Steady states plot for a three level lambda system

[Vishal Narnaulia]

Hi,

I am trying to plot the elements of the density matrix for a three level lambda type atom in a steady state.  There are two two lasers also which are interacting with the atom.

When I tried to use the steadystate() solver from the QuTiP , then it gave the error. Then , I found out that this solver only works for the Hamiltonian which is time independent. 

But here, I'm working in Schrodinger Picture, so the Hamiltonian is time dependent.

In that scenario, then I found an alternative way from GitHub to find the steady states , which uses a propagator for the density matrix.

I found it here https://github.com/qutip/qutip/issues/1377

My question is whether the plots for the elements of the density matrix in a steady state come out to be oscillatory or constant lines?

In my case, it's coming out to be a constant (3x3) matrix. So it's only giving the plots as simple straight lines. Should it be constant or vary with the time? I'd thought that it should be like an asymptotic solution in which the plot should have damped oscillations (because of the spontaneous emission; decay terms accounted in the collapse operator) and then after a certain time it should become constant, and that's what we call a steady state!

Please correct me if I'm wrong. Also in the above link, it was for a simple 2 level atom in which there was only a single laser. But here in my case I have 2 lasers, so how would the code get changed?  I have plotted the elements for the density matrix corresponding to laser 1 only. The other plots are for the two-sided power spectrum, correlation values and the angular frequencies.

Thank you so much if you've read up to here. I hope you'll help me in this problem as I've been trying to solve this problem since more than a month but couldn't get it. So I finally have to post it here. The concerned code and pdf are attached here.

Thanks in advance!

Regards,

Vishal Narnaulia

[Anna Naden]

I think your collapse operators are too strong. I ran your Hamiltonian and initital state with no collapse operators and I got the attached trajectory in the Bloch sphere. To get this result, I had to transform the return states to the interaction picture. If you have any questions, feel free to ask. I am attaching also the code I used.

[Anna Naden]

[Vishal Narnaulia]

Thank you for your response.

I didn't get what these trajectories are representing. The theory behind this is not clear to me.

I only need the plots for the elements of the density matrix in the steady state w.r.t. time. Is that possible to implement on the Bloch Sphere?

[Anna Naden]

There is a simple transformation that takes a density matrix and gives you the coordinates in the Bloch sphere. Here it is:

def to_bloch_qobj(dm):

matrix = np.array(dm.data.todense()[:2,:2])

[[a, b], [c, d]] = matrix

x = complex(c + b).real

y = complex(c - b).imag

z = complex(a - d).real

return x, y, z

dm is the density matrix. x,y and z are the Cartesian coordinates of the point on the shere. For a pure state, the point likes on the surface of the sphere but for a mixed state, it lies below the surface. For a maximally mixed state, it is in the center.

So what the trajectory represents is the time evolution of the quantum state. The reason I said your collapse operators were too strong is that there wasn't much movement on the sphere. The state decays very rapidly. I hope that helps.

[Vishal Narnaulia]

Yes, I got your point. Thanks a lot!

[Arjit]

I'm trying to plot the time evolution of two level atom by using mesolve but i was not able to get as I'm a beginner in the field , it would be very helpful if someone could help, 

i have taken random density matrix with unit trace and trying to see its evolution with time , of all the 4 states of a two two level system , 

taking Hamiltonian as [sigmaz() * omega /2] , take omega =1 and  every other constant in the simplest way possible to get the plot of all four states evolving in time .

im basically trying to plot all the four ode's given in equation 98 with respect to time.

please help!!