 
Summary: On the exact unitary integration of timevarying quantum Liouville equations
Claudio Altani
SISSAISAS
International School for Advanced Studies
via Beirut 24, 34014 Trieste, Italy y
In this paper, the Dyson series corresponding to the timevarying Hamiltonian of a nite dimen
sional quantum mechanical system is expanded in terms of products of exponentials of a complete
basis of commutator superoperators in the corresponding Liouville space. The CayleyHamilton
theorem and the WeiNorman formula allow to express explicitly the functional relation between
the Dyson series and the product of exponentials via a set of rst order dierential equations. Since
the method is structure preserving, it can be used for the exact unitary integration of the driven
Liouvillevon Neumann equation.
PACS numbers: 02.60.Cb, 32.80.Qk, 42.50.Hz
I. INTRODUCTION
The general solution of a quantum Liouville equation for timevarying Hamiltonians is given by the Dyson
series. Normal procedure for its practical use is to truncate this expansion and work with the corresponding
approximation. Beside providing approximate solutions, the main drawback of such truncations is that the
unitarity of the time evolution is not necessarily preserved [15]. The method we present here relies on the
formalism of the canonical coordinates of the rst and second kind of the adjoint representation of the unitary
group, and on the relation between them. In fact, the dierential operator governing the Liouville equation
