 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2019/1/c/iii/solution

The physical branch can be plotted without a nonlinear solver by parameterizing

r (x) = \frac{- l n ( 1 - x )}{x}, 0 < x < 1.

It begins at

(r, x) = (1, 0)

and increases monotonically to

x = 1

r \to \infty

. Expanding near threshold gives

r = 1 + x /2 + O (x^{2})

, so

x ≃ 2 (r - 1)

. For strong over-pumping the residual inversion fraction is exponentially small,

1 - x ≃ e^{- r}

. Thus the curve saturates at the upper bound

β η = 1

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