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Lectures on QuTiP: slides and ipython notebooks
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Jaynes Cumming model - sigma minus vs sigma z #3

Open mklilley opened 4 years ago

mklilley commented 4 years ago

Hi.

I'm very new to Qutip and still in the process of refreshing my quantum physics knowledge so please forgive me if this is a silly question.

In the notebook on the Jaynes Cumming model I cannot understand why the hamiltonian is written with the sm.dag()*sm for the atomic part in the following:

H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())

When I look at the description of Jaynes Cumming model that you give and also on Wikipedia it seems like we should be using something like sigmaz instead, i.e.

H = wc * a.dag() * a + wa *tensor(qeye(N),sigmaz())+ g * (a.dag() * sm + a * sm.dag())

I wondered whether they might be the same, but they do not appear to be when I calculate them.

Thanks for your time.

Matt

lukelbro commented 4 years ago

Hi Matt,

Did you get to the bottom of this? I am also confused.

Luke

mklilley commented 4 years ago

Hey. No one replied to me so never completely got to the bottom of it. I am however pretty confident now that it's just an error in their code.

lukelbro commented 4 years ago

Hi Matt,

I found Chapter 15 of Optical Coherence and Quantum Optics - Leonard Mandel, Emil Wolf very useful for figuring out what was going on.

Our confusion is about the atomic Hamiltonian which is written as either:

A harmonic oscillator:

Where and so that in the excited state = and in the ground state = 0.

Equivalent to a spin 1/2 particle in a magnetic field:

Where , is the inversion operator (which is often written as ). Be careful as this is subtly different to Pauli Z. Your second Hamiltonian should be:

H = wc * a.dag() * a - wa * tensor(qeye(N),sigmaz())+ g * (a.dag() * sm + a * sm.dag())

The reference energy level is usually dropped as a constant does not affect the dynamics. If is in the ground state we can see that , and the hamiltonians are equivalent.

Sorry about the weird \rangles and \langles, I can't figure out why they are rendering like that.

mklilley commented 4 years ago

Thanks @lukelbro . Forgive me, I can't see the difference between the Hamiltonian you wrote and my second one...maybe I just need some more coffee lol.