Dark state & schrödinger equation

[Mahdimatika]

Hello, this is my first thread here and i have a question regarding coherent population trapping and the dark state. I am from Germany, so please excuse me, when my english isn't very good. This is what I found on wikipedia:

http://en.wikipedia.org/wiki/Dark_state

in the section "Three-level systems" they give a semiclassical Hamiltonian and solve Schrödingers eq. However, in the end they give solutions for the functions c_i(t) and i think especially c_3(t) is interesting. In the text they say, that with the right initial conditions, we can gain c_3(t)=0, which is exactliy what is needed for electromagnetically Induced Transparency to work. When I try to find the initial conditions on my own, i would set c_3(t)=0, and would gain the solution

(1) c_3(0)=0
(2) c_1(0)*Ω_p=c_2(0)*Ω_c

The second equation allows the values c_1(0)=Ω_c/Ω and c_2(0)=Ω_p/Ω, which would give the state:

|ψ(t)> = (Ω_c*Exp(-i*ω_1*t)|1> + Ω_p*Exp(-i*ω_2*t)|2>)/Ω

Wikipedia though, as well as other literatures, states the dark state as:

|D> = (Ω_c*|1> + Ω_p*|2>)/Ω

Apperently i am missing something crucial here. my questions now are the followong:
1. Where have the exponential expressions gone? Or how do i get the dark state |D>

2. Equation (2) leaves space for more than one solution. does that mean, there is more than one way to set up two lasers, to induce coherent population trapping in a medium?

Thank you for your time.

 

[DrClaude]

1. Where have the exponential expressions gone? Or how do i get the dark state |D>

Set $t=0$

2. Equation (2) leaves space for more than one solution. does that mean, there is more than one way to set up two lasers, to induce coherent population trapping in a medium?

I don't see which other solutions you get.

I will try to write a more substantive answer tomorrow.

 

[Mahdimatika]

First, thanks for your answer.

I don't see which other solutions you get.

I forgot about normalization condition. That is, why i found more than one solution. But now it is clear to me.
I still have a few questions. I am trying to understand, how electromagnetically induced transparency works. I will try and write down, what i understood so far.

when using a laser with a frequency that matches the transition-frequency between a state |1> and a state |3>, you can prevent the atoms from absorbing the light, by using a second so called coupling laser, with a frequency that matches the transition frequency between a state |2> and |3>. State |1> and |2> are hyperfinestates, i don't know, if they have to be hyperfinstates though. if you do that, then the atoms will enter the dark state, and the Light with frequency $$\omega_p$$ (according to wikipedias notation), will propagate in the medium with hardly any inensity loss.
well, at this point i have many questions. first of all, i don't understand how the method of EIT influences light speed. There is a method to cool lasers into the scale of nano-Kelvin in which the Light speed can even be reduced to 17 m/s. Is that cooling process also done with the aid of EIT?

I find that subject really interesting, but also hard to understand. It would be nice, if someone would take his time to aid me with an explanation.

 

[Zarqon]

A couple of answers at least:

An explanation for how dark states work with words would be something like: In a three level system like you described, with two laser beams there are two possible ways for the system to go between the two ground states. 1) via a resonant absorption of 1 photon to the excited state followed by decay down to the other state and 2) via a simultaneous 2-photon interaction where one photon from each laser is absorbed together and the atom makes the ground state transition directly without passing through the excited state. These two paths can both happen, and in the quantum mechanical world, to figure out what actually happens you have to sum up the coherent amplitudes from both paths, and then you find interference. The dark state happens when you have destructive interference (determined by the relative phases) and the bright state is when you have constructive interference and you get an increased transition probability.

Then, no, it doesn't have to be hyperfine states, it works with any 3 levels where the conditions of two interfereing paths are fulfilled.

And about slow light, the slow light oocurs because the EIT transmission peak is usually a very narrow band phenomena, and wherever you have narrow absorption features you also get very steep slopes on the dispersive effects (see the Kramers-Kronig relations). This directly translates into a greatly changed group refractive index and hence either slow or fast light. To see this in more details check out the formula for group refractive index.

 

[Mahdimatika]

First thank you for your time. Now I understand a lot more. I must say, that I still don't find a difference in coherent population trapping and electromagnetically induced transparency. I read in reviews of physics, volume 77, April 2005, that CPT occurs in optical thin samples while EIT occurs in optical thick media. But I don't understand why that is.

 

[Zarqon]

I don't think there is a difference. I've seen people call it differernt things, sometimes for a reason, like they say it's CPT when the two lower levels are degenerate. But from my view it's exactly the same physics, so it's the same thing.