Qutip: Mesolve gives different and weird results with different fock state numbers

[安南Anand]

I have been trying to simulate the average number of particles at 3 sites of coupled harmonic oscillators.
I have used the code from the below tutorial:
https://notebook.community/ajgpitch/qutip-notebooks/examples/trilinear
I have modified the Hamiltonian as the following with decay parameters gamma being zero.

My Hamiltonian has two parts. Initial Hamiltonian and driving hamiltonian.
The driving Hamiltonian is given as following:
$H_{0}=\Omega(b_{1}^{\dagger}+b_{2}^{\dagger}+b_{3}^{\dagger})+\Omega^{*}(b_{1}+b_{2}+b_{3})$

I have modified the qutip tutorial of Trilinear oscillators to the form above mentioned Hamiltonian. As I change the number of Fock states considered N1,N2,N3 from 2,3,4...8 the average number particle at each site would change. They have some weird behavior. Can anyone tell me why the average population changes as we change N? How to choose right N to get the average particle at each site?

Any kind of help would be appreciated. My notebook attached. 

[安南Anand]

I have posted the question on stack exchange as well in case someone want to see the figures: 

https://quantumcomputing.stackexchange.com/questions/27563/qutip-mesolve-gives-different-and-weird-results-with-different-fock-state-numbe

[nwla...@gmail.com]

Hi Anand,

I had a quick look at your notebook, and it seems to converge for a cut-off of around 5 or 6, which seems like reasonable behavior to me...

As for why you need that cut-off, I think for  your particular example and parameters it's largely due to the choice of initial condition. If you reduce alpha you will see convergence for smaller N1 etc, because coherent states have  Poissonian distribution in Fock space. Conversely, I expect if you increase the drive term K, you will need a larger cutoff (essentially because in your model the drive causes a displacement  of each mode by a factor on order of something like ~2K/w0,  like  a coherent state, so to accurately capture that displacement term in the Hamiltonian you need more Fock states again).

Sometimes you can look at the parameters and properties of the Hamiltonian, dissipation, and initial state and make a guess at the cut-off you will need, but it's always good to check for convergence by trying different cut-offs, as you did.

all the best

neill

[安南Anand]

Dear Neill, 

I apologise for my late reply. I tried your suggestion and it worked very well. 

However, it makes me further think, if my initial state is just a Fock state rather than a coherent state, then how could we represent the initial state?

Say, for the Hamiltonian in my example above, I want to have just a 2 level system. How could I construct an initial Fock state which wouldn't diverge for n values 2? 

Any comments on this would be extremely helpful and appreciated. 

Sincerely, 

Anand