By Andrei B. Klimov

Written via significant participants to the sphere who're renowned in the neighborhood, this is often the 1st entire precis of the numerous effects generated by way of this method of quantum optics so far. As such, the ebook analyses chosen subject matters of quantum optics, targeting atom-field interactions from a group-theoretical viewpoint, whereas discussing the central quantum optics versions utilizing algebraic language. the final result's a transparent demonstration of the benefits of making use of algebraic ways to quantum optics difficulties, illustrated via a few end-of-chapter difficulties. a useful resource for atomic physicists, graduates and scholars in physics.

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**Example text**

C. 48, the operators describing a collection of A indistinguishable atoms with three energy levels. There are still three conﬁgurations, and appropriate operators are simply obtained by substituting jk jk sz ↔ Sz and E → AE. Then the atomic transition operators are : S12 ±, S23 ± : S13 ±, S23 ± V : S12 ±, S13 ± jk jk The matrix realization of each Sz , S± for each conﬁguration form a symmetric irreducible representation of the su(3) algebra of the dimension (A + 1)(A + 2)/2. 51) and all the Sz operators commute.

8) k which corresponds to a system on decoupled oscillators. 12) k The ﬁrst sum corresponds to an inﬁnite set of noninteracting quantum oscillators, which allows us to use all the techniques developed in the study of the quantum harmonic oscillator. Each term in the second sum represent the vacuum energy of the appropriate mode and will be omitted in future calculations. The total Hilbert space is a direct product of Hilbert spaces corresponding to each ﬁeld mode. 13) where the vectors |nk describe the states with a deﬁned energy, that is, with a ﬁxed number of photons in the kth ﬁeld mode.

20) where α = |α| eiϕ is a complex number. 22) 47 48 3 Quantized Electromagnetic Field we ﬁnd the recursion relation for the coefﬁcients cn : can be easily solved: √ n + 1 cn+1 = α cn , which αn cn = √ c 0 n! 23) where c0 is determined from the normalization condition α|α = 1. This yields c0 = exp −|α|2 /2 . Thus, we get |α = e−|α| 2 /2 n αn √ |n n! 24) which means that the coherent states can be considered as generating functions for the number states: 1 dn 2 e(|α| /2) |α |n = √ n n! 25) α=0 ˆ the relation between its matrix elements in Likewise, we obtain for any operator X, the basis of coherent states and in the basis of number states: ˆ m|X|n = √ dn dn 1 2 2 ˆ e(|α| /2)+(|β| /2) β|X|α √ n dα∗n dβ m!