Effective hamiltonians and the two-photon laser

Abstract

The authors derive the equation of motion for the field density matrix of the non-degenerate two-photon laser under conditions of two-photon resonance starting from the full microscopic hamiltonian. They calculate the mean photon numbers and establish which solutions are stable. The results are compared with the corresponding quantities obtained from the effective hamiltonian. In general, there are significant differences. It is shown how the full diagonal density matrix equations tend to the effective hamiltonian density matrix equations in an appropriate limit. The same limit applied to the microscopic off-diagonal elements does not however give the off-diagonal equations of motion obtained in the effective hamiltonian approach. The off-diagonal elements determine, for example, the laser linewidths and cross-correlation coefficient so that the effective hamiltonian approach leads to correct results for these quantities. The results show that for a full understanding of the two-photon laser, the microscopic approach is essential.