Quantum
coherent control represents a universal approach for predictable manipulation
of the properties of the quantum systems, such as atoms and molecules, most
recently applied to solid state systems. Coherent-optical control in
semiconductor nanostructures has recently attracted significant interest
because it allows coherent manipulation of the carrier wave functions on a time
scale shorter than typical dephasing times. This in turn is a prerequisite for
successful implementation of ultrafast optical switching and
quantum-information processing. A necessary condition for quantum coherence is
the use of sufficiently short optical pulses so that they can interact with the
quantum system before it can be affected by its environment.

We have developed a general dynamical model for
description of the coherent interactions of ultrashort linearly- and
circularly-polarised optical pulses with the resonant nonlinearities in planar
optical waveguides and semiconductor microcavities in multi-dimensions. The
adopted approach models the coherent light-matter interaction exploiting SU(N)
Lie group symmetries in a discrete multi-level quantum system and the full
vectorial treatment of the electromagnetic wave propagation. For the case of a
generic *N*-level system this approach
is based on the expansion of the density matrix and the system Hamiltonian in
terms of the SU(N) Lie algebra generators. The time-evolution of the quantum
system in the external dipole-coupling perturbation is described in terms of a
real pseudospin vector representation coupled to the vectorial Maxwell’s
equations for the optical field propagation. The latter permits a rigorous
description of pulse propagation and interactions on an ultrashort time scale

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