Calculate the group velocity in EIT (famous paper: light speed 17m/s)

[wangvivi]

TL;DR

calculation about the group velocity in electromagnetic induced transparency(EIT) in the paper:Light speed reductionto 17 metres per second in an ultracold atomic gas.my result is not agreement with the published one.

hello everyone!
Recently,i'm reading a paper about slow light,that's really a famous work published in Nature.[Light speed reduction to 17 metrespersecond in an ultracold atomicgas].
But I'm trouble with some calculation about the velocity of slow light.here are below:

i try to use the parameters in this paper and plug them into function(1) to caculate the group velocity of the slow light,the resulting light speed in this paper is 32.5m/s,but my result is 2.94m/s,i checked all the parameters but can't know where the problem is.
Other parameters are below:
function（1）is the group velocity of slow light. in this function,
h/bar is reduced Planck constant,
c is the light velocity in vaccum,
epsilon_0 is vacuum permittivity,
Omega_c is the rabi frequncy of coupling light,
omega_p is the frequncy of probing light,
u13 is the dipole matrix element between 1 and 3,
N is the atomic density.
paper has been attached.
by the way,maybe the dipole matrix element of sodium between zeeman state is not 2.988*10^-29 C·m？
can anybody tell me why my result is not in agreement with the experimental result in this paper?
thanks a lot!
hope everyone well!

 

[mfb]

Here is the paper

The peak density will be higher than the average density, which means the average speed is higher than the slowest speed. That could cause a discrepancy.

You could email the authors and ask if no one here finds a solution.

 

[wangvivi]

Here is the paper

The peak density will be higher than the average density, which means the average speed is higher than the slowest speed. That could cause a discrepancy.

You could email the authors and ask if no one here finds a solution.

thank you so much!actually I considered this situation.
I'll email the authers and post their answer if they reply me.
thank you!

 

[WernerQH]

can anybody tell me why my result is not in agreement with the experimental result in this paper?

It seems the experimental result is simply calculated from $v_g = L / \tau_\text{delay} = 229~{\rm \mu m} / 7.05~{\rm \mu s} = 32.5~{\rm m/s}$. The theoretical equation (1) should produce values in the right ballpark, but is perhaps better used to estimate the density $N$.

I noticed that you substituted $c/\lambda$ for $\omega_p$, but it should probably have been $2 \pi c/\lambda$. Unfortunately that change goes in the wrong direction. I think it's also unusual to express the Rabi frequency $\Omega_c$ in $\rm rad~s^{-1}$, rather than cycles per second, but I'm not an expert in the field and haven't studied the quoted publications.

 

[vanhees71]

But from the paper it seems that they exclusively work with angular frequencies rather than frequencies, as expected in the modern quantum-theoretical literature.

 

[wangvivi]

It seems the experimental result is simply calculated from $v_g = L / \tau_\text{delay} = 229~{\rm \mu m} / 7.05~{\rm \mu s} = 32.5~{\rm m/s}$. The theoretical equation (1) should produce values in the right ballpark, but is perhaps better used to estimate the density $N$.

I noticed that you substituted $c/\lambda$ for $\omega_p$, but it should probably have been $2 \pi c/\lambda$. Unfortunately that change goes in the wrong direction. I think it's also unusual to express the Rabi frequency $\Omega_c$ in $\rm rad~s^{-1}$, rather than cycles per second, but I'm not an expert in the field and haven't studied the quoted publications.

thanks for your reply!
Actually,I considered what you said.
First,the time delay is an experimental result,so 32.5m/s is also an experimental result,you are right.what I want to do is use the function（1）as a theoretical result to compare with 32.5m/s.
Secondly,I analysed the dimension of this function.you can see my picture:

In the function(1),if you use rad/s for omega_p and rabi frequncy,you get rad·m/s,not m/s.is that right?so I considered the SI units when I used these parameters.
parameters are from:https://steck.us/alkalidata/sodiumnumbers.1.6.pdf;the relationship between international system of units is from:https://en.wikipedia.org/wiki/Farad
I don‘t know much about this field either.so thank you for your suggestion.I will use rad/s for rabi frequency.and to my calculation,maybe the peak density will be higher than the average one.
If you find any problems in my answer,could you please contact me?I’m very gald to discuss this question with you.

 

[wangvivi]

But from the paper it seems that they exclusively work with angular frequencies rather than frequencies, as expected in the modern quantum-theoretical literature.

thank you.I noticed that.could you please read my reply to WernerQH?I just posted it.
If you find any problems in my answer,please contact me.thank you!

 

[vanhees71]

The only thing I see is that you attempt divide $\Omega_c$ by $2 \pi$. That's not in the formula, but if you think it must be there, you better put it in parenthesis to make sure that both factors are in the denominator. Also rad=1 of course. Then you have $\omega_{\text{P}}=c k_{\text{P}}=2 \pi c/\lambda_{\text{P}}$.

 

[WernerQH]

In the function(1),if you use rad/s for omega_p and rabi frequncy,you get rad·m/s,not m/s.is that right?so I considered the SI units when I used these parameters.

It is always wise to check the physical dimensions.
But people are not always consistent, and then factors of $2 \pi$ can slip in.
I think vanHees71 is right that in this case you should set $\text{rad} = 1$.

When in doubt, consult the given references (16-19).