Velocity selective coherent population trapping (VSCPT)

-Works for He atoms
Level scheme:

Two circularly polarized laser beams resonant on transitions as shown above.
Characterize state of atom by internal quantum numbers and momentum p: |e₀,p> coupled to $\vert g_{-},p-\hbar k>$ and $\vert g_{+},p+\hbar k>$ by laser light (closed family of states, neglecting spontaneous emission)
Transition matrix element:
$<g_{\pm},p\pm\hbar k\vert V_{int}\vert e_{0},p>\propto\mp\hbar\omega_{Rabi}$ ( $\hbar\omega_{Rabi}\propto eE<g\vert x\vert e>$),
sign from Clebsch-Gordan coefficent (given in graph)
Non-absorbing state: $\vert\psi_{NA}(p)>=\frac{1}{\sqrt{2}}(\vert g_{-},p-\hbar k>+\vert g_{+},p+\hbar k>)$
For $p\not=0$: Mixes with absorbing state
$\vert\psi_{A}>=\frac{1}{\sqrt{2}}(\vert g_{-},p-\hbar k>-\vert g_{+},p+\hbar k>)$
(state vector oscillates between $\vert\psi_{A}>$ and $\vert\psi_{NA}>$ with frequency $\Delta E/\hbar=2kp/M$, i.e. the difference in kinetic energy
$(p+\hbar k)^{2}/2M-(p-\hbar k)^{2}/2M$)
For p=0 this is a stationary state because $<e_{0}\vert V_{int}\vert\psi_{NA}>=0$, and also radiatively stable (mixture of ground states)
Absorption rate of ground state $\propto (kp/M)^{2}$.
Atoms with p=0 are trapped in this state indefinitely. Spontaneous emission randomizes p (recoil momentum along z-axis random variable between $-\hbar k$ and $+\hbar k$) and gives atoms chance to fall into the p=0 trap.
$\vert\psi_{NA}>$ is not a p-eigenstate $\Rightarrow$ two peaks in momentum distribution