Probabilities of n ′ ¹ P ₁ -state populations after radiative transitions from Rydberg n ¹ S ₀ states of alkaline-earth atoms

Once created, an ensemble of atoms in their identical Rydberg states immediately experiences significant variations of its structure and corresponding properties. The variation is caused by the inevitable radiation transitions of highly excited states into different states of higher or lower energies. In particular, the change of a uniform definite-state population for another kind of population of different states will change the long-range interaction between atoms, even transforming the van der Waals interaction, Δ E vdW =− C 6 / R 6 , into dipole–dipole interaction with a different dependence on the interatomic distance, Δ E d−d =− C 3 / R 3 . In this paper, the most important—dipole radiation transitions from Rydberg states to higher- or lower-energy | n ′ 1 P 1 ⟩ states—are considered, both spontaneous and induced by the blackbody radiation (BBR) of the environment. The total transition probability rate, which is usually determined as the natural width, equals the imaginary part of the Rydberg-state energy Γ n S /2=−Im E n S . The probabilities of the | n ′ 1 P 1 ⟩-state populations W n S ( n ′ P, T ), which depend on the BBR temperature T , on the lifetime scale are also rather important optical characteristics of the ensemble of the Rydberg-state atoms. The value of W n S ( n ′ P, T ) is determined as the ratio of a particular transition probability rate Γ n S→ n ′ P sp +Γ n S→ n ′ P BBR ( T ) to the sum of all possible radiative transition rates, which include spontaneous and BBR-induced rates, where the latter depend on the temperature T of the environment. Numerical values of W n S ( n ′ P, T ) are calculated and presented in wide regions of the principal quantum numbers n , n ′ , and BBR temperatures T .