7.3 Interaction with monochromatic radiation 131that
.
R·R=0so|R|^2 is constant—it is straightforward^15 to show that^15 |R|=1forthestateu=v=0,w=
1 and it always remains unity. This can
also be proved by writingu,vandwin
terms of|c 1 |^2 ,|c 2 |^2 ,etc.this constant is unity so that|R|^2 =|u|^2 +|v|^2 +|w|^2 =1. TheBloch
vector corresponds to the position vector of points on the surface of a
sphere with unit radius; thisBloch sphereis shown in Fig. 7.2.
It also follows from eqn 7.47 that
.
R·W=0andsoR·W=RWcosθ
is constant. For excitation with a fixed Rabi frequency and detuning
the magnitudeWis constant, and sinceRis also fixed the Bloch vector
moves around a cone withθconstant, as illustrated in Fig. 7.2(d). In
this caseρ 22 varies as in eqn 7.27 and
w=1− 2 ρ 22 =1−2Ω^2
W^2
sin^2(
Wt
2)
.
This motion of the Bloch vector for the state of the atom resembles that
of a magnetic moment in a magnetic field,^16 e.g. for adiabatic motion the
(^16) As described in Blundell (2001, Ap-
pendix G).
(a) (b)
(c) (d)
Fig. 7.2The Bloch sphere. The position vectors of points on its surface represent the states of a two-level system (in
Hilbert space). Examples of states are shown in (a) and (b). At the poles of the sphere the Bloch vector isR=ŵe 3 ,with
w=±1 corresponding to the states| 1 〉and| 2 〉, respectively. States that lie on the equator of the Bloch sphere have the
formR=ûe 1 +v̂e 2 , e.g. the states for whichR=v̂e 2 withu=0andv=±1 are shown in (b) and these correspond
to (| 1 〉±i| 2 〉)/
√
2, respectively (normalisation constants are not given in the figure for clarity). These examples illustrate an
interesting property of this representation of quantum states, namely that diametrically-opposite states on the Bloch sphere
are orthogonal. (c) The evolution of the Bloch vector for a system driven by a resonant field, i.e.δ=0sothatW=Ω̂e 1 in
eqn 7.49. The evolution follows a great circle from| 1 〉at the north pole to| 2 〉at the south pole and back again, as described
in Example 7.1. The Bloch vector remains perpendicular toW.(d)Whenδ = 0 the Bloch vector also has a fixed angle with
respect toW,sinceR·W=RWcosθis constant, butθis not equal toπ/2. (This quantum mechanical description of the
two-level atom is equivalent to that for a spin-1/2system.)