We consider the geometric phase and quantum tunneling in vicinity of diabolic and exceptional points. We show that the geometric phase associated with the degeneracy points is defined by the flux of complex magnetic monopole. In weakcoupling limit the leading contribution to the real part of geometric phase is given by the flux of the Dirac monopole plus quadrupole term, and the expansion for its imaginary part starts with the dipolelike field. For a two-level system governed by the generic non-Hermitian Hamiltonian, we derive a formula to compute the non-adiabatic complex geometric phase by integral over the complex Bloch sphere. We apply our results to to study a two-level dissipative system driven by periodic electromagnetic field and show that in the vicinity of the exceptional point the complex geometric phase behaves as step-like function. Studying tunneling process near and at exceptional point, we find two different regimes: coherent and incoherent. The coherent regime is characterized by the Rabi oscillations and one-sheeted hyperbolic monopole emerges in this region of the parameters. In turn with the incoherent regime the two-sheeted hyperbolic monopole is associated. The exceptional point is the critical point of the system where the topological transition occurs and both of the regimes yield the quadratic dependence on time. We show that the dissipation brings into existence of pulses in the complex geometric phase and the pulses are disappeared when dissipation dies out. Such a strong coupling effect of the environment is beyond of the conventional adiabatic treatment of the Berry phase.

Theory of pointlike magnetic monopole with an arbitrary magnetic charge is considered. It is shown that a proper description requires making use of nonunitary representations of the rotation group and the nonassociative generalization of the gauge group and fibre bundle theory.Comment: REVTEX4 file, 6 pages. To appear in Physics Letters

Degenerate four-wave mixing (DFWM) in excited atoms is investigated as a probe of the excited-state density produced by collision-assisted transitions. Sodium atoms are excited to the 3P state by absorption of light in the extreme blue wing of the D lines in the presence of He, Ne and Ar perturbers. The rare-gas pressure dependence of the DFWM signals, resonantly enhanced by the 3 P 4 D transition, is studied to determine the regime where the signals give an unambiguous measure of the excited-state (3P) density. The observed behaviour is compared with a simple model of DFWM in absorbing media. Good qualitative agreement is obtained and the results illustrate the role of absorption, saturation and finite laser bandwidth. The technique allows small excited-state densities (-IO9 ~m -~) to be detected with good temporal and spatial resolution.

Recently (Phys. Lett. A302 (2002) 253, hep-th/0208210; hep-th/0403146) employing bounded infinite-dimensional representations of the rotation group we have argued that one can obtain the consistent monopole theory with generalized Dirac quantization condition, 2κµ ∈ Z, where κ is the weight of the Dirac string. Here we extend this proof to the unbounded infinite-dimensional representations.