By Andrei B. Klimov

Written through significant participants to the sphere who're renowned in the neighborhood, this is often the 1st complete precis of the various effects generated through this method of quantum optics to this point. 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 types utilizing algebraic language. the final result's a transparent demonstration of the benefits of making use of algebraic how to quantum optics difficulties, illustrated by way of a couple of end-of-chapter difficulties. a useful resource for atomic physicists, graduates and scholars in physics.

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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! 1− ∞ = (−1)m n=0 m=0 b x1 n exp b x1 x2 x1 an bm n−m n−m x Lm (x1 x2 ) n! 71 the matrix elements of the displacement operator in the number state basis:  n! m−n −|γ|2 /2 m−n   |γ|2 m≥n γ e Ln  m!

Let us suppose that the initial state is |0, A (henceforth |0 ), that is, all the atoms are in their ground states. We calculate the probability P00 (t) of their remaining in the nonexcited state after a time t, P00 (t) = | 0|U (t) |0 |2 = | 0|eαS+ eβSz eγS− |0 |2 = | 0|eβSz |0 |2 Since Sz |0 = −A/2|0 , we obtain P00 (t) = e−ARe β = cos2A t Let us evaluate the probability P00 (t) averaged over time T P00 (t) = A C2A (2A)! = 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.

Thus, the Schr¨odinger equation is reduced to a system of two coupled first-order ordinary differential equations, with coefficients periodic in time and having the same period as the Hamiltonian. 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.

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