1 Diagonalisation of Hamiltonian

Harald J W Müller-Kirsten, Armin Wiedemann

P. Croxson

Diagonalisation of quadratic multi-mode, time-dependent Hamiltonians can be fraught with difficulties. We set about diagonalising Hamiltonians from quantum optics. The Hamiltonians we choose represent Raman scattering, parametric amplification and the frequency converter. We present a systematic approach to this problem. The process of diagonalisation allows us to find invariants of the Hamiltonians under time evolution, find ground-state energies, apply the theory of Greens functions to find the solution to Schrodinger's equation, examine the conditions for oscillation or growth, and squeezing.

L. M. Garrido, P. Pascual

‎Louis Marchildon

J.H.P. Colpa

It is well known that the spectrum of a homogeneous quadratic Hamiltonian in m fermion construction-operator pairs is characterised by m mode energies. In this paper it is proved that the spectrum of a quadratic Hamiltonian with a linear part presents the same feature. Simple methods are given for the calculation of the mode energies and of the ground-state energy of the Hamiltonian. It is shown that, in contrast to the boson case, the diagonalisation cannot be carried out by a linear transformation of the fermion construction operators. Application of such a transformation can only result in a diagonalisation after the introduction of a 'ghost' particle (fermion) and a corresponding pair of Hermitian-conjugate construction operators.

Stefan K. Kehrein