I want to transform an operator into the interaction Picture. Therefore, I have to calculate the following
$$\exp(i H_E t) (\sum_k g_k a_k )\exp(-i H_E t),$$ where
$$H_E=\sum \omega_k a_k^\dagger a_k.$$
Plugging $H_E$ in yields (I am not 100% sure if the notation that I chose is very clever, so corrections concerning my notation are really appreciated)
$$ \underbrace{\sum_k \sum_n \exp(i \omega_k n) |n\rangle \langle n|}_{\exp(i H_E t)} \, \underbrace{(\sum_l \sum_p g_l c_p |p-1\rangle \langle p| )}_{(\sum_k g_k a_k )} \, \underbrace{\sum_m \sum_j\exp(-i \omega_m j) |j\rangle \langle j|}_{\exp(-i H_E t)}=(*)$$
I chose to represent the creation and annihilation operators such that
$$a_k=\sum_l c_l |l-1\rangle \langle l|$$
$$a_k^\dagger=\sum_l c_l |l+1\rangle \langle l|$$
Besides that, I am not really sure how to Label everything correctly because the annihilation and creation operator only act on states in the correct mode ($\omega_k$), so I don't know how to indicate this in the following expression
$$\sum_k \omega_k a_k^\dagger a_k=\sum_k \omega_k \sum_l l |l \rangle \langle l | .$$
If I continue now my calculation for the representation in the interaction picture, I obtain the following Expression
$$(*)= \sum_k \sum_n \exp(i \omega_k n) \, (\sum_l \sum_p g_l c_p ) \, \sum_m \sum_j\exp(-i \omega_m j) |n\rangle \langle n| |p-1\rangle \langle p| |j\rangle \langle j| $$ $$= \sum_k \sum_n \exp(i \omega_k n) \, (\sum_l \sum_p g_l c_p ) \, \sum_m \sum_j\exp(-i \omega_m) |n\rangle \delta_{n,p-1} \delta_{p,j} \langle j |$$ $$= \sum_k \exp(i \omega_k(p-1)) \, (\sum_l \sum_p g_l c_p ) \, \sum_m \exp(-i \omega_m p) |p-1\rangle \langle p |$$ $$= \sum_k \exp(i \omega_k(p-1)) \sum_m \exp(-i \omega_m p) \, (\sum_l \sum_p g_l c_p |p-1\rangle \langle p | ) $$ $$= \sum_k \exp(i \omega_k(p-1)) \sum_m \exp(-i \omega_m p) \, (\sum_l g_l a_l ). $$
Now, the problem is that this is not correct, since the result is
$$\sum_l g_l a_l \exp(-i \omega_l t).$$
I don't know where the additional $\delta_{k,m}$ and $\delta_{l,m}$ should dome from. Made I started with too many sums and thats the mistake but I don't see why there should be too many of them.
EDIT: Following udrv's advice:
I split up the exponential function
$$ \exp(-i H_E t)= \prod \limits_{i=1} \exp(-i H_i t)$$, where $$ H_i=\omega_i a_i^\dagger a_i$$
Then performing the calculation for a generic $a_k$ yields $$\exp(i H_1 t) \exp(i H_2 t) \dots \exp(i H_k t) \dots g_k a_k \exp(-i H_1 t) \exp(-i H_2 t) \dots \exp(-i H_k t) \dots$$ $$=\exp(i H_1 t) \exp(-i H_1 t) \exp(i H_2 t) \exp(-i H_2 t) \dots \exp(i H_k t)g_k a_k \exp(-i H_k t) \dots$$ $$=\mathbb{I} \cdot \mathbb{I}\cdot \mathbb{I} \dots \sum_l \exp(-i \omega_k l) |l \rangle \langle l | g_k \sum_j c_j |j-1\rangle \langle j| \sum_n \exp(-i\omega_k n ) |n \rangle \langle n |$$ $$=\exp(i \omega_k (j-1)) \sum_j c_j |j-1\rangle \langle j | \exp(- i \omega_k j )$$ $$=\exp(-i \omega_k ) g_k a_k$$.
Now, the Summation over the different $a_k$ should yield the desired result. Correct?