System-Bath Interaction

Assume a `bath' of bosonic modes $Q$ (the index $Q$ contains all quantum numbers of that mode) with creation operator $a_Q^{\dagger}$. The simplest interaction between the two-level system and the bath is linear in $a_Q^{\dagger}$ and $a_Q$ and can be written with coupling constant vectors ${\bf g}_Q(t)$,

$$H_{SB}(t) \equiv \hat{\bf A}(t) \vec{\sigma}\equiv \sum_Q \left( {\bf g}_Q(t) a_Q^{\dagger} + {\bf g}_Q^{\dagger}(t) a_Q\right)\vec{\sigma}.$$ (110)

Note that $\hat{\bf A}(t)$ can be regarded as a fluctuating (quantum operator) pseudo magnetic field.