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By Shore B.W.

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51) For null phase this corresponds to a rotation of the statevector by 45 degrees (a rotation of the Bloch angle by π/2). Each of these objectives can be obtained by producing a predetermined portion of a Rabi oscillation. However, other procedures are possible, often providing advantages. In particular, alternative schemes based upon adiabatic changes are much less sensitive to details of the pulse; they are basically independent of the time integral of the Rabi frequency, for example, and are therefore termed robust.

78) and require that the resuting equation be fulfilled along each coordinate. We thereby obtain a set of N coupled ordinary differential equations for the probability amplitudes. For two states these are 43 i d c1 (t) = − [E1 c1 (t) + V (t)∗ c2 (t)] , (83) dt 43 Although the distinction between V and V ∗ for a classical field involves only a phase, when one treats quantized radiation the interaction V ∗ , originating with the negative-frequency part of the field, is associated with the creation of a photon, while V , based on the positive-frequency part of the field, is associated with photon annihilation.

2), and a statevector having 2S + 1 Hilbert-space coordinates (and hence a representation of total spin S) [61]. Applied to the present case the Majorana approach considers the N = 2S + 1 roots xn of the polynomial equation 37 n−1 ΣN = 0. (n + 1)! , (48) 37 The statevector coefficients c have indices n = 1, . . , N − 1. These are associated in the polynomial with powers n xN −1 , · · · , x0 . g. hn = 1). Each complex-valued root can be expressed as a pair of angles, xn = tan(θn /2)eiφn . (49) Each of these define a point on a unit sphere – the Bloch sphere The complete description of the statevector consists of N − 1 points on the Bloch sphere.

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Coherent manipulations of atoms using laser light by Shore B.W.

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