Perturbation theory and semiclassical approximation in the energy representation

Jeremy M. Moix, Eli Pollak, Jiushu Shao

A generalized time-dependent perturbation theory is derived for superoperators. Instead of using the ``standard'' breakup of the Hamiltonian into a known zeroth order term and a correction, we use the approximate superpropagator to define the correction superoperator which is then used to obtain a series representation of the exact Liouville operator. The theory reduces to known limits and may be used for a perturbation expansion of classical Wigner and Husimi dynamics as well as for recent phase-space-based semiclassical approximations. The theory is demonstrated for a model quartic potential.

Andrea Sacchetti

S. Graffi, Manfredi, V R, Salasnich, L

A test on the numerical accuracy of the semiclassical approximation as a function of the principal quantum number has been performed for the Pullen--Edmonds model, a two--dimensional, non--integrable, scaling invariant perturbation of the resonant harmonic oscillator. A perturbative interpretation is obtained of the recently observed phenomenon of the accuracy decrease on the approximation of individual energy levels at the increase of the principal quantum number. Moreover, the accuracy provided by the semiclassical approximation formula is on the average the same as that provided by quantum perturbation theory.

Vladislav G. Bagrov, Владимир Владимирович Белов, M. F. Kondrat’eva

Daniel Speed, Wenyang Lyu, Roman Schubert

Gaussian quantum channels are well understood and have many applications, e.g., in Quantum Information Theory and in Quantum Optics. For more general quantum channels one can in general use semiclassical approximations or perturbation theory, but it is not easy to judge the accuracy of such methods. We study a relatively simple model case, where the quantum channel is generated by a Lindblad equation where one of the Lindblad operators is a multiple of the internal Hamiltonian, and therefore the channel is not Gaussian. For this model we can compute the characteristic function of the action of the channel on a Gaussian state explicitly and we can as well derive a representation of the propagator in an integral form. This allows us to compare the exact results with semiclassical approximations and perturbation theory and evaluate their accuracy. We finally apply these results to the study of the evolution of the von Neumann entropy of a state.