# Alkali atom - photon interaction in zero magnetic field

An alkali atom has a single outer electron that interacts with incoming photons of the right wavelength (for alkalies it's in the visible & IR range). If there is an external magnetic field, the electron has a well defined quantization axis and the the incoming light can be separated into three components: linearly polarized and left/right-handed circular polarized parts. The interaction strength between the different atomic energy levels (levels described by different quantum numbers) and each of the polarization components is relatively straightforward to calculate (with the appropriate Clebsch-Gordan coefficients and 6-j symbols).

What is then the situation if there is no external magnetic field to provide the quantization axis? Is "polarization" meaningful from the atoms point of view? How one sets out to calculate the interaction strength for the different polarizations in the laboratory frame?

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Basically, if there is no magnetic field, you are free to chose any quantization axis.

In theory, the quantization axis choice can be arbitrary, various axis choice corresponding to different "coordinate choices" in the Hilbert space. However, when you have a magnetic field, the state corresponding to axis different than the magnetic field direction are not eigenstates of the energy, and choosing the "right" axis makes the maths simpler. When you have no field, the different states are degenerate, and every choice is as simple as the others.

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The choice of the axes is arbitrary, but if you're dealing with light incident from a well-defined direction, that naturally picks out a preferred axis. In the zero field case, you would take the direction of the incoming light beam as your quantization axis.

The polarization of the light is meaningful even without an external field, because photons carry angular momentum that depends on the polarization. That's why the different polarizations drive different transitions in the atoms: absorbing right-hand circular polarization increases the angular momentum of the electron by one unit (in addition to raising it to a higher energy state), and absorbing left-hand circular polarization decreases the angular momentum of the electron by one unit, with the angular momentum being directed along the direction of propagation of the light.

If you're dealing with light incident from multiple different directions, well, that's kind of a mess. Each individual interaction would have its own quantization axis, and you would need to do the appropriate projection of the state for the next interactions. And, of course, unless extreme care is taken to shield out stray magnetic fields, whatever small local field you may have will pick out a preferred direction, which also complicates matters. This is why people doing experiments where they really care about the distribution over Zeeman sublevels tend to apply a known field, just to make life easier.

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