Download A Group-Theoretical Approach to Quantum Optics by Andrei B. Klimov PDF

By Andrei B. Klimov

Written through significant individuals to the sphere who're popular in the group, this can be the 1st complete precis of the numerous effects generated via this method of quantum optics up to now. As such, the ebook analyses chosen issues of quantum optics, concentrating on atom-field interactions from a group-theoretical viewpoint, whereas discussing the primary quantum optics types utilizing algebraic language. the final result's a transparent demonstration of some great benefits of employing algebraic ways to quantum optics difficulties, illustrated via a couple of end-of-chapter difficulties. a useful resource for atomic physicists, graduates and scholars in physics.

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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.

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. We suppose that the transitions in atomic systems only take place in the dipole approximation, that is, there are no transitions between levelswith the same parity. Thus, there are only three possible configurations for systems with three energy levels: cascade ( ), lambda ( ), and (V) configurations. Enumerating energy levels in such a way that E1 ≤ E2 ≤ E3 , we see that the transitions allowed in the system are 1 ↔ 2 ↔ 3; in the system, 1 ↔ 3 ↔ 2; and in the V system 3 ↔ 1 ↔ 2.

Each term in the second sum represent the vacuum energy of the appropriate mode and will be omitted in future calculations. The total Hilbert space is a direct product of Hilbert spaces corresponding to each field mode. 13) where the vectors |nk describe the states with a defined energy, that is, with a fixed number of photons in the kth field mode. 18) Henceforth, we work mainly with one only field mode, so that only one term from the corresponding sums is needed. 19) and the representation space coincides with the space of a one-dimensional harmonic oscillator, nˆ |n = n|n .

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