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Additional resources for Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light

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The important characteristics of this function are the following: • it has its maximum value of T 2 at E = 0; • its width is of order 2πh¯ /T; • its area is proportional to T, or more precisely:7 +∞ −∞ dE gT (E) = 2πh¯ T. 37) 0 2πh T E Form of the function gT (E) = (T sin(ET/2--h)/(ET/2--h))2 . Its value at E = 0 is T2 and the first zeros are at E = ±2π--h/T. 6 As we pointed out in the introduction, this calculation applies equally to the case of a Hamiltonian H ˆ0 +W ˆ ˆ 0. independent of time, but when the system is prepared and subsequently detected in an eigenstate of H 7 Its value is half the product of its height and of the distance between the two first zeros, as though the function gT were triangular.

Perturbation theory consists of developing γk (t) as a power series in λ (which, we recall is much smaller than unity): (0) (1) (2) γk (t) = γk (t) + λγk (t) + λ2 γk (t) + ... 17) we can collect together terms of the same order in λ. 20) ˆ 1 (t) |n ei(Ek −En )t/h¯ γn(r−1) (t). 21) ih¯ • to order 1 ih¯ d (1) γ (t) = dt k n • to order r ih¯ d (r) γ (t) = dt k n This system of equations can be solved iteratively. In fact the zeroth-order terms are already known: they are constants determined by the initial state of the system.

T 14 The evolution of interacting quantum systems In the case in which ωki is negative, the condition for quasi-resonant excitation is fulfilled when |ωki + ω| ω. 47). 38) to within a factor of four if one replaces Ek −Ei by Ek −Ei ± h¯ ω. 49) ⎪ πh¯ πh¯ ⎪ ⎩Ei − hω − < E − hω + < E . ¯ ¯ k i T T For sufficiently long interaction times the only possible transitions will be those to states |k for which the energy separation Ek − Ei is exactly hω: ¯ the change in the atomic energy occurs by the absorption or emission of a quantum of energy h¯ ω.

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