Expand a creation operator as sum over frequencies

The Question

For a system with translational invariance, one can expand a position-space bosonic operator in a basis of momentum-space operators. For 1D this looks like

$$hat{a}_x(t) = sum_k hat{a}_k(t) text{e}^{text{i} k x}.$$

Now let’s consider a 0D bosonic operator. Is it reasonable/valid to expand the operator as

$$hat{a}(t)=sum_omega hat{a}_omega text{e}^{text{i} omega t}.$$

Additional context

My motivation is to treat a Fröhlich type Hamiltonian. In $k$-space this is normally written as

$$H=sum_k left[ omega_0 hat{a}_k^dagger hat{a}_k + sum_q left( G_q hat{b}_q hat{a}_{k+q}^dagger hat{a}_k + text{H.c.} right) right].$$

Here, the term $hat{b}_q hat{a}_{k+q}^dagger hat{a}_k$ describes wavevector transfer to the $hat{a}$-field from the $hat{b}$-field, and via the (not shown) dispersion relations, a transfer of energy. I would like to describe a similar process, but for a 0D system without a wavevector. Something like

$$H=sum_{omegamu} left[ omega_0 hat{a}_omega^dagger hat{a}_mu text{e}^{text{i}t(mu-omega)} + left( G_omega hat{b}_omega hat{a}_{mu+omega}^dagger hat{a}_mu + text{H.c.} right) right].$$