By propagating a series of ADOs, representing different arrangements of bath phonons, the HEOM accounts for a history of interactions such that non-Markovian effects are automatically included. The ADOs are interconnected via raising and lowering terms which are denoted by j ± = (j 10 , ..., j nk ± 1, ..., j N B K ) vectors. The ρ -dependent terms are raising operations. The action of i ℏ d nl B n , and its conjugate, is to destroy bath phonons, of coupling amplitude d nl , as they are absorbed by the system. This corresponds to an increase of ADO tier resulting in a 'raising' of the ADO number along a Matsubara axis. Subsequently, this process is associated with thermal fluctuations and the real part of the correlation function because of its temperature dependence. The final term, dependent on ρ + , is the corresponding lowering term. The action of i ℏ B × n is to destroy the system states, corresponding to the creation of bath phonons as they are emitted from the system into the bath. Destruction of system states in this manner are a consequence of the imaginary part of the correlation function associated with system dissipation.