Effective Hamiltonians in quantum optics: a systematic approach

Abstract

We discuss a general and systematic method for obtaining effective Hamiltonians that describe different nonlinear optical processes. The method exploits the existence of a nonlinear deformation of the usual su(2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter, dictated by the process under consideration, becomes small, we immediately get a diagonal effective Hamiltonian that correctly represents the dynamics for arbitrary states and long times. We extend the technique to su(3) and su(N), finding the corresponding effective Hamiltonians when some resonance conditions are fulfilled.Comment: 13 Pages, no figures, submitted for publicatio

Effective Hamiltonian theory and its applications in quantum information

“…In particular, Avan et al [8] examined the problem using the Schrödinger picture, and obtained an expression involving sums over matrix elements which is related to ours. More recently, the notion of an effective Hamiltonian for specific models of nonlinear optical interactions has been presented by Klimov et al [9]. Furthermore, there is a considerable body of work in the nuclear magnetic resonance literature dealing with the notion of averaged Hamiltonians, which is a related concept (see for example [10]).…”

“…Usually the dimensionless parameter η is small, hence we can make the "Lamb-Dicke" approximation, exp ik zẑ (t) ≈ 1 + ik zẑ (t). This allows us to make the identification of three harmonic terms in this Hamiltonianĥ 9) with ω 1 = ∆ − ω 0 , ω 2 = ∆ and ω 3 = ∆ + ω 0 . In this case the effective Hamiltonian iŝ…”

Effective damping in the Raman cooling of trapped ions

“…In Refs. [44] and [45] we have claimed that such a procedure has drawbacks and proposed instead an alternative technique involving nonlinear rotations.…”

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