QUANTUM THEORY I: ELEMENTS by D. R. Bates

By D. R. Bates

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6 Collective Atomic Systems Now let us consider the evolution of a collection of A two-level atoms in a classical external field. 13); 1 (t) = 2dEx (t) and 2 (t) = 2dEy (t). 20) can be rewritten as H = ω0 Sz + + S+ + In general, the parameters − S− , +, − + = 1 −i 2 2 = ∗ − may depend on time. 21 we obtain ˙ αS+ Sz e−αS+ U + γ˙ eαS+ eβSz S− e−βSz e−αS+ U i αS ˙ + U + βe = (ω0 Sz + + S+ + − S− ) U Using the following relations eαS+ Sz e−αS+ = Sz − αS+ , eβSz S− e−βSz = e−β S− eαS+ S− e−αS+ = S− + 2αSz − α2 S+ we obtain i αS ˙ + + β˙ (Sz − αS+ ) + γ˙ e−β S− + 2αSz − α2 S+ = ω0 Sz + + S+ + − S− The operators S±,z are linearly independent.

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

17) consisting of A indistinguishable two-level atoms. m! 4 Displacement Operator If we now introduce the new variables A = n + m and k = (n − m) /2 + A/2 |α1 |α2 = e−(|α1 | 2 +|α |2 /2) 2 ∞ A A=0 k=0 αk1 α2 A−k k! A − k ! 17) for A two-level atoms and pA = e −n/2 √ −iψ ne √ A! 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.

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