Derivation of multi-photon anharmonic oscillator model from its classical counterpart

Abstract

The electromagnetic field coupled to a nonlinear medium of having nonvanishing polarizations and magnetizations could be modeled as a classical anharmonic oscillator with velocity (p)- and position (q)-dependent anharmonicities. The position and velocity dependent anharmonic oscillators are also realized for center of mass motion of trapped particles in magneto optical trap. The Hamiltonian corresponding to the quantum anharmonic oscillator with velocity- and position-dependent anharmonicities is obtained from the knowledge of its classical counterpart. Under rotating wave approximation, the Heisenberg formalism are used to obtain the solution of the quantum oscillator with q-dependent and p-dependent anharmonicities. The analytical solution is used to obtain the dipole moment matrix elements and hence the shifts of the frequency of the oscillator. Interestingly, the shifts of the oscillator due to the q-dependent anharmonicity is opposite to those of the shifts due to the p-dependent anharmonicity. Therefore, the shifts of the frequency assert the presence of particular anharmonicity as well (i.e. p- or q-type).