7. Atomic Angular-Momentum Structure¶
PDF pages 301–398
7.1.1 Operators and Eigenstates¶
Chapter 7 Atomic Angular-Momentum Structure In this chapter, we will review and develop some of the formalism for handling angular momentum, in particular as it applies to the structure of simple (hydrogen-like) atoms. We will use these results to look at fine and hyperfine structure, and in particular how to handle light–matter interactions in the presence of Zeeman-degenerate states (degenerate angular-momentum sublevels). 7.1 Angular Momentum 7.1.1 Operators and Eigenstates The basics of the quantization of angular momentum is covered well enough in most introductory quantum- mechanics text, so we’ll just review them here so we can get on to applications of the theory to atoms and quantum optics. First, we will suppose that we have a set of operators Jx, Jy, and Jz, and we will take them to be defined by the commutation relation
(7.1) (angular-momentum commutator) where ϵ\alpha\beta\gamma is the Levi-Civita symbol (completely antisymmetric ‘‘tensor,’’ though not technically a tensor because it doesn’t transform correctly under rotations), having the values +1 if (\alpha\beta\gamma) is a cyclic permutation of (xyz), −1 if an odd permutation, and 0 otherwise. These operators will obviously represent angular momenta associated with the three Cartesian axes, and so it will also be useful to define an operator associated with the total angular momentum: J2 = J 2 x + J 2 y + J 2 z . (7.2) We assume these operators to correspond to observables, and are thus Hermitian. Out of the set of operators {J2, Jx, Jy, Jz}, the above relations (7.1) and (7.2) show that the full set can be expressed in terms of only two (e.g., Jx and Jy). Thus, to completely span the space of angular momentum states, we can choose to have simultaneous eigenstates of any two such operators. This strategy is useful in spherically symmetric systems, where such eigenstates should exist, and so any component J\alpha is as good as any other. However, in view of the fundamental commutation relation, we can’t have simultaneous eigenstates for J\alpha and J\beta if \alpha̸ = \beta. However, J2 commutes with J\alpha:
(7.3) (angular-momentum commutator) To see this, we can take J\alpha = Jx without loss of generality, in which case [Jx, J2] = [Jx, J 2 y ] + [Jx, J 2 z ] = [Jx, Jy]Jy + Jy[Jx, Jy] + [Jx, Jz]Jz + Jz[Jx, Jz]
= 0. (7.4)
7.1.2 Ladder Operators and Eigenvalues¶
Chapter 7. Atomic Angular-Momentum Structure Thus, we are free to construct simultaneous eigenstates of J2 and J\alpha. We make the arbitrary but conventional choice of taking simultaneous eigenstates of J2 and Jz. We will thus need two quantum numbers, which we call j and m, and define the eigenvalue \lambdaj of J2 to be some function of j,
(7.5) and the eigenvalue \lambdam of Jz will similarly be some function of m,
(7.6) Our goal in setting up the basic formalism will now be to work out the angular-momentum eigenvalues. 7.1.2 Ladder Operators and Eigenvalues It is also useful to define two non-Hermitian operators, the ladder operators J\pm := Jx \pm iJy, (7.7) which will turn out to be somewhat more convenient than Jx and Jy separately. Given the commutation relation (7.3), we immediately see that the ladder operators commute with J2: [J2, J\pm] = 0. (7.8) (ladder-operator commutator) The commutators with Jz is not hard to work out, [Jz, J\pm] = [Jz, Jx] \pm i[Jz, Jy] = i¯hJy \pm ¯hJx, (7.9) or [Jz, J\pm] = \pm¯hJ\pm. (7.10) (ladder-operator commutator) We can also readily compute the commutator of the two ladder operators as [J+, J−] = −2i[Jx, Jy] = 2¯hJz. (7.11) Now, to put the ladder operators to use, we can consider the action of J\pm on an eigenstate state |j m\rangle . In particular, notice that since the J\alpha commute with J2, they transform |j m\rangle to a state J\alpha|j m\rangle such that
(7.12) Thus, J\alpha|j m\rangle is an eigenstate of J2 with the same eigenvalue as |j m\rangle , implying that j is unchanged. The same conclusion of course holds for J\pm, and thus, since we will be considering the action of Jx,y,z,\pm on the states |j, m\rangle for the rest of this section, we can regard j as a fixed quantity. We can then use the commutator (7.10) on |j m\rangle to write
(7.13) This shows that J\pm|j m\rangle is an eigenstate of Jz with eigenvalue \lambdam \pm ¯h. Now we see the reason for the name ‘‘ladder operators,’’ since J+ acts to raise \lambdam by ¯h, and J−lowers it by the same amount. Now since m is an arbitrary label for the states, we may define it such that \lambdam = m¯h. That is, m represents the projection of angular momentum along the z-axis in multiples of ¯h. Then we may write
(7.14) (Jz eigenvalues) for the Jz eigenvalue equation, and for the ladder operators we have thus shown that
(7.15)
7.1 Angular Momentum To establish the proper normalization, we note that J∓J\pm = J 2 x + J 2 y \pm i[Jx, Jy] = J2 −J2
(7.16) and thus the norm of the raised/lowered state is
(7.17) Note that the right-hand side becomes negative for sufficiently large m, assuming \lambdaj to be fixed. However, since \langle j m|J∓J\pm|j m\rangle \ge 0, we can conclude that there is a maximum value of m, say mmax, such that
(7.18) Then applying Eq. (7.16) to |j mmax\rangle ,
(7.19) The left-hand side vanishes, so
(7.20)
with j \ge 0, so that
(7.21) (J2 eigenvalue equation) Repeating this argument, Eq. (7.17) implies a smallest (negative) value of m, say mmin, so that
(7.22) Again applying Eq. (7.16) to |j mmin\rangle ,
(7.23) and since the left-hand side vanishes,
(7.24) Thus,
(7.25)
Thus, m is constrained to be within a bounded range, −j \le m \le j. (7.26) (range constraint of m)
In particular, if we start with the state |j −j\rangle and repeatedly apply J+, we should eventually end up with (something proportional to) the |j +j\rangle state. How do we know this? Referring again to Eq. (7.17), which we may rewrite now as
h j(j + 1) −m(m \pm 1) i ¯h2. (7.27) we see that the only state that vanishes when hit by J+ is |j (+j)\rangle . Thus, the only way to avoid a contradiction (negative state norm) is for J n + |j (−j)\rangle ∝|j (+j)\rangle for some integer n. Further, we may conclude that every state |j m\rangle may be written as J n + |j (−j)\rangle (up to a scalar factor) for some integer n; otherwise we would have
7.1.3 Addition of Two Angular Momenta: Clebsch-Gordan Coefficients¶
Chapter 7. Atomic Angular-Momentum Structure a state that, when raised arbitrarily many times by J+, would not vanish. Thus, we may conclude that m takes on discrete, integer-separated values, according to m \in {−j, −j + 1, . . . , j −1, j} (2j + 1 possible values), (7.28) (range constraint of m) which means that there are 2j+1 possible values for m (i.e., because m+j ranges from 0 to 2j). Furthermore, 2j+1 must be an integer, because 2j is the number of times J+ must be applied to |j (−j)\rangle to obtain |j (+j)\rangle . This implies that j \in Z or j + 1 2 \in Z, (7.29) (integer/half-integer constraint) with j also nonnegative, as we discussed before. That is, j is either an integer or a half-integer. As we will discuss later, only integer j can correspond to coordinate-space angular momenta; half-integer j are restricted to representing intrinsic particle spin angular momenta (which can also have integer j). Finally, just to tidy up loose ends, we can use Eq. (7.27) to write down
p
= ¯h p
(7.30) (ladder operator effects) as the properly normalized action of the ladder operators on the angular-momentum eigenstates. 7.1.3 Addition of Two Angular Momenta: Clebsch–Gordan Coefficients 7.1.3.1 Basis States Suppose we have two angular momenta J1 and J2, and we want to consider their sum J = J1 + J2. We assume these angular momenta to correspond to independent degrees of freedom, and thus they commute:
(7.31) Treating the two angular momenta as separate entities, we can construct simultaneous eigenstates of J 2 1 , J 2 2 , J1z, and J2z, since everybody here commutes. We will denote these eigenstates by |j1 m1; j2 m2\rangle ≡ |j1 m1\rangle |j2 m2\rangle , so that J 2
J 2
J1z|j1 m1; j2 m2\rangle = m1¯h|j1 m1; j2 m2\rangle J2z|j1 m1; j2 m2\rangle = m2¯h|j1 m1; j2 m2\rangle . (7.32) Now note that the total angular momentum J has the characteristics of an angular momentum operator, since
(7.33) Thus, we may have simultaneous eigenstates of J2 and Jz. Also, it is easy to see that J 2 1 and J 2 2 both commute with J2 and Jz (but J1z and J2z don’t commute with J2), so that we can represent our states in terms of simultaneous eigenstates of J 2 1 , J 2 2 , J2, and Jz, which we will label by |j1, j2; j m\rangle , so that J2
J 2
Jz|j1, j2; j m\rangle = m¯h|j1, j2; j m\rangle . (7.34) Sometimes, the state |j1, j2; j m\rangle is written more succinctly as |j, m\rangle if j1 and j2 are clear from the context.
7.1 Angular Momentum 7.1.3.2 Transformation between Bases and Clebsch–Gordan Coefficients Now we have two distinct bases by which to represent a general state. The basic problem of angular- momentum addition is thus to express any basis state in terms of a superposition of states from the other basis. This is easy to do by using the representations of the identity in each basis:
X j′ 1j′ 2m1m2 |j′ 1 m1; j′
1 m1; j′ 2 m2|j1, j2; j m\rangle
X j′ 1j′ 2jm |j′ 1, j′
1, j′ 2; j m|j1 m1; j2 m2\rangle . (7.35) The inner products on the right-hand sides of the above equations are Clebsch–Gordan coefficients. Note that J 2 1 and J 2 2 are Hermitian, and thus \langle j′ 1, j′ 2; j m|J 2
1(j′
1, j′ 2; j m|j1 m1; j2 m2\rangle
1, j′ 2; j m|j1 m1; j2 m2\rangle , (7.36) so that the Clebsch–Gordan coefficient vanishes unless j1 = j′
2). Additionally, Jz = J1z + J2z, so
= m¯h\langle j1, j2; j m|j1 m1; j2 m2\rangle , (7.37) and thus we must have
(7.38) (angular-momentum conservation) for the Clebsch–Gordan coefficient to be nonvanishing. Thus, we may rewrite the transformation relations (7.35) as
X m1m2
|j1 m1; j2 m2\rangle \langle j1 m1; j2 m2|j1, j2; j m\rangle
X jm
|j1, j2; j m\rangle \langle j1, j2; j m|j1 m1; j2 m2\rangle , (7.39) or omitting the redundant labels,
X m1m2
|j1 m1; j2 m2\rangle \langle j1 m1; j2 m2|j m\rangle
X jm
(7.40) (transformation rules) The other important constraint is |j1 −j2| \le j \le j1 + j2 (7.41) (triangular condition)
maximum value of j, but is also given by j1 +j2. Thus, jmax = j1 +j2. To find the minimum value of j, note that in the |j1 m1; j2 m2\rangle basis, there are 2j1 + 1 states associated with the the |j1 m1\rangle space and 2j2 + 1 states associated with the the |j2 m2\rangle space, and thus the composite space is spanned by (2j1 + 1)(2j2 + 1)
Chapter 7. Atomic Angular-Momentum Structure states. In the other basis, we get the correct number of states if jmin = |j1 −j2|. That is, jmin is the solution to j1+j2 X j=jmin
(7.42) which we can see because, assuming without loss of generality that j1 \ge j2, j1+j2 X j=j1−j2
j2 X j=−j2
j2 X j=−j2
j2 X j=−j2
(7.43) The cases j = |j1 −j2| and j = j1 + j2 clearly correspond to antialigned and aligned constituent momentum vectors, respectively. The Clebsch–Gordan coefficients obey orthogonality relations as follows. From the second transforma- tion rule in Eqs. (7.40), \langle j1 m′ 1; j2 m′
X jm \langle j1 m′ 1; j2 m′
(7.44) The left-hand side is zero unless m′ 1 = m1 and m′ 2 = m2, so X jm \langle j1 m′ 1; j2 m′
1\deltam2m′ 2. (Clebsch–Gordan orthogonality relation) (7.45) Similarly, the other transformation rule leads to X m1m2
(Clebsch–Gordan orthogonality relation) (7.46) which are reasonably obvious applications of different representations of the identity operator. 7.1.3.3 Calculation of Clebsch–Gordan Coefficients To determine the Clebsch–Gordan coefficients, we make use of the raising and lowering operators J1\pm = J1x \pm iJ1y J2\pm = J2x \pm iJ2y
(7.47)
p
p j1(j1 + 1) −m1(m1 ∓1) \langle j1 m1∓1; j2 m2|j m\rangle + p j2(j2 + 1) −m2(m2 ∓1) \langle j1 m1; j2 m2∓1|j m\rangle (Clebsch–Gordan recursion relation) (7.48) This recursion relation, in addition to some initial conditions, is sufficient to compute the coefficients. The
p j1(j1 + 1) −m1(m1 −1) \langle j1 m1−1; j2 m2|j j\rangle + p
(7.49) which together with the special cases of Eq. (7.46) X m1m2
(7.50)
7.1 Angular Momentum that pin down the normalization, all coefficients of the form \langle j1 m1 −1; j2 m2|j j\rangle can be determined up to an arbitrary phase. It is conventional to take the coefficients \langle j1 j1; j2 j −j1|j j\rangle to be real and positive.1 The recursion relation (7.49) can then generate all the rest of the Clebsch–Gordon coefficients from these ‘‘basis cases,’’ and an important consequence of the recursion relation (which only involves real recursion coefficients) and the phase convention is that by convention all Clebsch–Gordon coefficients are real (though not necessarily positive). 7.1.3.4 Explicit Formula The above recursion procedure is fairly cumbersome, although sometimes useful. In a numerical calculation, it is convenient to have explicit formulae to implement any of the coupling coefficients. Fortunately, the Clebsch–Gordan coefficient may be computed according to the rather complicated formula2
s
\times p (2j3 + 1)(j1 + m1)!(j1 −m1)!(j2 + m2)!(j2 −m2)!(j3 + m3)!(j3 −m3)! \times nmax X n=nmin (−1)n (j1 −m1 −n)!(j3 −j2 + m1 + n)!(j2 + m2 −n)!(j3 −j1 −m2 + n)!n!(j1 + j2 −j3 −n)!, (Clebsch–Gordan coefficient: explicit formula) (7.51) where the summation limits nmin = max{j2 −j3 −m1, j1 + m2 −j3, 0} nmax = min{j1 −m1, j2 + m2, j1 + j2 −j3} (7.52) are chosen such that no factorial arguments are negative. For a nonzero result, we reiterate that we must
7.1.3.5 Symmetry Relations and Wigner 3-j Symbols Now that we can compute the Clebsch–Gordan coefficients, we can ask, what are the shortcuts to relating them if we just want to permute some symbols? For example, recall that the coupling of two angular momenta according to
(7.53) is represented by the coefficient \langle j1 m1; j2 m2|j3 m3\rangle . However, J1 and J2 are on equal footing in being added together to form J3, and so we should be able to switch them without a problem, at least up to an overall phase. It turns out that according to the sign convention we have chosen,
(7.54) (symmetry relation) We can see this by redefining the index n in the explicit formula (7.51) according to n −\rightarrow (j1 + j2 −j3) −n, with the limits redefined appropriately to avoid any negative factorials, together with the simultaneous exchanges j1 \leftarrow \rightarrow j2 and m1 \leftarrow \rightarrow m2. This transformation leaves the sum invariant, except for the sign (−1)j1+j2−j3 (the same exchanges leave the prefactor of the sum invariant as well). We can go even farther than this. The addition (7.53) is clearly equivalent to the addition J3 −J1 = J2, (7.55) and thus we expect
(7.56) 1This is known as the Condon–Shortley phase convention. See, e.g., D. M. Brink and G. R. Satchler, Angular Momen- tum, 2nd ed. (Oxford, 1968), Section 2.7.2, p. 33. 2D. M. Brink and G. R. Satchler, op. cit., p. 34, Eq. (2.34).
Chapter 7. Atomic Angular-Momentum Structure In fact, we can see that this is the case by noting that the recursion relation (7.48) p j3(j3 + 1) −m3(m3 \pm 1) \langle j1 m1; j2 m2|j3 m3\pm1\rangle = p j1(j1 + 1) −m1(m1 ∓1) \langle j1 m1∓1; j2 m2|j3 m3\rangle + p j2(j2 + 1) −m2(m2 ∓1) \langle j1 m1; j2 m2∓1|j3 m3\rangle , (7.57) upon the substitutions |j3 m3\rangle −\rightarrow |j2 m2\rangle , |j2 m2\rangle −\rightarrow |j1 −m1\rangle , |j1 m1\rangle −\rightarrow |j3 m3\rangle , and \pm \leftarrow \rightarrow ∓, and multiplying through by (−1)−m1, becomes (−1)−m1p j3(j3 + 1) −m3(m3 \pm 1) \langle j3 m3\pm1; j1 −m1|j2 m2\rangle
j1(j1 + 1) −(−m1)[−(m1 ∓1)] \langle j3 m3; j1 −(m1∓1)|j2 m2\rangle + (−1)−m1p j2(j2 + 1) −m2(m2 ∓1) \langle j3 m3; j1 −m1|j2 m2∓1\rangle . (7.58) This recursion relation has the same form as the original, and indicates that (−1)−m1\langle j3 m3; j1 −m1|j2 m2\rangle obeys the same recursion relation as \langle j1 m1; j2 m2|j3 m3\rangle . Since the recursion relation determines the m-dependence of the Clebsch–Gordan coefficients, we conclude that these two coefficients are proportional, \langle j1 m1; j2 m2|j3 m3\rangle ∝(−1)−m1\langle j3 m3; j1 −m1|j2 m2\rangle , (7.59) with the remaining proportionality constant to be determined depends only on the j’s. To get the j-dependent amplitude, note that from Eq. (7.46) we may write X m1m2
X m1m3
(7.60) but since m1 + m2 = m3 holds in either case, the sums simplify to X m2
X m3
(7.61) Noting that the coefficients are equivalent in each case and that we have already taken care of the m- dependence, we count 2j2 + 1 terms in the first sum and 2j3 + 1 in the second. Thus, for the sums to be equivalent, we require \langle j1 m1; j2 m2|j3 m3\rangle ∝(−1)−m1 s 2j3 + 1
(7.62) where the remaining proportionality constant is a j-dependent phase. This we establish by noting the convention we already mentioned that \langle j1 j1; j2 (j3 −j1)|j3 j3\rangle is always positive. In this case, we need a factor of (−1)j1 to cancel the factor of (−1)−m1 for this case, and so we can finally write the symmetry relation
s 2j3 + 1
(Clebsch–Gordan symmetry rule) (7.63)
7.1.4 Addition of Three Angular Momenta: Racah Coefficients and Wigner 6-j Symbols¶
7.1 Angular Momentum We can then take \langle j1 m1; j2 m2|j3 m3\rangle , apply Eq. (7.54), apply Eq. (7.63), and apply Eq. (7.54) again to find
s 2j3 + 1
s 2j3 + 1
s 2j3 + 1
(7.64)
s 2j3 + 1
(Clebsch–Gordan symmetry rule) (7.65) Noting that this rule amounts to a cyclic permutation of the angular momenta while flipping the orientation of one, we can apply this rule three times to find
s 2j3 + 1
s 2j3 + 1
(7.66) Noting that j3 −m3 and m1 + m2 −j3 are both integers, we can rewrite this as the final symmetry rule
(Clebsch–Gordan symmetry rule) (7.67) A nice way to summarize the symmetry relations here is to define the Wigner 3-j symbol in terms of the Clebsch–Gordan coefficient as j1 j2 j3 m1 m2 m3
(7.68) (Wigner 3-j symbol) Then the symmetries are as follows: The symbol on the left-hand side of Eq. (7.68) is invariant under even permutations of the columns, but odd permutations are accompanied by a factor (−1)j1+j2+j3. The simultaneous, triple replacement m1,2,3 −\rightarrow −m1,2,3 is similarly accompanied by the same factor. Finally, the symbol is only nonvanishing if m1 + m2 + m3 = 0, and if j1, j2, and j3 obey the usual triangle condition. 7.1.4 Addition of Three Angular Momenta: Racah Coefficients and Wigner 6-j Sym- bols Suppose now that we want to couple three angular momenta, J = J1 + J2 + J3. We can use the formalism we have just developed for adding together two angular momenta, and simply iterate it. Unfortunately, the result of doing this turns out not to be unique: it depends on which two angular momenta are coupled first.
X m1m2
|j1 m1; j2 m2\rangle \langle j1 m1; j2 m2|j12 m12\rangle . (7.69)
Chapter 7. Atomic Angular-Momentum Structure Now we can add J = J12 + J3 to obtain
X m12m3
|j12 m12; j3 m3\rangle \langle j12 m12; j3 m3|j m\rangle . (7.70) Combining these two relations, we find the composite state in terms of the three original angular momenta as
X m1m2m12m3
|j1 m1; j2 m2; j3 m3\rangle \langle j12 m12; j3 m3|j m\rangle \langle j1 m1; j2 m2|j12 m12\rangle . (7.71) On the other hand, suppose we instead first added J23 = J2 + J3, and then J = J1 + J23, Then we instead obtain
X m1m23
|j1 m1; j23 m23\rangle \langle j1 m1; j23 m23|j m\rangle = X m1m2m3m23
|j1 m1; j2 m2; j3 m3\rangle \langle j1 m1; j23 m23|j m\rangle \langle j2 m2; j3 m3|j23 m23\rangle . (7.72) This expression is clearly different from Eqs. (7.70) and (7.71), but equally valid. Since we have proceeded via the established procedure of adding two angular momenta, where the composite states form a complete basis for the uncoupled states, we know that both |j12; j3; j m\rangle and |j1; j23; j m\rangle form alternate, complete bases for the original space spanned by |j1 m1; j2 m2; j3 m3\rangle . Thus, there exists a unitary transformation between the bases, which we can write in the same way as for Clebsch–Gordan coefficients as
X j23 |j1, j23; j m\rangle \langle j1, j23; j m|j12, j3; j m\rangle . (7.73) (No sum over m is required, due to the orthogonality of the composite basis vectors in either addition
X j23 |j1, j23; j j\rangle \langle j1, j23; j j|j12, j3; j j\rangle , (7.74) Now apply the lowering operator J−from Eq. (7.30)
X j23 |j1, j23; j j −1\rangle \langle j1, j23; j j|j12, j3; j j\rangle , (7.75) since the normalization coefficient will be the same on either side of the equation. We can continue to apply the lowering operator to obtain the relation between states of any m, which are transformed with the same coefficient as for m = j; thus, the coefficient is m-independent. Finally, we may write this ‘‘recoupling equation’’ as
X j23 |j1, j23; j m\rangle p (2j12 + 1)(2j23 + 1) W(j1 j2 j j3; j12 j23) = X j23
j1 j2 j12 j3 j j23 , (recoupling relation) (7.76)
7.1 Angular Momentum where the m-independent Racah W-coefficient3 is
p
(7.77) (Racah W-coefficient) for any m, and the m-independent Wigner 6-j symbol4 is j1 j2 j12 j3 j j23
p
. (7.78) (Wigner 6-j symbol) Evidently, the two symbols are related by j1 j2 j3 l1 l2 l3
(relation between Racah and Wigner 6-j symbols) (7.79) The two symbols are equivalent up to a sign, with the Wigner 6-j symbol being somewhat more symmetric in terms of permutation relations, as we discuss below. Both symbols are commonly used, though we will generally stick with the 6-j symbol. Note that from the definition in Eq. (7.73), the (real) inner product \langle j1; j23; j m|j12; j3; j m\rangle is a unitary and thus orthogonal matrix, with rows and columns labeled by j12 and j23. In particular, this means that p
j1 j2 j′ j3 j4 j′′ (7.80) represents an orthogonal matrix with indices j′ and j′′. Then matrix multiplication with its transpose leads to the identity matrix, X j p
j1 j2 j j3 j4 j′ p
j1 j2 j j3 j4 j′′
(7.81) Since the result is nonzero only when j′ = j′′, we may instead write X j
j1 j2 j j3 j4 j′ j1 j2 j j3 j4 j′′
(orthogonality relation) (7.82) as an orthogonality relation in terms of 6-j symbols. 7.1.4.1 Explicit Forms To obtain a more useful expression for the 6-j symbol, we can first invert Eq. (7.72) to obtain
X j23m23jm |j1; j23; j m\rangle \langle j1 m1; j23 m23|j m\rangle \langle j2 m2; j3 m3|j23 m23\rangle . (7.83) Putting this into Eq. (7.71),
X m1m2m3m12 j23m23j′m′ |j1; j23; j′ m′\rangle \langle j12 m12; j3 m3|j′ m′\rangle \langle j1 m1; j2 m2|j12 m12\rangle
(7.84) 3Giulio Racah, ‘‘Theory of Complex Spectra. II,’’ Physical Review 62 438 (1942) (doi: 10.1103/PhysRev.62.438); D. M. Brink and G. R. Satchler, Angular Momentum, 2nd ed. (Oxford, 1968), Section 3.2, p. 40. 4A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton, 1957), Section 3.3, p. 40.
Chapter 7. Atomic Angular-Momentum Structure and then projecting with \langle j1; j23; j m|, we find
X m1m2m3 m12m23
(7.85) Thus, we have the explicit form j1 j2 j12 j3 j j23 =
p
X m1m2m3 m12m23
(Wigner 6-j symbol, connection to Clebsch–Gordan coefficients) (7.86) A somewhat simpler explicit formula due to Racah5 comes from putting in the explicit formula (7.51) for the Clebsch–Gordan coefficients into the above expression, with the result j1 j2 j3 l1 l2 l3
\times nmax X n=nmin (−1)n(n + 1)! (n −J)!(n −k1)!(n −k2)!(n −k3)!(m1 −n)!(m2 −n)!(m3 −n)!, (Wigner 6-j symbol: explicit formula) (7.87) where we have used the shorthand symbols
nmin = max{j, k1, k2, k3} nmax = min{m1, m2, m3}
s
. (7.88) The 6-j symbol must satisfy the triangular constraints for four sets of ordered triples, (j1, j2, j3) : |j1 −j2| \le j3 \le j1 + j2, |j3 −j1| \le j2 \le j3 + j1, |j2 −j3| \le j1 \le j2 + j3, (j1, l2, l3) : |j1 −l2| \le l3 \le j1 + l2, |l3 −j1| \le l2 \le l3 + j1, |l2 −l3| \le j1 \le l2 + l3, (l1, j2, l3) : |l1 −j2| \le l3 \le l1 + j2, |l3 −l1| \le j2 \le l3 + l1, |j2 −l3| \le l1 \le j2 + l3, (l1, l2, j3) : |l1 −l2| \le j3 \le l1 + l2, |j3 −l1| \le l2 \le j3 + l1, |l2 −j3| \le l1 \le l2 + j3, (7.89) which follow from the constraint (7.41) for the Clebsch–Gordan coefficients [where we may exchange j1 \leftarrow \rightarrow j23 and j12 \leftarrow \rightarrow j3 in view of the definition (7.85)], the expression (7.86) for the 6-j symbol, and the permutation symmetries in the following section. (The permutation symmetries lead to yet more triangle
5A. R. Edmonds, op. cit., p. 99; Giulio Racah, op. cit.
7.1 Angular Momentum to represent angular-momentum eigenvalues (either integer or half-integer spin). For the 6-j symbol to be nonzero, the elements of the above triples must also add up to an integer,
(7.90) due again to the expression (7.86) for the Clebsch–Gordan coefficients, which effectively add any two elements of each triple to obtain the third: all integer momenta are okay, but two half-integer momenta must add to produce an integer momentum. Of course, once the 6-j symbols are computed this way, the Racah coefficient can then be found using Eq. (7.79). 7.1.4.2 Symmetry Relations As we mentioned above, the 6-j symbols are simpler than the Racah coefficients under permutations of the elements. We will simply summarize the symmetry relations, which follow from the above formulae, in particular by recasting Eq. (7.86) in a very symmetric form as a product of four 3-j symbols. The 6-j symbols are invariant under any exchange of columns, such as j1 j2 j3 l1 l2 l3 = j2 j1 j3 l2 l1 l3 = j3 j2 j1 l3 l2 l1 = j1 j3 j2 l1 l3 l2 , (7.91) and so on. The 6-j symbols are also invariant under the following interchanges, j1 j2 j3 l1 l2 l3 = l1 l2 j3 j1 j2 l3 = l1 j2 l3 j1 l2 j3 = j1 l2 l3 l1 j2 j3 , (7.92) where the upper and lower values are interchanged in any two columns. The Wigner 6-j symbols will be useful in decomposing the reduced matrix elements for the dipole operator that we will derive. We will thus consider the 6-j symbols again below. 7.1.4.3 Addition of Four Angular Momenta: Wigner 9-j Symbols Now let’s consider the coupling of four angular momenta,6 J = J1 + J2 + J3 + J4. Obviously, from our discussion of adding three angular momenta, there will be no unique way to add these together. For example, suppose that we add J12 = J1 + J2 and J34 = J3 + J4, and then finally J = J12 + J34. We can denote an eigenstate coupled in this fashion as |j12, j34; j m\rangle , (7.93) where we keep only the last addition of uncoupled momenta to make it obvious how we arrived at the result (i.e., the dependence on j1, j2, j3, and j4 is implied). We could also couple the angular momenta in alternate pairs, adding J13 = J1+J3 and J24 = J2+J4, and then finally J = J13+J24, where we denote the eigenstate |j13, j24; j m\rangle . (7.94) Again, there is an m-independent, orthogonal transformation between these two bases, which we use to define the Wigner 9-j symbol:7
p
j1 j2 j12 j3 j4 j34 j13 j24 j . (Wigner 9-j symbol) (7.95) 6‘‘Four, four angular momentum vectors, ah ah ah…’’ –Count von Count 7Edmonds, op. cit., Section 6.4, p. 100.
Chapter 7. Atomic Angular-Momentum Structure Note that we have dropped the m quantum number, since the result is m-independent anyway. To obtain an expression for the 9-j symbol, we can perform the recoupling of the angular momenta in multiple steps as
X j234
= X j234
= X j234
(7.96) where we have used parentheses in the subscripts in cases where the order of coupling is ambiguous— though note that the numerical values are the same for the same set of subscripts, as in j24 = j42 or
(7.97) while in the last step of Eq. (7.96), we used the symmetry rule (7.54) to change the order of two couplings. These three coefficients each represent the coupling of three angular momenta, and thus we can use the definition (7.78) of the 6-j symbol three times to obtain
X j234
j1 j2 j12 j34 j j234
j3 j4 j34 j2 j234 j24
j1 j3 j13 j24 j j234 = X j234 (−1)2j234(2j234 + 1) p
\times j1 j2 j12 j34 j j234 j3 j4 j34 j2 j234 j24 j13 j24 j j234 j1 j3 , (7.98) where in the last step we used j1 + j + j234 \in Z and j3 + j234 + j24 \in Z, according to the constraints of the 6-j symbols, to simplify the expression for the sign. Comparing to the definition (7.95) of the 9-j symbol, we find the explicit formula j1 j2 j3 k1 k2 k3 ℓ1 ℓ2 ℓ3 = X s (−1)2s(2s + 1) j1 j2 j3 k3 ℓ3 s k1 k2 k3 j2 s ℓ2 ℓ1 ℓ2 ℓ3 s j1 k1 , (Wigner 9-j symbol in terms of 6-j symbols) (7.99) in terms of a sum over products of 6-j symbols, after changing to a more symmetric notation. We will be able to accomplish what we want in terms of angular-momentum structure without having to resort to the 9-j symbol. However, it will help to consider the coupling of four angular momenta in a
7.2 Static Angular-Momentum Structure of Atoms¶
7.2 Static Angular-Momentum Structure of Atoms slightly different way. First, we can couple the four angular momenta in two stages as
\times j1 j2 j12 j3 j123 j23 j23 j1 j123 j4 j j14 , (7.100) while we can make the same coupling in three stages, as we did for the 9-j symbol:
X j124
= X j124
= X j124
j4 j12 j124 j3 j j123
j4 j1 j14 j2 j124 j12
j14 j2 j124 j3 j j23 . (7.101) Equating these two expressions and permuting some of the 6-j symbol elements, we find the Biedenharn– Elliott sum rule 8 j1 j2 j12 j3 j123 j23 j23 j1 j123 j4 j j14 = X j124
\times j3 j2 j23 j14 j j124 j2 j1 j12 j4 j124 j14 j3 j12 j123 j4 j j124 (Biedenharn–Elliott sum rule) (7.102) after using the usual tricks to simplify the sign factor. 7.2 Static Angular-Momentum Structure of Atoms In the standard textbook version of the nonrelativistic, quantum-mechanical hydrogen-like atom,9 an electron of reduced mass m = memn me + mn \approx me, (7.103) where me is the electron mass, and mn is the nuclear mass, moves in the central potential
4\piϵ0r, (7.104) 8L. C. Biedenharn, ‘‘An Identity Satisfied by the Racah Coefficients,’’ Journal of Mathematics and Physics 31, 287 (1953); J. P. Elliott, ‘‘Theoretical Studies in Nuclear Structure. V. The Matrix Elements of Non-Central Forces with an Application to the 2p-Shell,’’ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 218, 345 (1953); A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton, 1957), p. 97, Eq. (6.2.12). 9See, e.g., John L. Powell and Bernd Crasemann, Quantum Mechanics (Addison–Wesley, 1961), Section 7-7, p. 220.
7.2.1 Fine Structure¶
Chapter 7. Atomic Angular-Momentum Structure where e is the fundamental charge, and the nuclear charge is Ze. The standard result for the energies is En = − mc2 2 (Z\alpha)2 1 n2 , (7.105) where n is the radial (principle) quantum number,
e2 4\piϵ0¯hc \approx (7.106) is the fine-structure constant, and the coefficient of 1/n2 has the approximate value −Z2(13.6 eV). This energy expression says that, at this crude level of approximation, the hydrogen-like-atom energies do not depend on any angular-momentum quantum numbers. In what follows, we will use the standard notation of n, l, and m as the usual quantum numbers referring to the single-electron state |n l m\rangle . However, to be a bit more general, for multielectron atoms we will refer to the total quantities using capital letters. That is, L is the quantum number for the total electron orbital angular momentum, S is the quantum number for the total electron spin, and so on. 7.2.1 Fine Structure At the next level of approximation, we find that angular momentum does contribute some energy shifts, splitting some of the degenerate lines in the above simplistic treatment. Because of the relatively small splittings, at least in lighter atoms—for example 0.58 nm for the common yellow 589 nm line of sodium (seen in sodium lamps everywhere)—this splitting goes by the name of fine structure.10 We can treat this effect to lowest order as follows. Because the electron orbits the nucleus, it moves through the nuclear Coulomb field and thus ‘‘sees’’ in its rest frame an effective magnetic field B = −v c2 \times E (7.107) via the Lorentz transformation for electromagnetic fields.11 The Coulomb force on the electron is
r\partial rV (r), (7.108) and so with the orbital angular momentum
(7.109) the effective magnetic field becomes
mc2er L. (7.110) The electron’s magnetic moment due to its intrinsic spin is µS = −µBgS S ¯h , (7.111)
10Fine structure was first described by A. Sommerfeld, ‘‘Zur Quantentheorie der Spektrallinien,’’ Annalen der Physik 356, 1 (1916) (doi: 10.1002/andp.19163561702). 11See David J. Griffiths, Introduction to Electrodynamics, 2nd ed. (Prentice-Hall, 1989), Eq. (10.120), p. 497. 122006 CODATA recommended value; see P. J. Mohr, B. N. Taylor, and D. B. Newell, ‘‘The 2006 CODATA Recommended Values of the Fundamental Physical Constants, Web Version 5.1,’’ available at http://physics.nist.gov/constants (National Institute of Standards and Technology, Gaithersburg, MD 20899, 31 December 2007).
7.2 Static Angular-Momentum Structure of Atoms
field (7.110) with the magnetic moment (7.111) gives the fine-structure Hamiltonian
mc2¯her
Ze2 4\piϵ0 gS 2m2c2r3 L \cdot S. (7.112) The problem with this expression is that we still need to transform back into the lab frame, which is a noninertial transformation. This correction is Thomas precession, and the correction amounts to adding the Thomas-precession Hamiltonian14
(7.113) so that the real fine-structure Hamiltonian is
mc2¯her
Ze2 4\piϵ0 (gS −1) 2m2c2r3 L \cdot S. (7.114) The coupling is thus proportional to L \cdot S. The uncoupled states are of the form |n L mL mS\rangle , where L are represent eigenvalues of the L2 operator, mL represent eigenvalues of the Lz operator, and mS represent eigenvalues of the Sz operator. (We suppress dependence on the S quantum number, since it is always 1/2.) Under the L \cdot S coupling, these are no longer good quantum numbers. We can thus introduce the composite quantum number
(7.115) (fine-structure angular momentum) where from the triangularity condition (7.41) we have the new quantum number in the range |L −S| \le J \le L + S (7.116) The magnitude of J is
(7.117) or solving for the dot product,
J2 −L2 −S2 . (7.118) Thus, under the interaction we may still have eigenstates of L, S, J and mJ. In particular, the fine-structure shift due to this interaction is given to by simply taking the expectation value of the interaction Hamltonian in the coupled basis: ∆Efs = \langle n; L, S; J mJ|Hfs|n/L, S; J mJ\rangle
2mc2e
(7.119) The fine-structure shift then depends on J, breaking the degeneracy of different L levels. Actually, in writing down this expression, we are ignoring a relativistic correction of similar order, which also depends on L.15 However, the form of the perturbation is the important issue; when dealing with heavier alkali atoms it is difficult to obtain a quantitatively accurate expression anyway. Thus the main point is the introduction of the composite angular momentum J to label the fine-structure states, where the energy levels can be taken to be experimentally known, generally to high accuracy. 132006 CODATA recommended value. 14John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 11.8, p. 548. 15David J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, 1995), Section 6.3.1, p. 236.
7.2.2 Hyperfine Structure¶
Chapter 7. Atomic Angular-Momentum Structure 7.2.1.1 Spectroscopic Notation for Simple Atoms Now that we have introduced the basics of atomic structure, we briefly describe the common spectroscopic labels, of the form n 2S+1LJ. (7.120) The n is the principal quantum number of the active electron. The 2S + 1 gives the multiplicity of the electron spin, or the number of possible electron-spin states. For a single-electron (hydrogen-like) atom, 2S + 1 is always 2, since S = 1/2. The L quantum number is represented by a letter: S for L = 0, P for L = 1, D for L = 2, F for L = 3, G for L = 4, H for L = 5, and so on. The first four letters stand for descriptors for lines in alkali spectra (sharp, principal, diffuse, fundamental), and the rest are alphabetic continuations. Since the scheme is based on abbreviations, the letters should, in the author’s opinion, be set in roman, not italics, as is commonly the case in the literature. Finally, the subscript indicates the J quantum number. For example, the principle |g\rangle −\rightarrow |e\rangle laser-cooling transition for cesium is the D2 line, which is written
(7.121) The principal quantum number is 6 for both states, being the lowest available for the valence electron. Again for a single active electron in alkali atoms, 2S + 1 = 2. The ground and excited L quantum numbers are 0
As a slightly more complicated example, we consider strontium, which has two valence electrons, and a slightly more general notation. Here, for example, the narrow, second-stage laser-cooling transition is the ‘‘intercombination line’’ 5 s2 1S0 −\rightarrow 5 p 3P1. (7.122) (The 1S0 here would be read as ‘‘singlet S zero,’’ and the 3P1 would be read as ‘‘triplet P one.’’) Note that the configurations for the two electrons are also given in lower case. In the ground state, both electrons are
unexcited electron is implied). In these two levels the respective total orbital quantum number is given also
reflects the other possible S = 1 value, where S now represents the total electron spin S1 + S2. Hence, the ‘‘intercombination line’’ name, since the transition flips an atomic spin, which is electric-dipole-forbidden (a
state, where S = 1 and L = 1, J could be 0, 1, or 2. 7.2.2 Hyperfine Structure The hyperfine structure of an atom arises from the interaction between the total atomic angular momen- tum J and the nuclear angular momentum I. We will develop this a bit more carefully, as in atomic physics and quantum optics a single laser could interact almost resonantly with hyperfine-split states, which is not as often the case with fine structure. The basic idea is essentially the same as for the fine-structure case. The nuclear magnetic moment is µI = −µBgI I ¯h, (7.123) where I is the nuclear spin operator and gI is the nuclear spin g-factor.16 Again, the electron is effectively a current loop, and generates a magnetic field of the form B = −bJ, (7.124) where b is some positive constant, since B and J should be antiparallel for an electron where the charge is negative. The interaction is then given by (ignoring hyperfine couplings between different J, and thus 16Experimentally measured values for gI for the alkali atoms are given by E. Arimondo, M. Inguscio, and P. Violino, ‘‘Ex- perimental determinations of the hyperfine structure in the alkali atoms,’’ Reviews of Modern Physics 49, 31 (1977) (doi: 10.1103/RevModPhys.49.31).
7.2 Static Angular-Momentum Structure of Atoms assuming J is still a good quantum number) Hhfs = −µI \cdot B = −µBgIb ¯h
I \cdot J ¯h2 , (7.125) where, since we have considered the interaction of the nuclear and electron magnetic dipoles, Ahfs is called the magnetic dipole hyperfine constant and has the dimension of energy. For all abundant alkali atoms, Ahfs > 0, since gI < 0 (with the exception of 40K, where the sign is opposite). The form of this interaction (I \cdot J) is very similar to the fine-structure (L \cdot S) interaction. As in the fine-structure case, we can add the angular momenta to obtain the total atomic angular momentum
(7.126) (hyperfine-structure angular momentum) Under this interaction, we can use the new hyperfine quantum number F to label the new eigenstates; as in the fine-structure case, we square Eq. (7.126) to obtain the operator equation
(7.127) and thus when the operator I \cdot J acts on a hyperfine state |JIF\rangle we find that it is an eigenstate,
|JIF\rangle , (7.128) where the eigenvalue is written in terms of the combination
(7.129) of angular-momentum quantum numbers. Thus, the energy shift due to this interaction is simply ∆Ehfs = 1 2AhfsK. (7.130) Here Ahfs can be computed, though the calculation can be complex, or simply experimentally measured. The hyperfine shift is much smaller than the fine-structure shift. This is because of the weak nuclear magnetic moment: while the electron moment was of the order of µB = e¯h/2me, the nuclear moment is of the order of µN = e¯h/2mp, where mp is the proton mass. The nuclear moment is thus smaller by a factor on the order of me/mp \approx 1/1836, and so the hyperfine interaction should be smaller than the fine-structure interaction by a factor on the same order. The higher-order corrections to this simple theory become quite involved, and so we summarize the main points here.17 In general, the interaction between the nuclear and electron angular momenta can be expanded in a multipole series, Hhfs = X k T(k) e \cdot T(k) n , (7.131) where T(k) e and T(k) n are spherical tensor operators of rank k (defined below in Section 7.3.3), that respectively operate on only the electronic and nuclear Hilbert spaces. The k = 0 monopole term has already been included in the fine-structure calculation. We have treated the k = 1 magnetic-dipole term above. The k = 2 and k = 3 terms correspond respectively to the electric-quadrupole and magnetic-octupole terms. Because we will compute expectation values with respect to |J I F\rangle states, due to parity considerations either the electric or magnetic interaction will alternatingly vanish at each multipole order (with the electric dipole operator coupling only states of opposite parity, for example). The interaction between the electron and 17Charles Schwartz, ‘‘Theory of Hyperfine Structure,’’ Physical Review 97, 380 (1955) (doi: 10.1103/PhysRev.97.380). The electric hexadecapole term is given here in addition to the ones we have shown. See also Lloyd Armstrong, Jr., Theory of the Hyperfine Structure of Free Atoms (Wiley-Interscience, New York, 1971), Eqs. (IV-25), (IV-28), and (IV-31).
Chapter 7. Atomic Angular-Momentum Structure nuclear angular momenta is given by evaluating these operators, with the result up to the magnetic-octupole contribution reading Hhfs = Ahfs I \cdot J ¯h2 + Bhfs
2I(2I −1)J(2J −1) + Chfs
I(I −1)(2I −1)J(J −1)(2J −1) . (7.132) Again, the three terms on the right-hand side respectively represent magnetic-dipole (applicable for I, J > 0), electric-quadrupole (applicable for I, J > 1/2), and magnetic-octupole (applicable for I, J > 1) couplings. Thus, Ahfs is the magnetic-dipole hyperfine constant, Bhfs is the electric-quadrupole hyperfine constant, and Chfs is the magnetic-octupole hyperfine constant. We can see which terms are applicable to a given transition from the general product rule (7.267) that we prove later for commuting tensor operators, \langle J, I; F mF|T(k) e \cdot T(k)
F I J k J I \langle J||T(k)
n ||I\rangle , (7.133) where to satisfy the triangle inequalities for the 6-j symbol, we must have J \ge k/2 and I \ge k/2. The eigenenergies under the hyperfine interaction may then be written in terms of the shift ∆Ehfs = 1 2AhfsK + Bhfs
4I(2I −1)J(2J −1) + Chfs
I(I −1)(2I −1)J(J −1)(2J −1) . (7.134) Generally the effect of the last octupole term is quite difficult to observe, but it has been observed in
the octupole interaction contributes in cesium to the hyperfine splittings only at the kHz level, a very difficult level of accuracy to achieve in observing optical transitions. To illustrate the hyperfine structure, the hyperfine structure of the cesium D2 (laser-cooling) transition is shown here.19 Note that the ground-state hyperfine splitting of 133Cs is particularly significant, as it defines our measure of time: the second is defined such that the ground-state hyperfine splitting of an isolated 133Cs atom is exactly 9.192 631 770 GHz. 18Vladislav Gerginov, Andrei Derevianko, and Carol E. Tanner, ‘‘Observation of the Nuclear Magnetic Octupole Moment of 133Cs,’’ Physical Review Letters 91, 072501 (2003) (doi: 10.1103/PhysRevLett.91.072501). 19For sources of the measured values, along with more compiled data and a more terse description of hyperfine structure, see Daniel A. Steck, ‘‘Cesium D2 Line Data,’’ available online at http://steck.us/alkalidata.
7.3.1 Rotation Operator¶
7.3 Rotations and Irreducible Tensor Operators 62S1/2 62P3/2 852.347 275 82(27) nm 351.725 718 50(11) THz 11 732.307 104 9(37) cm-1 1.454 620 563(35) eV 4.021 776 399 375 GHz (exact) 5.170 855 370 625 GHz (exact) 9.192 631 770 GHz (exact) F = 4 F = 3
(0.35 MHz/G)
263.8906(24) MHz 12.798 51(82) MHz 188.4885(13) MHz 339.7128(39) MHz 251.0916(20) MHz 201.2871(11) MHz 151.2247(16) MHz F = 5 F = 4 F = 3 F = 2
(0.56 MHz/G)
(0.37 MHz/G) gFo=o0 (0.00 MHz/G)
7.3 Rotations and Irreducible Tensor Operators 7.3.1 Rotation Operator What is the operator that induces a rotation in quantum mechanics? Rather than deduce it directly, we will ‘‘cheat’’ and simply quantize the classical version of a rotation. Consider the classical angle (generalized
Thus, time evolution according to this Hamiltonian is equivalent to a rotation through an angle t. Quantum mechanically, since this Hamiltonian is time-independent, the time-evolution operator is
−iHt ¯h = exp −iJ\zeta ¯h
(7.135) then we have the rotation operator (for a two-dimensional system) for a rotation through angle \zeta. General-
Chapter 7. Atomic Angular-Momentum Structure angle \zeta) is induced by the unitary rotation operator
¯h . (7.136) (rotation operator) Note that a rotation of an angular-momentum state |j m\rangle about the z-axis (the quantization axis) is partic- ularly simple, as
−iJz\zeta ¯h
(7.137) (rotation operator) However, a rotation about any other axis is more complicated, as the result will in general be a superposition of angular-momentum states. Being a rotation, the j quantum number must be left unchanged (following
m values. This is a good way to see that the angular orientation of a state is encoded in the m quantum number; the j quantum number by itself doesn’t tell you about orientation. 7.3.1.1 Rotation Matrix To formalize the transformation of |j m\rangle into a superposition of states |j m′\rangle by a rotation we can write out an explicit rotation matrix in the basis of angular-momentum states. Wigner’s convention is to write such a matrix as
j X m′=−j |j m′\rangle d(j) m′m(\zeta), (7.138) (action of rotation matrix) where d(j)
(7.139) (rotation matrix) Note the ‘‘backwards’’ convention for the matrix indices for the matrix-vector product in Eq. (7.138). The point is that there is a (2j + 1) \times (2j + 1) rotation matrix d(j)(\zeta) associated with the rotation operator R(\zeta) when it acts on a subspace of angular-momentum states with fixed quantum number j. If we follow one rotation R(\alpha) by another rotation R(\beta), we can represent the total rotation by a composite rotation operator R:
(7.140) Projecting into the angular-momentum representation and using the completeness relation,
X m′′
(7.141) The corresponding rotation matrices thus compose by normal matrix multiplication, so long as the first rotation to operate is the rightmost:
(7.142) (composition of rotations) This property is very useful in decomposing arbitrary rotations, as we now discuss. 7.3.1.2 Euler Angles As in classical mechanics, a general rotation may be represented as a composition of rotations through the three Euler angles: first rotate about the z-axis by angle \alpha, then rotate about the new y-axis by angle \beta, and finally rotate about the new z-axis by angle \gamma. These angles are illustrated in the diagrams below. Note that the rotation operators act on the state, not the coordinate system; however, we are also considering rotations of the axes solely to define the second and third rotations.
7.3 Rotations and Irreducible Tensor Operators z y x xoo' yoo' a zoo' y x xoo' yoo' xoo'o' zoo'oo' a b zoo' y x xooo' yoo'oo' xoo'oo' zoo'o' a b g Thus an arbitrary rotation R may always decomposed in the form
(7.143) where again \alpha = \alphaˆz, \beta = \betaˆy′, where ˆy′ is along the new y-direction after the \alpha rotation, and \gamma = \gammaˆz′′, where ˆz′′ is along the new z-direction after the \beta rotation. Clearly, R(\alpha) is written to the right since it is the first rotation, and thus must operate first on the state vector. (The order is important, because in general rotation operators for different axes do not commute.) Writing these operators out explicitly,
(7.144) But now, since R(\beta) is written in terms of the coordinate system after the R(\alpha) rotation, we can write this rotation as a rotated version of the operator in the original coordinate system:
(7.145) Similarly, for the last rotation, we can write
(7.146) and putting this into Eq. (7.144), we find
(7.147) Now putting in Eq. (7.145) and the analogous result with Jz′,
(7.148) (rotation operator, Euler angles) Conveniently, then, a rotation according to the Euler angles may be implemented solely in the original coordinate system, if the order of the rotations is reversed. Now to return to the rotation matrix. Using the definition (7.139) for the matrix corresponding to this rotation operator, d(j)
(7.149) The first and last rotations are thus easy to represent, leaving the second rotation as the only nontrivial one: d(j)
(7.150) (rotation matrix, Euler angles) Wigner’s explicit expression for the remaining rotation matrix is20 d(j)
p (j + m)!(j −m)!(j + m′)!(j −m′)! \times X s (−1)s (j −m′ −s)!(j + m −s)!(s + m′ −m)!s! cos \beta 2j+m−m′−2s −sin \beta m′−m+2s , (middle rotation matrix, explicit form) (7.151) 20M. E. Rose, Elementary Theory of Angular Momentum (Wiley, 1957), p. 52.
Chapter 7. Atomic Angular-Momentum Structure where the sum is over all values of s where the factorials are nonnegative. This form is particularly useful for computer implementation of the rotation matrices. We can also see from this formula that under the replacement \beta −\rightarrow −\beta, only the sin factor changes sign, so that d(j)
m′m(\betaˆy), (7.152) since the 2s part never contributes a minus sign. Furthermore, this formula is invariant under the replace- ments m −\rightarrow −m′ and m′ −\rightarrow −m, d(j)
m′m(\betaˆy). (7.153) Finally, since the rotation by −\beta is the transpose of the rotation by \beta (this rotation matrix is orthogonal), d(j)
m′m(\betaˆy). (7.154) Combining these last three expressions, we find d(j)
−m,−m′(\betaˆy). (7.155) This last expression may be generalized to arbitrary axes. Combining it with Eq. (7.150) gives d(j)
−m′,−m(\zeta), (7.156) (rotation matrix conjugation) where the complex conjugation ‘‘undoes’’ the minus signs of m and m′ in the exponents of the general rotation matrix. 7.3.1.3 Clebsch–Gordan Series One other useful relation comes by considering the rotation matrix for an arbitrary rotation operator R: d(j)
(7.157) If we regard the vector J associated with j to be the sum J1 + J2, we may write d(j) m′m = X m1m′ 1m2m′ \langle j m′|j1 m′ 1, j2 m′
1, j2 m′ 2|R|j1 m1, j2 m2\rangle \langle j1 m1, j2 m2|j m\rangle . (7.158) Since the rotation acts on each subspace, d(j) m′m = X m1m′ 1m2m′ \langle j m′|j1 m′ 1, j2 m′ 2\rangle d(j1) m′ 1m1d(j2) m′ 2m2\langle j1 m1, j2 m2|j m\rangle . (Clebsch–Gordan-series inverse) (7.159) This relation acts as a recursion relation by which rotation matrices can be constructed from other rotation matrices of smaller angular momentum. It is also easy to write down the inverse relation, where we find d(j1) m′ 1m1d(j2) m′ 2m2 = X jmm′ \langle j1 m′ 1, j2 m′ 2|j m′\rangle d(j) m′m\langle j m|j1 m1, j2 m2\rangle . (Clebsch–Gordan series) (7.160) This relation is called the Clebsch–Gordan series.21 Obviously, the summations in both this relation and its inverse are constrained heavily by the triangularity of the Clebsch–Gordan coefficients. 21M. E. Rose, op. cit., Eq. (4.25), p. 58.
7.3.2 Spherical Harmonics¶
7.3 Rotations and Irreducible Tensor Operators 7.3.2 Spherical Harmonics Now we consider the physical-space representation of angular-momentum states |j m\rangle , in particular the projection into angular states \langle \theta, \phi|j m\rangle . First, consider what happens under a simple rotation, say about the z-axis. This rotation corresponds to \beta = \gamma = 0 in the Euler angles above, giving a rotation operator of simply
(7.161) or a rotation matrix d(j)
(7.162)
d(j)
(7.163) but if j is a half-integer, then so is m, and the rotation operator amounts to a factor of −1. On the other hand, if j is an integer, then so is m, and the rotation operator is just the identity. The latter corresponds to what we expect for a vector in coordinate space: a rotation by 2\pi should amount to nothing. However, this is not the case for half-integer angular momenta, and so we conclude that these do not represent angular momenta of, say, particles (i.e., orbital angular momentum). However, for intrinsic particle spins, half- integer angular momenta are just fine, since we don’t require 2\pi-periodicity in that case. Nonetheless, it seems rather strange that, say a qubit (spin-1/2 particle), under a 2\pi-rotation, flips its sign; only under a 4\pi-rotation is it invariant.22 Thus, for coordinate representations of angular-momenta, we will only consider the case of integer j. We will thus use the alternate notation for such ‘‘orbital’’ angular momenta of |ℓm\rangle being a simultaneous eigenstate of L2 and Lz, with L := r \times p. We can thus define the spherical harmonic as the projection onto the usual spherical angles Y m
(7.164) (spherical harmonic) Later, in Section 8.4.4.1, we show that the spherical harmonics have the form Y m
s (2ℓ+ 1)(ℓ−m)!
P m
(7.165) (spherical harmonic) where P m ℓ(cos \theta) is an associated Legendre function, by solving the scalar wave equation in spherical coordi- nates, which applies to the present case of the Schrödinger equation. Some examples of low-order spherical harmonics are Y 0
\sqrt 4\pi , Y 0
r
Y \pm1
r
(7.166) corresponding to monopole and dipole angular patterns. The spherical harmonics are orthonormal, being representations of |ℓm\rangle . Thus, using \langle ℓ′ m′|ℓm\rangle = \deltaℓℓ′\deltamm′, we can insert the identity Z
(7.167) in terms of angular states to obtain Z dΩY m
(7.168) (orthonormality relation) which is simply the expicit statement of orthonormality of the spherical harmonics. 22Thus when the qubit is mapped to the sphere, as in the Bloch sphere, you really have to keep track of whether the particle is ‘‘inside’’ or ‘‘outside’’ the sphere, which is one representation of the minus sign. See, e.g., F. De Zela, ‘‘Topological phase for entangled two-qubit states and the representation of the SO(3) group,’’ Journal of Optics B: Quantum and Semiclassical
Chapter 7. Atomic Angular-Momentum Structure 7.3.2.1 Sum Rule and Addition Theorem Another important relation comes from considering the sum X m
X m Y m∗ ℓ
(7.169) for two spherical angles (\theta1, \phi1) and (\theta2, \phi2). We now intend to show that this expression is independent of orientation (i.e., it is a scalar under rotations) by showing it is equivalent to the rotated version X m
X m Y m∗ ℓ (\theta′ 1, \phi′ 1) Y m ℓ(\theta′ 2, \phi′ 2), (7.170) for some rotation operator R, where R|ℓ, m\rangle is the rotated state. Recall that there are two ways to think of a rotation: the first is that the rotation operator acts on (and rotates) the state vector, while the other is that the rotation operator acts on the basis vectors |\theta, \phi\rangle and rotates the coordinate system in the opposite sense. Thus, the rotated angles (\theta′ 1, \phi′ 1) and (\theta′ 2, \phi′ 2) are defined by |\theta′
operator has a matrix representation that we will denote by d(ℓ) m′m, which is unitary matrix. Thus, Eq. (7.170) becomes X m Y m∗ ℓ (\theta′ 1, \phi′ 1) Y m ℓ(\theta′ 2, \phi′
X mm′m′′ d(ℓ) m′m d(ℓ)∗
(7.171) where we have used the unitarity of the rotation matrix. We can carry out the sum over m by again using the unitarity of the rotation matrix, which we may write as X m d(ℓ) m′m d(ℓ)∗
(7.172) so that we arrive at X m Y m∗ ℓ (\theta′ 1, \phi′ 1) Y m ℓ(\theta′ 2, \phi′
X m
(7.173) after dropping primes from the remaining dummy index. Then comparing to Eq. (7.169), we now see the independence of the sum under rotations: X m Y m∗ ℓ (\theta′ 1, \phi′ 1) Y m ℓ(\theta′ 2, \phi′
X m Y m∗ ℓ
(7.174) In particular, we may choose the rotation such that (\theta′ 1, \phi′ 1) point along the z-axis, and \phi′ 2 = 0. Now we use the fact that P m
Y m
r 2l + 1 4\pi \deltam0. (7.175) Thus we arrive at the spherical-harmonic addition theorem Y 0
r 4\pi 2l + 1 X m Y m∗ ℓ
(7.176) (addition theorem)
2 is the angle between the radial vectors corresponding to the two directions (\theta1, \phi1) and (\theta2, \phi2). Taking \theta1 = \theta2 and \phi1 = \phi2, so that \theta = 0 in the addition theorem, we can drop the subscripts and write the sum rule ℓ X m=−ℓ |Y m
4\pi , (7.177) (sum rule) where we have again used Eq. (7.175). This sum rule is essentially just another statement of the rotational invariance of products of spherical harmonics when summed over m. This statement indicates indirectly that the m quantum number determines the orientation of the modes; summing over it results in an isotropic angular distribution.
7.3 Rotations and Irreducible Tensor Operators 7.3.2.2 Relation to the Rotation Matrix As in the previous section, when a spherical harmonic
(7.178) is rotated, we can express the result in primed coordinates
(7.179) and then expressing the rotation operator as a matrix and using the first expression, X m′ Y m′
m′m = Y m
(7.180) In general, the rotation matrix can be specified in terms of the Euler angles. Writing this explicitly while
Y m
X m′ d(ℓ)
(7.181) Now we set \theta −\rightarrow \theta2, \phi −\rightarrow \phi2, m = 0, \alpha −\rightarrow \theta1, \beta −\rightarrow \theta2, \theta′ −\rightarrow \theta, and we take the rotation to be such
Y 0
X m′ d(ℓ)
(7.182) We can now compare this result to the spherical-harmonic sum rule (7.176) and see that they have the same form if we identify Y m∗ ℓ
r 2ℓ+ 1 4\pi d(ℓ)
(spherical harmonic as rotation matrix) (7.183) Indeed, in our setup here, \theta is still the angle between the vectors along (\theta1, \phi1) and (\theta2, \phi2). We have chosen the rotation \phi1 to bring the vector along (\theta2, \phi2) to the x-z plane, and thus with this particular orientation of the problem, \phi1 = \phi2. The remaining rotation indicated by \theta1 determines the separation angle between
opposite rotation as on the states.) In particular, this representation of the spherical harmonics implies the conjugation relation Y m∗ l
l
(7.184) (spherical-harmonic conjugation) as a direct consequence of Eq. (7.156). Furthermore, if we use the Clebsch–Gordan series (7.160) with the second indices set to zero, d(ℓ1) m10 d(ℓ2) m20 = X ℓm
(7.185) we can then use the representation (7.183) to write Y m1
X ℓm s
(recoupling relation) (7.186) after complex conjugation. This is the recoupling relation for spherical harmonics. Using Eq. (7.68) to relate the Clebsch–Gordan coefficients to 3-j symbols, we find the alternate form23 Y m1
X ℓm (−1)m r
4\pi ℓ1 ℓ2 ℓ m1 m2 m ℓ1 ℓ2 ℓ Y −m ℓ
(recoupling relation) (7.187) 23Note that this relation is referred to as the ‘‘addition theorem’’ by A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton, 1957), p. 63, Eq. (4.6.5).
7.3.3 Irreducible Tensor Operators¶
Chapter 7. Atomic Angular-Momentum Structure after letting m −\rightarrow −m. Again, ℓranges from |ℓ1 −ℓ2| to ℓ1 + ℓ2 and m1 + m2 = −m for the 3-j symbols to be nonvanishing. 7.3.3 Irreducible Tensor Operators 7.3.3.1 Spherical Basis As a prelude to introducing irreducible tensor operators, we will examine the spherical basis, which will be important in treating dipole interactions with angular momentum. The spherical basis is simply an alternative to the Cartesian vector basis that is especially convenient when dealing with angular momentum. In terms of the Cartesian basis vectors ˆx, ˆy, and ˆz, the spherical basis vectors are defined as ˆe\pm1 := ∓1 \sqrt
(7.188) (spherical basis vectors) Likewise, if the Cartesian components of a vector A are defined such that A = Axˆx + Ayˆy + Azˆz, then the components of A in the spherical basis are given in the same way by A\pm1 = ∓1 \sqrt 2(Ax \pm iAy) A0 = Az, (7.189) (vector components in spherical basis) which is to say Aq := ˆeq \cdot A, and A = X q
X q ˆe∗ qAq. (7.190) (vector in spherical basis) This funny form for the vector in spherical components comes about because ˆe∗ q \cdotˆeq′ = \deltaqq′, so we really need to think about products of vectors, where one of them is conjugated (which is, of course, a natural idea in quantum mechanics). Inverting Eqs. (7.189) gives Ax = −1 \sqrt 2(A1 −A−1) Ay = i \sqrt 2(A1 + A−1) Az = A0, (7.191) with, of course, the same relations for the Cartesian basis vectors in terms of the spherical basis vectors. In the spherical basis, the dot product of two vectors is given by
X q
X q
X q Aq(B∗)∗ q. (dot product in spherical basis) (7.192) The conjugation in the last expression should be read as: conjugate the full vector B [by conjugation in Cartesian coordinates or via the conjugate of Eq. (7.190)], then project the qth coordinate via the inner product with ˆeq, so that (B∗)q ≡ˆeq \cdot(B∗), and then conjugate the (scalar) result. It follows from Eqs. (7.192) that
X q (Aq)∗Bq, |A|2 = X q |Aq|2. (dot product and norm in spherical basis) (7.193)
7.3 Rotations and Irreducible Tensor Operators These expressions are a bit more sensible because A∗\cdot B is a more natural inner product in this basis than A \cdot B. Finally, we note that the components of the position vector r can be written r\pm1 = ∓r \sqrt
(7.194) (position operator in spherical basis) or more compactly, rq = r r 4\pi 3 Y q
(7.195) (position operator as spherical harmonic) These forms will be useful, for example, when evaluating the dipole radiation pattern, and they show explicitly the connection of spherical-basis vector operators to the dipolar spherical harmonics. 7.3.3.2 General Definition The position vector here (as with any Cartesian three-vector) in the spherical basis is a vector operator, because of the way the three components transform among each other under rotations. We will now generalize this notion of sets of operators that are closed under rotations. An irreducible tensor operator of rank k (specifically, a spherical tensor operator), which we denote by T(k) is a set of 2k+1 operators that transform among themselves under rotations in the same way as the angular-momentum states |j m\rangle , where j = k: R(\zeta)T (k)
k X q′=−k T (k) q′ d(k) q′q(\zeta). (7.196) (spherical tensor operator) Equivalently, they transform in the same way under rotations as the spherical harmonics, as in Eq. (7.180). In this context, irreducible means that there is no proper subset of the component operators that transform among themselves in a similar way. This is already guaranteed by the definition, as the set of angular- momentum basis states |j m\rangle is irreducible in the same sense. This definition actually introduces spherical tensors in general: that is, sets of components that transform into each other. Since again we require that they transform as the spherical harmonics, then the spherical harmonics Y m ℓ give a particular example of a spherical tensor of rank ℓ. The tensor operator comes about when we take each component of the tensor to be an operator. Since we have already seen that the position operator in the spherical basis is proportional to Y m 1 , as in Eq. (7.195), we know that r transforms as a spherical tensor of rank 1. Thus, r is an example of a rank-1 irreducible tensor operator according to the definition here, which is again also a vector operator. 7.3.3.3 Cartesian Tensors The more familiar type of tensor is the Cartesian tensor, of the form M\alpha\beta, for example, for a rank-2 tensor, where \alpha and \beta range from 1 to 3 (or x to z). A rank-k Cartesian tensor is generally represented by k indices, and transforms under rotations according to ˜
(7.197) where
(7.198) is the rotation operator expressed in Cartesian coordinates. That is, the rotation operator is applied to each dimension, represented by each index. How is the Cartesian tensor related to the irreducible tensors? Well, returning to the rank-2 example, the Cartesian tensor operator has nine independent component operator, whereas the irreducible, rank-2
Chapter 7. Atomic Angular-Momentum Structure tensor has only five. The Cartesian tensor must be reducible, and we can reduce it as follows. We may construct a scalar, or rank-0 operator, by computing the trace,
(7.199) (scalar part) This is invariant under rotations, since computing the trace after a rotation gives Tr[RMR\dagger] = Tr[M] after cyclic permutation under the trace. We can then form a vector (rank-1) operator as M (1) µ
(7.200) (vector part) which has three independent components and is clearly related to the antisymmetric part of M\alpha\beta. To see that it transforms as a vector under rotations, we can compute the vector after rotation of the tensor, with the result ˜ M (1) µ
(7.201) Now note that the cross product A\timesB of two vectors, after rotating each vector, is the same as the rotation of the cross product itself, or
(7.202) where R is the rotation matrix. Expressed in components, this becomes
(7.203) Since this holds for any A and B, we may drop them and write
(7.204) Putting this into Eq. (7.201), ˜ M (1) µ
(7.205) which is the proper vector rotation of Eq. (7.200). Obviously, this vector operator is still expressed in Cartesian components, but can be transformed to a spherical tensor by Eqs. (7.189). Finally, the reduced (now irreducible) rank-2 tensor is what remains, or is in other words the original tensor with the trace and antisymmetric parts subtracted away: M (2)
(7.206) (irreducible tensor part)
symmetric and traceless, and has only 5 independent components, as is consistent with the irreducible rank-2 form. It is also still obviously a rank-2 tensor, since it is a linear combination of M\alpha\beta, M\beta\alpha, and \delta\alpha\beta, which are all rank-2 tensors. However, the transformation of the remaining components to a spherical rank-2 tensor is more complicated than for the vector-operator case. In any case, we may now write the original tensor in terms of its irreducible components as
4M (1)
(7.207) (reduced Cartensian tensor)
7.3 Rotations and Irreducible Tensor Operators We can see this by using Eqs. (7.199), (7.200), and (7.206) to write
4M (1)
h
i
= 1
= 1
= 1 h
i + 1
(7.208) where we have used the relation
(7.209)
notation (Problem 7.2). 7.3.3.4 Products of Tensors With Cartesian tensors, taking the product of two tensors to form a higher-rank tensor is straightforward: just multiply them as usual. For example, to take two vectors to form a tensor, we write
(7.210) (Cartesian tensor product) In general, the tensor product of two Cartesian tensors of rank k1 and k2 will be of rank k = k1 + k2. However, the case of spherical tensors is a bit more complicated. If we take the addition rules (7.40) for two angular momenta, and then project the first one into angular states |\theta, \phi\rangle , we find that the combination of two spherical harmonics is Y m
X m1m2
Y m1
(7.211) where |ℓ1 −ℓ2| \le ℓ\le ℓ1 +ℓ2. Spherical harmonics are an example of spherical tensors, and in fact we defined spherical tensors to transform in the same way as spherical harmonics. Thus, we conclude that T (k) q = X q1q2
T (k1) q1 T (k2) q2
(7.212) (spherical tensor product) where |k1 −k2| \le k \le k1 + k2. This is how products of spherical tensors work: spherical tensors of rank k1 and k2 can be combined to form a spherical tensor with a range of different ranks. For example, suppose we want to take a product of two vector operators A and B. The resulting product tensor T(k) could have a rank of k = 0, 1, or 2. The rank-0 combination is T (0) = X q=−1
X q=−1 (−1)q \sqrt 3 AqB−q, (7.213) which we see is the scalar product of the two vectors, up to a constant factor. T (0)
\sqrt 3 , (7.214) (rank-0 vector product)
Chapter 7. Atomic Angular-Momentum Structure The rank-1 combination is T (1) q = X q′=−1 Aq′Bq−q′ \langle 1 q′; 1 q −q′|1 q\rangle . (7.215) Writing out the three components of the resulting vector, T (1) = \sqrt 2(A1B0 −A0B1) T (1) = \sqrt 2(A1B−1 −A−1B1) T (1) −1 = \sqrt 2(A0B−1 −A−1B0), (7.216) and then putting in the definitions of the spherical-vector components, we see that the vector product is the usual cross product, expressed in the spherical basis: T (1) q = i \sqrt 2(A \times B)q. (7.217) (rank-1 vector product) Finally, the rank-2 combination is T (2) q = X q′=−1 Aq′Bq−q′ \langle 1 q′; 1 q −q′|2 q\rangle . (7.218) Writing out the resulting five tensor components, T (2) \pm2 = A\pm1B\pm1 T (2) \pm1 = \sqrt 2(A\pm1B0 + A0B\pm1) T (2) = \sqrt
(7.219) (rank-2 vector product) In fact, what we have rederived here is the reduction of the previous section of the rank-2 Cartesian tensor
trace (7.199), the cross product (7.217) is the antisymmetric part (7.200) of the tensor, and the rank-2 tensor (7.219) is the traceless, symmetric part (7.206) of the tensor, but here written out in spherical components
Finally, we note that with Cartesian tensors, tensor products of lower rank than we have already considered are possible via contraction, or making two indices the same and summing over the result. For example, the scalar product of two Cartesian vectors is A\alphaB\alpha, which is of course lower rank than the tensor product A\alphaB\beta. The usual matrix product M\alpha\gamma = A\alpha\betaB\beta\gamma is the same idea, giving a rank-2 tensor as the product of two rank-2 tensors, which could give instead a rank-4 tensor without contraction. A scalar can then be obtained by a second contraction, M\alpha\alpha = A\alpha\betaB\beta\alpha. In general, the product of a rank-k1 tensor and a rank-k2 tensor is of rank k1 + k2, and this composite rank can be reduced by 2 at a time by contraction. We have already seen how this works for spherical vectors above, and in fact we have also seen that it is possible to reduce the rank by only one, by multiplying by ϵ\alpha\beta\gamma and then contracting the resulting tensor product (i.e., to give a cross product between vectors). We will simply note here that given two spherical tensors of the same rank k, it is always possible to construct a scalar product. Using Eq. (7.212), T (0) = X q T (k) q U (k)
X q (−1)k+q \sqrt 2k + 1 T (k) q U (k) −q , (7.220)
7.3 Rotations and Irreducible Tensor Operators after evaluating the Clebsch–Gordan coefficient. Usually we move the invariant factor out of the sum (−1)−k\sqrt 2k + 1 T (0) = X q (−1)q T (k) q U (k) −q , (7.221) and then define the result to be the scalar product of the two spherical tensors:
k X q=−k (−1)q T (k) q U (k) −q . (spherical tensor product) (7.222) This extra factor is precisely the factor of −1/ \sqrt 3 from Eq. (7.214) beyond the usual scalar product of two
as we see from Eq. (7.192) in our discussion of spherical vectors. 7.3.3.5 Commutation Rules Consider the operator for an infinitesimal rotation \delta\phi:
(7.223) The tensor operator T (k) q transforms under this rotation as in Eq. (7.196), where the rotation matrix corre- sponding to the rotation is d(k)
1 −i
|k q\rangle . (7.224) Thus, Eq. (7.196) becomes 1 −i
T (k) q 1 + i
= X q′ T (k) q′ \langle k q′| 1 −i
|k q\rangle . (7.225) Multiplying this out and dropping second-order terms in \delta\phi gives h
q i = X q′ T (k)
(7.226) Setting \delta\phi −\rightarrow ˆz\delta\phi then gives h Jz, T (k) q i = X q′ T (k) q′ \langle k q′|Jz|k q\rangle , (7.227) and using Eq. (7.14) leads to the commutation rule h Jz, T (k) q i
q . (7.228) (Jz commutator) On the other hand, setting \delta\phi −\rightarrow [∓(ˆx \pm iˆy)/ \sqrt
h J\pm, T (k) q i = X q′ T (k)
(7.229) Then using Eq. (7.30) leads to the commutation rule h J\pm, T (k) q i = p (k \pm q + 1)(k ∓q) T (k) q . (7.230) (J\pm commutator) These commutation rules are the analogous relations to the effects of Jz and J\pm on kets |j m\rangle in Eqs. (7.14) and (7.30).
7.3.4 Wigner-Eckart Theorem¶
Chapter 7. Atomic Angular-Momentum Structure 7.3.4 Wigner–Eckart Theorem Now we come to an extremely important result in angular momentum algebra. Consider the action of a tensor-operator component on an angular-momentum state, T (k) q
(7.231) where \alpha′ represents other (i.e., radial) quantum numbers that do not represent angular dependence of the state. How does this state transform under a rotation? Since we may write the rotated form as RT (k) q
(7.232) evidently T (k) q and |\alpha′ j′ m′\rangle transform separately. In particular, by definition the state |\alpha′ j′ m′\rangle transforms as the ket |j′ m′\rangle , while by comparing Eq. (7.196) to Eq. (7.138) we recall that T (k) q transforms via the rotation matrix in the same way as the angular-momentum ket |k q\rangle . Thus, the state T (k) q |\alpha′ j′ m′\rangle transforms as the composite state |k q\rangle |j′ m′\rangle , or in the way we’ll set it up, |j′ m′\rangle |k q\rangle (there is in principle a phase in making this rearrangement, and ignoring it amounts to absorbing it into the definition of the |\alpha′\rangle ). We can then consider the usual angular-momentum-addition relation
X k′q′
(7.233) and write in analogy to it the same superposition T (k) q
X k′q′
(7.234) where ˜\alpha is some set of transformed radial quantum numbers, since the states in the two relations transform equivalently. This is the crux of the argument: since we know that the tensor operator transforms like angular-momentum states, the action of the tensor operator on an angular-momentum state is just like a mixing of two angular momenta. In fact, the form of the transformed vector is quite constrained: Note that the vector in Eq. (7.234) is an eigenvector of Jz with eigenvalue ¯h(m′+q)—although it is a linear combination of vectors of the form |k′ q′\rangle , the Clebsch–Gordan coefficients constrain these such that q′ = m′ + q. Now we can operate from the left on Eq. (7.234) with \langle \alpha j m|, we then find the matrix element
q
X k′q′
(7.235) where we have used the orthogonality of the angular-momentum states to obtain the second equality. Now we note that the inner product \langle \alpha j m|˜\alpha j m\rangle is, in fact, independent of m, just as the inner product \langle j m|j m\rangle = 1 is m-independent. We may thus define the m-independent reduced matrix element
(7.236) (reduced matrix element) where the dependence on j, j′, and T(k) comes in via the way \alpha transforms into ˜\alpha. (The transformation \alpha −\rightarrow ˜\alpha of course introduces no m-dependence because by assumption \alpha represented the radial and thus orientation-independent part of the quantum state.) Note that the reduced matrix element, while using the notation of a tensor, is in fact a scalar quantity, as is clear from the right-hand side of the definition. Finally, using the reduced matrix element in Eq. (7.235), we arrive at the Wigner–Eckart theo- rem24
q
(Wigner–Eckart theorem) (7.237) 24Carl Eckart, ‘‘The Application of Group Theory to the Quantum Dynamics of Monatomic Systems,’’ Reviews of Modern Physics 2, 305 (1930) (doi: 10.1103/RevModPhys.2.305); Eugene P. Wigner, ‘‘Group Theory and Its Application to Quantum Mechanics of Atomic Spectra,’’ (Academic Press, 1959).
7.3 Rotations and Irreducible Tensor Operators Many sign and normalization conventions abound, particularly for the Wigner–Eckart theorem and the reduced matrix elements. By using the orthogonality relation (7.46), we can invert (7.237) to give
m′q
q
(reduced matrix element) (7.238) as an expression for the reduced matrix element in terms of a sum over matrix elements. The Wigner–Eckart theorem thus factors a matrix element of a component of an irreducible tensor operator into an orientation- independent part (the reduced matrix element) and a Clebsch–Gordan coefficient (which encapsulates all the orientation dependence of the matrix element). Note that for the reduced matrix elements, we are following here the normalization convention of Brink and Satchler.25 A common alternate convention for the Wigner–Eckart theorem may be written as26
q
(7.239) where the alternate reduced matrix element is related to the first one by
(7.240) However, we shall stick exclusively to the matrix element \langle \alpha j∥T(k)∥\alpha′ j′\rangle . This normalization convention is thus defined by
X m′q
q
X m′
(normalization convention for reduced matrix element) (7.241) which follows from squaring the Wigner–Eckart theorem (7.237) and then summing over all m′ and q, along with the orthogonality relation (7.46) to eliminate the Clebsch–Gordan coefficient. The Clebsch–Gordan coefficient in Eq. (7.237) requires that j take values between |k−j′| and k+j′. In particular, this indicates that T (k) q can only be of integer rank k. A tensor of half-integer rank would have the awkward consequence of inducing transitions between integer and half-integer states (i.e., between bosonic and fermionic states, which would only be acceptable if a particles are changing or being created/destroyed). 7.3.4.1 Dipole Operator As we mentioned above, the Wigner–Eckart theorem is so powerful, because it completely pins down the angular part of a matrix element of a tensor operator: the angular dependence of the matrix element can be factored out completely and written solely in terms of a Clebsch–Gordan coefficient (or equivalently, a Wigner 3-j symbol). Of course, this is because of an implicit spherical symmetry to the problem, since we assumed the existence of radial and angular-momentum quantum numbers. The real utility of the Wigner–Eckart theorem in quantum optics comes from its application to the dipole operator. Recalling that the dipole operator is proportional to the position operator, we know that it
\langle J mJ|dq|J′ m′
J; 1 q\rangle
r 2J + 1
(Wigner–Eckart theorem, dipole operator) (7.242) 25D. M. Brink and G. R. Satchler, Angular Momentum, 2nd ed. (Oxford, 1968), Section 4.7, p. 56 (ISBN: 0198514190). 26See, for example, Gordon Baym, Lectures on Quantum Mechanics (Westview Press, 1969); or A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton, 1957).
Chapter 7. Atomic Angular-Momentum Structure where the second form follows upon application of the symmetry relation (7.65) followed by an application of (7.54). Again, the orientation dependence of the dipole matrix element appears simply as a Clebsch– Gordan coefficient, while the radial dependence appears in the reduced matrix element. The reduced matrix elements of an atom may, as indicated above, be calculated from the radial parts of the atomic wave functions. However, the simplest way to obtain the reduced matrix element is via the following relation to the atomic spontaneous decay rate from the Je fine-structure level to the Jg level \GammaJgJe = \omega 3 3\piϵ0¯hc3 2Jg + 1
(spontaneous decay rate and reduced dipole matrix element) (7.243) as we show later in Chapter 11. This relates the reduced matrix element to a quantity readily accessible to experiment. We will return to the implications of the Wigner–Eckart theorem in detail below in Section 7.3.7. 7.3.4.2 Projection Theorem A useful special case of the Wigner–Eckart theorem arises if we consider the scalar product of a vector operator A with angular momentum J.27 Considering a matrix element of the scalar combination and expanding in the spherical basis using the scalar product (7.192),
X q
J0A0 −J1A−1 −J−1A1
(7.244) Then comparing Eqs. (7.7) and (7.189) to identify the spherical components of J in terms of the ladder operators, J\pm1 = ∓1 \sqrt 2J\pm, (7.245) we can use the action (7.30) of the ladder operators on angular-momentum eigenstates (with J\dagger
write
\sqrt p
−¯h \sqrt p
(7.246) Although this result is actually straying a bit from the main point, it shows how the matrix elements here can be expressed solely in terms of matrix elements of A. For our purposes, the more useful result comes from applying the Wigner–Eckart theorem (7.237) to the scalar operator J\cdotA, for which the matrix element must be orientation-independent (and thus m-independent):
(7.247) Here, cj is again independent of m, and also independent of A (except that it is a vector operator) and \alpha and \alpha′. Now to compute cj, we take advantage of the A-independence of this last relation, setting A −\rightarrow J and \alpha′ −\rightarrow \alpha to obtain
(7.248) Using this relation to eliminate cj in Eq. (7.247), we have
¯h2j(j + 1)
(7.249) 27Here we are following J. J. Sakurai and Jim Napolitano, Modern Quantum Mechanics, 2nd ed. (Cambridge, 2017), pp. 254-5 (ISBN: 9781108422413).
7.3.5 Hermitian Conjugates of Tensor Operators¶
7.3 Rotations and Irreducible Tensor Operators¶
¯h2j(j + 1)
(7.250) assuming that the Jq matrix element does not vanish. Rearranging, we come to
¯h2j(j + 1)
(7.251) (projection theorem) which is known as the projection theorem. This is sometimes a handy special case of the Wigner–Eckart theorem where the reduced matrix element has already been eliminated. This will be useful later, for example, when we discuss the Zeeman effect (Section 7.4.1). 7.3.5 Hermitian Conjugates of Tensor Operators Now we can ask, what is the Hermitian conjugate of an irreducible tensor operator? This is not too hard to establish, given the commutation relations (7.228) and (7.230). First, we can establish the commutator of Jz with the conjugate of T (k) q , using the fact that Jz is Hermitian and Eq. (7.228): Jz, T (k) q \dagger = − h Jz, T (k) q i\dagger = −¯hq T (k) q \dagger . (7.252) Similarly, we find the commutator of J\pm with the conjugate of T (k) q , using the fact that J\dagger \pm = J∓is Hermitian and Eq. (7.230): J\pm, T (k) q \dagger = − h J∓, T (k) q i\dagger = − p (k ∓q + 1)(k \pm q) T (k) q∓1 \dagger . (7.253) Notice that if we introduce the operator ˜T (k) q
T (k) −q \dagger , (7.254) then the above commutation relations take the form (after letting q −\rightarrow −q) h Jz, ˜T (k) q i
q h J\pm, ˜T (k) q i = p (k \pm q + 1)(k ∓q) ˜T (k) q\pm1. (7.255) These are precisely the commutation rules (7.228) and (7.230) for T (k) q . From our derivation of the commu- tators, we recall that they determine the behavior of the operators under rotations, and since the operators are irreducible we can identify ˜T (k) q with T (k) q . Thus, the Hermitian conjugate of T (k) q is (up to an arbitrary phase) T (k) q \dagger
−q . (7.256) (tensor operator conjugate) Evidently, only the q = 0 component of a tensor operator is Hermitian. 7.3.5.1 Conjugates of Reduced Matrix Elements By considering the Wigner–Eckart theorem from both Eqs. (7.237) and (7.242), we can write \langle J m|T (k) q
r 2J + 1
(7.257)
7.3.6 Relations Between Reduced Matrix Elements of Tensor Operators¶
Chapter 7. Atomic Angular-Momentum Structure Clearly the reduced matrix element is not symmetric in J and J′, for we may exchange the primed and unprimed numbers and let q −\rightarrow −q to write \langle J′ m′|T (k)
(7.258) Noting from Eq. (7.256) that \langle J′ m′|T (k)
q |J′ m′\rangle ∗, we can compare the above two expressions, using q = m −m′, to write the following relation between the reduced matrix elements
r 2J + 1
(reduced matrix element conjugate) (7.259) Of course, this relation applies as well to reduced matrix elements of the dipole operator, and thus when using reduced matrix elements to compute transition probabilities, it is important to pay attention to the ordering of the J and J′ (F and F ′ for a hyperfine transition) quantum numbers. 7.3.6 Relations Between Reduced Matrix Elements of Tensor Operators 7.3.6.1 Tensor Operator Acting on One Component Suppose we have a reduced matrix element \langle j∥T(k)∥j′\rangle ≡\langle j1, j2; j∥T(k)∥j′ 1, j′ 2; j′\rangle (7.260) between angular-momentum states of the composite angular momentum J = J1 + J2. Suppose further that T(k) acts only on the states associated with J1, but not those of J2. We can then reduce this matrix element to a form in terms of an uncoupled matrix element:
j1 j′ k j′ j j2 \langle j1∥T(k)∥j′ 1\rangle . (reduced matrix element, single subsystem) (7.261) Obviously, if T(k) doesn’t couple at all to the J2 space, the matrix element should only be determined in terms of J1 matrix elements. Further, this result sensibly says that states of different j2 are not coupled by this operator. To prove this result, we start with the expression (7.238) for the reduced matrix element, and then transform into the uncoupled states:
m′q \langle j m|T (k) q
m′q m1m2m′ 1m′ \langle j m|j1 m1; j2 m2\rangle \langle j1 m1; j2 m2|T (k) q |j′ 1 m′ 1; j′ 2 m′
1 m′ 1; j′ 2 m′ 2|j′ m′\rangle
X m′q m1m2m′ 1m′
1 m′ 1; j′ 2 m′
q |j′ 1 m′
X m′q m1m2m′ 1m′
1 m′ 1; j2 m2|j′ m′\rangle \langle j m|j′ m′; k q\rangle
q |j′ 1 m′ 1\rangle . (7.262)
7.3 Rotations and Irreducible Tensor Operators Now applying the Wigner–Eckart theorem (7.237) to the matrix element,
X m′q m1m2m′ 1m′
1 m′ 1; j2 m2|j′ m′\rangle \langle j m|j′ m′; k q\rangle
1 m′ 1; k q\rangle
X m′q m1m2m′ 1m′
1 m′ 1; j2 m2|j′ m′\rangle
1 m′ 1; k q|j1 m1\rangle \langle j1∥T(k)∥j′ 1\rangle
X m′q m1m2m′ 1m′
1 m′ 1; j2 m2|j′ m′\rangle
1 m′
1\rangle , (7.263) where in the last step we used the symmetry relation (7.54) to exchange the first two angular momenta in each of the last two Clebsch–Gordan coefficients. The combination of Clebsch–Gordan coefficients here, if we make the identifications k −\rightarrow j1, j′ 1 −\rightarrow j2, j2 −\rightarrow j3, j −\rightarrow j, j1 −\rightarrow j12, and j′ −\rightarrow j23, has the same form as in the expression (7.86) for the 6-j symbol, and thus
2(−1)j′+k−j(−1)j′ 1+k−j1(−1)−k−j′ 1−j2−jp
k j′ j1 j2 j j′ \langle j1∥T(k)∥j′ 1\rangle
2(−1)j′−j1+k−j2−2jp
j1 j′ k j′ j j2 \langle j1∥T(k)∥j′ 1\rangle , (7.264) after exchanging the first and last rows of the 6-j symbol. Finally, we use the fact from Eq. (7.90) that j1 + j2 + j is an integer, and thus we can add 2(j1 + j2 + j) to the exponent of the (−1), and thus we arrive at the result (7.261). 7.3.6.2 Scalar Products of Tensor Operators Suppose we have two tensor operators, T(k) and U(k). We will assume that components of the different tensors commute, [T (k) q , U (k) q ] = 0, so that the two tensors represent independent systems and thus can support simultaneous eigenstates of each system. However, we can suppose that the two systems are coupled according to the product of the two operators,
X q (−1)qT (k) q U (k) −q . (7.265)
|j1 m1\rangle of J 2 1 and J1z, U(k) is diagonal in the eigenstates |j2 m2\rangle of J 2 2 and J2z, and the interaction Hint is diagonal in the coupled eigenstates |j m\rangle of J2 and Jz. We can treat this problem essentially just as in the previous section. But first, if we apply the Wigner–Eckart theorem (7.237), we obtain
(7.266) Thus, we need only consider diagonal matrix elements of the scalar product. What we will show is the result
j j2 j1 k j1 j2
(matrix element of scalar product) (7.267) where the interaction is represented by a product of reduced matrix elements on each subspace and then coupled by a 6-j symbol.
Chapter 7. Atomic Angular-Momentum Structure To prove this, we start by taking matrix elements of the interaction in the coupled basis and trans- forming to the uncoupled basis,
X q (−1)q\langle j m|T (k) q U (k) −q |j m\rangle = X qm1m2m′ 1m′ (−1)q\langle j m|j1 m1; j2 m2\rangle
q U (k) −q |j1 m′ 1; j2 m′
1; j2 m′ 2|j m\rangle = X qm1m2m′ 1m′
1; j2 m′ 2|j m\rangle
q |j1 m′
−q |j2 m′ 2\rangle . (7.268) Applying the Wigner–Eckart theorem (7.237) twice,
X qm1m2m′ 1m′
1; j2 m′ 2|j m\rangle
(7.269) Permuting the symbols in the last Clebsch–Gordan coefficient via (7.65),
X qm1m2m′ 1m′ (−1)q(−1)k−q\langle j1 m1; j2 m2|j m\rangle \langle j1 m′ 1; j2 m′ 2|j m\rangle
1; k q|j1 m1\rangle \langle k q; j2 m2|j2 m′
(7.270) Now again if we identify j1 −\rightarrow j12, j2 −\rightarrow j3, j′ 1 −\rightarrow j1, j′ 2 −\rightarrow j23, j −\rightarrow j, and k −\rightarrow j2, we can again use Eq. (7.86) for the 6-j symbol, with the result
j1 k j1 j2 j j2
(7.271) Using the fact that j1 + j2 + j \in Z, as required for the 6-j symbol, and permuting the elements of the 6-j symbol as permitted by its symmetries, we obtain the result (7.267). The general case of the tensor product of two commuting tensor operators is more complicated, as it involves a 9-j symbol.28 Since we will not use this case, we will avoid it here. 7.3.6.3 Matrix Elements of Tensor Products Operating on the Same System One last variation on the above theme is to consider a tensor product
(7.272)
angular-momentum space of J. In this case, we have the reduced matrix element29
J′′ p
k1 k2 k J′ J J′′
(matrix element of operators on same space) (7.273) so that we have a rule for splitting reduced matrix elements of operator products into products of reduced matrix elements. 28D. M. Brink and G. R. Satchler, Angular Momentum, 2nd ed. (Oxford, 1968), Section 5.3, p. 80. 29Brink and Satchler, op. cit., Eq. (5.5).
7.3.7 Application to Atomic Transitions¶
7.3 Rotations and Irreducible Tensor Operators To prove this, we start with the matrix elements of T(k), as given by the inverse (7.238) of the Wigner– Eckart theorem:
m′q \langle J m|T (k) q |J′ m′
J; k q\rangle . (7.274) Now using Eq. (7.272) in the product-component form of Eq. (7.212), T (k) q = X q1q2
U (k1) q1 V (k2) q2
(7.275) we can replace T (k) q and introducing the identity to find
X m′qq1q2 \langle J mJ|U (k1) q1 V (k2) q2 |J′ m′
J; k q\rangle
X m′qq1q2 J′′m′′ \langle J mJ|U (k1) q1 |J′′ m′′
J|V (k2) q2 |J′ m′
J; k q\rangle . (7.276) Using the Wigner–Eckart theorem (7.237) twice, and introducing the identity to find
m′qq1q2 J′′m′′
J|J′ m′ J; k2 q2\rangle
J; k q\rangle
m′qq1q2 J′′m′′ \langle J′′ m′′ J; k1 q1|J mJ\rangle \langle J′ m′ J; k2 q2|J′′ m′′ J\rangle
X m′qq1q2 J′′m′′ \langle J′′ m′′ J; k1 q1|J mJ\rangle \langle J′ m′ J; k2 q2|J′′ m′′ J\rangle
(7.277) where we used the symmetry rule (7.54) for the last Clebsch–Gordan coefficient. Again identifying J′ −\rightarrow j1, J′′ −\rightarrow j12, J −\rightarrow j, k1 −\rightarrow j3, k2 −\rightarrow j2, and k −\rightarrow j23, we can again use Eq. (7.86) to introduce the 6-j symbol, with the result
\times X J′′ p
J′ k2 J′′ k1 J k
J′′ p
k1 k2 k J′ J J′′
(7.278)
7.3.7 Application to Atomic Transitions 7.3.7.1 Decomposition and Calculation of Reduced Matrix Elements The Wigner–Eckart theorem (7.237) and the decomposition rule (7.261) for reduced matrix elements apply immediately to the matrix elements of the dipole operator that govern atomic electric-dipole transitions.
Chapter 7. Atomic Angular-Momentum Structure transition between two fine-structure sublevels |J mJ\rangle −\rightarrow |J′ m′ J\rangle , we find \langle J mJ|dq|J′ m′
J; 1 q\rangle
J−mJ r 2J + 1
J|J mJ; 1 −q\rangle , (Wigner–Eckart theorem, fine-structure transition) (7.279) where to write the last expression, we used the symmetry rules (7.65) and (7.54) to write \langle J′ m′
r 2J + 1 2J′ + 1\langle 1 −q; J mJ|J′ m′
r 2J + 1 2J′ + 1\langle J mJ; 1 −q|J′ m′ J\rangle , (7.280)
J−mJ for nonvanishing coefficients. The Wigner–Eckart theorem applies in exactly the same way to a hyperfine transition |F mF\rangle −\rightarrow |F ′ m′ F\rangle , so that \langle F mF|dq|F ′ m′
F; 1 q\rangle
F −mF r 2F + 1
F|F mF; 1 −q\rangle . (Wigner–Eckart theorem, hyperfine transition) (7.281) In both cases, the Wigner–Eckart theorem gives the rather convenient result that the dependence on the two m levels of the matrix element (which, for example, measures the relative transition rate of the transition) is given entirely by a Clebsch–Gordan coefficient. Stated another way: the entire angular dependence of the dipole matrix elements is given simply by a Clebsch–Gordan coefficient. Of course, the dependence on the J or F quantum numbers still appears in what remains in the reduced matrix element. However, for the reduced matrix element we can make further progress according to Eq. (7.261). The crucial point is that the dipole operator refers to the position of the electron. However, a hyperfine transition is a coupling between two states corresponding to different F = J+I, where in terms of the uncoupled states |J mJ\rangle |I mI\rangle , the dipole operator acts only on the electron angular-momentum state |J mJ\rangle , not the nuclear state |I mI\rangle . Thus, applying the decompostion (7.261) to the reduced hyperfine matrix element,
J J′ F ′ F I . (decomposition of hyperfine reduced matrix element) (7.282) Note again that since the dipole operator doesn’t refer to the nucleus, the nuclear spin I is preserved in the transition. Thus we see that the hyperfine reduced matrix element is just given in terms of the fine-structure (electronic) reduced matrix element, multiplied by a factor that essentially represents the orientation of the electron with respect to the nucleus. Recalling that the 6-j symbol represents a transformation between two different ways to couple three angular momenta, the interpretation in this sense is a bit more murky. However, the basic idea is that one can view the photon as either changing J or F; this amounts to coupling the photon (of unit angular momentum) to the electron either before or after coupling the electron to the nucleus, and thus the appearance of the 6-j symbol. By exactly the same procedure, the fine-structure reduced matrix element can be further factored into another 6-j symbol and a reduced matrix element involving only the L quantum number:
L L′ J′ J S . (decomposition of fine-structure reduced matrix element) (7.283) This, of course, works out because the dipole operator again represents the atomic position, but does not refer to its spin. Thus, the dipole operator couples states of different orbital angular momentum L, but doesn’t touch the spin S of the electron. Essentially the same interpretations as for the hyperfine case apply here, with the nuclear spin replaced by the electron spin.
7.3 Rotations and Irreducible Tensor Operators 7.3.7.2 Fine-Structure Selection Rules The Wigner–Eckart theorem immediately leads to selection rules for ‘‘dipole-allowed’’ transitions. That is, for a transition |J mJ\rangle −\rightarrow |J′ m′ J\rangle represented by the matrix element \langle J mJ|dq|J′ m′ J\rangle to be a non- vanishing matrix element, several conditions are required. In particular, the Clebsch–Gordan coefficient \langle J mJ|J′ m′ J; 1 q\rangle , representing the angular dependence of the matrix element according to the Wigner– Eckart theorem, represents the addition of angular momenta |J′ m′ J\rangle and |1 q\rangle to form the composite state |J mJ\rangle . Recall from the triangular condition (7.41) that for such an addition the allowed range of J is bounded below by |J′ −1| and above by J′ + 1. This leads to the first selection rule J′ = J or J′ = J \pm 1. (7.284) (first selection rule) Next, the addition of angular momentum requires first requires mJ = m′ J + q (7.285) (angular-momentum conservation) to conserve angular momentum. Since the dipole component index q can take on the values −1, 0, or +1 (we will see these correspond to interactions with different polarizations of the electromagnetic field), the second selection rule becomes m′ J = mJ or m′ J = mJ \pm 1. (7.286) (second selection rule) Finally, one can show from Eq. (7.67) that the Clebsch–Gordan coefficient \langle J 0|J 0; 1 q\rangle vanishes for any J and q, leading to the final selection rule J′̸ = J if m′
(7.287) (third selection rule) In particular, J = J′ = 0 represents a forbidden transition; intuitively, this is because the atom must
and after absorbing the photon. In all other cases, the corresponding dipole matrix element vanishes, or in other words the transition is dipole forbidden. As a final note, the dipole interaction couples the electric field to the electron’s position and not to its spin, and thus the electron spin S (and mS) should not change in an electric-dipole transition. S′ = S, m′ S = mS. (7.288) (electric-dipole spin selection rules) Furthermore, we can note that the above selection rules for J also apply to the orbital angular momentum L, so we may write L′ = L or L′ = L \pm 1 L̸ = 0 or L′̸ = 0. (7.289) (orbital selection rules) In particular, from the second rule any fine-structure transition of the form nS1/2 −\rightarrow n′S1/2 is dipole forbidden. Of course, analogous rules should hold for mL, but this often isn’t a useful quantum number, so we’ll skip it. We recall also that the dipole operator only couples states of opposite parity (Section 5.1.1). A hydrogen-atom state |nlm\rangle has an angular dependence given by the spherical harmonic Y m ℓ(\theta, \phi). But under a parity transformation Y m
(7.290) so evidently ∆ℓ= \pm1 (7.291) (single-electron orbital selection rules)
Chapter 7. Atomic Angular-Momentum Structure for exactly one electron in the atom. (Recall that l is the orbital quantum number for a single electron, while L is the combined orbital quantum number for all the electrons.) This selection rule is often interpreted as conservation of angular momentum when a photon is absorbed or emitted. Thus for a single electron atom, or in two-electron atoms in low-energy transitions where only one electron is active, this rule implies that ∆L = \pm1 as well. One final selection rule is a bit more subtle, and is specific to fine-structure transitions where S = 1/2. Because J = L + S, we have
(7.292) and since J′ = L′ + S, we have
(7.293) Now consider a transition where L′ = L + 1 but J′ = J −1. Then the second condition becomes
(7.294)
becomes
(7.295)
rule if L′ = L \pm 1 then J′̸ = J ∓1. (7.296) (fourth selection rule) This argument assumed S = 1/2, and so will not in general carry over to, for example, the hyperfine transitions we consider later. In any case, the selection rules for l, L, and S are approximate, since they assume that these are good quantum numbers. In heavy, many-electron atoms, this may not be the case, and so these rules may be violated to some extent.30 7.3.7.3 Hyperfine Selection Rules Just as in Section 7.3.7.2, the constraints on the Clebsch–Gordan coefficient in the hyperfine Wigner–Eckart theorem (7.281) induce selection rules on the hyperfine quantum numbers. Of course, the selection rules on J and mJ of Section 7.3.7.2 still apply, but because the involved Clebsch–Gordan coefficients are the same, the same constraints apply to F and mF. In particular, the rules F ′ = F or F ′ = F \pm 1 m′ F = mF or m′ F = mF \pm 1 F ′̸ = F if m′
(7.297) (hyperfine selection rules) apply to the hyperfine transition |F mF\rangle −\rightarrow |F ′ m′ F \rangle . Finally, since the dipole interaction couples only the field to the electron dipole, there is no nuclear coupling and thus the nuclear spin I should not change in an electric-dipole transition (nor should mI, if that is the preferred basis). Thus, we may write the extra selection rules I′ = I, m′ I = mI, (electric-dipole nuclear-spin selection rules) (7.298) Of course, the nuclear spin may effectively change relative to the electron angular momentum, which is why it is meaningful to have hyperfine states and, say, to optically pump into particular hyperfine sublevels. 30See Alan Corney, Atomic and Laser Spectroscopy (Oxford, 1977), Chapters 5-7.
7.3 Rotations and Irreducible Tensor Operators 7.3.7.4 Decay Rate and the Reduced Matrix Element But now the question remains, how do we compute the dipole matrix elements? The basic answer is to know the decay rate \Gamma (equivalently, the lifetime) of the excited level, and here we will relate the decay rate to the reduced matrix elements. For the two level atom, the spontaneous emission rate from the quantum treatment of the atom–field interaction, Eq. (11.29), from |e\rangle −\rightarrow |g\rangle (with transition frequency \omega0) is
3\piϵ0¯hc3 . (7.299) This result is only for two levels, but now we are confronted with the physically important case of decay between levels with angular-momentum degeneracy. Consider the decay of the Jg −\rightarrow Je fine-structure transition (with Je being the excited state as usual). Then the decay rate from sublevel |Je me\rangle −\rightarrow |Jg mg\rangle is just given by Eq. (7.299) with the appropriate matrix element: \GammaJg,mg;Je,me = \omega 3 3\piϵ0¯hc3 |\langle Jg mg|d|Je me\rangle |2. (7.300) The quantum vacuum is isotropic, and so spontaneous emission is as well. Thus, all magnetic sublevels of the excited manifold decay at the same rate, and we can drop the explicit dependence on me. The total decay rate out of any excited sublevel (and thus of the whole excited manifold) is this decay rate summed over all ground sublevels: \GammaJgJe = \omega 3 3\piϵ0¯hc3 X mg |\langle Jg mg|d|Je me\rangle |2. (7.301) Summing over the excited sublevels as well, we can write \GammaJgJe = \omega 3 3\piϵ0¯hc3 2Je + 1 X mg,me |\langle Jg mg|d|Je me\rangle |2. (7.302) Recalling the normalization convention (7.241) for the reduced dipole matrix element, X me
X meq
(7.303) we can eliminate the excited-state sum: \GammaJgJe = \omega 3 3\piϵ0¯hc3 2Je + 1 X mg
(7.304) The summand no longer depends on mg, so we arrive at the final result: \GammaJgJe = \omega 3 3\piϵ0¯hc3 2Jg + 1
(spontaneous decay rate, fine-structure transition) (7.305) This expression can also effectively define the normalization convention of the matrix elements. More impor- tantly, this is generally how you compute the dipole matrix elements, since lifetimes are a commonly measured quantity used to define the strength of transitions (and even better, the lifetimes are free of ambiguities from normalization conventions!). A similar argument applies for hyperfine levels. However, when summing the dipole matrix elements over all ground sublevels |Fg mg\rangle connected to an excited sublevel, the explicit dependence on all hyperfine quantum numbers vanishes: X Fgmg
(7.306)
7.4.1 Static Magnetic Fields: Zeeman Effect¶
The Schrödinger Equation in a Time-Dependent Potential¶
The time-dependent Schrödinger equation for the harmonic oscillator with frequency \(\Omega(t)\) is given by \(i\hbar{\partial}{t}\ket{p}=\hat{H}(t)\ket{p}\) where \(x\) and \(\xi\) are operators satisfying the commutation relation \((,)=1\). The Hamiltonian can be written as \(\hat{H}= \frac { 1 }{2m} (p-m\alpha(t)x)^2+U(\Omega_{0}^{b}(t), t) + c(x,t)\) with \(c(\xi,t) = e\phi(\xi, -ct)\) where the potential is of the form \(\Phi(\eta)=e^{-i\eta}\int _{\mathbb{R}} \frac {d\mu(t)}{\Omega_{0}^{b}(t)^2-4m^2}\) in polar coordinates \((r,z)\).
The time-dependent harmonic oscillator solutions are given by \(x(\tau) = q_1(\lambda,\xi)\sinh\xi + iq_3(\alpha;\beta)(\cosh \xi )\) where \(\hat{x},t)\) is a Gaussian wave packet with width determined by the initial conditions. The operator algebra relations satisfy \((a,t)=e^{it} [A(\omega,0),B(t')] = e^{\frac {i}{2}(t-\tau)}\).
For \(q_1\) we have \(\lambda(x,q) + \alpha(q)x^{-\beta}\left[ q_{\mu,\nu}-c\xi\right] - a_n= b\sigma+ 4k(\eta,x)\) in the case of anisotropic squeezing. The equation for small displacements \((r,t+\delta r)=(a(t)+O(r^3))\) reduces to \(\frac {1}{2} m \dot{\beta} \xi^{-5/6}\) near stationary points where the phase space volume element is \(dV = 4q_2(\theta,\phi)\).
The operator \(x^\alpha q_\mu + i\partial_\tau - k\xi^3\) acts on states with energy expectation value \(\langle H \rangle = \frac {1}{m} (p-m v(t))^2+U(\xi)\) satisfying the Virial theorem. The Gaussian representation reads as $\(f(x) = c_1 x^{-n/4} + i^n e^{i n\pi}/(n!)\)$ with norm \(||x^\alpha \hat{x}_{t,t'} f || L^p(dx') < (\frac {2m}{3})^{\beta}\).
7.4 Interaction with Static Fields where we used Jz = Lz + Sz and of course Jz|J mJ\rangle = mJ¯h|J mJ\rangle . To evaluate the second term, we first note that since J = L + S, squaring this relation gives
J2 + S2 −L2 . (7.310) Using the projection theorem (7.251), we can write
mJ
= mJ
J2 + S2 −L2 |J mJ\rangle
2J(J + 1) mJ¯h, (7.311) where we used Eq. (7.310) for the dot product, and we recall that |J mJ\rangle ≡|L S J mJ\rangle . Putting this expectation value into Eq. (7.309), we obtain the perturbative shift ∆E(fs) B = µBgJmJB, (7.312) (fine-structure Zeeman shift, small B) where the Landé gJ factor31 is
2J(J + 1) . (7.313) (Landé gJ factor) Note that this expression does not include corrections due to multielectron32 and QED33 effects, and thus measured values may differ slightly from this. Note also that since gL \approx 1 and gS \approx 2, the gJ factor is commonly written as
2J(J + 1) . (7.314) The shift in this regime is thus proportional to both the mJ quantum number and the magnetic field. Recall that this is only valid when the magnetic field is along the ˆz direction; otherwise, you must compute the energy shifts according to this method for the states quantized along the magnetic field, and then use the rotation operators below to obtain the states with the desired quantization axis, which will no longer be eigenstates of the system. Thus, with other quantization axes, a given state will ‘‘remix’’ with the others due to precession along the magnetic-field axis. For the hyperfine case, if the energy shift due to the magnetic field is small compared to the fine- structure splitting, then as we just argued J is a good quantum number. Then the interaction Hamiltonian can be written as the fine-structure interaction plus the magnetic-dipole interaction of the nuclear magnetic moment with the magnetic field: H(hfs) B
B
¯h (gJJz + gIIz)Bz. (7.315) Again, if the energy shift due to the magnetic field is small compared to the hyperfine splittings, then the fine-structure treatment carries through with J −\rightarrow F, S −\rightarrow I, and L −\rightarrow J, so that ∆E(hfs) B = µBgFmFB, (7.316) (hyperfine Zeeman shift, small B) 31S. Goudsmit, ‘‘Nuclear Magnetic Moments,’’ Physical Review 43, 636 (1933) (doi: 10.1103/PhysRev.43.636); Alfred Landé, ‘‘The Magnetic Moment of the Proton,’’ Physical Review 44, 1028 (1933) (doi: 10.1103/PhysRev.44.1028); Alfred Landé, ‘‘Nuclear Magnetic Moments and Their Origin,’’ Physical Review 46, 477 (1934) (doi: 10.1103/PhysRev.46.477); Paul Forman, ‘‘Alfred Landé and the anomalous Zeeman Effect, 1919-1921,’’ Historical Studies in the Physical Sciences 2, 153 (1970). 32Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, 1957). 33Leonti Labzowsky, Igor Goidenko, and Pekka Pyykkö, ‘‘Estimates of the bound-state QED contributions to the g-factor of valence ns electrons in alkali metal atoms,’’ Physics Letters A 258, 31 (1999).
Chapter 7. Atomic Angular-Momentum Structure where the gF factor is
2F(F + 1) , (7.317) or in a more symmetric form, gF := gJ
2F(F + 1) + gI
2F(F + 1) . (Landé gF factor) (7.318) Recalling that gI is much smaller than gJ, this is commonly written gF \approx gJ
2F(F + 1) , (7.319) which is correct at the 0.1% level. The shifts proportional to the magnetic fields in this weak-field regime were historically referred to as the anomalous Zeeman effect, after Zeeman’s observation of the splitting of spectral lines.34 The ‘‘normal’’ case was based on the predicted splittings due only to orbital angular momentum. However, ‘‘anomalous’’ cases—which included spin effects—were observed before spin was known. 7.4.1.2 Paschen–Back Effect: Strong Fields In the case of fine structure, when the field is large enough that the shifts due to the magnetic-field interaction Hamiltonian (7.307) dominate the fine-structure splittings, the interaction is again simple, but J is no longer a good quantum number. Ignoring the fine-structure Hamiltonian, the eigenstates of the interaction are the uncoupled, or ‘‘high-field’’ fine-structure states |L mL; S mS\rangle . The energy shifts are thus given by the expectation value of the interaction Hamiltonian, or ∆E|L mL;S mS\rangle = D H(fs) B E = µB
(7.320) The energies again shift linearly with the applied field amplitude, but now the shifts have contributions proportional to mS and mL rather than simply being proportional to mJ. The shift in this large-field regime is called the Paschen–Back effect.35 We will consider in a bit more detail the hyperfine case, since it is easier to enter the Paschen–Back regime for the much smaller hyperfine splittings. For strong fields where the appropriate interaction is described by Eq. (7.315), the interaction Hamiltonian dominates the hyperfine Hamiltonian (7.132), so that the hyperfine Hamiltonian perturbs the strong-field eigenstates |J mJ; I mI\rangle . For this treatment to hold, the energy perturbations must still be small compared to the fine-structure splitting, otherwise we need to account for that effect as well. We can compute the energies to first order in perturbation theory (lowest order in 1/B) by computing the expectation value E|J mJ;I mI\rangle = D Hhfs + H(hfs) B E (7.321) with respect to the strong-field states |J mJ I mI\rangle . To do this, we first invert the defining relations (7.7) for
I \cdot J = IzJz + IxJx + IyJy
. (7.322) 34P. Zeeman, ‘‘The Effect of Magnetisation on the Nature of Light Emitted by a Substance,’’ Nature 55, 347 (1897) (doi: 10.1038/055347a0); P. Zeeman, ‘‘On the influence of Magnetism on the Nature of the Light emitted by a Substance,’’ Philo- sophical Magazine 43, 226 (1897); P. Zeeman, ‘‘Doubles and triplets in the spectrum produced by external magnetic forces,’’ Philosophical Magazine 44 55 (1897). 35Named for F. Paschen and E. Back, ‘‘Normale und anomale Zeemaneffekte,’’ Annalen der Physik 344, 897 (1912) (doi: 10.1002/andp.19123441502).
7.4 Interaction with Static Fields In the expectation value, the second term vanishes, leaving
(7.323) Squaring Eq. (7.322) then gives
, (7.324) and then computing the expectation value, the middle terms vanish, while the last term can be computed from Eq. (7.27) to obtain36
4 [I(I + 1) −mI(mI −1)][J(J + 1) −mJ(mJ + 1)] + ¯h4 4 [I(I + 1) −mI(mI + 1)][J(J + 1) −mJ(mJ −1)]
2 [I(I + 1) −m 2 I ][J(J + 1) −m 2 J ] −¯h4 2 mImJ. (7.325) Now we can evaluate Eq. (7.321) by putting the above expectation values into (7.132), while dropping the small Chfs term for (relative) simplicity, with the result E|J mJ;I mI\rangle \approx AhfsmImJ + Bhfs 9(mImJ)2 −3J(J + 1)m 2 I −3I(I + 1)m 2
4J(2J −1)I(2I −1) + µB(gJ mJ + gI mI)B. (hyperfine Paschen–Back effect) (7.326) The expectation of the interaction Hamiltonian is trivial in this case. Clearly, the hyperfine Hamiltonian, while not contributing a B-dependent energy, is important in determining the correct splittings between the states, even for strong fields. The energy shift in this regime is called the hyperfine Paschen-Back effect. Note that for both instances of the Paschen–Back effect, we are considering fields that cause large shifts on the scale of the unperturbed splittings. However, we don’t want the fields to be too large, where for
as we recall from Eq. (9.119), which leads to a quadratic Zeeman effect. 7.4.1.3 Incomplete Paschen–Back Effect: Intermediate Fields For intermediate fields, where for example in hyperfine structure the interaction Hamiltonian neither weakly perturbs nor dominates the hyperfine Hamiltonian, the energy shift is more difficult to calculate, and in general one must numerically diagonalize Hhfs + H(hfs) B . A notable exception comes about in hyperfine
the magnetic-dipole term, Hhfs = Ahfs I \cdot J ¯h2 . (7.327) In the strong-field basis, we have from Eq. (7.322) again the diagonal matrix elements \langle J mJ; I mI|Hhfs|J mJ; I mI\rangle = AhfsmImJ (7.328) 36This expression differs from that of E. B. Alexandrov, M. P. Chaika, and G. I. Khvostenko, Interference of Atomic States (Springer–Verlag, 1993), p. 222, Eq. (5.160), where the authors neglected the contributions from the last term in Eq. (7.324). Comparison of the two expressions to energies from numerical diagonalization confirms that the expression shown here is a better approximation.
Chapter 7. Atomic Angular-Momentum Structure and also from Eq. (7.30) the off-diagonal matrix elements
p
p
(7.329) with all other matrix elements vanishing. The interaction Hamiltonian is diagonal in the strong-field basis, with diagonal matrix elements \langle J mJ; I mI|H(hfs) B
(7.330) Suppose now for concreteness that J = 1/2. Then states with mJ = 1/2 are only coupled to states with
(and mI decreased by 1). Thus, the combined Hamiltonian is block diagonal, with blocks of the form Ahfs mI 2 + µB gJ 2 + gImI B Ahfs p
Ahfs p
−Ahfs (mI + 1) + µB −gJ
B . (7.331) Then we have a matrix with eigenvalues of the form A C C D −\rightarrow eigenvalues: A + D \pm 1 p (A −D)2 + 4C2, (7.332) so that the new hyperfine eigenvalues are
- µBgI mI + 1 B \pm 1 Ahfs 2 (2mI + 1) + µB(gJ −gI)B 2 + A 2 hfs[I(I + 1) −mI(mI + 1)] 1/2 . (7.333) Introducing the notations ∆Ehfs = Ahfs I + 1
∆Ehfs , (7.334) where ∆Ehfs is the hyperfine splitting and x is a scaled magnetic-field strength,
∆Ehfs
mI + 1 B \pm ∆Ehfs (2mI + 1)
2
(2I + 1)2 1/2 = − ∆Ehfs
mI + 1 B \pm ∆Ehfs
1/2 = − ∆Ehfs
mI + 1 B \pm ∆Ehfs
1/2 (7.335)
for the dressed states of the two-level atom, will simply be spread further due to the interaction. Thus, we can associate the \pm energy with the mJ = \pm1/2 state. Further, we can define m = mI + 1/2; noting that we labeled the upper (+) state as having quantum number mI while the lower (−) state has quantum number
7.4 Interaction with Static Fields mI + 1, we can thus also interpret m = mI \pm mJ, where mI and mJ are the literal quantum numbers for each state, and thus we obtain the Breit–Rabi formula37
∆Ehfs 2(2I + 1) + gIµBmB \pm ∆Ehfs 1 + 4mx
1/2 . (Breit–Rabi formula) (7.336)
formula
I 2I + 1 \pm 1 2(gJ + 2IgI)µBB (7.337)
the shifts of the ground states of the alkali atoms, which are of the form n2S1/2. For example, recall above that the ground-state hyperfine F = 2 −\rightarrow F ′ = 3 splitting of 133Cs defines the second, but of course the hyperfine transitions can depend on the local magnetic field. For this reason the mF = 0 −\rightarrow m′ F = 0 transition is used, because both states have no Zeeman shift to lowest order. However, they do shift to higher order, and the small frequency shift is
B 2¯h∆Ehfs B2 (7.338) to second order in the field strength, an important systematic effect to keep in mind when designing an atomic clock. As an example, shown below is the magnetic-field-dependent hyperfine structure of the ground (62S1/2) state of 133Cs, ranging from the weak-field (Zeeman) regime through the hyperfine Paschen-Back regime. 15000 magnetic field (G) 10000 -25 E/h (GHz) F = 3 F = 4
-10 -20 For the ground state, the Breit–Rabi formula applies and can be used to compute the energy levels. How- ever, it does not apply to the D2 excited (62P3/2) state, where the level structure, shown below, is more complicated. 37G. Breit and I. I. Rabi, ‘‘Measurement of Nuclear Spin,’’ Physical Review 38, 2082 (1931) (doi: 10.1103/PhysRev.38.2082.2).
7.4.2 Static Electric Fields: Stark Effect¶
Chapter 7. Atomic Angular-Momentum Structure magnetic field (G) -1500 E/h (MHz) F = 2 F = 3 F = 4 F = 5
-500 -1000 In both cases, the features we have discussed are visible. For small magnetic fields, the levels cluster around the hyperfine energies and have shifts proportional to B. For large magnetic fields, the states instead cluster according to their mJ value, with small splittings induced according to the value of mI. In the incomplete Paschen–Back regime, there is a smooth crossover between the two types of level spacings. 7.4.2 Static Electric Fields: Stark Effect Like static magnetic fields in the Zeeman effect, static electric fields also shift the fine- and hyperfine-structure sublevels. However, the details of the electric-field shifts, or dc Stark shifts38 turn out to be quite different from the case of the Zeeman effect. We will take the atom–field interaction Hamiltonian to be the usual electric-dipole interaction,
(7.339) where E is a static electric-field amplitude, and d is the atomic dipole operator as usual. The atomic energy-level shifts in level |\alpha\rangle is given up to second order in perturbation theory by
X j
, (7.340) where the |\betaj\rangle label all the other atomic states, and E\alpha and E\betaj are the atomic level energies. Recalling that the dipole operator only couples states of opposite parity (Section 5.1.1), the first-order shift vanishes and we are left only with the second-order term. Thus, the remaining effect is second order in E, and is thus called the quadratic Stark effect. (A notable exception to this occurs in hydrogen, where degeneracy of opposite-parity states leads to a first-order shift and thus a linear Stark effect.) 38J. Stark, ‘‘Beobachtungen über den Effekt des elektrischen Feldes auf Spektrallinien I. Quereffekt,’’ Annalen der Physik 43, 965 (1914).
7.4 Interaction with Static Fields 7.4.2.1 Effective, First-Order Interaction It is conventional to define an effective Stark interaction Hamiltonian by39 HStark := X j
= X j
EµE\nu, (7.341) where Eµ and E\nu are the electric-field components (make sure to keep track of which E’s are energies and which are fields!). With this effective Hamiltonian, we get the same shift, but now it looks like a first-order shift:
(7.342) However, it’s important to realize that this result is really second order in perturbation theory. Now we note that the Stark Hamiltonian has the form of a rank-2 tensor operator, contracted twice with the electric-field vector:
(7.343) where the tensor operator is
X j
. (7.344) This is a symmetric Cartesian tensor of rank 2. Recall from Section (7.3.3.3) that a rank-2 Cartesian tensor may be decomposed into irreducible parts of rank 0, 1, and 2, where the rank-0 part is related to the trace, the rank-1 part is related to the antisymmetric part of the tensor, and the rank-2 part is effectively what is left. Writing the decomposition as in Eq. (7.207),
µ\nu , (7.345) where the vector term vanishes since (7.344) is obviously a symmetric tensor, the scalar part is
(7.346) as in Eq. (7.199), and the irreducible tensor part is S(2)
(7.347) as in Eq. (7.206). Thus, the Stark shift from Eq. (7.342) becomes
(7.348) which is now separated into scalar and tensor parts: the first is the orientation-independent part of the shift, while the second is the anisotropic part. 7.4.2.2 Scalar Shift: Fine Structure The scalar part of the shift is given by the first term of Eq. (7.348). For a fine-structure state |J mJ\rangle , we may write the shift as ∆E(0)
X J′ m′ J \langle J mJ|dµ|J′ m′
J|dµ|J′ m′ J\rangle 3(EJ −EJ′) E2. (7.349) 39For other treatments and more details, see J. R. P. Angel and P. G. H. Sandars, ‘‘The Hyperfine Structure Stark Effect. I. Theory,’’ Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 305, 125 (1968); Abbas Khadjavi, Allen Lurio, and W. Happer, ‘‘Stark Effect in the Excited States of Rb, Cs, Cd, and Hg,’’ Physical Review 167 128 (1968) (doi: 10.1103/PhysRev.167.128); and Robert W. Schmieder, ‘‘Matrix Elements of the Quadratic Stark Effect on Atoms with Hyperfine Structure,’’ American Journal of Physics 40, 297 (1972) (doi: 10.1119/1.1986513).
Chapter 7. Atomic Angular-Momentum Structure Using the Wigner–Eckart theorem (7.237) for the matrix elements, we find ∆E(0) J = X J′ m′ J µ
3(EJ −EJ′)\langle J mJ|J′ m′ J; 1 µ\rangle 2E2. (7.350) Note that we transposed the second matrix element so that we could apply the Wigner–Eckart theorem in exactly the same form on each matrix element, ending up with one form of the reduced matrix element and the Clebsch–Gordan coefficient. From the orthogonality relation (7.46), we can evaluate the double sum X m′ J µ \langle J mJ|J′ m′
(7.351) Then, by analogy with the classical polarizability, as in Eq. (1.60), we then define a scalar polarizability for the fine-structure level J by
X J′
EJ −EJ′ , (7.352) (scalar polarizability) so that the scalar shift becomes ∆E(0) J = −1 2\alpha(0)(J) E2. (7.353) (scalar Stark shift) The sum in the polarizability here extends over the other (i.e., radial) quantum numbers as well as J′. 7.4.2.3 Tensor Shift: Fine Structure The remaining (tensor) part of the Stark shift is given by the second term of Eq. (7.348). Again, for a fine-structure state |J mJ\rangle , we may write the shift as ∆E(2)
X q=−2 (−1)q\langle J mJ|S(2) q |J mJ\rangle EE −q, (7.354) where we have switched to the scalar product (7.222) of spherical-tensors from the scalar product of the irreducible Cartesian tensors S(2) µ\nu and the traceless part of EµE\nu. (Note that the scalar part of EµE\nu has a vanishing interaction with S(2) µ\nu , since its trace has been removed.) Recall that the spherical-tensor components for the rank-2 tensors are given in Eq. (7.219). Again using the Wigner–Eckart theorem (7.237) for the matrix element, we can write ∆E(2)
X q=−2
−q \langle J mJ|J mJ; 2 q\rangle , (7.355) where the only nonvanishing Clebsch–Gordan coefficient is
3m 2 J −J(J + 1) p
(7.356) where the constraint on J ensures that the triangularity constraint of the Clebsch–Gordan coefficient is
the relevant component of the field tensor is EE = \sqrt 6(E1E−1 + 2E 2
\sqrt 6(3E 2 z −E2). (7.357) Putting these pieces together, we can define a tensor polarizability
q ∥J\rangle s 8J(2J −1)
(7.358) (tensor polarizability)
7.4 Interaction with Static Fields so that the tensor shift is ∆E(2)¶
4\alpha(2)(J) (3E 2 z −E2) 3m 2 J −J(J + 1) J(2J −1) . (7.359) (tensor Stark shift) Combining this with the scalar shift from Eq. (7.353), we can write the total shift as
2\alpha(0)(J) E2 −1 4\alpha(2)(J) (3E 2 z −E2) 3m 2 J −J(J + 1) J(2J −1) . (fine-structure Stark shift) (7.360)
the Stark shift simplifies to
2\alpha(0)(J) E 2 z −1 2\alpha(2)(J) E 2 z 3m 2 J −J(J + 1) J(2J −1) . (fine-structure Stark shift) (7.361) This latter form explains the normalization of the tensor term: for mJ = \pmJ, the shift becomes
2\alpha(0)(J) E 2 z −1 2\alpha(2)(J) E 2 z , (7.362) where the tensor term has the same form as the scalar term. Because of these forms for the Stark shift, we can write down yet another effective Stark Hamiltonian,
2\alpha(0)(J) E 2 z −1 4\alpha(2)(J) (3E 2 z −E2) 3J 2
J(2J −1) , (effective Stark Hamiltonian) (7.363) which has the same eigenvalues as the effective Hamiltonian (7.341) for fine-structure states |J mJ\rangle . Again, we have taken the quantization axis to coincide with the electric-field direction. This is because the matrix \langle J mJ|HStark|J m′ J\rangle for the original Hamiltonian is only diagonal for E = Ezˆz. Only for this choice of E is EE the only nonvanishing tensor component, as is evident from Eqs. (7.219), and the off-diagonal
J, since q = 0. Obviously, this effective applies so long as the shift is weak enough that states of different J do not mix. Again, we reiterate that the tensor polarizability vanishes if J = 0 or J = 1/2. More intuitively, we can understand this by noting that if J = 0, then there is only one mJ level, and so there can be no orientation dependence; the entire Stark shift is accounted for by the scalar term. Similarly, if J = 1/2, the tensor Stark shift as we have seen only depends on |mJ|, so the two sublevels are degenerate. Again there is no orientation dependence, and thus no tensor shift. A more explicit expression for the tensor polarizability comes about if we factorize the reduced matrix element \langle J∥S(2)∥J\rangle into reduced matrix elements of dipole operators. We wish to apply Eq. (7.273) to perform the factorizaton, regarding S(2) q to be the rank-2 product of the vector dµ and the vector X j
d\nu (7.364) from the definition in Eq. (7.344). The factorized matrix element then becomes
J′ p 5(2J′ + 1) 1 J J J′
X j
EJ −E\betaj d∥J\rangle
J′ p 5(2J′ + 1) J J J′
EJ −EJ′ = X J′ (−1)J+J′p 5(2J + 1) J J J′
EJ −EJ′ , (7.365)
Chapter 7. Atomic Angular-Momentum Structure where in the last step we used Eq. (7.259) to conjugate the second matrix element:
r 2J + 1
(7.366) Putting this matrix element into the expression (7.358) for the tensor polarizability, we arrive at the expres- sion40
X J′
s 40J(2J + 1)(2J −1)
J J J′
EJ −EJ′ , (tensor polarizability) (7.367) thus giving the tensor polarizability as a direct sum over dipole matrix elements. Again, EJ is the energy of level J, and in the sum over J′ we also implicitly sum over any other necessary quantum numbers to enumerate all possible states of the same J′. 7.4.2.4 Hyperfine Structure: Weak Fields In the case of hyperfine structure, we can make exactly the same arguments as we did for fine structure, but using the hyperfine quantum number F instead of J. Thus, the effective hyperfine Stark Hamiltonian as in Eq. (7.363) is
2\alpha(0)(F) E 2 z −1 2\alpha(2)(F) (3E 2 z −E2) 3F 2
F(2F −1) , (effective hyperfine Stark Hamiltonian) (7.368) such that the quadratic Stark shift as in Eq. (7.360) is
2\alpha(0)(F) E 2 z −1 4\alpha(2)(F) (3E 2 z −E2) 3m 2 F −F(F + 1) F(2F −1) , (hyperfine-structure Stark shift) (7.369) the hyperfine scalar polarizability defined as in Eq. (7.352) by
X F ′
EF −EF ′ , (7.370) (hyperfine scalar polarizability) and the hyperfine tensor polarizability is given as in Eq. (7.367) by
X F ′
s 40F(2F + 1)(2F −1)
1 F F F ′
EF −EF ′ . (tensor polarizability) (7.371) These expressions suffice to compute the Stark shift due to static electric fields in the case of hyperfine structure. However, recalling that the electric field coupled only to the electron and not to the nucleus, the effective Stark Hamiltonian (7.363) in terms of the electron angular momentum J must also be a perfectly good Hamiltonian for the hyperfine Stark shift. For example, we may illustrate this by relating the fine- structure polarizabilities to the hyperfine versions. Starting with the scalar hyperfine polarizability, we can start with Eq. (7.282) to reduce the hyperfine matrix element to a fine-structure matrix element in 40cf. Khadjavi et al. and Angel et al., noting the difference in convention for the reduced dipole matrix element.
7.4 Interaction with Static Fields Eq. (7.370),¶
X F ′J′
J J′ F ′ F I
EF −EF ′ \approx −2 X F ′J′
J J′ F ′ F I
EJ −EJ′ = −2 X J′
EJ −EJ′ , (7.372) where we used the orthogonality relation (7.82) for the 6-j symbols in the last step, and we assumed that
scalar polarizability is the same in either case,
(hyperfine and fine-structure scalar polarizabilities) (7.373) at least to the extent that the hyperfine splittings lend a negligible contribution to the polarizability (which is generally true to within modern experimental error in precision measurements of polarizabilities). Similarly, for the tensor polarizability,
X F ′J′
s 40F(2F + 1)(2F −1)
F F F ′
J J′ F ′ F I
EF −EF ′ . (7.374) Again, making the approximation EF \approx EJ, we can then carry out the sum over F ′ via the Biedenharn– Elliott sum rule (7.102), which gives X F ′ (−1)F ′(2F ′ + 1) 1 F F F ′ J J′ F ′ F I 2
1 J J J′ J J F F I . (7.375) Putting this into the above expression for the tensor polarizability, we have
X J′
s 40F(2F + 1)(2F −1)
1 J J J′ J J F F I
EJ −EJ′ , (7.376) and on comparison to Eq. (7.367), we can relate the hyperfine and fine-structure polarizabilities by
s
J J F F I \alpha(2)(J). (7.377) From the 6-j symbol, the combination J + F + I \in Z, and since J and J′ differ by 0 or 1, we conclude that
s
J J F F I \alpha(2)(J). (7.378) We can evaluate the 6-j coefficient here, with the result
\alpha(2)(J), (hyperfine and fine-structure tensor polarizabilities) (7.379)
Chapter 7. Atomic Angular-Momentum Structure where
(7.380) Then the total hyperfine Stark shift is ∆E|F mF \rangle \approx −1 2\alpha(0)(J) E 2 z −\alpha(2)(J) (3E 2 z −E2) [3m 2
, (hyperfine-structure Stark shift) (7.381) Again, these expression makes the (generally good) approximation of neglecting hyperfine shifts in computing the relevant transition energies. Also, note that we have assumed in writing down these expressions that F is a good quantum number, and thus we assume the tensor shifts to be much smaller than the hyperfine splittings. Recalling that the tensor polarizability vanished in the fine-structure cases of J = 0 and J = 1/2, we can see by the coefficient in this expression that the tensor hyperfine polarizability also vanishes for F = 0
where J = 1/2, there is no tensor Stark shift. This statement predicts that for the cesium-clock hyperfine transition, for example, the transition frequency is independent of a dc electric field, because the levels shift together. Actually, this statement is only true in second-order perturbation theory, as we have used here; in third-order perturbation theory, it turns out that, with hyperfine structure, there is still a small tensor Stark shift.41 7.4.2.5 Hyperfine Structure: Stronger Fields In the case of stronger fields, when F is no longer a good quantum number, the formulae of the previous section no longer apply. However, the effective Stark interaction Hamiltonian (7.363) for the interaction of the electron with the static electric field is still valid. In the limit of a very strong electric field, the interaction HStark(J) will dominate the hyperfine Hamiltonian Hhfs, and as in the Paschen–Back (strong- field) interaction for magnetic fields, the appropriate basis is |J mJ; I mI\rangle , where HStark(J) is diagonal. In this case, ignoring the hyperfine splittings, the energies are given by the fine-structure expression (7.360). Also, in the same way as in Eq. (7.326) for the Paschen–Back effect, we can keep the lowest-order contribution of Hhfs by taking its expectation value in the strong-field basis, with the result E|J mJ;I mI\rangle \approx AhfsmImJ + Bhfs 9(mImJ)2 −3J(J + 1)m 2 I −3I(I + 1)m 2
4J(2J −1)I(2I −1) −1 2\alpha(0)(J) E2 −1 4\alpha(2)(J) (3E 2 z −E2) 3m 2 J −J(J + 1) J(2J −1) . (hyperfine Stark shift, strong field) (7.382) Thus, in this ‘‘electric Paschen–Back’’ regime, we expect the hyperfine sublevels to split into major groups according to the value of |mJ|, with smaller splittings according to mI. Obviously this only works if the shifts are not so large that they mix state of different J. However, for general (intermediate) electric fields, we must in general diagonalize Hhfs + HStark(J), as is the case in general for magnetic fields. For example, using \langle F mF|J 2 z |F ′ m′
X mJmIm′ Jm′ I \langle F mF|J mJ; I mI\rangle \langle J mJ; I mI|J 2 z |J m′ J; I m′
J; I m′ I|F ′ m′ F\rangle = X mJmI ¯h2m 2 J \langle J mJ; I mI|F mF\rangle \langle J mJ; I mI|F ′ m′ F\rangle , (7.383) 41A. Weis and S. Ulzega ‘‘The Stark effect of the hyperfine structure of cesium,’’ Proceedings of SPIE 6604, 660408 (2007) (doi: 10.1117/12.726805).
7.4 Interaction with Static Fields we can write the matrix elements of the Stark interaction in the hyperfine basis as \langle F mF|HStark(J)|F ′ m′
2\alpha(0)(J) E 2
F −1 4\alpha(2)(J) (3E 2 z −E2) J(2J −1) X mJmI m 2 J \langle J mJ; I mI|F mF\rangle \langle J mJ; I mI|F ′ m′ F\rangle
F , (7.384) and then diagonalize the resulting matrix numerically. A somewhat nicer expression comes from using the fact that (3J 2
Eckart theorem (7.237), \langle F mF|(3J 2 z −J2)|F ′ m′
z −J2)∥F ′\rangle \langle F mF|F ′ m′ F; 2 0\rangle
F \langle F∥(3J 2
J J′ F ′ F I
J J F ′ F I
(7.385) where we used Eq. (7.261) to change to the fine-structure reduced matrix element, and we also used the Wigner–Eckart theorem (7.237) to evaluate the reduced matrix element: \langle J∥(3J 2
z −J2)|J′ mJ\rangle
J −J(J + 1)]
p
(7.386) Thus, the hyperfine matrix elements of the Stark Hamiltonian become \langle F mF|HStark(J)|F ′ m′
2\alpha(0)(J) E 2 z −1 4\alpha(2)(J) (3E 2 z −E2)\deltaJJ′\deltamF m′ F
s
2J(2J −1) \times J J F ′ F I
(7.387) After adding the diagonal matrix Hhfs, the result can be diagonalized to obtain the energies. Alternately, of course, the hyperfine Hamiltonian can be written in the strong-field basis |J mJ; I mI\rangle and then added to the Stark Hamiltonian in the same way; this method also conveniently carries over to the magnetic-field case. Shown below are the energies of the 6 2P3/2 hyperfine manifold of 133Cs, as in the magnetic-field example above, as obtained via numerical diagonalization. The overall scalar shift is visible as a quadratic downward trend of all the energy levels. The tensor part of the Stark shift is visible as a splitting of the initially degenerate hyperfine sublevels, which break up at the largest fields according to |mJ| in the ‘‘electric Paschen–Back’’ regime that we mentioned above.
7.5.2 Dipole and Atomic Lowering Operators¶
Chapter 7. Atomic Angular-Momentum Structure electric field (kV/cm) 0.5 -5.5 E/h (GHz) F = 2 F = 3 F = 4 F = 5
-1 -2 -3 -4 -5 7.5 Interactions with Optical Fields 7.5.1 Atomic Fine-Structure Hamiltonian We will now consider the interaction of a monochromatic laser field with an atomic Zeeman-degenerate fine-structure transition Jg −\rightarrow Je, making the simplifying assumption of a closed transition (i.e., no other
The atomic Hamiltonian is
X me |Je me\rangle \langle Je me|, (7.388) if we assume degenerate magnetic sublevels and a transition frequency of \omega0, while choosing the ground level to have zero energy. Since we will further consider an interaction with a field of frequency \omega, we can follow the example from the two-level atom and transform into the rotating frame of the laser field, which we recall amounts to shifting the excited states down in energy by ¯h\omega. The rotating-frame Hamiltonian is thus ˜HA = −¯h∆ X me |Je me\rangle \langle Je me|, (7.389) where the field detuning from the atomic resonance is ∆:= \omega−\omega0 as usual. Effects that break the degeneracy of the sublevels are easily accounted for here by including extra m-dependent shifts, such as to include Zeeman or dc Stark shifts. 7.5.2 Dipole and Atomic Lowering Operators We turn now to the dipole interaction between atom and field. Just as in the case of the two-level atom, if we assume the ground and excited levels to be of opposite parity, we can now decompose the q component of the dipole operator in the spherical basis into positive- and negative-rotating parts. Recall that we did this equivalently in terms of the time dependence of expectation values or the lowering/raising character of the operators. Denoting the projection operators for the excited and ground levels by Pe := X me |Je me\rangle \langle Je me|, Pg := X mg |Jg mg\rangle \langle Jg mg|, (7.390)
7.5 Interactions with Optical Fields respectively, we can conveniently separate the two parts by writing
= PgdqPe + PedqPg
q + d(−) q , (7.391) since the dipole operator does not couple states within the same level as a consequence of its parity properties. Then the positive-rotating part becomes d(+) q = PgdqPe = X memg \langle Jg mg|dq|Je me\rangle |Jg mg\rangle \langle Je me| = X memg \langle Jg∥d∥Je\rangle \langle Jg mg|Je me; 1 q\rangle |Jg mg\rangle \langle Je me|, (7.392) where we have used the Wigner–Eckart theorem (7.242) in the second step. In each term of the sum, the projection quantum numbers are of course subject to the constraints mg = me + q for the term to be nonvanishing. Introducing the notation
(7.393) for individual lowering operators, we have d(+) q = X memg \langle Jg∥d∥Je\rangle \langle Jg mg|Je me; 1 q\rangle \sigma(mg, me). (7.394)
q and d(−) q are not Hermitian conjugates unless q = 0, but rather d(−) q = PedqPg = X memg \langle Je me|dq|Jg mg\rangle |Je me\rangle \langle Jg mg| = X memg \langle Je∥d∥Jg\rangle \langle Je me|Jg mg; 1 q\rangle |Je me\rangle \langle Jg mg| = X memg (−1)q\langle Jg∥d∥Je\rangle \langle Jg mg|Je me; 1 −q\rangle |Je me\rangle \langle Jg mg| = X memg (−1)q\langle Jg∥d∥Je\rangle \langle Jg mg|Je me; 1 −q\rangle \sigma\dagger(mg, me), (7.395) where we switched the forms of the Clebsch–Gordan coefficient as in Eq. (7.242) along with the reduced-
so the transition is not the reverse of that in d(+) q
q is the Hermitian conjugate of (−1)qd(+) −q , as we expected from Section 7.3.5. Introducing the weighted lowering operators for the entire Jg −\rightarrow Je transition by \Sigmaq := X mgme \langle Jg mg|Je me; 1 q\rangle |Jg mg\rangle \langle Je me| = X mgme \langle Jg mg|Je me; 1 q\rangle \sigma(mg, me), (7.396)
7.5.3 Dipole Interaction¶
Chapter 7. Atomic Angular-Momentum Structure we can additionally implement the constraint mg = me + q on the sublevels to write \Sigmaq = X me \langle Jg me + q|Je me; 1 q\rangle |Jg me + q\rangle \langle Je me| = X mg \langle Jg mg|Je mg −q; 1 q\rangle |Jg mg\rangle \langle Je mg −q|. (atomic lowering operators, spherical basis) (7.397) Thus, we see that when \Sigmaq lowers an excited sublevel me, it ‘‘returns’’ the sublevel labeled by me + q. Note, however, that \Sigmaq is not a proper tensor operator, in the sense that \Sigma\dagger
Then using Eq. (7.396), we may rewrite the dipole-operator part of Eqs. (7.394) and (7.395) as
q + d(−) q
−q . (dipole operator in terms of lowering operators) (7.398) This form is the generalization of the two-level-atom expression for the dipole operator [cf. Eq. (5.12)] to a fine-structure transition in a physical atom. 7.5.3 Dipole Interaction Now considering the usual interaction Hamiltonian
X q (−1)qdqE−q (7.399) for the atom with a monochromatic optical field of frequency \omega, we can make the rotating-wave approximation and implement the spherical-basis dot product as in Eq. (7.192) to obtain
X q (−1)q d(+) q E(−) −q + d(−) q E(+) −q . (7.400) Then using Eq. (7.398) for the dipole operator, the interaction Hamiltonian becomes HAF = − X q
h (−1)qE(−)
−q (t) \Sigma\dagger −q i . (7.401) Defining the vector Rabi frequency
q (0) ¯h , (7.402) (vector Rabi frequency) and noting that this implies
−q (0) ¯h , (7.403) we can write the atom–field interaction as HAF = ¯h X q h Ω∗
−qe−i\omegati , (7.404) or letting q −\rightarrow −q in the second term, HAF = ¯h X q Ω∗
qe−i\omegat . (atom–field interaction, fine-structure transition) (7.405)
7.5 Interactions with Optical Fields As usual, we transform into the rotating frame of the laser field, say, by moving the excited states down in energy by ¯h\omega, the time dependence of the interaction goes away: ˜HAF = ¯h X q Ω∗
q . (atom–field interaction, rotating frame) (7.406) We can similarly write out the interaction Hamiltonian in terms of all pairs of coupled sublevels as ˜HAF = ¯h X mgme Ω∗(mg, me) \sigma(mg, me) + Ω(mg, me) \sigma\dagger(mg, me) , (atom–field interaction, rotating frame) (7.407) where
(sublevel Rabi frequencies) (7.408) is the Rabi frequency for the |Jg mg\rangle −\rightarrow |Je me\rangle sublevel transition in terms of the vector Rabi frequency. 7.5.3.1 Magnetic-Sublevel Transitions: Notation To illustrate the couplings in the above dipole Hamiltonian, we will consider as an example the possible transitions occurring in a Jg = 1/2 −\rightarrow Je = 3/2 transition. Recall that the selection rules in this case dictate that m can change by at most 1 in a dipole-allowed transition. The transitions where ∆m = 0 are
Correspondingly these transitions are coupled by the \Sigma0 operator. There are two possible \pi transitions in this example atom, shown here.
p transitions:
The transitions where me = mg + 1 are referred to as \sigma+ transitions, and they correspond to q = −1 and
x-y plane). Correspondingly these transitions are coupled by the \Sigma−1 operator. There are two possible \sigma+ transitions in this example atom, shown here.
s ô transitions:
Finally, the transitions where me = mg −1 are referred to as \sigma−transitions, and they correspond to
light in the x-y plane, but rotating in the opposite sense to \sigma−light). Correspondingly these transitions are coupled by the \Sigma+1 operator. There are of course two possible \sigma−transitions in this example atom, shown here.
s — transitions:
7.5.4 Dipole Interaction: Hyperfine Structure¶
Chapter 7. Atomic Angular-Momentum Structure To reiterate, because the notation is a bit strange: \pi transitions are coupled by the E0 = Ez component of the field, or linear polarization along ˆz. (Of course, z here defines the ‘‘quantization axis’’ for the angular- momentum states, since m is the quantum number for Jz.) The \sigma\pm transitions are coupled by the E∓1 components, respectively, of the electric field, corresponding to the two circular polarizations orthogonal to ˆz. 7.5.4 Dipole Interaction: Hyperfine Structure 7.5.4.1 Atomic Hyperfine Hamiltonian The treatment of hyperfine structure is much the same as for fine structure, but there is some extra compli- cation in typically having more hyperfine levels to deal with and in further decomposing the dipole matrix elements. We will consider transitions from a manifold of ground hyperfine levels to a manifold of excited hyperfine levels. The relevant hyperfine levels are again essentially determined by the quantum numbers
nuclear quantum number I. We still assume the fine-structure transition to be closed, but we include all possible hyperfine ground states |Jg −I| \le Fg \le Jg + I and excited states |Je −I| \le Fe \le Je + I, noting that only transitions satisfying |Fe −Fg| \le 1 will occur. The atomic Hamiltonian is HA = ¯h X Fgmg
X me
(7.409) if we assume degenerate magnetic sublevels within each hyperfine level. Here \omega0 is the transition frequency, which we can choose, for example, to correspond to the frequency difference between the centers of gravity of the hyperfine manifolds. Then \delta\omegaFg and \delta\omegaFe are the hyperfine shifts from each respective center of gravity. (Alternately, \omega0 could be chosen to correspond to a particular hyperfine transition of interest such as a laser-cooling transition, with the \delta\omegaFg and \delta\omegaFe corresponding to shifts from these levels within each manifold.) Again, with a coupling to a field of frequency \omega, it is convenient to work within the rotating frame of the field, where the Hamiltonian is ˜HA = ¯h X Fgmg
X me
(7.410) where ∆:= \omega −\omega0 is the usual field detuning from the atomic resonance. 7.5.4.2 Atom–Field Interaction Then the appropriate projection operators require summations over the hyperfine quantum numbers Fe and Fg as well as the sublevel indices me and mg: Pe := X Feme |Fe me\rangle \langle Fe me|, Pg := X Fgmg |Fg mg\rangle \langle Fg mg|. (7.411) Then again in our restricted Hilbert space, Pe +Pg is the identity, and we can again write the dipole operator as
= PgdqPe + PedqPg
q + d(−) q , (7.412)
7.5 Interactions with Optical Fields since the dipole operator does not couple states within the same hyperfine manifold. Then the positive- rotating part becomes d(+) q = PgdqPe = X FemeFgmg \langle Fg mg|dq|Fe me\rangle |Fg mg\rangle \langle Fe me| = X FemeFgmg \langle Fg∥d∥Fe\rangle \langle Fg mg|Fe me; 1 q\rangle |Fg mg\rangle \langle Fe me| = X FemeFgmg
Jg Je Fe Fg I \times\langle Fg mg|Fe me; 1 q\rangle |Fg mg\rangle \langle Fe me|, (7.413)
me + q must still hold for the nonvanishing terms in the sum. Similarly, we may verify directly using Eqs. (7.91), (7.259), and (7.281), that d(−) q = PedqPg = X FemeFgmg
Je Jg Fg Fe I \times\langle Fe me|Fg mg; 1 q\rangle |Fe me\rangle \langle Fg mg| = X FemeFgmg
Jg Je Fe Fg I \times\langle Fg mg|Fe me; 1 −q\rangle |Fe me\rangle \langle Fg mg|
d(+) q \dagger . (7.414) We can then indicate the rather complicated dependence in the dipole operator here by defining as before the weighted lowering operator \Sigmaq := X FgmgFeme
(2Fe + 1)(2Jg + 1) \langle Fg mg|Fe me; 1 q\rangle Je Jg Fg Fe I |Fg mg\rangle \langle Fe me| = X FgmgFeme
(2Fe + 1)(2Jg + 1) \langle Fg mg|Fe me; 1 q\rangle Je Jg Fg Fe I \sigma(Fg, mg; Fe, me), (hyperfine lowering operator) (7.415) where the individual lowering operators are
(7.416) We can then write the dipole operator in the same form as for the fine-structure transition as
q + d(−) q
−q . (hyperfine dipole operator) (7.417)
7.6 Angular Distribution of Dipolar Resonance Fluorescence¶
Chapter 7. Atomic Angular-Momentum Structure Then the dipole-interaction Hamiltonian becomes
= − X q (−1)q d(+) q E(−) −q + d(−) q E(+) −q = − X q
h (−1)qE(−)
−q (t) \Sigma\dagger −q i , (7.418) and defining the same vector Rabi frequency as before,
q (0) ¯h , (7.419) (vector hyperfine Rabi frequency) we switch to the rotating frame of the laser field and write the atom–field interaction in the same form as before: ˜HAF = ¯h X q Ω∗
q . (atom–field interaction, hyperfine transition) (7.420) The same form, that is, except that the raising and lowering operators are considerably more complicated. Putting these in explicitly yields the expression ˜HAF = ¯h X FgmgFeme Ω∗(Fg, mg; Fe, me) \sigma(Fg, mg; Fe, me) + Ω(Fg, mg; Fe, me) \sigma\dagger(Fg, mg; Fe, me) , (hyperfine atom–field interaction, rotating frame) (7.421) where Ω(Fg, mg; Fe, me)
(2Fe + 1)(2Jg + 1) \langle Fg mg|Fe me; 1 −(me −mg)\rangle Je Jg Fg Fe I Ω−(me−mg). (sublevel Rabi frequencies) (7.422) is the Rabi frequency for the |Fg mg\rangle −\rightarrow |Fe me\rangle hyperfine sublevel transition in terms of the vector Rabi frequency. 7.6 Angular Distribution of Dipolar Resonance Fluorescence Recall from Eq. (5.256) that the scattered intensity can be represented in terms of the scattered field as
\pi\eta Z \infty −\infty D
E
(7.423) when written in terms of the vector electric field. Recall that the classical field due to an oscillating dipole in the radiation zone is
¨d(+)(tr) r , (7.424) for a dipole orientation ˆ\epsilon. This carries over to the quantum atom here, as we are still treating the field classically. However, we need to be careful in the spherical basis. Specifically, in the above notation, we mean
h
ˆr −d(+)i , (7.425) and in the spherical basis, the dipole operator may be resolved into components as
X q ˆe∗ qdq = X q ˆ\epsilon∗ qdq, (7.426)
7.6.1 Angular-Distribution Tensor¶
7.6 Angular Distribution of Dipolar Resonance Fluorescence if we choose to represent the polarization-vector components as in terms of the usual basis vectors
(7.427) Thus, the polarization vector becomes h
ˆr −d(+)i = (ˆ\epsilon∗
q d(+), (7.428) and so the radiated electric field becomes
4\piϵ0c2 X q (ˆ\epsilon∗
q ¨d(+) q (tr) r (7.429) in terms of the dipole-vector components. We can then label the radiated field due to each dipole-vector component as E(+) q
4\piϵ0c2 (ˆ\epsilon∗
q ¨d(+) q (tr) r , (7.430) where the polarization vectors are again
\sqrt (7.431) for linear and circular polarizations, respectively. That is, ˆ\epsilonq is just the unit spherical basis vector ˆeq. Note that Eq is not a spherical vector, but dq is, so we should be careful to observe that E(−) q
−q (tr) r , (7.432) to ensure that we appropriately conjugate the dipole operator. The fields that appear in (7.423) are, as we defined them, sums of all three component fields, and so we may write the sum explicitly as
\pi\eta X qq′ Z \infty −\infty D E(−) q
E
(7.433) Paralleling our previous treatment in Section 5.7, we may now write the scattered intensity as
\omega 4 6\pi2ϵ0c3r2 X qq′
Z \infty −\infty D d(−) −q (r, t) d(+)
E
(7.434) where we have defined the angular-distribution tensor
(ˆ\epsilon∗
q′ = 8\pi
q′ \cdot ˆr) (7.435) We will now reduce this to a more explicit form in terms of the angle variables \theta and \phi. 7.6.1 Angular-Distribution Tensor We can first use the orthogonality relation ˆ\epsilonq \cdot ˆ\epsilon∗
second, we can use the relation
r 4\pi 3 Y q
(7.436)
7.6.2 Spectral Tensor and Total Scattered Power¶
Chapter 7. Atomic Angular-Momentum Structure for the projection of ˆr into basis vectors in terms of the spherical harmonics, which can be verified directly (but certainly makes sense since Y q 1 is the right tensor for representing three-vectors). Along with the conjugation rule [Y m
ℓ
(7.437) we can write the angular scattering tensor as
8\pi
3 (−1)q′ Y q
. (7.438) To reduce this yet more, we may use the recoupling relation (7.187), which in this special case reads Y q
X ℓ=0 (−1)q′−q r 9(2ℓ+ 1) 4\pi 1 ℓ q −q′ q′ −q 1 ℓ Y q−q′ ℓ
(7.439) Writing the terms out explicitly and using the 3-j symbols 1 = r 15; 1 = 0; 1 = r 3; 1 q −q′
\sqrt 3 \deltaqq′, (7.440) as well as the symmetry rule J1 J2 J m1 m2 m
J1 J2 J −m1 −m2 −m , (7.441) we find Y q
4\pi
r 2\pi (−1)q−q′ −q q′ q −q′ Y q−q′ ℓ
(7.442) Putting these together, we find the form
4\pi " \deltaqq′ − \sqrt 6\pi (−1)q Y q−q′
−q q′ q −q′ # , (angular-scattering tensor) (7.443) for the scattering tensor. Written out explicitly, the components are shown in the following table.
q′ −1 −1
− \sqrt
q − \sqrt
\sqrt
\sqrt
(7.444) 7.6.2 Spectral Tensor and Total Scattered Power We now continue with the intensity spectrum by writing the dipole-operator components using Eq. (7.398), assuming we are dealing with a single J −\rightarrow J′ transition. Then defining the spectral tensor
2\pi Z \infty −\infty
\Sigma\dagger
, (7.445)
7.6 Angular Distribution of Dipolar Resonance Fluorescence we can write the intensity spectrum (7.434) as
r2 2Je + 1 2Jg + 1 X qq′
(7.446) Here again, from Eq. (7.396), \Sigmaq is the lowering operator for all transitions coupled by polarization q, corresponding to the radiative decay |J′ m′\rangle −\rightarrow |J m′ + q\rangle . The expression (7.446) for the intensity spectrum is valid in the resonant approximation for a single J −\rightarrow J′ Zeeman-degenerate transition. In principle, it should then be summed over all possible transitions (accounting for the various detunings in each case).
oriented dipole, that is, with an orientation ˆ\epsilon = ˆe∗ q. In general, the atom will oscillate in some mixture of the various components, and the off-diagonal terms with q̸ = q′ in the sum represent interference due to coherence between the different dipole components. These interference terms do not change the total radiated power, but rather they change the angular distribution of the radiated power, as we can see by integrating the scattered intensity over all angles:
Z
2Je + 1 2Jg + 1 X q Sqq(\omegas). (7.447) Here, we have used Z
(7.448) which follows from Eq. (7.443), where the second term always vanished under the angular integration in view of the orthogonality of the spherical harmonics. Of course, integrating the power spectrum over all frequencies gives Psc = Z \infty
2Je + 1 2Jg + 1 X q
\Sigma\dagger q \Sigmaq
. (7.449) We can compute the operator sum as X q \Sigma\dagger q\Sigmaq = X qmgmem′e \langle Jg mg|Je m′ e; 1 q\rangle \langle Jg mg|Je me; 1 q\rangle |Je m′
= X qmgme \langle Jg mg|Je me; 1 q\rangle 2 |Je me\rangle \langle Je me| = 2Jg + 1 2Je + 1 X qmgme \langle Je me|Jg mg; 1 −q\rangle 2 |Je me\rangle \langle Je me| = 2Jg + 1 2Je + 1 X me |Je me\rangle \langle Je me|, (7.450) where we used the conservation constraint mg = m′ e + q = me + q to get to the second expression, and we used completeness of the angular-momentum states to get to the last expression. Thus, we may rewrite Eq. (7.449) as
(7.451) where recall that Pe, defined in Eq. (7.390), is the sum over projection operators for all excited sublevels, so that \langle Pe\rangle is the total excited-level population. Thus, the total photon scattering rate is sensibly given by \Gamma times the total excited-state population.
7.7 Optical Stark Shifts¶
Chapter 7. Atomic Angular-Momentum Structure 7.6.2.1 Hyperfine Structure and Interference Of course, this assumes that only one excited level Je is substantially populated. To see this, note that in the case of a hyperfine transition, the formulae in the last section are valid, except that the lowering operators \Sigmaq are given by the hyperfine expression (7.415), and thus we have X q \Sigma\dagger q\Sigmaq = X qFgmgFemeF ′e (−1)Fe−F ′ ep
e me; 1 q\rangle \langle Fg mg|Fe me; 1 q\rangle Je Jg Fg F ′ e I Je Jg Fg Fe I |F ′ e me\rangle \langle Fe me| = X qFgmgFemeF ′e (−1)2Fe−2Fg+2me−2mg(2Fg + 1) (2Jg + 1)
e me|Fg mg; 1 −q\rangle \langle Fe me|Fg mg; 1 −q\rangle Je Jg Fg F ′ e I Je Jg Fg Fe I |F ′ e me\rangle \langle Fe me| = X FgFemeF ′e
Je Jg Fg F ′ e I Je Jg Fg Fe I |F ′ e me\rangle \langle Fe me| = X FgFeme
Je Jg Fg Fe I 2 |Fe me\rangle \langle Fe me|. (7.452) In the second step we used the Clebsch–Gordan symmetry relations [see the Wigner–Eckart theorem in the forms of Eq. (7.281)], and then we used the orthogonality relation (7.46). Then using the orthogonality relation (7.82), we find X q \Sigma\dagger q\Sigmaq = 2Jg + 1 2Je + 1 X Feme |Fe me\rangle \langle Fe me|, (7.453) which is essentially the same result as for the fine-structure case above: this operator is diagonal in the hyperfine basis, and every state decays at the same rate \Gamma. Note that it is important to work this out to see if the fluorescence operator (7.453) contains off-diagonal terms, representing interference between two states with the same me, but different Fe. This is possible in principle because such decays are indistinguishable: if the atom starts in a superposition of these states, we can’t tell by the decay which state the atom ‘‘came from.’’ This is true even if the states are not degenerate, since, for example, with a steady-state drive, the dipoles corresponding to |Fg mg\rangle −\rightarrow |Fe me\rangle , |F ′ e m′ e\rangle oscillate at the same frequency. Of course, decay from states with different me don’t interfere: we can infer ‘‘which-state’’ information by analyzing the polarization of the fluorescence and the final state of the atom to determine what the initial atomic state was. These interferences correspond to quantum beats between particular excited states, as we discussed before in Section (6.2.4). Evidently, however, while such interferences may influence the decay rates to individual ground states, and hence the angular distribution of light, they do not affect the total decay rate from any excited state. 7.7 Optical Stark Shifts Now we would like to handle the Stark shifts due to laser light. In terms of hyperfine states, we may write the Kramers–Heisenberg formula for the polarizability tensor (assuming linear polarization) as see Eq. (14.147)
X F ′m′ F
F|dµ|F mF\rangle ¯h(\omega 2 F ′F −\omega2) . (hyperfine polarizability tensor, linear polarization) (7.454)
7.7.1 Irreducible Parts¶
7.7 Optical Stark Shifts For circular polarization, the formula for the polarizability tensor is \alpha(1)
X F ′m′ F
F|dµ|F mF\rangle ¯h(\omega 2 F ′F −\omega2) . (hyperfine polarizability tensor, circular polarization) (7.455) The tensor polarizability is defined such that to lowest order (i.e., for weak field intensities), the mean induced dipole moment vector is D d(+) µ (\omega) E
)\nu, (7.456) and thus according to the electric-dipole interaction, the energy shift (ac Stark shift) due to the optical field is [cf. Eq. (1.65)]
D
E \cdot E(−) −1 D d(−)(\omega) E
)µ(E(+) )\nu. (7.457) (ac Stark shift) In principle, this is the expression we’re after, but now we will break this expression down into parts according to its symmetry and express the result explicitly in terms of the quantum numbers of the states. 7.7.1 Irreducible Parts Given that the polarizability is a rank-2 tensor, it is convenient to decompose it into its irreducible parts. To simplify notation, we will write the polarizability as
X F ′
¯h(\omega 2 F ′F −\omega2) \alpha(1)
X F ′
¯h(\omega 2 F ′F −\omega2), (7.458) where we have defined the dipole-product tensor
X m′ F \langle F mF|dµ|F ′ m′
(7.459) Note that we include the sum over m′ F here to avoid any orientation dependence in this tensor, since this is what we will decompose into its irreducible components. Recall from Eqs. (7.199), (7.200), (7.206), and (7.207) that we may write Tµ\nu in terms of its scalar, vector, and tensor parts as
4T (1)
µ\nu
T (1) \sigma
T (2)
(7.460) We will handle each irreducible component here separately.42 7.7.1.1 Scalar Part The scalar part is simply the trace,
= X m′ F \langle F mF|dµ|F ′ m′
F|dµ|F mF\rangle
(7.461) 42The treatment here more or less follows Ivan H. Deutsch and Poul S. Jessen, ‘‘Quantum measurement and dynamics of atomic spins in polarization spectroscopy,’’ (2007, to be published). However, the notation here is somewhat different, as it is designed to be close to that of the dc case.
Chapter 7. Atomic Angular-Momentum Structure where we used the normalization formula (7.241) for the reduced matrix element. That was simple enough, but let’s redo this a more complicated way to see a couple details that will make the other components easier to calculate. First, recall that when two rank-1 spherical tensors (vectors) A and B are multiplied to form a rank-0 (scalar) tensor, the result from Eq. (7.214) is T (0) = −1 \sqrt 3A \cdot B. (7.462) It is important to keep the extra overall factors to make use of factorization formulae for matrix elements. Thus, we can rewrite the dipole-vector dot product as a rank-0 spherical tensor and use the Wigner–Eckart theorem (7.237) to find
X m′ F \langle F mF|dµ|F ′ m′
F|dµ|F mF\rangle = − \sqrt 3 \langle F mF| X m′ F d|F ′ m′
F|d (0) |F mF\rangle = − \sqrt 3 \langle F∥ X m′ F d|F ′ m′
F|d (0)
= − \sqrt 3 (−1)2F \sqrt 2F ′ + 1 F F F ′
= − \sqrt
2F + 1 1 F F F ′
(7.463) where we used the factorization formula (7.273) for the first matrix element, and then the conjugation
the 6-j symbol is (−1)−F −F ′−1/ p 3(2F + 1). 7.7.1.2 Vector Part The vector part is related to the cross product of the dipole vectors. T (1) \sigma
(7.464) In the case of the dc Stark shift, this component vanished due to the symmetry of the tensor Hamiltonian; however, here this component does not necessarily vanish, because the electric fields and dipole vectors may be complex (corresponding to circular polarization, which is meaningless in the dc limit). Again, to use the Wigner–Eckart theorem and the subsequent decomposition formula, we must express the vector product of the dipole operators as a rank-1 spherical tensor. From Eq. (7.217), we can express this in terms of the vector cross product as T (1) q = i \sqrt 2(A \times B)q, (7.465)
7.7.2 Total Shift¶
7.7 Optical Stark Shifts where Roman indices to indicate spherical components and Greek indices to indicate Cartesian components. The procedure is otherwise as outlined for the scalar case, with the result T (1) q = −i2 \sqrt 2 \langle F mF| X m′ F d|F ′ m′
F|d (1) q |F mF\rangle = −i2 \sqrt 2 \langle F∥ X m′ F d|F ′ m′
F|d (1)
\sqrt p F(F + 1) p 3(2F ′ + 1) 1 F F F ′
s 24(2F + 1) F(F + 1) 1 F F F ′
(7.466) where we used \langle F mF|F mF; 1 q\rangle = \deltaq0 mF/ p F(F + 1). 7.7.1.3 Tensor Part Finally, the tensor part is T (2)
(7.467) Converted to the spherical basis, there is no scaling factor in this case, and thus, with the usual procedure, T (2) q
X m′ F d|F ′ m′
F|d (2) |F mF\rangle
X m′ F d|F ′ m′
(2)
s 5(2F + 1)
1 F F F ′
(7.468) where we used \langle F mF|F mF; 2 0\rangle = \deltaq0 [3m 2
p
7.7.2 Total Shift Now we can write the total ac Stark shift (7.457) using the polarizability tensor (7.458), along with Eq. (7.460), we can write
X F ′ ¯h(\omega 2 F ′F −\omega2) \times \omegaF ′F T (0) h E(−) E(+) i(0)
4 T (1) (−i \sqrt 2) h E(−)
i(1)
h E(−) E(+) i(2) . (7.469) Note that the tensor products in spherical form are particularly simple because only the q = 0 components are involved. Writing out the relevant field components,
X F ′ ¯h(\omega 2 F ′F −\omega2) \times \omegaF ′F T (0)|E(+)
4 T (1) (E(−)
\sqrt 6 T (2) 3|E(+) 0z |2 −|E(+) |2 . (7.470)
7.7.3 Example: Stark Shifts of the F = 1 −→F ′ = 0 Transition¶
Chapter 7. Atomic Angular-Momentum Structure Now using Eqs. (7.463), (7.466), and (7.468) for the irreducible tensors, we can write the shift as
)z mF F
3|E(+) 0z |2 −|E(+) |2 3m 2 F −F(F + 1) F(2F −1) , (7.471) (ac Stark shift) where we have defined the scalar, vector, and tensor polarizabilities as
X F ′
3¯h(\omega 2 F ′F −\omega2)
X F ′
r 6F(2F + 1) F + 1 1 F F F ′
¯h(\omega 2 F ′F −\omega2)
X F ′ (−1)F +F ′ s 40F(2F + 1)(2F −1)
1 F F F ′
¯h(\omega 2 F ′F −\omega2) , (scalar, vector, and tensor polarizabilities) (7.472) respectively. As in the dc case, the scalar shift causes a level-independent shift, while the tensor shift has the same quadratic dependence on mF. The vector field causes a shift linear in mF, which has the same form as a weak-field Zeeman shift. We will see below that the vector shift is ‘‘activated’’ by circular polarizations, and thus in terms of the level shift, circularly polarized light acts as an effective magnetic field. Linearly polarized light drives the tensor shift, and thus acts as an effective dc electric field. Also as in the dc case, we have chosen the normalization of the tensor polarizability such that the maximum shift (where E(+)
|2. Indeed, both the scalar and tensor polarizabilities here reduce to the respective dc polarizabilities (7.370) and (7.371) in the dc limit \omega = 0. The vector polarizability is similarly normalized such that the maximum shift for mF = F and \sigma+ polarization, E(+)
)−1, has the same form −\alpha(1)(F; \omega)|E(+) |2. To see this, we can write out the field-vector cross product as (iE(−)
0x E(+) 0y −iE(−) 0y E(+) 0x = 1 E(−) 0,1 −E(−) 0,−1 E(+)
0,−1 −1 E(−) 0,1 + E(−) 0,−1 E(+) 0,1 −E(+) 0,−1
0,1 E(+) 0,−1 −E(−) 0,−1E(+) 0,1
0,1 |2 −|E(+) 0,−1|2, (7.473)
0,1 ) light lead to contributions of opposite sign. 7.7.2.1 Excited States Note that these formulae also apply to the ac Stark shifts of excited states, so long as \omegaF ′F = \omegaF ′ −\omegaF is interpreted with the proper sign: the excited state of a two-level atom has opposite Stark shifts for the two levels, and the sign of \omegaF ′F keeps appropriate track of this in the sum over all levels. 7.7.3 Example: Stark Shifts of the F = 1 −\rightarrow F ′ = 0 Transition As a simple example of the formalism we have presented so far, we will consider an F = 1 −\rightarrow F ′ = 0 transition, such that the ground level has three sublevels, but there is only one excited level. We will assume the field to be sufficiently close to resonance with this transitions that other terms in the polarizability sum are negligible in comparison. In this case, we may drop the summations over F ′ in Eqs. (7.472) and write
7.7 Optical Stark Shifts the polarizabilities as¶
3¯h(\omega 2 F ′F −\omega2)
3¯h∆F ′F
s 27F(2F + 1) 2(F + 1) 1 F F F ′
s 30F(2F + 1)(2F −1)
1 F F F ′
(polarizabilities for single hyperfine transition) (7.474)
making the rotating-wave approximation), as is consistent with the two-level approximation. Plugging in F = 1, F ′ = 0, we can write the vector and tensor polarizabilities for this example as
(7.475) and thus the ac Stark shift (7.474) becomes
|E(+) |2 −3 2(iE(−)
)zmF − 3|E(+) 0z |2 −|E(+) |2 3 2m 2 F −1
|E(+) |2 −3 |E(+) 0,−1|2 −|E(+) 0,1 |2 mF − 3|E(+) 0z |2 −|E(+) |2 3 2m 2 F −1 , (7.476) where we have explicitly written the dependence on the three spherical components in the last expression. For circularly polarized light, say \sigma+ light, with E(+)
, this expression reduces to
|2 mF(mF −1) . (7.477) Note that this shift is zero unless mF = −1, where the bracketed expression is unity, which is consistent with our expectation from noting that the mF = −1 state is the only state coupled to the excited state by \sigma+-polarized light. Fo=o1 s ô light: mo=o0 mo=o1
Fo=o0 For linearly (\pi) polarized light, where E(+) 0z
, the ac Stark shift (7.476) becomes
|2 1 −m 2 F
, (7.478) which vanishes unless mF = 0, when the bracketed factor is again unity. This is again in accordance with our expectation that the only level coupled to the excited level by the \pi-polarized light is Stark shifted. Fo=o1 p light: mo=o0 mo=o1
Fo=o0
7.7.5 Large Detuning¶
Chapter 7. Atomic Angular-Momentum Structure Note that the magnitudes of the Stark shifts in both cases are the same, which is a consequence of the simple level structure of this transition. 7.7.4 Polarizability Tensor Revisited As in the dc case, the expression (7.471) for the ac Stark shift allows us to write down an effective Hamiltonian for the ac Stark shift:
)z Fz F
3|E(+) 0z |2 −|E(+) |2 3F 2 z −F2 F(2F −1) . (effective ac Stark Hamiltonian) (7.479) Here, Fz and F2 are the operators, an obviously this Hamiltonian applies to a single hyperfine level, where we have defined the polarizabilities as in (7.472), and the energy shift for the level |F mF\rangle is given by the expectation value of the effective Hamiltonian,
(shift in terms of effective Hamiltonian) (7.480) Another, basis-independent way to write the Stark shift comes about if we define the tensor polarizability operator43
F\sigma
F(2F −1) 1
, (tensor polarizability operator) (7.481) such that the effective Hamiltonian (7.479) becomes
0µ E(+) 0\nu . (effective ac Stark Hamiltonian) (7.482) Note that the operator parts of the polarizability in terms of F are the scalar, vector, and tensor reductions of FF; when computing the expectation value with respect to the |F mF\rangle state, the Wigner–Eckart theorem guarantees that only the q = 0 spherical-tensor components are projected out, in agreement with the basis- dependent expression (7.479). 7.7.5 Large Detuning A crucial aspect of the ac polarizabilities lies in the relative detunings of the optical field with respect to the relevant atomic transitions. In particular, depending on the detunings involved, cancellations may occur. To see examples of this, we can take the polarizabilities (7.472) and factor the reduced matrix elements based 43Deutsch and Jessen, op. cit.
7.7 Optical Stark Shifts on the decoupling rule (7.282):¶
X F ′
3¯h(\omega 2
J J′ F ′ F I 2
X F ′
r 6F(2F + 1) F + 1 F F F ′
¯h(\omega 2 F ′F −\omega2)
J J′ F ′ F I 2
X F ′ (−1)F +F ′ s 40F(2F + 1)(2F −1)
1 F F F ′
¯h(\omega 2 F ′F −\omega2)
J J′ F ′ F I 2 . (7.483) In the limit of large detunings compared to the hyperfine splittings, we can use \omegaF ′F \approx \omegaJ′J, so that the only dependence on F ′ in the sums is in the sign and the 6-j symbols. We can then use the orthogonality relation (7.82) in the scalar case and the Biedenharn–Elliott sum rule (7.102) in the forms [the second expression has the form of (7.375)] X F ′ (−1)F ′(2F ′ + 1) 1 F F F ′ J J′ F ′ F I 2
1 J J J′ J J F F I X F ′ (−1)F ′(2F ′ + 1) 1 F F F ′ J J′ F ′ F I 2
1 J J J′ J J F F I (7.484) for the vector and tensor cases, with the somewhat simpler result
X J′
3¯h(\omega 2 J′J −\omega2)
X J′ (−1)−2J−J′−F −I r 6F(2F + 1) F + 1
¯h(\omega 2 J′J −\omega2) J J J′ J J F F I
X J′ (−1)−2J−J′−F −I s 40F(2F + 1)(2F −1)
¯h(\omega 2 J′J −\omega2) \times 1 J J J′ J J F F I . (far-detuned polarizabilities) (7.485) Thus, as in the dc case, the hyperfine polarizabilities may be expressed for large detunings directly in terms of the fine-structure dipole matrix elements. 7.7.5.1 Effective Dipole Moment Returning to the ac Stark shift (7.471), the scalar part is
|2. (7.486) For detunings large compared to the hyperfine splitting, we can use Eq. (7.485) for the scalar shift so that
X J′
3¯h(\omega 2 J′J −\omega2) |E(+) |2. (7.487) Suppose that although the detuning is large compared to the hyperfine splitting, only one fine-structure level J′ is dominantly resonant. Then we can keep only one term in the sum and make the rotating-wave
Chapter 7. Atomic Angular-Momentum Structure approximation via \omega 2
3¯h∆J′J |E(+) |2. (7.488)
to the ground-state Stark shift from the two-level atom from Section 5.8, Eq. (5.415), ∆E = ¯h|Ω|2 4∆, (7.489) where Ωis the local Rabi frequency
¯h , (7.490) we see that the expressions are equivalent provided we identify
, (7.491) or simply
\sqrt . (7.492) (effective dipole moment, large detuning) The interpretation here is that the field interacts directly with the fine-structure transition if the hyperfine shifts are negligible. The factor of 1/3 simply comes from representing the dipole as d2 = d 2 x +d 2 y +d 2 z = 3d 2 z if the atom is spherically symmetric. Since the polarization vector ˆ\epsilon picks out a particular direction, the field interacts with only one of three possible components of the dipole operator, and thus the factor 1/3. Of course, the scalar polarizability only represents the average behavior for the transition, ignoring any vector or tensor (and thus mF-dependent) shifts. However, as we will see below, the vector and tensor shifts disappear anyway in some important cases for large detunings. In any case, it is best to regard this effective matrix element as being for linearly polarized light, where there is no approximation in neglecting the vector shift. In this case, the saturation intensity is defined as usual by I Isat = 2Ω2 \Gamma2 , (7.493)
|2 [Eq. (1.68)], and Ωfrom Eq. (7.490), so that we have Isat = 3cϵ0\Gamma2¯h2
(saturation intensity, linear polarization, far detuned) (7.494) for the far-detuned saturation intensity of the fine-structure transition (without a hyperfine-resolved excited level). For the shifts where the atom–field interaction is dominated by one hyperfine transition, we can instead compare Eq. (7.483) to the two-level atom, so that the effective dipole moment is
SF F ′, (7.495) (hyperfine effective dipole moment) where the hyperfine transition-strength factor is
J J′ F ′ F I 2 . (hyperfine relative transition strength) (7.496)
7.7 Optical Stark Shifts Or, including the proper sign from Eq. (7.282),¶
\sqrt p SF F ′, (hyperfine effective dipole moment) (7.497) From the orthogonality relation (7.82), we have X F ′ SF F ′ = 1. (7.498) (hyperfine strength sum rule) Thus, the factor SF F ′ acts as a ‘‘relative oscillator strength’’ for the hyperfine transitions in a particular fine- structure line. Again, except in certain special cases, this effective dipole moment only captures orientation- averaged behavior, as for example happens with excitation by isotropic light. 7.7.5.2 Alkali Ground States The expressions above are still rather complicated, so we can get a bit more insight by considering the specific case of the ground state of alkali atoms, where L = 0 and J = 1/2. Using the 6-j symbols 1/2 1/2 J′ = 0 1/2 1/2 J′ = − 3(2J + 1) J J F F I
p
, (7.499) the ground-state polarizabilities become
X J′
3¯h(\omega 2 J′J −\omega2)
X J′
s
\omega \omegaF ′F
(far-detuned polarizabilities) (7.500) Here, we have used Eq. (7.319) for the Landé gF factor gF \approx gJ
2F(F + 1) , (7.501) ignoring the term proportional to gI. Thus, in the regime of large detuning compared to the excited-state hyperfine splitting—when the excited states are effectively degenerate—the tensor component of the ground-
polarizability vanished for J = 0 or J = 1/2. Note, however, that the vector polarizability remains in this regime. In the yet-farther-detuned regime, where the two excited fine-structure states J′ = 1/2, 3/2 are ef-
(7.502) (far-detuned vector polarizability) Again, we also expected this for the dc case, where the vector polarizability vanished for any atomic configu- ration. (This is because the dc field must be real, and thus the \sigma\pm components must always be present with equal weight.) For the alkali-atom ground state, in the regime of far detuning compared to the fine-structure aplitting, the shift is purely scalar.
7.8.1 Fine Structure¶
Chapter 7. Atomic Angular-Momentum Structure 7.8 Atomic Master Equation 7.8.1 Fine Structure To consider a fine structure transition |Jg\rangle −\rightarrow |Je\rangle , we will of course need to consider all possible transitions between sublevels |Jg mg\rangle −\rightarrow |Je me\rangle . We showed in Section (7.3.7.4) that the decay rate (7.305) of a fine-structure transition is \Gamma = \omega 3 3\piϵ0¯hc3 2Jg + 1
(7.503) while the rate for the |Je me\rangle −\rightarrow |Jg mg\rangle decay process is [Eq. (11.38)] \Gammamg,me = \omega 3 3\piϵ0¯hc3 |\langle Jg mg|d|Je me\rangle |2, (7.504) which using the Wigner–Eckart theorem (7.242) gives
(7.505) However, note that any decay corresponding to emission into polarization q is indistinguishable as far as measurement of the radiated field is concerned. Thus, amplitudes for such decays should be added together, while decay rates for decays of different polarization should be added together. We have already concluded this from our above discussion of the spontaneous decay, where in particular from Eq. (7.449) we may conclude that the rate at which photons are scattered is Rsc = \Gamma 2Je + 1 2Jg + 1 X q
\Sigma\dagger q \Sigmaq
. (7.506) This expression has precisely the form we want. The master equation that properly accomplishes this decay along with the appropriate Hamiltonian evolution is
¯h h
i + \Gamma 2Je + 1 2Jg + 1 X q D[\Sigmaq]˜\rho, (master equation, fine-structure transition) (7.507) where ˜HA is defined in Eq. (7.389), where ˜HAF is defined in Eq. (7.407), \Sigmaq is defined in Eq. (7.396) as \Sigmaq = X mgme \langle Jg mg|Je me; 1 q\rangle |Jg mg\rangle \langle Je me|, (7.508) and \Gamma is the total decay rate of any excited sublevel. To verify the action of the decay term here, consider the evolution of the matrix elements ˜\rho\alpha m\alpha,\beta m\beta ≡\langle J\alpha m\alpha|˜\rho|J\beta m\beta\rangle due only to the decay term:
= \Gamma 2Je + 1 2Jg + 1 X q
q −1 2\Sigma\dagger q\Sigmaq ˜\rho −1
q\Sigmaq
= \Gamma 2Je + 1 2Jg + 1 " X q
−1
2Jg + 1 2Je + 1
2Jg + 1 2Je + 1
= \Gamma " X q¶
−1
. (7.509)¶
7.8.2 Hyperfine Structure¶
7.8 Atomic Master Equation Thus, we can see that the excited-state populations (and coherences) decay at rate \Gamma, excited-ground coher- ences decay at rate \Gamma/2, and ground-state populations (and coherences) increase according to the decay of the excited states and the branching factors (Clebsch–Gordan coefficients) into each state. Working out the similar terms for Hamiltonian evolution is relatively straightforward, and thus we can write out the evolution equations for the density matrix elements according to the master equation (7.507) as \partial
X mg
X me
X me
X mg
(pump field)
\Gamma
\Gamma
X q=−1 h
i (dissipation) +
) (free evolution) (master equation, fine-structure transition) (7.510) The Rabi frequencies Ω(mg, me) here were defined before in Eq. (7.408) by
(7.511) The form here is rather complicated but is suited for numerical computations. 7.8.2 Hyperfine Structure For an atom with hyperfine structure, the interaction still occurs between the atom and the atomic dipole, and thus the master equation still has exactly the same form as for fine structure:
¯h h
i + \Gamma 2Je + 1 2Jg + 1 X q D[\Sigmaq]˜\rho. (master equation, fine-structure transition) (7.512) However, the dipole-related symbols here must be interpreted in terms of hyperfine structure. The atomic Hamiltonian is given in the rotating frame by shifting the center of gravity of the excited state down by ¯h\omega, where \omega is the laser frequency, combined with the hyperfine energy shifts expressed in Eq. (7.134) in the hyperfine basis,
2AhfsK + Bhfs
4I(2I −1)J(2J −1) + Chfs
I(I −1)(2I −1)J(J −1)(2J −1) , (7.513)
7.8.3 Rate-Equation Limit¶
Chapter 7. Atomic Angular-Momentum Structure so that for a single fine-structure transition, ˜HA = X Feme ∆Ehfs(Je, I, Fe) −¯h∆ |Je, I; Fe me\rangle \langle Je, I; Fe me| + X Fgmg ∆Ehfs(Jg, I, Fg) |Jg, I; Fg mg\rangle \langle Jg, I; Fg mg|, (7.514) where as usual ∆:= \omega −\omega0, but now \omega0 is the transition frequency for the transition center of gravity (i.e., corresponding to the energy difference in the absence of the hyperfine interaction). The atom–field interaction Hamiltonian in the rotating frame is given in Eq. (7.420) by ˜HAF = ¯h X q Ω∗
q , (7.515) where the lowering operator from Eq. (7.415) is \Sigmaq = X FgmgFeme
(2Fe + 1)(2Jg + 1) \langle Fg mg|Fe me; 1 q\rangle Je Jg Fg Fe I |Fg mg\rangle \langle Fe me| = X FgmgFeme
SFgFe \langle Fg mg|Fe me; 1 q\rangle |Fg mg\rangle \langle Fe me|, (7.516) where SFgFe is defined by Eq. (7.496), and the vector Rabi frequency is given in Eq. (7.402) by
0q ¯h (7.517) for a field of the form [cf. Eq. (7.190)]
X q (−1)qˆe−qE(+)
(7.518) Often, it is most useful to consider the interaction of multiple fields with hyperfine transitions (as in laser cooling of alkali atoms). From Eq. (7.405) we can get the time-dependent form of the interaction Hamiltonian for an additional ‘‘probe’’ field of frequency \omegap and transform to the rotating frame by shifting the frequency by \omega, obtaining H′ AF = ¯h X q Ω∗ q\Sigmaqei∆pt + Ωq\Sigma\dagger qe−i∆pt , (7.519)
extra fields as necessary may be added in this way. 7.8.3 Rate-Equation Limit In certain cases, the atom–field master equation (optical Bloch equations) including Zeeman degeneracy may be excessively difficult to solve. This happens, for example, when the detuning from a fine-structure level, or with well-resolved hyperfine structure, where the hyperfine splittings are much larger than the natural linewidths, necessarily leading to large detunings of the field from some levels. If we don’t care about the fast oscillations due to the large detunings, but choose instead to focus on other, relatively slow dynamics, we can make an adiabatic approximation to obtain a rate-equation formalism to replace the full master equation. First, we will consider a transition Jg −\rightarrow Je between two fine-structure levels, ignoring any hyperfine structure. From Eq. (7.510), the equations of motion for the excited-state populations and Zeeman coherences (coherences between different excited states) are
X mg Ω(mg, me) ˜\rhog mg,e m′e −Ω∗(mg, m′ e) ˜\rhoe me,g mg −\Gamma\rhoe me,e m′e, (7.520)
7.8 Atomic Master Equation while the equations of motion for the ground-state populations and Zeeman coherences are
X me Ω(m′ g, me) ˜\rhog mg,e me −Ω∗(mg, me) ˜\rhoe me,g m′g + \Gamma X q
g + q|Jg m′ g; 1 q\rangle . (7.521) Similarly, the optical coherences (coherences between ground and excited states) evolve as
X m′g Ω(m′ g, me) \rhog m′g,g mg + i X m′e Ω(mg, m′ e) \rhoe me,e m′e − \Gamma 2 −i∆ ˜\rhoe me,g mg
X m′g Ω∗(m′ g, me) \rhog mg,g m′g −i X m′e Ω∗(mg, m′ e) \rhoe m′e,e me − \Gamma 2 + i∆ ˜\rhog mg,e me. (7.522) Now to make the adiabatic approximation, we set \partial t˜\rhoe me,g mg \approx \partial t˜\rhog mg,e me \approx 0, so that we assume the coherences to be always in equilibrium with respect to the populations. This is justified, for example, when the detuning is much larger than the damping rate, where the large separation of time scales justifies ignoring the fast rotations caused by the terms proportional to ∆(recall this argument for the two-level atom in Section 5.8.3). It also may be that the optical coherences are damped quickly, as would be the case for strong collisional dephasing (Section 5.6.2). Then solving Eqs. (7.522) for the coherences in this approximation, we find that we can write the optical coherences in terms of populations and Zeeman coherences as
i 2(\Gamma/2 −i∆) X m′g Ω(m′ g, me) \rhog m′g,g mg − X m′e Ω(mg, m′ e) \rhoe me,e m′e
i
X m′g Ω∗(m′ g, me) \rhog mg,g m′ g − X m′e Ω∗(mg, m′ e) \rhoe m′ e,e me . (7.523) Putting these expressions into Eqs. (7.520) and (7.521), we obtain the following ‘‘rate equations’’ for the populations and Zeeman coherences of a fine-structure transition coupled to a single field:
X mgm′g Ω∗(mg, m′ e) Ω(m′ g, me) 4(\Gamma/2 −i∆) + Ω(mg, me) Ω∗(m′ g, m′ e)
\rhog mg,g m′g − X mgm′′ e Ω∗(mg, m′ e) Ω(mg, m′′ e ) 4(\Gamma/2 −i∆) \rhoe me,e m′′ e + Ω(mg, me) Ω∗(mg, m′′ e )
\rhoe m′′ e ,e m′e −\Gamma\rhoe me,e m′e
X mem′e Ω∗(mg, me) Ω(m′ g, m′ e) 4(\Gamma/2 −i∆) + Ω(m′ g, m′ e) Ω∗(mg, me)
\rhoe me,e m′e − X mem′′ g Ω∗(mg, me) Ω(m′′ g, me) 4(\Gamma/2 −i∆) \rhog m′′ g ,g m′g + Ω(m′ g, me) Ω∗(m′′ g, me)
\rhog mg,g m′′ g + \Gamma X q
g + q|Jg m′ g; 1 q\rangle . (fine-structure rate equations) (7.524) These equations greatly reduce the complexity of the full master equation, since now the optical coherences are eliminated, and only the populations and Zeeman coherences must be tracked.
Chapter 7. Atomic Angular-Momentum Structure 7.8.3.1 Single Field Polarization At first glance, it may seem strange that the rate equations (7.524) should still contain the Zeeman coherences, since rate equations explicitly ignore coherences. In fact, for a single polarization of the field in the spherical basis (i.e., not more than one of Ω−1,0,1 is nonzero), the rate equations simplify. Noting that due to the Clebsch–Gordan coefficients involved, products such as Ω(mg, me) Ω∗(mg, m′ e) are proportional to Kronecker symbols like \deltamem′e, since as in this example, a single polarization couples a ground state to at most one excited state. Then the equation of motion for the excited-state populations reduces to
X mg |Ω(mg, me)|2 \Gamma(1 + 4∆2/\Gamma2)(\rhoe me,e me −\rhog mg,g mg) −\Gamma\rhoe me,e me. (fine-structure rate equations, single polarization) (7.525) We can perform a similar reduction for the ground-state populations, but to simplify the decay terms we
e). The resulting equation is
X me |Ω(mg, me)|2
X me \rhoe me,e me\langle Je me|Jg mg; 1 (me −mg)\rangle 2. (fine-structure rate equations, single polarization, no Zeeman coherence) (7.526) Under these conditions, the rate equations for the populations only are closed, and thus we need not consider the Zeeman coherences. Note that from the original rate equations (7.524), we can see that the excited- state coherences, if initially zero, will remain so if the ground-state coherences are also zero, and vice versa. This is because the field transfers coherence between the excited and ground levels via Rabi flopping. Thus, for a single polarization, our neglecting Zeeman coherences is justified if they all start out as zero (including the ground-state coherences). This may be due to an unoriented atom, but note that since the Zeeman coherences decay with time but are not otherwise excited by the field (for a single polarization), this assumption is eventually justified anyway. Thus we see the importance of the Zeeman coherences in the rate equations (7.524): they are necessary to represent an arbitrary orientation of the atom in the ground or excited state. When the atom is in an initial state with no Zeeman coherence, such as a single ground sublevel, light with an arbitrary polarization will in general put the atom in a coherent superposition of excited states, thus inducing Zeeman coherence that represents the field-induced orientation of the atom. 7.8.3.2 Multiple Fields We already indicated above in Eq. (7.519) that adding a second field introduces an extra Hamiltonian interaction with explicit time dependence if the second field is of a different frequency from the main field. This is because the rotating-frame transformation can only eliminate explicit time dependence at a single frequency; a second frequency must be dealt with directly. This is difficult to handle analytically, as the nonlinear response of the atom will in general generate slowly varying dynamics as well as dynamics at the probe detuning ∆p, and multiples thereof. We have already studied the interaction of atoms with bichromatic fields, for example, in the probe absorption by a driven two-level atom (Section 5.7.6, Problem 5.20), in the Autler-Townes doublet (Section 5.7.6.1 Problem 5.16), in stimulated Raman scattering (Section 6.1), and in coherent population trapping (Section 6.2). We have seen in these cases that the nonlinear mixing of the two fields can lead to strong, coherent effects such as level splittings and population transfer, and in general these effects can only be fully caputured in a full master-equation treatement. Under conditions where these effects are negligible or unimportant, however, the interaction of an atom with multiple fields can still be treated within a rate-equation formalism. The basic assumption required here is that all fields perturb the atom in the linear-response regime, and thus there are no cooperative effects induced by the multiple fields. This requires that the fields are weak (with either intensities well below the saturation intensities, or far detuned compared to the Rabi frequency from any transition), and that no multiphoton resonance (e.g., Raman resonance) occurs. Also, we should assume that any beat frequencies between the multiple fields are fast on time scales of interest (slow beats can be crudely modeled using a slowly varying Rabi frequency).
7.8 Atomic Master Equation At this level of approximation, rate-equation terms involving Ω, for example as in Eqs. (7.524), are simply repeated for each field. 7.8.3.3 Hyperfine Structure The hyperfine case is somewhat more complicated by the proliferation of states and the fact that not all sublevels are degenerate in view of the hyperfine shifts. From the master equation (7.512), the equation of motion for the excited-state populations and coherences is
e m′ e = −
e ˜\rhoFe me,F ′ e m′ e −i X Fgmg Ω(Fg, mg; Fe, me) ˜\rhoFg mg,F ′e m′e −Ω∗(Fg, mg; F ′ e, m′ e) ˜\rhoFe me,Fg mg , (7.527) where the hyperfine splittings are given in terms of the hyperfine shifts (7.134) as
e) ¯h , (7.528) the hyperfine Rabi frequencies are defined by Eq. (7.422), and we used the form (7.453) for the decay operator to work out the decay term. Using essentially the same procedure leading to Eq. (7.453) we can also work out the remaining decay term \Sigmaq\rho\Sigma\dagger q to find the equation of motion \partial t˜\rhoFg mg,F ′g m′g = −i\omegaFgF ′g ˜\rhoFg mg,F ′g m′g + X FeF ′e X q=−1 \Gamma(Fg, mg; F ′ g, m′ g; Fe; F ′ e; q) ˜\rhoFe mg+q,F ′e m′g+q + i X Feme Ω(F ′ g, m′ g; Fe, me) ˜\rhoFg mg,Fe me −Ω∗(Fg, mg; Fe, me) ˜\rhoFe me,F ′g m′g , (7.529) where we have defined \Gamma(Fg, mg; F ′ g, m′ g; Fe; F ′
g−Fg q
Je Jg F ′ g F ′ e I Je Jg Fg Fe I
e m′ g + q|F ′ g m′
(7.530) as the return rate for the ground-state populations and coherences, and the ground-state hyperfine splittings are defined in the same way as for the excited states:
g) ¯h . (7.531) The equations of motion for the optical coherences are
\Gamma 2 −i∆(Fg, mg; Fe, me) ˜\rhoFe me,Fg mg −i X F ′gm′g Ω(F ′ g, m′ g; Fe, me) ˜\rhoF ′g m′g,Fg mg + i X F ′em′e Ω(Fg, mg; F ′ e, m′ e) ˜\rhoFe me,F ′e m′e
\Gamma 2 + i∆(Fg, mg; Fe, me) ˜\rhoFg mg,Fe me + i X F ′gm′g Ω∗(F ′ g, m′ g; Fe, me) ˜\rhoFg mg,F ′g m′g −i X F ′em′e Ω∗(Fg, mg; F ′ e, m′ e) ˜\rhoF ′e m′e,Fe me, (7.532) where ∆(Fg, mg; Fe, me) is the detuning from the |Fg\rangle −\rightarrow |Fe\rangle hyperfine transitions,
\omega0 + ∆Ehfs(Je, I, Fe) −∆Ehfs(Jg, I, Fg) ¯h
¯h , (7.533)
Chapter 7. Atomic Angular-Momentum Structure where again \omega0 is the transition frequency of the center of gravity of the hyperfine transition, and ∆is the laser detuning with respect to \omega0. In the adiabatic approximation, Eqs. (7.532) become ˜\rhoFe me,Fg mg = − i 2[\Gamma/2 −i∆(Fg, mg; Fe, me)] " X F ′gm′g Ω(F ′ g, m′ g; Fe, me) ˜\rhoF ′g m′g,Fg mg − X F ′em′e Ω(Fg, mg; F ′ e, m′ e) ˜\rhoFe me,F ′e m′e # ˜\rhoFg mg,Fe me = i 2[\Gamma/2 + i∆(Fg, mg; Fe, me)] " X F ′gm′g Ω∗(F ′ g, m′ g; Fe, me) ˜\rhoFg mg,F ′g m′g − X F ′em′e Ω∗(Fg, mg; F ′ e, m′ e) ˜\rhoF ′e m′e,Fe me # , (7.534) and putting these into the equations of motion for the populations and hyperfine coherences, we obtain the rather complicated rate equations
X FgmgF ′gm′g Ω∗(Fg, mg; F ′ e, m′ e) Ω(F ′ g, m′ g; Fe, me) 4[\Gamma/2 −i∆(Fg, mg; Fe, me)] + Ω(Fg, mg; Fe, me) Ω∗(F ′ g, m′ g; F ′ e, m′ e) 4[\Gamma/2 + i∆(Fg, mg; F ′e, m′e)] ˜\rhoFg mg,F ′g m′g − X FgmgF ′′ e m′′ e Ω∗(Fg, mg; F ′ e, m′ e) Ω(Fg, mg; F ′′ e , m′′ e ) 4[\Gamma/2 −i∆(Fg, mg; Fe, me)] ˜\rhoFe me,F ′′ e m′′ e + Ω(Fg, mg; Fe, me) Ω∗(Fg, mg; F ′′ e , m′′ e ) 4[\Gamma/2 + i∆(Fg, mg; F ′e, m′e)] ˜\rhoF ′′ e m′′ e ,F ′e m′e −
˜\rhoFe me,F ′e m′e
X FemeF ′em′e Ω∗(Fg, mg; Fe, me) Ω(F ′ g, m′ g; F ′ e, m′ e) 4[\Gamma/2 −i∆(F ′g, m′g; Fe, me)] + Ω(F ′ g, m′ g; F ′ e, m′ e) Ω∗(Fg, mg; Fe, me) 4[\Gamma/2 + i∆(Fg, mg; Fe, me)] ˜\rhoFe me,F ′e m′e − X FemeF ′′ g m′′ g Ω∗(Fg, mg; Fe, me) Ω(F ′′ g , m′′ g; Fe, me) 4[\Gamma/2 −i∆(F ′g, m′g; Fe, me)] ˜\rhoF ′′ g m′′ g ,F ′g m′g + Ω(F ′ g, m′ g; Fe, me) Ω∗(F ′′ g , m′′ g; Fe, me) 4[\Gamma/2 + i∆(Fg, mg; Fe, me)] ˜\rhoFg mg,F ′′ g m′′ g + X FeF ′eq \Gamma(Fg, mg; F ′ g, m′ g; Fe; F ′ e; q) ˜\rhoFe mg+q,F ′e m′g+q −i\omegaFgF ′g ˜\rhoFg mg,F ′g m′g. (hyperfine-structure rate equations) (7.535)
g and Fe̸ = F ′ e) that rotate at the hyperfine splittings. If the hyperfine structure is well-resolved, so that the hyperfine splittings are much larger than \Gamma, then it may be that we are similarly uninterested in these fast oscillations, and we can adiabatically eliminate these hyperfine coherences as well. We can obtain the adiabatic relations for
7.8 Atomic Master Equation F ′ e̸ = Fe by setting \partial t˜\rhoFe me,F ′e m′e \approx 0, with the result ˜\rhoFe me,F ′e m′e = X FgmgF ′gm′g Ω∗(Fg, mg; F ′ e, m′ e) Ω(F ′ g, m′ g; Fe, me) 4(\Gamma + i\omegaFeF ′e)[\Gamma/2 −i∆(Fg, mg; Fe, me)] + Ω(Fg, mg; Fe, me) Ω∗(F ′ g, m′ g; F ′ e, m′ e) 4(\Gamma + i\omegaFeF ′e)[\Gamma/2 + i∆(Fg, mg; F ′e, m′e)] ˜\rhoFg mg,F ′g m′g − X FgmgF ′′ e m′′ e Ω∗(Fg, mg; F ′ e, m′ e) Ω(Fg, mg; F ′′ e , m′′ e ) 4(\Gamma + i\omegaFeF ′e)[\Gamma/2 −i∆(Fg, mg; Fe, me)] ˜\rhoFe me,F ′′ e m′′ e + Ω(Fg, mg; Fe, me) Ω∗(Fg, mg; F ′′ e , m′′ e )
e m′′ e ,F ′e m′e , (7.536) and the relations for F ′ g̸ = Fg follow by setting \partial t˜\rhoFg mg,F ′g m′g \approx 0, with the result ˜\rhoFg mg,F ′g m′g = X FemeF ′em′e Ω∗(Fg, mg; Fe, me) Ω(F ′ g, m′ g; F ′ e, m′ e) 4i\omegaFgF ′g[\Gamma/2 −i∆(F ′g, m′g; Fe, me)] + Ω(F ′ g, m′ g; F ′ e, m′ e) Ω∗(Fg, mg; Fe, me) 4i\omegaFgF ′g[\Gamma/2 + i∆(Fg, mg; Fe, me)] ˜\rhoFe me,F ′e m′e − X FemeF ′′ g m′′ g Ω∗(Fg, mg; Fe, me) Ω(F ′′ g , m′′ g; Fe, me) 4i\omegaFgF ′g[\Gamma/2 −i∆(F ′g, m′g; Fe, me)] ˜\rhog m′′ g ,g m′g + Ω(F ′ g, m′ g; Fe, me) Ω∗(F ′′ g , m′′ g; Fe, me) 4i\omegaFgF ′g[\Gamma/2 + i∆(Fg, mg; Fe, me)] ˜\rhoFg mg,F ′′ g m′′ g + i\omegaFgF ′g X FeF ′eq \Gamma(Fg, mg; F ′ g, m′ g; Fe; F ′ e; q) ˜\rhoFe mg+q,F ′e m′g+q. (7.537) Putting these two relations into the rate equations (7.535) leads to a yet more complicated set of rate equa- tions, but that only retain coherences between degenerate levels (which properly accounts for the orientation of the atom in the various levels). Note that the terms generated by this adiabatic elimination are quartic in the field Ω, whereas there are also quadratic terms in the field. If we work in the weak-field approximation, where all Rabi frequencies are small compared to the hyperfine splittings, \Gamma, or the detunings from the hyperfine resonances (or even better, small compared to all of these), then we can ignore these higher-order terms. This corresponds to taking ˜\rhoFe me,F ′e m′e \approx 0 ˜\rhoFg mg,F ′g m′g \approx i\omegaFgF ′g X Feq \Gamma(Fg, mg; F ′ g, m′ g; Fe; Fe; q) ˜\rhoFe mg+q,Fe m′g+q, (7.538)
7.9 Whither has Wandered the Two-Level Atom?¶
Chapter 7. Atomic Angular-Momentum Structure again for Fg̸ = F ′ g and Fe̸ = F ′ e. Putting these into the rate equations (7.535), we obtain the low-intensity rate equations
X Fgmgm′g Ω∗(Fg, mg; Fe, m′ e) Ω(Fg, m′ g; Fe, me) 4[\Gamma/2 −i∆(Fg, mg; Fe, me)] + Ω(Fg, mg; Fe, me) Ω∗(Fg, m′ g; Fe, m′ e) 4[\Gamma/2 + i∆(Fg, mg; Fe, m′e)] \times ˜\rhoFg mg,Fg m′g + X FgmgF ′ gm′ gFeq
Ω∗(Fg, mg; Fe, m′ e) Ω(F ′ g, m′ g; Fe, me) 4[\Gamma/2 −i∆(Fg, mg; Fe, me)] + Ω(Fg, mg; Fe, me) Ω∗(F ′ g, m′ g; Fe, m′ e) 4[\Gamma/2 + i∆(Fg, mg; Fe, m′e)] \times i\omegaFgF ′g \Gamma(Fg, mg; F ′ g, m′ g; Fe; Fe; q) ˜\rhoFe mg+q,Fe m′g+q − X Fgmgm′′ e Ω∗(Fg, mg; Fe, m′ e) Ω(Fg, mg; Fe, m′′ e ) 4[\Gamma/2 −i∆(Fg, mg; Fe, me)] ˜\rhoFe me,Fe m′′ e + Ω(Fg, mg; Fe, me) Ω∗(Fg, mg; Fe, m′′ e ) 4[\Gamma/2 + i∆(Fg, mg; Fe, m′e)] ˜\rhoFe m′′ e ,Fe m′e −\Gamma˜\rhoFe me,Fe m′e (hyperfine-structure rate equations, small intensity) (7.539) for the excited states and \partial t˜\rhoFg mg,Fg m′ g = X Femem′e Ω∗(Fg, mg; Fe, me) Ω(Fg, m′ g; Fe, m′ e) 4[\Gamma/2 −i∆(Fg, m′g; Fe, me)] + Ω(Fg, m′ g; Fe, m′ e) Ω∗(Fg, mg; Fe, me) 4[\Gamma/2 + i∆(Fg, mg; Fe, me)] ˜\rhoFe me,Fe m′e − X Femem′′ g Ω∗(Fg, mg; Fe, me) Ω(Fg, m′′ g; Fe, me) 4[\Gamma/2 −i∆(F ′g, m′g; Fe, me)] ˜\rhoFg m′′ g ,Fg m′g + Ω(Fg, m′ g; Fe, me) Ω∗(Fg, m′′ g; Fe, me) 4[\Gamma/2 + i∆(Fg, mg; Fe, me)] ˜\rhoFg mg,Fg m′′ g − X FemeF ′′ g m′′ g F ′ eq (F ′′
Ω∗(Fg, mg; Fe, me) Ω(F ′′ g , m′′ g; Fe, me) 4i\omegaF ′′ g Fg[\Gamma/2 −i∆(Fg, m′g; Fe, me)] \times \Gamma(F ′′ g , m′′ g; Fg, m′ g; F ′ e; F ′ e; q) ˜\rhoF ′e m′′ g +q,F ′e m′g+q + Ω(Fg, m′ g; Fe, me) Ω∗(F ′′ g , m′′ g; Fe, me) 4i\omegaFgF ′′ g [\Gamma/2 + i∆(Fg, mg; Fe, me)] \times \Gamma(Fg, mg; F ′′ g , m′′ g; F ′ e; F ′ e; q) ˜\rhoF ′e mg+q,F ′e m′′ g +q + X Feq \Gamma(Fg, mg; Fg, m′ g; Fe; Fe; q) ˜\rhoFe mg+q,Fe m′g+q. (hyperfine-structure rate equations, small intensity) (7.540) for the ground states. 7.9 Whither has Wandered the Two-Level Atom? As we have seen, the real situation with atomic angular-momentum structure is considerably more com- plicated than the idealized model of a two-level atom. So to what extent is the two-level atom a useful model? Actually, there are some important situations under which atoms with angular-momentum degen- eracy behave as two-level atoms. One obvious candidate is the a transition of the form J = 0 −\rightarrow J′ = 0
7.9.1 Optical Pumping to Stretched States¶
7.9 Whither has Wandered the Two-Level Atom?
numbers), where each level has only one sublevel. Unfortunately, we have already seen according to the dipole selection rules that this transition is forbidden. The most direct realization of the two-level atom comes in the form of a J = 0 −\rightarrow J′ = 1 transition, where depending on the polarization, the ground state can be coupled to one of three excited states. Jo=o0 mo=o0 mo=o1
Jo=o1 The important thing to realize is that, given an arbitrary polarization of the field, the ground state is coupled to some linear combination of the three excited states, while two other linear combinations are not coupled. The transition matrix elements from the Wigner–Eckart theorem (7.237) are
\sqrt \deltam′ J,−q (7.541) Thus, we see that the amplitude of the matrix element for every polarization is identical, and in fact equal to the effective matrix element (7.492) for linear polarization that we wrote down before, up to a minus sign. The point is that an any polarization couples to the atom with the same strength, and so except for the atomic orientation in the excited state (corresponding to the orientation of the induced dipole moment) there is no dependence of any of the physics on the polarization. Further, since only one excited state is coupled, we may regard this system as a two-level atom. Any decay will be back to the ground state, corresponding to dipole radiation with the same orientation as the inducing field. Of course, this argument breaks down if there are two fields present with different orientation, such as a second laser with another polarization, or a static magnetic or electric field. Then one field induces an orientation that modifies the interaction with the other field. 7.9.1 Optical Pumping to Stretched States Another important situation comes in the form of a J −\rightarrow J′ = J + 1 transition pumped by circularly polarized light. As a concrete example, we can consider a J = 1 −\rightarrow J′ = 2 transition coupled by \sigma+=po- larized light. Recall that this light drives sublevel transitions of the form mJ −\rightarrow m′ J = mJ + 1. However, spontaneous decay occurs from any possible m′ J −\rightarrow mJ = m′ J \pm 1, 0. Jo=o1 mo=1 mo=o2 mo=o0
Jo=o2
J = 2 transition is thus closed in this scheme. Atoms starting in any other state will eventually become pumped into this cycling transition (on the ‘‘stretched states’’), and thus, at least in steady state, we effectively have a two-level atom. Jo=o1 mo=1 mo=o2 mo=o0
Jo=o2
Chapter 7. Atomic Angular-Momentum Structure The dipole matrix element for this transition is given by the Wigner–Eckart theorem (7.237) as
s 2J + 1
r 2J + 1 2J′ + 1. (7.542) The matrix element is thus the reduced matrix element for the hyperfine transition multiplied by a degeneracy ratio for the transition. Notice that this is precisely the same degeneracy ratio that appears in the decay-rate formula (7.305). That is, if we define the effective dipole moment dstretch := \langle J∥d∥J′\rangle r 2J + 1 2J′ + 1, (7.543) (effective stretched-transition dipole) for the stretched-state transition, then we obtain the two-level-atom formula (7.299) when we write the decay rate in terms of this transition: \GammaJgJe = \omega 3 3\piϵ0¯hc3 |dstretch|2. (7.544) Physically, this is because the stretched excited state has only one decay path, which decays at the full rate \GammaJgJe. Thus, no summation—as is implied in the reduced matrix element—is necessary to compute the full decay rate \GammaJgJe. The same thing happens in a closed hyperfine transition between stretched states, in the case where the excited state also has only one decay path. This happens again for the fine structure transition J −\rightarrow J′ = J +1, in particular for the \sigma+ hyperfine transition F = J +I, mF = F −\rightarrow F ′ = J′ +I = F +1, m′ F = F ′. In
transition. The transition here is closed because the F ′ excited level can only decay to a ground level with F = F ′ \pm 1, 0, and thus only has one decay option. As in the fine-structure case, the stretched excited state only decays to the stretched ground state. The dipole moment for this hyperfine transition is
r 2F + 1 2F ′ + 1
\times p
J J′ F ′ F I r 2F + 1 2F ′ + 1
r 2J + 1 2J′ + 1
r 2J + 1 2J′ + 1 (7.545) after using the hyperfine Wigner–Eckart theorem (7.281), the decomposition rule (7.282), the 6-j symbol J J′ F ′ F I = J J + 1
I + J I = (−1)−2(I+J) p
, (7.546) and F ′ −J + 1 −I = F −J −I + 2 = 2. Thus, exactly the same effective dipole moment applies to the hyperfine stretched-state transition as to the similar fine-structure transition. Typically, the effective dipole moment defined here is larger than that for large detunings, as in Eq. (7.492), because the optical pumping of the atom towards the stretched state produces an atomic
7.9.2 Optical Pumping with Linearly Polarized Light¶
7.9 Whither has Wandered the Two-Level Atom? orientation that is aligned with the field. For J = 0, of course, the stretched-state transition has an effective squared dipole moment of |\langle J∥d∥J′\rangle |2/3, while in the limit as J −\rightarrow \infty, the effective squared dipole approaches |\langle J∥d∥J′\rangle |2, corresponding to perfect alignment with the field (so that the full weight of the atomic dipole is represented by only one spherical component). While the effect here is restricted to transitions of certain forms, obviously some alignment occurs when an atom is pumped by a circularly polarized field, even for transitions of arbitrary form. However, for J −\rightarrow J′ = J transition interacting with circularly polarized light, the atoms still become completely aligned in a stretched state, which is a dark state, in a way that is essentially equivalent to the mechanism described below. 7.9.2 Optical Pumping with Linearly Polarized Light When pumping atoms with linearly polarized light, alignment phenomena occur that are similar to the circular-polarization case. Consider a J −\rightarrow J′ = J transition pumped by linearly polarized light, where J is some integer. For concreteness, we can consider a J = 2 −\rightarrow J′ = 2 transition. Jo=o2 mo=1 mo=o2 mo=o0
Jo=o2 Because the mJ = 0 −\rightarrow m′ J = 0 transition is forbidden, the mJ = 0 ground sublevel is not excited, but all the other sublevels are. However, other states can decay into the mJ = 0 sublevel by \sigma\pm transitions. Jo=o2 mo=1 mo=o2 mo=o0
Jo=o2 Since the mJ = 0 sublevel has no excitation path, but there are excitation/decay routes from any other state into mJ = 0, in steady state the atom will end up entirely in mJ = 0. Again, this state is not coupled to an excited level by the linearly polarized light, so the atoms are in a dark state, no longer interacting with the light. Of course, particularly in the hyperfine case, there are other levels around for which transitions are not dipole-forbidden, so there will be leakage to some extent; furthermore there may be multiple ground hyperfine levels, in which case repumping from the other ground level(s) is necessary to pump the atoms into the dark state. In a general J −\rightarrow J′ fine-structure transition, some degree of alignment towards mJ = 0 tends to occur. The exception is the case of J −\rightarrow J′ = J −1, where the mJ = \pmJ stretched ground states are dark, and thus the atoms tend to accumulate in those states. However, the less trivial cases are J −\rightarrow J′ = J + 1 for arbitrary J or J −\rightarrow J′ = J for half-integer J. In steady state, the transition takes on a well-defined alignment, and behaves as a two level atom, so long as the transition is taken to have an appropriate (geometry-dependent) dipole moment. To find the explicit steady-state solutions in this case,44 we can start with the rate equations (7.525) and (7.526). In deriving these equations, we assumed a single polarization and that the coherences were in quasi-steady state; thus, for computing steady states of the full master equation (7.507) or (7.510), they will produce exact results. Starting with the ground-state rate equation
|Ω(m, m)|2
(7.547) 44These solutions were derived with the resolvent method by Bo Gao, ‘‘Effects of Zeeman degeneracy on the steady-state properties of an atom interacting with a near-resonant laser field: Analytic results,’’ Physical Review A 48, 2443 (1993) (doi: 10.1103/PhysRevA.48.2443). For further results, see also Bo Gao, ‘‘Effects of Zeeman degeneracy on the steady-state properties of an atom interacting with a near-resonant laser field: Probe spectra,’’ Physical Review A 49, 3391 (1994) (doi: 10.1103/PhysRevA.49.3391); Bo Gao, ‘‘Effects of Zeeman degeneracy on the steady-state properties of an atom interacting with a near-resonant laser field: Resonance fluorescence,’’ Physical Review A 50, 4139 (1994). (doi: 10.1103/PhysRevA.50.4139).
Chapter 7. Atomic Angular-Momentum Structure while the same condition \partial tPg,m = 0 for the population Pg,m := \rhog m,g m = 0 in the ground state |Jg m\rangle gives |Ω(m, m)|2
X m′ Pe,m′\langle Je m′|Jg m; 1 (m′ −m)\rangle 2. (7.548) Combining these two relations to eliminate the Rabi frequency, we find a closed equation for the excited-state populations: Pe,m = X m′ Pe,m′ \langle Je m′|Jg m; 1 (m′ −m)\rangle 2. (7.549) This is somewhat surprising: for a linearly polarized pump, the relative populations of the excited states (i.e., the orientation of the atomic excitation) is completely independent of the driving intensity. If we define the tridiagonal matrix
(7.550) then Eq. (7.549) amounts to the homogeneous, tridiagonal linear system
(7.551) which we must now solve for the populations xm. Note that the matrix Amm′ has the extra constraints
symmetry of the problem, the populations satisfy xm = x−m. The solution for m > 0 is given by the recursion formula (Problem 7.3)
Am,m+1 xm, (7.552) which we may explicitly iterate to find xm = m−1 Y m′=0 Am′+1,m′ m−1 Y m′=0 Am′,m′+1 x0 = m−1 Y m′=0
m−1 Y m′=0
x0 (7.553) for m > 0. Since we will explicitly normalize these populations anyway, we can take a convenient normal- ization by writing Jg−1 Y m′=0
xm x0 = m−1 Y m′=0
! Jg−1 Y m′=m
! , (7.554) then using Eq. (7.67) to reverse the mJ quantum numbers in the first factor while letting m′ −\rightarrow m′ −1 in the second, Jg−1 Y m′=0
xm x0 = m−1 Y m′=0
! Jg Y
! , (7.555) then letting m′ −\rightarrow −m′ in the first factor, Jg−1 Y m′=0
xm x0 =
Y
! Jg Y
! , (7.556)
7.9 Whither has Wandered the Two-Level Atom? and finally multiplying through by the same factor on the left, Jg−1 Y m′=0
xm x0 =
Jg Y
! Jg Y
! . (7.557) This expression is explicitly invariant under m −\rightarrow −m, and so we can write out these weights explicitly normalized by defining a new symbol for the left-hand side of the above equation and then normalizing it:
Jg Y
! Jg Y
! we,m := \chie,m Jg X m′=−Jg \chie,m′ . (relative excited-state weighting factors) (7.558) Then the relative excited-state populations are given by the normalized weights we,m, Pe,m = we,mPe, (7.559) (excited-state populations) where the total excited-state population is Pe := Jg X m′=−Jg Pe,m′. (7.560) To find the total excited-state population, we can write Eq. (7.547) in the form |Ω(m, m)|2
Pe,m = |Ω(m, m)|2
(7.561) or
|Ω(m, m)|2 we,mPe = Pg,m. (7.562) Summing over m, we find "
X m we,m |Ω(m, m)|2 !# Pe = Pg, (7.563) where the total ground-state population is Pg := Jg X m=−Jg Pg,m. (7.564) Now using Eq. (7.408) to factor the Rabi frequencies, we can define the geometric factor g := Jg X m=−Jg we,m
−1 , (7.565) (geometric factor) so that
gΩ2 Pe = Pg, (7.566)
Chapter 7. Atomic Angular-Momentum Structure
we can solve this equation to write the total excited-state population
gΩ2 \Gamma2 1 + 2∆ \Gamma 2 + 2gΩ2 \Gamma2 . (steady-state excitation, linearly polarized drive) (7.567) Note that this expression is exactly the same as the corresponding expression (5.137) for the excited-state population of the two-level atom if we identify the two-level-atom Rabi frequency via Ω2 −\rightarrow gΩ2 0 . Alter- nately, we can write the excitation in the standard form [as in Eq. (5.250) for the two-level atom]
1 I/Isat
, (7.568) so long as we identify I Isat = 2gΩ2 \Gamma2 . (7.569) Using Eq. (7.402), we can solve this to obtain the effective saturation intensity Isat = cϵ0\Gamma2¯h2
(effective saturation intensity, linear polarization) (7.570) in terms of the reduced dipole matrix element for linearly-polarized excitation. Surprisingly, the entire effect of the atomic Zeeman-degenerate structure is wrapped up in the single geometric factor g, at least as far as the total excitation is concerned. For some representative values, for a Jg = 0 −\rightarrow Je = 1 transition, g = 1/3;
the case of integer angular momenta), though this is certainly not obvious from its definition. As in the circular-polarization case, the effective squared dipole moment of the transition is |\langle Jg∥d∥Je\rangle |2/3 for Jg = 0 and increases with Jg. However, it saturates at a smaller value than in the circular case, indicating that the atomic alignment is not as complete. This is not entirely surprising, as when excited with linearly polarized light the emitted photons may have any polarization, whereas with a circular pump, the emitted photons may have only one polarization. Explicitly, then, the individual excited-state populations in steady state are
we,m gΩ2 \Gamma2 1 + 2∆ \Gamma 2 + 2gΩ2 \Gamma2 , (excited-state populations, linearly polarized drive) (7.571) and using Eq. (7.562), the ground-state populations are
we,m g Ω2 \Gamma2 +
1 + 2∆ \Gamma 2 + 2gΩ2 \Gamma2 . (ground-state populations, linearly polarized drive) (7.572) Note that unlike the excited-state case, the relative ground-state populations depend on the intensity of the field.
difference in conventions for the Rabi frequency (and hence the reduced dipole matrix element).
7.10 Exercises 7.10 Exercises Problem 7.1¶
transition due to a constant applied magnetic field is given by
B 2¯h∆Ehfs B2 (7.573) for small fields, to second order the in the field B. Put in numbers appropriate for the transition
in Hz/G2. Problem 7.2 Prove the identity
(7.574)
C(A \cdot B), in terms of components. Problem 7.3 Given the homogeneous linear system from (7.551),
(7.575) where m, m′ are either integers or half-integers, −Jg \le m, m′ \le Jg, and the tridiagonal matrix Amm′ satisfies Amm′ = A−m,−m′ and P m′ Amm′ = 1, prove by induction that
Am,m+1 xm (7.576) for m > 0. Problem 7.4 (a) Consider the following expression for the ground-state dipole shift of an atom with fine structure in a linearly polarized laser field of frequency \omega, Vdip = ¯h X J′ Ω2
−
, (7.577) where J is the angular-momentum quantum number of the ground state, the sum is over the excited- state quantum numbers J′ (with an implied sum over other quantum numbers labeling relevant excited states), the frequency of the J −\rightarrow J′ transition is
¯h , (7.578) and the Rabi frequency is given in terms of the reduced dipole matrix element by
¯h , (7.579) where E(+) is the positive-rotating electric-field amplitude. Argue that this expression is correct to lowest order in the field intensity, and interpret all the factors. Note that the atom is assumed to be in the state |J m\rangle , but the shift is independent of m.
Chapter 7. Atomic Angular-Momentum Structure Note that in the case of a ground state with J = 0 or J = 1/2, the expression above simplifies, since
Vdip = ¯h X J′ Ω2 JJ′
−
. (7.580) The reduction in complexity is sensible in these cases because for such simple ground states there can be no dependence on the m quantum number. (b) Consider the following estimate for the scattering rate for the above atom–field system, Rsc = 1 X q X J′
p \GammaJ′\langle J′ m|J m −q; 1 q\rangle \times \omega \omegaJ′J 3/2
−
2 (7.581) where we sum over the (spherical) polarization index q for the scattered light, and \GammaJ′ is the total decay rate of level J′. Note that this expression assumes the atom to be in the particular state |J m\rangle , and thus the scattering rate should be averaged over all populated ground states, weighted by their steady-state population. This expression assumes that all spontaneous-scattering events return the atom to one of the J levels. Argue that this expression is correct to lowest order in the field intensity, subject to the above assumptions. (c) Argue that for an atom with hyperfine structure, the expression for the ac Stark shift should be modified to read Vdip = ¯h X F ′ Ω2 F F ′|\langle F mF |F ′ mF ; 1 0\rangle |2
−
, (7.582) where the overall hyperfine Rabi frequency is
¯h
¯h
J J′ F ′ F I , (7.583) where I is the nuclear angular momentum, and the hyperfine states F and F ′ are also labeled by J and J′, respectively, while the scattering rate becomes Rsc = 1 X q F ′′ X F ′ ΩF F ′\langle F mF |F ′ mF ; 1 0\rangle p \GammaF ′F ′′\langle F ′ mF |F ′′ mF −q; 1 q\rangle \times \omega \omegaF ′F 3/2
−
2 (7.584) where the decay-rate factor, with the proper emission phase factor, is p \GammaF ′F ′′ = p
J′ J′′ F ′′ F ′ I (7.585) for small hyperfine splittings. Note that we now sum over all possible final hyperfine levels F ′′, in the case of hyperfine-changing Raman scattering events. This expression also ignores cascading transitions (i.e., it only accounts for two-photon processes), and assumes that the final states |F ′′ mF −q\rangle are nearly degenerate with |F mF \rangle , so that \omegaF ′F \approx \omegaF ′F ′′.
7.10 Exercises Problem 7.5 The D-line fine-structure doublet in hydrogen and hydrogen-like (alkali) atoms is a transition doublet from an S1/2 ground state to a pair of excited states, P1/2 and a P3/2, with the P3/2 at higher energy.
states, in terms of the fine-structure reduced matrix element \langle J∥d∥J′\rangle . Exclude the contributions due to states outside the D line.
vanishes. Also estimate the shift of the |J, mF = +J\rangle state at this wavelength. (c) Derive an expression for the ac Stark shift for \sigma+ light for the hyperfine ground states, assuming the hyperfine splittings are negligibly small. Exclude the contributions due to states outside the D line.
vanish. What is the (intensity-dependent) shift of the other |F, mF\rangle states at these wavelengths? Problem 7.6 Work out the reduced matrix element \langle F = 1∥µ∥F ′ = 2\rangle as well as the transition matrix element
F = 0\rangle for the 6.8 GHz ground-state hyperfine ‘‘clock’’ transition in 87Rb
µ = −µB ¯h (gSS + gLL + gII) (7.586) [cf. Eqs. (7.307) and (7.315)]. Problem 7.7
value, the splitting is insensitive to first order to fluctuations in the field. Derive an expression for the field strength and the minimum splitting.