By Andrei B. Klimov

Written through significant individuals to the sector who're popular in the group, this is often the 1st entire precis of the various effects generated via this method of quantum optics thus far. As such, the ebook analyses chosen issues of quantum optics, targeting atom-field interactions from a group-theoretical viewpoint, whereas discussing the crucial quantum optics types utilizing algebraic language. the final result's a transparent demonstration of the benefits of utilizing algebraic ways to quantum optics difficulties, illustrated by way of a few end-of-chapter difficulties. a useful resource for atomic physicists, graduates and scholars in physics.

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**Extra resources for A Group-Theoretical Approach to Quantum Optics**

**Sample text**

If the atoms are identical (the Hamiltonian is invariant under permutations of atoms), it is sufﬁcient to consider only the symmetric representation. A useful method for obtaining the matrix elements of operators acting in the space of this representation of su(n) is the so-called Schwinger construction. 5 Atoms with n > 2 Energy Levels † We may think of bi and bi as the creation and annihilation operators for the † excitation at the ith atomic level (i = 1, 2, . . , n). In terms of bi and bj , the collective atomic operators have the form † S ij = bj bi , i, j = 1, 2, .

49 ij S+ |k1 , . . , ki , . . , kj , . . , kn = ki (kj + 1)|k1 , . . , ki − 1, . . , kj + 1, . . , kn , ij S− |k1 , . . , ki , . . , kj , . . , kn = kj (ki + 1)|k1 , . . , ki + 1, . . , kj − 1, . . , kn , 1 ij Sz |k1 , . . , ki , . . , kj , . . , kn = (kj − ki )|k1 , . . , ki , . . , kj , . . 50) ij The operators {Sz , S+ , S− }i=j span a representation of the su (n) algebra. 2 Systems with Three Energy Levels Let us consider a particular case of systems with three energy levels.

Kj , . . , kn = kj (ki + 1)|k1 , . . , ki + 1, . . , kj − 1, . . , kn , 1 ij Sz |k1 , . . , ki , . . , kj , . . , kn = (kj − ki )|k1 , . . , ki , . . , kj , . . 50) ij The operators {Sz , S+ , S− }i=j span a representation of the su (n) algebra. 2 Systems with Three Energy Levels Let us consider a particular case of systems with three energy levels. A pure state is a superposition of bare states | j , j = 1, 2, 3, |ψ = cos ϑ1 |1 + eiϕ1 sin ϑ1 cos ϑ2 |2 + eiϕ2 sin ϑ2 sin ϑ1 |3 where 0 ≤ ϕ1,2 ≤ 2π, 0 ≤ ϑ1,2 ≤ π/2.