Transformation design and nonlinear Hamiltonians

Abstract

We study a class of nonlinear Hamiltonians, with applications in quantum optics. The interaction terms of these Hamiltonians are generated by taking a linear combination of powers of a simple ‘beam splitter’ Hamiltonian. The entanglement properties of the eigenstates are studied. Finally, we show how to use this class of Hamiltonians to perform special tasks such as conditional state swapping, which can be used to generate optical cat states and to sort photons.

Keywords:

• quantum optics
• nonlinear optics
• quantum information

Acknowledgements

We acknowledge financial support from Grants MSM6840770039 and MSMT LC06002 of the Czech Republic. The authors are obliged to Professor A. Odzijewicz for helpful comments and remarks.

Notes

Notes

1. One could imagine that we are working in an interaction picture.

2. The spectrum of Ĥ ₀ on the whole Hilbert space will have degeneracies due to the fact that the spectrum of Ĥ ₀ on the different subspace is not disjointed and thus can share eigenvalues.

3. The name ‘cat state’ is a reference to the thought experiment developed by Schrödinger 23.