By Andrei B. Klimov

Written through significant members to the sphere who're popular in the neighborhood, this is often the 1st finished precis of the numerous effects generated via this method of quantum optics up to now. As such, the booklet analyses chosen subject matters of quantum optics, targeting atom-field interactions from a group-theoretical point of view, whereas discussing the valuable quantum optics types utilizing algebraic language. the final result's a transparent demonstration of some great benefits of making use of algebraic how to quantum optics difficulties, illustrated through a couple of end-of-chapter difficulties. a useful resource for atomic physicists, graduates and scholars in physics.

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**Example text**

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. 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.

We begin with the simplest example: the evolution in a constant magnetic ﬁeld. Let us denote the magnetic ﬁeld vector by H and the magnetic moment of the spin by µ = µσ, where σ = {σx , σy , σz }. 1) The solution of the Schr¨odinger equation ( = 1), i ∂t |ψ(t) = H|ψ(t) has the form | (t) = U(t)| (0) , U(t) = exp(−iHt) where U(t) is the evolution operator. In this case, the exponential exp(−iHt) can be directly evaluated by expanding it in a Taylor series. Since H2 = 2 I, where I≡ 1 0 0 1 , = µ H2x + H2y + H2z A Group-Theoretical Approach to Quantum Optics: Models of Atom-Field Interactions Andrei B.

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. In terms of the diagonal projectors | j j|, the free Hamiltonian for a single atom has the form H0 = E1 |1 1| + E2 |2 2| + E3 |3 3| so that |1 1| + |2 2| + |3 3| = I.