Nonunitary connection between explicitly time-dependent and nonlinear approachesfor the description of dissipative quantum systems

Abstract

Caldirola [Nuovo Cimento 18, 393 (1941)] and Kanai's [Prog. Theor. Phys. 3, 440 (1948)] time-dependent approach for dissipative systems can be traced back to the conventional system-plus-reservoir approach; however, it apparently violates the uncertainty principle. This discrepancy can be avoided by considering a consistent transition between the Caldirola-Kanai formalism and a logarithmic nonlinear Schr\"odinger equation that requires not only a transformation of the operators, but also a nonunitary transformation of the wave function. This procedure also shows the equivalence of three at first sight incompatible approaches to describe dissipative quantum systems.