Linearization of the Hamiltonian

William Thomas Payne

Stefan K. Kehrein

Avia Rosenhouse, Jacob Katriel

The multiphoton single-atom Dicke Hamiltonian in the dipole and the rotating-wave approximations is shown to reduce to the one-photon single-atom Dicke Hamiltonian via an approximate linearization process. For the special case of the Jaynes-Cummings Hamiltonian, it is shown that it is possible to derive a linearized form that is exactly equal to the original Hamiltonian.

L. M. Garrido, P. Pascual

Ulf Leonhardt, Arnold Neumaier

Linear optical networks are devices that turn classical incident modes by a linear transformation into outgoing ones. In general, the quantum version of such transformations may mix annihilation and creation operators. We derive a simple formula for the effective Hamiltonian of a general linear quantum network, if such a Hamiltonian exists. Otherwise we show how the scattering matrix of the network is decomposed into a product of three matrices that can be generated by Hamiltonians.

Igor G. Vladimirov, Ian R. Petersen

This paper extends the energy-based version of the stochastic linearization method, known for classical nonlinear systems, to open quantum systems with canonically commuting dynamic variables governed by quantum stochastic differential equations with non-quadratic Hamiltonians. The linearization proceeds by approximating the actual Hamiltonian of the quantum system by a quadratic function of its observables which corresponds to the Hamiltonian of a quantum harmonic oscillator. This approximation is carried out in a mean square optimal sense with respect to a Gaussian reference quantum state and leads to a self-consistent linearization procedure where the mean vector and quantum covariance matrix of the system observables evolve in time according to the effective linear dynamics. We demonstrate the proposed Hamiltonian-based Gaussian linearization for the quantum Duffing oscillator whose Hamiltonian is a quadro-quartic polynomial of the momentum and position operators. The results of the paper are applicable to the design of suboptimal controllers and filters for nonlinear quantum systems.