Optimal number of cavity fock states

[Joan Agusti]

Hello eveybody,

I am here to clarify a small issue I found while working on a simulation: simulating the mean photon number in a cavity in a parametric amplifier. 

The simulation by itself is easy, there is even one similar problem solved by QuTip: https://nbviewer.jupyter.org/github/jrjohansson/qutip-lectures/blob/master/Lecture-5-Parametric-Amplifier.ipynb

The isse is that the results are highly dependent on the number of fock basis N! I understand that the result may change from N=3 to N=5, but if we are working in vacuum, we should not observe any difference between N=1000 and N=1200.

The difference is not huge, it has similar behavioir, but it is there, mostly in its peak value. How it is possible that truncating the Hamiltonian, at so large values, give differences?

Let me attach the code:

E=1.0  #electric field amplitude, consider real

hbar=1.0

####Bosonic operator

N=1200

a=destroy(N)

psi0 =fock(N, 0)   #initial state, vacuum state

Hd=1j*hbar/2.0*(E*a.dag()*a.dag()-E.conjugate()*a*a)  #squeeze generator

#Dynamics

T_max=100

N_steps=T_max*50

tlist = np.linspace(0, T_max, N_steps)

mean_values = [a.dag()*a]

result = mesolve(Hd, psi0, tlist, [], mean_values)

Regards,

Joan.

[Neill Lambert]

Hi Joan,

I guess the problem is a parametric oscillator, without any damping, will diverge in the number of photons at sufficiently long times, so in your example, you can see that first of all you start to get a very large number of photons (400 - 500), and that occupation oscillates, probably partly because it is '''bouncing'' off the cut-off.  The maximum increases as you increase N, as well, another indicator.

The example in Robert's notebook has no damping but I guess he only looks at short times, before the occupation approaches the cut-off too much

hope this helps!

kind regards

neill

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[Joan Agusti]

Hi Neill,

thanks for your answer. 

There is still some question I have: Even if I put some damping, such that the mean number of photons saturate to a certain value, whenever I increase N , both for small values N=5 to N=7 and large values N=100 to N=120, the saturation value increases. This shouldn't happen, right?

There should be a constant value for N going to infinity, right?

If not, how does one choose the optimal N for the fock states? Small N truncate the Hamiltonian too brute, and large N gives wanton dynamics.
Regards,

Joan.

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[Neill Lambert]

Hi Joan,

I am not 100% sure, but I remember in the classical parametric oscillator there is a threshold in terms of E and damping above which the instability still occurs?  I guess this is reflected in your solution as well (you need to put pretty large gamma to inhibit some exponential growth)?

Looking at the literature, the typical approach seems to be to limit the growth with an additional (quartic) non-linearity

https://journals.aps.org/prx/abstract/10.1103/PhysRevX.9.021049  

kind regards

neill

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