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In order to see the essential physics happening in the three-level Lambda system under bichromatic illumination one can choose a more suitable basis {| 3 >, | + >, | - >}, consisting of the excited state and of what is essentially the symmetric and the antisymmetric superposition of the two ground states, respectively. In particular, state | - > looks like this in the standard energy eigenstate basis:
| - > = ( R2 | 1 > - R1 | 2 > ) / Reff
where R1 and R2 are the Rabi frequencies on the transitions | 1 > -> | 3 > and | 2 > -> | 3 >, respectively, with the usual definition of the Rabi frequency via the matrix elements d12 or d13 of the electric dipole operator:
R1 = -2 p d13 E1/ h, R2 = -2 p d23 E2 /h.
Here E1 and E2 are the electric field amplitudes of the light waves with frequencies w1 and w2, and Reff is simply the Pythagorean mean of the two Rabi frequencies:
Reff = (R12+R22)1/2.
State | - >, which is in fact the coherent dark state, is not an energy eigenstate of the system, so it develops in time basically like
| - > = ( R2 | 1 > exp(i W1t) - R1 | 2 > exp(i W2 t ) / Reff
where W1 and W2 are the energies of the respective levels divided by h/2p. In order to prove that | - > really is the coherent dark state, one can calculate the transition amplitude into the excited state:
< 3 | d • E | - > = R1 R2 (exp(i w1t + i W1t) - exp(i w2t + i W2t)) / Reff2
This expression is identically zero for all times when the two exponents are equal, i.e., when the laser difference frequency matches the ground state splitting W1 - W2: the Raman resonance!
How does the system find this lucky "dark" superposition? Well, it reaches the coherent dark state through optical pumping: it keeps being excited into | 3 > until from there it spontaneously decays into the coherent superposition of ground states that is the coherent dark state. In this state the population remains trapped, and the capture process is called "coherent population trapping". The result is that a medium which is absorbing for each of the two laser fields individually becomes transparent through the electromagnetic interaction: "electromagnetically induced transparency". However, here (if not earlier) the simple mathematical picture breaks down because one has to include relaxation. In fact, without dephasing between the two ground states the system can never reach the dark state! On the other hand, with ground state dephasing the simple mathematical picture presented above is untenable.
A fully satisfying explanation of coherent population trapping is beyond the scope of this web page. You can find all the details you want in a nice review article by E. Arimondo, Progress in Optics 35, 257 (1996).
You can also find a small QuickTime movie that visualizes the behavior of the coherent dark state as a function of the relative laser intensities, although only for the case of negligible relaxation.