Geometrical phases in self-pulsing lasers

As observed in detuned lasers [1, 2] the arbitrary constant phase of the laser field in the CW region starts to drift linearly besides pulsations in the self-pulsing region. The fact that this drift might have similarities with the Berry phase [3] was pointed out in [2, 4] and a comparative study was given by us [2]. It has been not easy to establish an exact mathematical relation between the two, however. Thus a compelling analogy is still lacking. The main obstacle is that the original formulation [3] of the Berry phase was given for linear Schrodinger systems, whereas we have here an essentially non-linear and dissipative system. Fortunately we have succeeded in borrowing the geometrical formulation of the Berry phase for linear systems [5] and essentially generalizing it to a certain kind of nonlinear dissipative systems, to which detuned one- and two-photon lasers belong. An exact analogy is therefore established. We show that the whole phase accumulation of the laser field in a period of the intensity pulsation consists of two parts: a dynamical part given directly by the equation of movement and a geometrical part given by the path-integral along the trajectory of limit cycles in a certain phase space. This later part has the same origin as that due to parallel transportations of vectors in a curved space.