcal{K}}$ [36]. The linear transformation between the operators $(a,a^{†},\tilde{a},{\tilde{a}}^{†})\mathrm{a}\mathrm{n}\mathrm{d}(b,b^{‡},\tilde{b},{\tilde{b}}^{‡})$ is given by

$\left\{\begin{array}{l}a=b+\overline{n}{b}^{‡}\\ a^{†}=\tilde{b}+(\overline{n}+1)b^{‡}\\ \tilde{a}=\tilde{b}+\overline{n}{b}^{‡}\\ {\tilde{a}}^{†}=b+(\overline{n}+1){\tilde{b}}^{‡}\end{array}\right.\left\{\begin{array}{l}b=(\overline{n}+1)a-\overline{n}{\tilde{a}}^{†}\\ b^{‡}=a^{†}-\tilde{a}\\ \tilde{b}=(\overline{n}+1)\tilde{a}-a^{†}\\ {\tilde{b}}^{‡}={\tilde{a}}^{†}-a.\end{array}\right.$

(12.226)

Then the time-evolution generator $\hat{\mathcal{K}}$ becomes

$\hat{\mathcal{K}}=-\text{κ}(b^{‡}b+{\tilde{b}}^{†}\tilde{b}).$

(12.227)

Here we introduce a Fock-like state |m, n) by

$|m,n)=\frac{1}{\sqrt{m!n!}}{b}^{‡m}{\tilde{b}}^{‡n}|0,0)(m,n=0,1,$