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We study the hamiltonian and dissipative dynamics of a system undergoing a sequence of level crossings. The resulting Landau-Zener effect makes a new implementation of a general single qubit gate possible. In the dissipative case, with a periodic bias, the level crossing counteracts the interlevel relaxation and drives the system toward a two dimensional attractor. This feature can in principle be used to implement quantum memory devices of new type.

As the first useful Quantum Computers presumably will be quantum simulators, here the minimum number of qubits necessary for the solution of Schrödinger's equation in simple test problems is evaluated. From the present preliminary results it appears that it is possible to realize a useful quantum simulator with a register of only 10-15 qubits. The intrinsic sensitivity to some errors appears to be moderate without the need of error correcting methods. At present there is at least some indication that the amplitude errors are more dangerous than phase and decoherence errors. So corrections limited to amplitude errors may be very useful.

We review the main results obtained recently by our group in the field of quantum noise reduction in nonlinear optical systems. We illustrate the possibility of obtaining significant effects in degenerate parametric oscillators, intracavity second-harmonic generation, nondegenerate four-wave mixing, and one- and two-photon optical bistability.

We describe the dynamics of a laser with saturable absorber by a two-level model, which assumes a Gaussian radial profile for the electric field. Both the amplifier and the absorber are exactly resonant with the cavity mode frequency. We calculate all the stationary solutions of the system, including those detuned from the resonance frequency. We perform the linear stability analysis of the normal lasing state in the rate equation approximation. The instability range turns out to be somehow smaller in the Gaussian than in the plane wave case.

We consider the process of second-harmonic generation with the nonlinear crystal placed in a resonant cavity. Both the fundamental and the second-harmonic mode are shown to exhibit squeezing. An analysis is made for the general values of the ratio between the damping constants of the two modes. We suggest that second-harmonic generation is the most suitable system to use for an experimental observation of the squeezing effect.

We discuss the possibility of setting upper limits to deviations of the dimensionD of space from the standard integer value of 3. The method we suggest — a reanalysis of the Planck law for the black-body radiation in aD-dimensional space — is less sensitive than other methods previously suggested. It offers, however, the possibility of probing the fractal dimension of space-time over cosmological distances.