Preservation of Bosonic commutation relation: Explicit evaluation of quantum Langevin operator products

The commutation relation for the field creation and annihilation operators is shown to be preserved explicitly if the transient operators products are evaluated from the quantum Langevin equations that contain noise operators with non-vanishing expectation (mean) values. The well-known Einstein relation, which is valid for the vanishing mean value, is inapplicable here. For a two-level atom initially in arbitrary state interacting with quantum vacuum radiation, detailed steps are provided that may be useful for calculations involving transient dynamics using the full quantum formalism, particularly in evaluating products involving the noise operators, the initial (time) field operator and initial atomic operators. The cross products of these operators contribute to the preservation of the commutation relation.