By Andrei B. Klimov

Written via significant individuals to the sphere who're renowned in the group, this is often the 1st accomplished precis of the various effects generated through this method of quantum optics so far. As such, the publication analyses chosen issues of quantum optics, targeting atom-field interactions from a group-theoretical standpoint, whereas discussing the primary quantum optics versions utilizing algebraic language. the general result's a transparent demonstration of some great benefits of using algebraic the right way 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 info for A Group-Theoretical Approach to Quantum Optics**

**Example text**

The operators sz , s ij = |i j|, j = i, i, j = 1, . . , n form the n-dimensional representation of the su(n) algebra. In the case of A identical atoms with n energy levels, the collective operators A S ij = |a, i a, j|, i, j = 1, . . 48) a=1 are introduced where |a, i is a state of the ath atom. 45. Thus the problem is reduced to the construction of the irreducible representations for the su(n) group. If the atoms are identical (the Hamiltonian is invariant under permutations of atoms), it is sufﬁcient to consider only the symmetric representation.

19) is time independent. 4). 19. This is done by applying in a perturbative way a series of small Lie-type transformations. 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.

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, . . 5 Dynamics of the Two-level Atom without the RWA Here, E and E † are ﬁeld phase operators, E0 is the shifted photon-number operator, and m = n − n. 16) are approximately satisﬁed if n. 17. 15). Furthermore, we can, without loss of generality, set the global common phase θ equal to 0.