  Lie algebraic techniques are powerful and widely-used tools for studying
dynamics and metrology in quantum optics. When the Hamiltonian generates a Lie
algebra with finite dimension, the unitary evolution can be expressed as a
finite product of exponentials using the Wei-Norman expansion. The system is
then exactly described by a finite set of scalar differential equations, even
if the Hilbert space is infinite. However, the differential equations provided
by the Wei-Norman expansion are nonlinear and often have singularities that
prevent both analytic and numerical evaluation. We derive a new Lie algebra
expansion for the quantum Fisher information, which results in linear
differential equations. Together with existing Lie algebra techniques this
allows many metrology problems to be analysed entirely in the Heisenberg
picture. This substantially reduces the calculations involved in many metrology
problems, and provides analytical solutions for problems that cannot even be
solved numerically using the Wei-Norman expansion. We provide detailed examples
of these methods applied to problems in quantum optics and nonlinear
optomechanics.
