Fisher information of quantum damped harmonic oscillators

Abstract

We calculate the time-dependent Fisher information in position and momentum for the lowest lying state of two classes of quantum damped (Lane–Emden (LE) and Caldirola–Kanai (CK)) harmonic oscillators. The expressions of and are written in terms of a c-number quantity satisfying a nonlinear differential equation. Analytical solutions of were obtained. For the LE and CK oscillators, we observe that increases while decreases with increasing time. The product increases and tends to a constant value in the limit for the LE oscillator, while it is time-independent for the CK oscillator. Moreover, for the CK oscillator the product decreases as the damping 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.