By Andrei B. Klimov

ISBN-10: 3527408797

ISBN-13: 9783527408795

Written by way of significant individuals to the sphere who're popular in the neighborhood, this is often the 1st entire precis of the numerous effects generated through this method of quantum optics to this point. As such, the e-book analyses chosen themes of quantum optics, concentrating on atom-field interactions from a group-theoretical standpoint, whereas discussing the relevant quantum optics types utilizing algebraic language. the final result's a transparent demonstration of the benefits of utilizing 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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**Additional resources for A Group-Theoretical Approach to Quantum Optics: Models of Atom-Field Interactions**

**Sample text**

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. 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.

19). Indeed, by introducing the notations H0 = ωSz + E0 , Hr = g E † S− + ES+ , Ha = g E † S+ + ES− we have † U1 H0 U1 = H0 − ε(ω + )(E † S+ + ES− ) − ε2 2(ω + )Sz E † E † U1 Hr U1 = Hr + 2εg(E † 2 + E 2 )Sz −ε2 g E † 3 S+ + E 3 S− + E 2 E † S+ + EE † 2 S− † U1 Ha U1 = Ha + 4εgE † ESz − 2ε2 g EE † 2 S+ + E † E 2 S− 31 32 2 Atomic Dynamics where the second term in the ﬁrst equation cancels the antirotating term. 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.

We start by rewriting the Hamiltonian in a time-independent form in the spirit of Floquet’s theory. 4) we reveal the possible resonant transitions that can be observed in the system. Actually, by slightly generalizing the problem, we consider here a collection of A two-level atoms driven by a linearly polarized monochromatic ﬁeld of frequency . 15) Here, S+,−,z are collective spin operators. If S± → σ± , Sz → σz /2, the Hamiltonian is reduced to the one-atom case. 15). ) These operators act in a Hilbert space spanned by eigenstates of the operator † E0 = E0 : E0 |m = m|m , m = · · · − 1, 0, 1, .

### A Group-Theoretical Approach to Quantum Optics: Models of Atom-Field Interactions by Andrei B. Klimov

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