Rabi frequency and "optical resonance and two level atoms" by Allen

[Paintjunkie]

TL;DR Summary

While reading a book by L. Allen "optical resonance and two level atoms" a factor of two seems to have made its way into the Rabi frequency

This might not really be that tuff a question, so the Rabi frequency...

the definition that I seem to find in multiple locations seems to be in agreement with what's on wiki here: "https://en.wikipedia.org/wiki/Rabi_frequency"

but when I am reading a book by L. Allen "optical resonance and two level atoms" I seem to be getting a factor of two worked in. Hopefully someone here has access to the text and can give me some help. The development is like what's on this wiki page under the semiclassical approach: "https://en.wikipedia.org/wiki/Rabi_problem" but its missing the development on that kE value in the first set of equations. In the Allen book it seems to have a factor of 2 there.

In the book On page 40 he defines a quantity, for kE in equations 2.27. This was part of the optical Bloch equations he derived earlier on page 39. Then later on page 54 I think he says that same quantity is the Rabi frequency. The only difference being that he has made the assumption on page 54 that the quantity is constant in time.

anyways there is a factor of 2 in there... it seems to agree with what I found here on this wiki page "https://en.wikipedia.org/wiki/Vacuum_Rabi_oscillation"

clarification would be awesome

Ill include pictures of the pages.

 

[Cthugha]

Well, if you have a look at the energy levels of the dressed states at zero detuning where the bare states have the same energy, you will find a certain splitting between these states. Now some authors define the energy difference between the dressed states as the Rabi energy. Others define the energy difference between the bare states and one of the dressed states as the Rabi energy. The difference is exactly a factor of 2.

One can see the definition in the equations afterwards. In the former definition, the time-dependent terms scale as $\Omega_R/2$, while in the second defintion they scale as $\Omega_R$. What definition one uses is mainly a matter of taste and convenience. In the latter case the probability amplitudes oscillate with $\Omega_R$, while in the former case the actual occupation probabilities oscillate with $\Omega_R$. Both definitions are commonly used.

 

[Paintjunkie]

hmmm ok. seems to make sense maybe.

If I look at this online thesis "http://www.bgu.ac.il/atomchip/Theses/Amir_Waxman_MSc_2007.pdf" starting on page 12 and going to page 13. He defines his rabi frequency in equation 2.16, and that's because his energy difference, stated at the top of page 12, is ħω₀?

So then looking at "https://www.physik.hu-berlin.de/de/nano/lehre/copy_of_quantenoptik09/Chapter8" their rabi frequency is defined at the bottom of page 90. It is twice that what Waxman has put. Seems to me that their unperturbed Hamiltonian, defined on page 90, has energy levels of H_A = 1/2*ħω₀σ_zħ doesn't that make the energy difference the same ħω₀.

Is there a contradiction here somewhere? I think I might just be misunderstanding something.

 

[Cthugha]

I am not sure I get what you mean. The $\hbar \omega_0$ is the energy difference between the bare states. This is usually called the detuning and called $\delta$ in the script by Benson on page 90. This energy scale is not directly related to the Rabi energy. It just tells you whether the laser is on resonance with the transition or not.
The Rabi energy is related to the strength of the light-matter coupling only, and thus corresponds to the off-diagonal elements of the matrix. If you now set the diagonal elements of the matrix to zero or some constant value, this means that the laser is on resonance with the two-level system. Now, if you diagonalize the matrix you will get new eigenenergies of the so-called dressed states. One state will be at higher energy and one will be at lower energy. This splitting of the diagonalized matrix is the Rabi splitting and it depends only on the light-matter interaction. Based on that, the definition of the Rabi energy in both scripts is the same.
It is given by Waxman by the definition $V_{01}=\frac{1}{2}\hbar \Omega$ and by Benson by $\Omega=\frac{2}{\hbar}|\langle|e|H_1|g\rangle|$, where $V_{01}$ corresponds to $|\langle|e|H_1|g\rangle|$.

 

[Paintjunkie]

Ok I think I was misunderstanding, probably still am. so Waxman definition in equation 2.11for V₀₁ is consistent with equation 2.16? or should there be a factor of 2 in equation 2.16?

 

[Cthugha]

Ah, now I see what you mean. No, this is not really consistent. I think he noticed the mismatch and invoked his polarization argument to explain the difference, but that is sloppy.

 

[Paintjunkie]

Well I don't know what to do... because when follow Allen and do my own calculations to arrive at what Waxman has as equation 2.28, I am off by a factor of 2. I think its cause of this inconsistency.

 

[Paintjunkie]

to be a little more clear I get the same answer for equation 2.26, Waxman has a negative in the wrong place, but it does not matter. The answer in 2.26 is correct