By Andrei B. Klimov

Written through significant individuals to the sphere who're popular in the neighborhood, this is often the 1st complete precis of the numerous effects generated by means of this method of quantum optics thus far. As such, the ebook analyses chosen subject matters of quantum optics, concentrating on atom-field interactions from a group-theoretical viewpoint, whereas discussing the critical quantum optics types utilizing algebraic language. the general result's a transparent demonstration of the benefits of utilizing algebraic easy methods 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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**Extra resources for A Group-Theoretical Approach to Quantum Optics: Models of Atom-Field Interactions**

**Sample text**

1 Spin in a Constant Magnetic Field As we have already mentioned, a two-level atom can be described in terms of isotopic spin. Thus, the dynamics of a two-level atom in a classical external ﬁeld is equivalent to the dynamics of a spin 1/2 particle in a corresponding magnetic ﬁeld. 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.

3 For the case of A = 2 atoms, ﬁnd the evolution of the Bloch vector in the circularly polarized ﬁeld if initially one of the atoms was excited. 4 Find the evolution operator of a collection of A two-level atoms for the Hamiltonian ∞ H = ωSz + δT (t) Sx , δT (t) = χ δ (t − nT) n=0 at the instant NT + ε, ε → +0 and analyze the evolution of the initial nonexcited atomic state. 5 Find the evolution operator for a three-level atom with conﬁguration whose dynamics is described by the Hamiltonian (E1 ≤ E2 ≤ E3 ) H = E1 S11 + E2 S22 + E3 S33 + g13 (e−iω1 t S12 + −iω2 t 23 + eiω1 t S12 S+ + eiω2 t S23 − ) + g23 (e −) Analyze the case: E2 − E1 = E3 − E2 .

Note also that the term 2g 2 Sz E † E ω+ appears in the transformed Hamiltonian. 18), θ(t)|EE † |θ(t) = 1; we see that this term describes the effective shift of the atomic transition frequency, 2g 2/(ω + ), which is known as the Bloch--Siegert shift. ˜ Let us now apply a second transformation to H: U2 = eδε E †2 −E 2 Sz , δ= g 1 This removes the term εg2(E † 2 + E 2 )Sz . Among the different terms that appear at this level in the transformed Hamiltonian, there is a term g3 E 3 S+ + E † 3 S− ( + ω) − which describes the triple resonance and becomes important if the condition ω = 3 is satisﬁed.