How to represent the frequency domain information of excitation light field

[Junyong]

Dear All,  when we use QuTip to slove light-matter interaction , some parameters of the light field need to be considered in the simulation, such as pulse duration, pulse center frequency ,pulse frequency FWHM and so on.

Pulse duration can be definited in the Gaussian profile of light field. The enter frequency can also be reflected in Hamiltonian no matter in Lab frame or rotating frame.

Just like in this example https://nbviewer.jupyter.org/github/qutip/qutip-notebooks/blob/master/examples/pulse-wise-second-order-coherence-g2.ipynb

However，How can I definite the pulse frequency FWHM in simulation if I need a Gaussian pulse of with FWHM=100 GHz？Can that included in pulse time domain profile？

[Jake Lishman]

It's a difficult to point you in the right direction without some knowledge of the Hamiltonian you're trying to simulate.  If you can write down your pulse Hamiltonian in the form (or transform to a rotating frame where this is the case)

  H = sum_k f_k(t) A_k

where the `f_k(t)` are scalar functions of times and the `A_k` are constant operators, then you can simulate it efficiently with QuTiP.  The `f_k(t)` can be arbitrary functions of time; if you're dealing with finite-width continuous pulses, this may involve an integral over a range of frequencies, so the function might involve you performing your own integration over the frequency (analytically or numerically).

Here's another notebook which uses this "functional" form of time-dependence, and here is the current documentation on defining time-dependent Hamiltonians.

I hope that helps you find the right way forwards.

Jake

[Junyong]

Thanks.

I may not have made myself clear.

Just like the code in the attachment，I am simulating monochromatic light-two level atom interaction. But there is no true monochromatic light field in real experimental conditions, a 5 ps FWHM Gaussian transform-limited pulse always has ~0.2 nm spectral FWHM .

I mean How can the  spectral  width of the pulse be taken into account in the simulation, so that the simulation can be close to the real experimental situation.

One of the pieces of code in the attachment:

def rabi_frequency(t, args): # Gaussian profile
return 0.5 * area / (w * sqrt(pi)) * exp(-((t - t_center) / w) ** 2)
H0 = delta * e * e.dag()
H = [H0, [e*g.dag()+g*e.dag(), rabi_frequency]]

[Jake Lishman]

The larger question I think you need to answer for yourself is what the Hamiltonian you're actually trying to simulate is, in mathematical terms.  I can maybe offer some advice on how to convert a Hamiltonian you have written down into a programmable form, but I can't really offer quantum optics teaching.  For example, you seem to have the two-level Hamiltonian described by

so you need to work out what your Rabi frequency is as an equation.  If it involves an integral, then you either need to evaluate it analytically before-hand, and programme in the result, or you can use numerical integration methods such as numpy's `trapz` to approximate it yourself.

I will suggest that if you're dealing with detuning (like in a pulse with a continuous frequency distribution), the Rabi frequency will likely be complex so there will be a phase inequality between the e->g and g->e transitions.

As a side note: in your code you use variables such as the speed of light being 3e8, and the wavelengths being 800e-9.  You may have problems with numerical accuracy if you do this; it's safer to redefine your units so all the numbers you use are much closer in magnitude (e.g. using natural units where c=\hbar=1, and measuring frequencies in GHz or whatever is sensible).

Jake

[Junyong]

Thank you Jake!!  

[Christian Rentschler]

Hi Junyong,

I think by now you might have figured it out by yourself. But to answer directly your initial question:

The spectral shape of your pulse is automatically be set by its time domain profile. So if you want to have 100 GHz bandwidth you can easily calculate the corresponding pulse duration via a Fourier transform.

However, keep in mind that your simulation time is finite so your spectrum will have some Sinc like features at its borders (depending on the pulse duration), as it is a convolution of your pulses spectrum and the spectrum of a square function.

Chris