Symplectic invariant method for time-dependent quadratic Hamiltonians
Université Pierre et Marie Curie - Paris 6, UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée 75252, Paris cedex 05, France, EU
Received: 2 August 2012
Accepted: 15 October 2012
Using the ABCD formalism of atom optics, the generating function method and the invariant operator method (Liouville-von Neumann picture), I show in this letter how the quantum time evolution of any wave packet can be entirely determined by a natural symplectic invariant, which provides both the (non-Hermitian) invariant operator and the adequate eigenvalue. Within this framework, the quantum propagator does not appear any more as the fundamental kernel of integration but merely as a particular property of this invariant operator. The symplectic invariant method is interpreted in terms of creation/annihilation operators of N-dimensional Hermite-Gaussian modes.
PACS: 03.65.Ca – Formalism / 02.30.Ik – Integrable systems / 11.30.-j – Symmetry and conservation laws
© EPLA, 2012