CUSP-SINGULARITY OF THE TRAJECTORY OF THE QUBIT’S BLOCH VECTOR UNDER AN EXTERNAL PERIODIC LIGHT FIELD

Abstract

The quantum dynamics of a two-level quantum-mechanical system subjected to the external monochromatic action beyond the rotating wave approximation was investigated. It was shown that under the condition of exact resonance on the trajectories of the Bloch vectors, special points are manifested under different initial conditions. These points are classified as cusps singularities. It is revealed that at such points, the instantaneous rotation axis, relative to which the Bloch vector rotates, reverses its direction. There is a movement stop. For a nonzero frequency detuning, the cusp singularities vanish. A numerical analysis of the singularities of the trajectories of the Bloch vector without rotating wave approximation was supplemented by a study based on the use of the Floquet methods. Within the framework of this approach, recurrence relations for the spectral components of the probability amplitudes were obtained and analyzed. An analytic expression was found for the two values of quasi-energies within a fourth order of magnitude in the interaction energy. It was shown that to obtain a singular behavior of the trajectories of the Bloch vector, it is sufficient to confine by four spectral harmonics in the Floquet expansion. The results obtained are important for achieving the accuracy when performing coherent transformations with two-level systems in the cases where the rotating wave approximation is inapplicable.