Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 5 / c

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- Solution c

Solution

Put

C = cos (χ /2)

and

S = sin (χ /2)

. Commutativity of the two creation operators and the binomial theorem give

∣ 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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