State Space Clock Model & Control
This is more of an extended comment than an answer.
The system may be inherently discrete-time. It may not make sense to find the continuous time plant model, as it may not exist. I am not familiar with atomic clock plant modeling, but the following points in the references indicate that the system is inherently discrete-time and that the input to the synthesizer is the incremental frequency steps.
Page 11. The word synthesizer and the word step.

What is the smallest step you can use on your synthesizer to correct the frequency?

Page 2. The input seems to be the frequency step by which the synthesizer needs to be adjusted. So with each input pulse, the frequency seems to step by a fixed amount. i.e., \$f_{k+1} = f_k + u_k\$.

The control vector ... corresponding to the fractional frequency change of the synthesizer ...

same page. The system seems to be inherently discrete-time.

is the time interval between measurements

However, ref 4 of the above paper provides a continuous time plant model.
page 2. confirms the above.

assumes  frequency  steps  are  used  to  implement  the  control

page 2

we  consider only those in which the clock is controlled by shifting its  frequency  as  fractions  (gains)  of  its  phase  and  frequency  deviations from the reference standard

The above 2 references indicate that you have to just accept that this is the model.
Perhaps, References 3,4 to this paper may show the model derivation. I dont have access to them to be sure.