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Precision Dark Resonance Spectroscopy
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An intuitive (?) explanation

Here is a three-step explanation of coherent dark states and how they relate to "conventional" dark states and optical pumping. We assume that there is no decay or dephasing between the two ground states | 1 > and | 2 >.


Step 1: Just laser frequency 1 present: conventional dark state

Lambda's right leg ripped out

Let us start with a ground state population initially in thermal equilibrium (equal populations in | 1 > and | 2 >, state | 3 > unoccupied). Atoms in state | 1 > can absorb a photon from laser 1 and be excited into state | 3 > from where they decay spontaneously into | 1 > or | 2 >. If they decay into | 1 > they can be excited again. If they end up in state | 2 >, however, they cannot absorb photons any more because laser 1 is not resonant on the transition 2->3. Population in | 2 > therefore remains trapped in this "dark state". It is called a "dark state" because after a few absorption-emission cycles on transition 1->3 all atoms end up in (have been "optically pumped into") state | 2 > and no more absorption and emission is possible: the atoms do not scatter photons any more and therefore appear dark.

The existence of dark state | 2 > is not a property of the atom alone but only of the combination of atom plus the light fields, as can be seen in the next step.


Step 2: Just laser frequency 2 present: conventional dark state

Lambda's left leg ripped out

Here everything works the same as in step 1, except that now atoms can be pumped out of state | 2 > while in | 1 > they cannot be excited any more. Now state | 1 > is a dark state.


Step 3: Both laser frequencies present: the coherent dark state

The situations depicted in steps 1 and 2 with one laser intensity zero are just limiting cases of the experimental situation (where both laser beams are present simultaneously). In fact, one can go continuously from situation 1 to situation 2 by gradually increasing laser intensity 2 and decreasing laser intensity 1. In the intermediate steps, the light field is a superposition of the two laser fields with electric field strengths E1 and E2. One can therefore guess that the dark state is also a coherent superposition of the two limiting cases, something like

| coherent dark state > = A | 1 > + B | 2 >

where A and B are some suitable scaling factors, mainly containing the strengths of the two laser fields (more specifically: the Rabi frequencies on the two transitions).

The QuickTime movie below visualizes the behavior of the coherent dark state as a function of the relative laser intensities. The radii of the green circles and the thickness of the red and blue lines are proportional to the respective Rabi frequency, and the green dotted line symbolizes the strength of the ground state coherence created by the two lasers.

Just play with the movie controls:
stop the movie, step through it
frame by frame, whatever...
In order to play this movie you need the QuickTime-Plugin of Apple Computer for your browser

Caution: this simple picture is only strictly true when there is no relaxation between | 1 > and | 2 >! If this is no longer a valid assumption the full theoretical treatment has to be used (E. Arimondo, Progress in Optics 35, 257 (1996)).