 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2014/paper-1/5/c/solution

Put

C = cos (χ /2)

and

S = sin (χ /2)

∣ χ ⟩ = \sum_{j = 0}^{N} (j N) C^{N - j} S^{j} ∣ N - j, j ⟩ .

Thus

P (N_{1} = j) = (j N) (1 - p)^{N - j} p^{j}, p = S^{2}, \overline{N}_{1} = Np, \overline{N}_{0} = N (1 - p) .

The state is a fixed-number two-mode condensate with binomial mode occupations; it is not a product of two independent Poisson-distributed Glauber coherent states.

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