It is convenient to write the interaction operators with the broadband fields as
$$\begin{gathered} {{V}_{c}}(t) = {{\gamma }_{c}}\sum\limits_{\forall \omega \ne 0}^{} {(c{{e}^{{ - i{{\omega }_{c}}t}}} + {{c}^{\dag }}{{e}^{{i{{\omega }_{c}}t}}}){{a}_{\omega }}{{e}^{{ - i\omega t}}},} \\ {{a}_{{ - \omega }}} = a_{\omega }^{\dag },\quad \omega > 0, \\ \end{gathered} $$
$$\begin{gathered} {{V}_{r}}(t) = {{\gamma }_{r}}\sum\limits_{\forall \omega \ne 0}^{} {(r{{e}^{{ - i{{\omega }_{r}}t}}} + {{r}^{\dag }}{{e}^{{i{{\omega }_{r}}t}}}){{a}_{\omega }}{{e}^{{ - i\omega t}}},} \\ {{b}_{{ - \omega }}} = b_{\omega }^{\dag },\quad \omega > 0. \\ \end{gathered} $$
The terms changing slowly with time are then
$$V_{c}^{'}(t) = {{\gamma }_{c}}\sum\limits_{\omega \in ({{\omega }_{c}})}^{} {(ca_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{c}})t}}} + {{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}})} ,$$
$$V_{r}^{'}(t) = {{\gamma }_{r}}\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {(rb_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{r}})t}}} + {{r}^{\dag }}{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}})} .$$
They define the operators of the first order in coupling constants.
The rapidly changing terms can be represented as
$$V_{c}^{{''}}(t) = {{\gamma }_{c}}\sum\limits_{\forall \omega \notin ( - {{\omega }_{c}})}^{} {c{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}} + {{\gamma }_{c}}\sum\limits_{\forall \omega \notin ({{\omega }_{c}})}^{} {{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}},} $$
$$V_{r}^{{''}}(t) = {{\gamma }_{r}}\sum\limits_{\forall \omega \notin ( - {{\omega }_{r}})}^{} {r{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}} + {{\gamma }_{r}}\sum\limits_{\forall \omega \notin ({{\omega }_{r}})}^{} {{{r}^{\dag }}{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}},} $$
where (ω
c) is the region with sizes |ω – ω
c| ≪ ω
c near ω = ω
c, but no less than the reciprocal rate of relaxation via this channel of coupling with the broadband field. The regions of the continuous spectrum (–ω
c), (ω
r), and (–ω
r) are defined similarly.
Identifying the rapidly and slowly changing terms in the interaction operator between the oscillators causes no difficulties at all in the case where the oscillator frequencies are equal, ωc = ωr:
$$V_{{c - r}}^{'}(t) = g(c{{r}^{\dag }} + {{c}^{\dag }}r),$$
$$V_{{c - r}}^{{''}}(t) = g({{c}^{\dag }}{{r}^{\dag }}{{e}^{{i({{\omega }_{c}} + {{\omega }_{r}})t}}} + cr{{e}^{{ - i({{\omega }_{c}} + {{\omega }_{r}})t}}}).$$
The first-order transformation generators are found from the relations
$$\frac{{d{{S}^{{(1,0,0)}}}(t)}}{{dt}} = - {{\hbar }^{{ - 1}}}V_{{c - r}}^{{''}}(t),$$
$$\frac{{d{{S}^{{(0,1,0)}}}(t)}}{{dt}} = - {{\hbar }^{{ - 1}}}V_{c}^{{''}}(t),$$
$$\frac{{d{{S}^{{(0,0,1)}}}(t)}}{{dt}} = - {{\hbar }^{{ - 1}}}V_{r}^{{''}}(t).$$
They are fast functions of time. Under the assumption of adiabatic interaction switching, the first-order transformation generators are defined by the expressions
$${{S}^{{(1,0,0)}}}(t) = cr\frac{{g{{e}^{{ - i({{\omega }_{c}} + {{\omega }_{r}})t}}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}} - {{c}^{\dag }}{{r}^{\dag }}\frac{{g{{e}^{{i({{\omega }_{c}} + {{\omega }_{r}})t}}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}},$$
$${{S}^{{(0,1,0)}}}(t)\, = \,{{\gamma }_{c}}\sum\limits_{\omega \notin ( - {{\omega }_{c}})}^{} {\frac{{c{{a}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{c}})t}}}}}{{i\hbar (\omega + {{\omega }_{c}})}}} \, + \,{{\gamma }_{c}}\sum\limits_{\omega \notin ({{\omega }_{c}})}^{} {\frac{{{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}}}{{i\hbar (\omega - {{\omega }_{c}})}}} ,$$
$${{S}^{{(0,0,1)}}}(t)\, = \,{{\gamma }_{r}}\sum\limits_{\omega \notin ( - {{\omega }_{r}})}^{} {\frac{{r{{b}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{r}})t}}}}}{{i\hbar (\omega + {{\omega }_{r}})}}} \, + \,{{\gamma }_{r}}\sum\limits_{\omega \notin ({{\omega }_{r}})}^{} {\frac{{{{r}^{\dag }}{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}}}{{i\hbar (\omega - {{\omega }_{r}})}}} .$$
Then,
$$\begin{gathered} \text{[}{{S}^{{(1,0,0)}}}(t),V_{c}^{{''}}(t)]' = \frac{{g{{\gamma }_{c}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}} \\ \times \sum\limits_{\omega \in ({{\omega }_{r}})}^{} {ra_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{r}})t}}}} + \frac{{g{{\gamma }_{c}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}}\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {{{r}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}} , \\ \end{gathered} $$
$$\begin{gathered} \text{[}{{S}^{{(0,1,0)}}}(t),V_{{c - r}}^{{''}}(t)]' \\ = {{\gamma }_{c}}g\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {\frac{{a_{\omega }^{\dag }r{{e}^{{i(\omega - {{\omega }_{r}})t}}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}}} + {{\gamma }_{c}}g\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {\frac{{{{a}_{\omega }}{{r}^{\dag }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}}}{{i\hbar ({{\omega }_{c}} + {{\omega }_{r}})}}} . \\ \end{gathered} $$
As a result, the term that describes the coupling of the single-model oscillator with frequency ωr to the foreign heat bath appears in the effective Hamiltonian:
$$\begin{gathered} {{{\tilde {H}}}^{{(1,1,0)}}}(t) = - \frac{{g{{\gamma }_{c}}}}{{2\hbar {{\omega }_{c}}}}\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {{{r}^{\dag }}c{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{r}})t}}}} \\ - \frac{{g{{\gamma }_{c}}}}{{2\hbar {{\omega }_{c}}}}\sum\limits_{\omega \in ({{\omega }_{r}})}^{} {ra_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{r}})t}}}} . \\ \end{gathered} $$
Here, we set ωc = ωr ≈ ω.
Similarly, it is easy to obtain
$$\begin{gathered} {{{\tilde {H}}}^{{(1,0,1)}}}(t) = - \frac{{g{{\gamma }_{r}}}}{{2\hbar {{\omega }_{c}}}}\sum\limits_{\omega \in ({{\omega }_{c}})}^{} {{{c}^{\dag }}{{b}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}} \\ - \frac{{g{{\gamma }_{r}}}}{{2\hbar {{\omega }_{c}}}}\sum\limits_{\omega \in ({{\omega }_{c}})}^{} {bc_{\omega }^{\dag }{{e}^{{i(\omega - {{\omega }_{c}})t}}}} . \\ \end{gathered} $$
The operator \({{\tilde {H}}^{{(0,1,1)}}}\)(t) describes the second-order coupling between the heat baths that does not affect the open system and, therefore, below it is disregarded.
Other, possibly relevant, second-order terms are
$$\begin{gathered} {{{\tilde {H}}}^{{(2,0,0)}}}(t) = - \frac{i}{2}[{{S}^{{(1,0,0)}}}(t),V_{{c - r}}^{{''}}(t)]' \\ = - \frac{{{{g}^{2}}}}{{2\hbar {{\omega }_{c}}}}({{c}^{\dag }}c + {{r}^{\dag }}r + 1), \\ \end{gathered} $$
$${{\tilde {H}}^{{(0,2,0)}}}(t) = - \frac{i}{2}[{{S}^{{(0,1,0)}}}(t),V_{c}^{{''}}(t)]',$$
$$\begin{gathered} \text{[}{{S}^{{(0,1,0)}}}(t),V_{c}^{{''}}(t)]' \\ = \left[ {{{\gamma }_{c}}\sum\limits_{\omega \notin ( - {{\omega }_{c}})}^{} {\frac{{c{{a}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{c}})t}}}}}{{i\hbar (\omega + {{\omega }_{c}})}}} } \right. + {{\gamma }_{c}}\sum\limits_{\omega \notin ({{\omega }_{c}})}^{} {\frac{{{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}}}{{i\hbar (\omega - {{\omega }_{c}})}}} , \\ \left. {{{\gamma }_{c}}\sum\limits_{\forall \omega \notin ( - {{\omega }_{c}})}^{} {c{{a}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{c}})t}}}} + {{\gamma }_{c}}\sum\limits_{\forall \omega \notin ( - {{\omega }_{c}})}^{} {{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}} } \right] \\ = \left[ {{{\gamma }_{c}}\sum\limits_{\omega \notin ({{\omega }_{c}})}^{} {\frac{{{{c}^{\dag }}{{a}_{\omega }}{{e}^{{ - i(\omega - {{\omega }_{c}})t}}}}}{{i\hbar (\omega - {{\omega }_{c}})}},{{\gamma }_{c}}\sum\limits_{\forall \omega ' \notin ( - {{\omega }_{c}})}^{} {c{{a}_{{\omega '}}}{{e}^{{ - i(\omega ' + {{\omega }_{c}})t}}}} } } \right] \\ + \left[ {{{\gamma }_{c}}\sum\limits_{\omega \notin ( - {{\omega }_{c}})}^{} {\frac{{c{{a}_{\omega }}{{e}^{{ - i(\omega + {{\omega }_{c}})t}}}}}{{i\hbar (\omega + {{\omega }_{c}})}},{{\gamma }_{c}}\sum\limits_{\forall \omega ' \notin ({{\omega }_{c}})}^{} {{{c}^{\dag }}{{a}_{{\omega '}}}{{e}^{{ - i(\omega ' - {{\omega }_{c}})t}}}} } } \right]. \\ \end{gathered} $$
As a result, we have the contributions
\({{\tilde {H}}^{{(0,2,0)}}}\)(
t) and
\({{\tilde {H}}^{{(0,0,2)}}}\)(
t) to the effective Hamiltonian:
$$\begin{gathered} {{{\tilde {H}}}^{{(0,2,0)}}}(t) = - \gamma _{c}^{2}({{c}^{\dag }}c + 1)\sum\limits_{\forall \omega \notin ( - {{\omega }_{c}})}^{} {\frac{1}{{\hbar (\omega + {{\omega }_{c}})}}} \\ - \gamma _{c}^{2}\sum\limits_{\omega \notin ( - {{\omega }_{c}})}^{} {\frac{{a_{\omega }^{\dag }{{a}_{{\omega '}}}{{e}^{{i\omega t}}}{{e}^{{ - i\omega 't}}}}}{{2\hbar (\omega + {{\omega }_{c}})}} - \gamma _{c}^{2}\sum\limits_{\substack{ \omega ' \notin ( - {{\omega }_{c}}) \\ \forall \omega ' \notin ( - {{\omega }_{c}})} } ^{} {\frac{{a_{{\omega '}}^{\dag }{{a}_{\omega }}{{e}^{{i(\omega ' - \omega )t}}}}}{{2\hbar (\omega + {{\omega }_{c}})}},} } \\ \end{gathered} $$
while the operator
\({{\tilde {H}}^{{(0,0,2)}}}\)(
t) is derived from
\({{\tilde {H}}^{{(0,2,0)}}}\)(
t) by the simultaneous substitutions of the subscripts and the like operators
c →
r and broadband field operators
aω →
bω. The operators
\({{\tilde {H}}^{{(0,2,0)}}}\)(
t) and
\({{\tilde {H}}^{{(0,0,2)}}}\)(
t) can be represented by quantum counting processes similarly to [
30,
31,
33]. These processes obey a different algebra of Ito differentials [
33] that generalizes the well-known Hudson–Parthasarathy algebra [
34]. If we use the results [
30,
31,
33] of an analysis of their role, along with Wiener processes, then it can be stated that the terms
\({{\tilde {H}}^{{(0,2,0)}}}\)(
t) and
\({{\tilde {H}}^{{(0,0,2)}}}\)(
t) in the statement of the problem considered here should be neglected.
