I Lifetime and Linewidth: Solving the 2π Factor

[Malamala]

I know this is a simple formula, but I got confused online about the $2\pi$ factor. If I have, say $100$ ns lifetime, what is the associated linewidth? Is it $1/(100 \times 10^{-9} s) = 10$ MHz? Or is it $10/(2\pi) = 1.6$ MHz? Thank you!

 

[gleem]

All derivations that I have seen give lifetime = τ = h /2π Γ

 

[Malamala]

All derivations that I have seen give lifetime = τ = h /2π Γ

So the formula would be $\tau = \frac{1}{\Gamma}$, given that usually $\hbar = 1$, right? So the linewidth is 10 MHz?

 

[gleem]

Usually for calculations ħ = 1.05457 . . .10^-34 joules⋅sec

anyway Γ = h /2πτ = h ν → ν = 1 / 2πτ = 1.5916 MHz

 

[Malamala]

Usually for calculations ħ = 1.05457 . . .10^-34 joules⋅sec

anyway Γ = h /2πτ = h ν → ν = 1 / 2πτ = 1.5916 MHz

Thanks for this! I get the same result as you, but I am a bit confused by the values quoted in literature. For example in the paper attached below (this is how my confusion started), in the last paragraph of the first page, they claim a FWHM of 30 MHz, while in the last paragraph on the second page they mention a lifetime (of the same levels) of 46.1 and 56 ns. Applying the formula above I would get a FWHM of ~3 MHz? Where is this factor of 10 difference coming from. Are there multiple definitions of the used terminology?

 

[gleem]

I am not familiar with this research and its nomenclature. However, as a guess, I think the relationship between the first and second page is that the 30 Mhz linewidth on page one may have been from an earlier experiment of lower resolution which would have a lifetime of 100 nsec. The second page relates to a later higher resolution experiment which resolved the previous 30 Mhz peak into three peaks (lines) one with a lifetime of 46 nsec and one with 56 nsec. lifetime the sum of which adds to 100 nsec. The third was too short to measure. This is probably wrong but the best I can do at this time.

 

[Malamala]

I am not familiar with this research and its nomenclature. However, as a guess, I think the relationship between the first and second page is that the 30 Mhz linewidth on page one may have been from an earlier experiment of lower resolution which would have a lifetime of 100 nsec. The second page relates to a later higher resolution experiment which resolved the previous 30 Mhz peak into three peaks (lines) one with a lifetime of 46 nsec and one with 56 nsec. lifetime the sum of which adds to 100 nsec. The third was too short to measure. This is probably wrong but the best I can do at this time.

Thanks for your reply! Actually the 2 transitions they are talking about are very far apart (on the order of THz). This is a molecular paper so it is a bit more complicated than atoms, but basically the 2 transitions they are talking about, they claim to both have ~30 MHz, then they say the first one has a 46.1 ns and the second one 56 ns lifetime (in this case there is no 3rd transition). So I am quite sure that the resolving power was not an issue, but I will look more in the literature (whether is atoms or molecules the formula should be the same).

 

[Cthugha]

This is a molecular paper so it is a bit more complicated than atoms, but basically the 2 transitions they are talking about, they claim to both have ~30 MHz, then they say the first one has a 46.1 ns and the second one 56 ns lifetime (in this case there is no 3rd transition).

They claim to use laser-induced fluorescence detection in the first paragraph and then state that they can do so at near the natural linewidth limit of 30 MHz. This is the resolution of the laser-induced fluorescence detection and should therefore be the linewidth of the laser. It is not related to the linewidth of the transitions at all (except that it is an upper limit for the actual linewidth of the transitions), but it is the resolution limit they can achieve using this method.