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

By Andrei B. Klimov

Written through significant participants to the sector who're renowned in the neighborhood, this is often the 1st entire precis of the numerous effects generated through this method of quantum optics so far. As such, the booklet analyses chosen issues of quantum optics, concentrating on atom-field interactions from a group-theoretical viewpoint, whereas discussing the imperative quantum optics versions utilizing algebraic language. the final result's a transparent demonstration of some great benefits of utilising algebraic the right way to quantum optics difficulties, illustrated through a few end-of-chapter difficulties. a useful resource for atomic physicists, graduates and scholars in physics.

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2A 2A 2 2 A! 2π/ 2 A where C2A are the binomial coefﬁcients. Thus, for one atom, we have P00 (t) = 1/2 and, for two atoms, we have P00 (t) = 3/8. √ In the case of a large number of atoms A 1, using the Stirling formula n! ∼ 2nπ nn e−n , we obtain P00 (t) ∼ √ 1 Aπ −→ 0 that is, the system with a large number of atoms has a greater probability of becoming excited. 35 36 2 Atomic Dynamics Next, we calculate the probability P01 (t) of a transition from the ground state to the ﬁrst excited state. k!

54) so that |pA |2 = PA is a Poisson distribution. 53) has a sharp maximum at A = n, so that |α1 |α1 ≈ e−iψn |ϑ, ϕ; n . 55) where α is a complex number, is called the displacement operator. 58) so that the value of the right hand side of the above equation at t = 1 gives us the desired result. |n , we immediately obtain ∞ 2 /2 αn †n a |0 n! n=0 αn √ |n = |α n! e. the application of the displacement operator to the vacuum state generates a coherent state. 71) Using the above formula, it is easy to calculate the trace of the operator D(γ).

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.