Fisher information of quantum damped harmonic oscillators

We calculate the time-dependent Fisher information in position $\left(F_x\right)$ and momentum $\left(F_p\right)$ for the lowest lying state $(n=0)$ of two classes of quantum damped (Lane–Emden (LE) and Caldirola–Kanai (CK)) harmonic oscillators. The expressions of $F_x$ and $F_p$ are written in terms of $\rho ,$ a c-number quantity satisfying a nonlinear differential equation. Analytical solutions of $\rho$ were obtained. For the LE and CK oscillators, we observe that $F_x$ increases while $F_p$ decreases with increasing time. The product $F_x{F}_p$ increases and tends to a constant value in the limit $t\to \infty$ for the LE oscillator, while it is time-independent for the CK oscillator. Moreover, for the CK oscillator the product $F_x{F}_p$ decreases as the damping $\left(\gamma \right)$ increases. Relations among the Fisher information, Leipnik and Shannon entropies, and the Stam and Cramer–Rao inequalities are given. A discussion on the squeezing phenomenon in position for the oscillators is presented. (paper)