Time evolution of an open lambda type system
Arpita Das
Barker, Daniel S. (Fed)
Hello Arpita,
I believe that pylcp can handle this case. I would start with the STIRAP example: https://python-laser-cooling-physics.readthedocs.io/en/latest/examples/basics/08_stirap.html. You can then add another H_0 block, (say ‘d’), by hand and couple it to the excited state by adding a corresponding d_q_block (this is all in the 2nd cell of the example). As long as there is no laser coupling ‘d->e’, the system should behave in the way that you want.
If you need a more complicated level structure that, say, includes hyperfine couplings, the approach of adding a new manifold should still work. However, it’ll be a bit more cumbersome to implement.
Daniel
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Arpita Das
Eckel, Stephen P. (Fed)
Hello Arpita,
I just went through your code and have two suggestions:
- Try to keep the g->e and r->e gamma = 1, and scale the l->e gamma accordingly. By changing the g->e and r->e, you are also changing the duration of your STIRAP pulse, since that is measured in units of gamma). This way you can more easily separate out the effect of the “l” state.
- I think most of your trouble was in plotting the results. I changed around your plotting code to make it a little more clear what it is that you are seeing. Rather than defining the factors before the loop, as was done in the example, I calculate them just before plotting each of the populations and update a legend entry appropriately.
fig, ax = plt.subplots(2, 1, figsize=(3.25, 2.))
linespecs = ['-', '--', '-.',':']
labels = ['gg','rr','ll','ee']
for ii in range(4):
factor = 10**-np.ceil(np.log10(np.amax(np.real(sol1.rho[ii, ii]))))
ax[0].plot(sol1.t/1e3, factor*np.real(sol1.rho[ii, ii]),
linespecs[ii], color='C%d'%ii, linewidth=0.75,
label='$\\rho_{%s}$'%labels[ii] if factor==1. else '$10^{%d}\\rho_{%s}$'%(np.log10(factor),labels[ii]))
factor = 10**-np.ceil(np.log10(np.amax(np.real(sol2.rho[ii, ii]))))
ax[1].plot(sol2.t/1e3, factor*np.real(sol2.rho[ii, ii]),
linespecs[ii], color='C%d'%ii, linewidth=0.75,
label='$\\rho_{%s}$'%labels[ii] if factor==1. else '$10^{%d}\\rho_{%s}$'%(np.log10(factor),labels[ii]))
[ax[ii].set_ylabel('$\\rho_{ii}$') for ii in range (2)];
[ax[ii].set_xlim((0, 1)) for ii in range(2)];
ax[0].legend(fontsize=7)
ax[1].legend(fontsize=7)
ax[1].set_xlabel('$\Gamma t$')
ax[0].xaxis.set_ticklabels('')
fig.subplots_adjust(bottom=0.18, left=0.14)
plt.show()
With this plotting I get the attached plots for the l->e gamma equal to 30 (and r->e and g->e gamma=1), which, at least at first glance, look correct to me. When STIRAP is done in the incorrect order with r->e pulse applied first (top plot), you get a lot of population into the excited state which get dumps mostly into the l state (note the \rho_{gg} population is scaled by a factor of 10). Then the g->e pulse comes along and takes the small amount of population that ended up in g and dumps it into l. In the correct order (bottom plot) you avoid accumulating population in the excited state and get roughly 50 % transfer to the g state. What little population you had in the excited state mostly gets dumped to the “l” state, ruining your transfer efficiency. But the correct order is significantly better than the incorrect order.
Hope that helps!
-Steve
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