By Andrei B. Klimov

Written via significant members to the sector who're renowned in the group, this can be the 1st finished precis of the numerous effects generated through this method of quantum optics up to now. As such, the publication analyses chosen subject matters of quantum optics, concentrating on atom-field interactions from a group-theoretical point of view, whereas discussing the significant quantum optics versions utilizing algebraic language. the general result's a transparent demonstration of some great benefits of making use of algebraic tips on how to quantum optics difficulties, illustrated by way of a few 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: Models of Atom-Field Interactions**

**Example text**

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). 26) and the generating function for the Laguerre polynomials, ∞ exp [ax1 + bx2 − ab] = n=0 ∞ (ax1 )n n!

For such a system, it is known that there exist 29 30 2 Atomic Dynamics solutions of the form | β (t) = |φβ (t) e−iλβ t , β = 0, 1 such that the vectors |φβ (t) are periodic in time: |φβ (t+T) = |φβ (t) The existence of periodic (up to the phase factor) solutions is known as Floquet’s theorem. It allows us to expand the functions φβ (t) in Fourier series and rewrite the Schr¨odinger equation as an equation for the corresponding Fourier coefﬁcients. This leads to an effective Hamiltonian in the inﬁnite-dimensional space of Fourier coefﬁcients.

18 1 Atomic Kinematics transitions between levels are s ij = | j i|, (j = i), i, j = 1, . . 1). Let us recall that any element of the U(n) group can be represented as the product of an element of the su(n) group by an element of the U(1) group. Obviously the element of the U(1) group is related to a global phase of the system state. To exclude this phase, we use the traceless combinations j j+1 = sz 1 | j + 1 j + 1| − | j j| , 2 j = 1, . . , n − 1 These operators are interpreted as atomic inversions between levels j + 1 and j.