A. Show that the beam splitter transformation equations of Eq. (6.10) do indeed follow from Eq. B. Consider the operator U(_) = exp[i _ J 1] where J 1 is defined as the operator J 1 _ (a 0a1 + a 1a0)/2. Obviously for _ = _/2 we obtain the transformation for the 50:50 beam splitter of Eqs. (6.10) and (6.11). For the more general case where the beam splitter is not 50:50, obtain the corresponding transformation of the mode operators. Relate the angle _ to the parameters r, t,r , t .
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