Jaynes-cummings Model with Degenerate Atomic Levels

The Jaynes-Cummings model describing the interaction of a single linearlypolarized mode of the quantized electromagnetic field with an isolated two-level atom is generalized to the case of atomic levels degenerate in the projections of the angular momenta on the quantization axis, which is a usual case in the experiments. This generalization, like the original model, obtains the explicit solution. The model is applied to calculate the dependence of the atomic level populations on the angle between the polarization of cavity field mode and that of the laser excitation pulse in the experiment with one-atom micromaser. The Jaynes-Cummings model [1] describes the interaction of a single linearly-polarized mode of the quantized electromagnetic field with an isolated two-level atom. The full set of states of the system atom+field is |n, α >= |n > ·|α >, , n = 0, 1, ..., α = b, c, where n is the number of photons in the field mode, while b and c denote the upper and lower atomic levels correspondingly. This model is applied successfully to analyse the results of the experiments with one-atom micromasers (see, e.g., [2]). However, the levels of an isolated atom are degenerate in the projections of the total elctronic angular momenta on the quantization axis, so that the original Jaynes-Cummings model becomes, in general, invalid. Now, let us take into account the degeneracy of atomic levels. Then, the full set of states of the system becomes |n, Jα,mα >= |n > ·|Jα,mα >, n = 0, 1, ..., mα = −Jα, ..., Jα, α = b, c, where Jb and Jc are the values of the total electronic angular momenta of resonant levels, whilemb andmc are their projections on the quantization axis the cartesian axis Z, which is directed along the polarization vector of the field mode. The Hamiltonian of the system may be written as Ĥ = ĤF + ĤA + V̂ , (1) where ĤF = h̄ωâ â is a free-field Hamiltonian, ĤA = 1 2 h̄ω0(n̂b − n̂c) is a free-atom Hamiltonian, V̂ = −(D̂Ê) 1 is an operator of field-atom interaction, while â and â are the operators of the creation and annihilation of photons with the frequency ω in the field mode, n̂α = Jα ∑ mα=−Jα |Jα,mα >< Jα,mα|, α = b, c, are the operators of total populations of resonant atomic levels b and c, ω0 is the frequency of the opticallyallowed atomic transition Jb → Jc, Ê = eâ+ eâ, e = ılz √ 2πh̄ω V , is the electric field intensity operator, V and lz being the resonator cavity volume and the unit vector of the cartesian axis Z, D̂ = d̂ + d̂ + , d̂ = ∑ mb,mc d JcJb mcmb · |Jc,mc >< Jb,mb|, is the dipole moment operator of the atomic transition Jb → Jc, which matrix elements are defined through Wigner 3j-symbols (see, e.g., [3]): (dq) JbJc mcmb = d(−1)bb ( Jb 1 Jc −mb q mc ) , d = d(JbJc) -being a reduced matrix element and dq (q = −1, 0, 1) are the circular components of vector d. In the interaction representation f̂I = exp (