More specifically, how do you physically prepare the superposition of two beams

Question

More specifically, how do you physically prepare the superposition of two beams of polarized light?

Suppose beam A and beam B, the two input beams, each have unknown polarizations.
Beam C, the output beam, must be the superposition of beams A and B.
How do you do it?

You can create an overlap of two beams with a half silvered mirror at 45%.
You can also creat a localized overlap by crossing the beams with a small angle.
In both cases you get a mixture, not a superposition.

How do you get the superposition?

Bonus Points: How do you prepare a|A?+b|B??

Explanation / Answer

Your problem is ill-posed, becaus the state of a beam is not described by a state vector ?, but by the corresponding density matrix ???, which determines ? only up to a phase. Thus it is meaningless to talk about ''the'' superposition of two beams.

If the two beams are known to have a fixed, but unknown, relative phase, the question becomes well-defined. The only way to gain this knowledge is to check that the beams have been coherently generated from a common source. In this case, a half-silvered mirror will produce the required superposition, and one can change the relative phase by a rotator (as described in the book by Mandel and Wolf).

For preparing arbitrary states of a collection of beams from a single coherent beam see my paper http://de.arxiv.org/abs/quant-ph/0306123. See also
M.Reck, A. Zeilinger, H.J. Bernstein and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett. 73 (1994), 58

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