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Lectures on LightNonlinear and Quantum Optics using the Density Matrix$

Stephen C. Rand

Print publication date: 2016

Print ISBN-13: 9780198757450

Published to Oxford Scholarship Online: August 2016

DOI: 10.1093/acprof:oso/9780198757450.001.0001

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(p.370) Appendix J Irreducible Representation of Magnetic Dipole Interactions

(p.370) Appendix J Irreducible Representation of Magnetic Dipole Interactions

Source:
Lectures on Light
Author(s):

Stephen C. Rand

Publisher:
Oxford University Press

Magnetic dipole moments and magnetic fields are examples of axial vectors. Their irreducible representations and scalar products differ from those of electric dipoles and fields which are polar vectors. To extend the treatment of Appendix H to magnetic interactions, the corresponding representations for axial vectors and their products are derived here.

An axial vector is a pseudovector that transforms as the cross product of two polar vectors. Let us write

(J.1)
Vˉaxial=Vˉ1Vˉ2,

where Vˉ1 and Vˉ2 represent polar vectors. Each polar vector transforms according to the rules discussed in Appendix H, namely

(J.2)
(V1)=j=13Rij(V1)j,

(J.3)
(V2)=m=13Rkm(V2)m.

Hence the individual vectors Vˉ1 and Vˉ2 have irreducible representations of the form determined by the previous analysis to be

(J.4)
V1(1)=Vx+iVy2,

(J.5)
V0(1)=Vz,

(J.6)
V1(1)=VxiVy2,

and inverse expressions which are

(J.7)
Vx=V1(1)V1(1)2,

(J.8)
Vy=iV1(1)+V1(1)2,

(J.9)
Vz=V0(1).

(p.371) Using Eqs. (J.7)–(J.9), Vˉaxial can be written in terms of spherical tensors immediately as

(J.10)
V¯axial=V¯1V¯2=x^[ V1yV2zV1zV2y ]y^[ V1xV2zV1zV2x ]+z^[ V1xV2yV1yV2x ]=x^i2[ [ (V1(1))1+(V1(1))1 ](V0(1))2(V0(1))1[ (V1(1))2+(V1(1))2 ] ]+y^12[ [ (V1(1))1(V1(1))1 ](V0(1))2(V0(1))1[ (V1(1))2(V1(1))2 ] ]z^i2[ [ (V1(1))1(V1(1))1 ][ (V1(1))2+(V1(1))2 ][ (V1(1))1+(V1(1))1 ][ (V1(1))2(V1(1))2 ] ]=i{ (V1(1))1(V0(1))2ε^(V1(1))1(V0(1))2ε^+(V0(1))1(V1(1))2ε^+(V0(1))1(V1(1))2ε^+ }i{ (V1(1))1(V1(1))2ε^0(V1(1))1(V1(1))2ε^0 }.

If we now simplify the notation by relabeling the vectors Vˉ1=Aˉ and Vˉ2=Bˉ, Eq. (J.10) can be written

(J.11)
Vˉaxial=iA1B0εˆA1B0εˆ+A0B1εˆ+A0B1εˆ++iA1B1+A1B1εˆ0.

Upon grouping the coefficients for each spherical basis vector, it becomes evident that this expression is already in irreducible form. Introducing the definitions

(J.12)
C±B0A±,

(J.13)
D±A0B±,

(J.14)
F±AB±,

one obtains

(J.15)
Vˉaxial=AˉBˉ=iC+εˆCεˆ+D+εˆ+Dεˆ++iF+Fεˆ0=i(C+D+)εˆ(CD)εˆ++iF++c.c.εˆ0.

The last result makes use of the replacement F±=AB±=(A±)(B)=(F), valid for real vectors Aˉ and Bˉ. The irreducible form of the vector can therefore be written as

(J.16)
Vˉaxial=iV+εˆiVεˆ++V0εˆ0,

where V±(C±D±) and V0(iF++c.c.) is entirely real. Because Vˉaxial is defined here in terms of a cross product, there is a sign ambiguity associated with whether its definition should be AˉBˉ or BˉAˉ=AˉBˉ. Consequently the irreducible representation has an overall sign (p.372) ambiguity that must be determined by other considerations, such as the direction of positive flow of energy associated with an axial field or the convention for positive energy itself.

As an example of the determination of sign for axial vectors, consider the magnetic flux density Bˉ. Given the irreducible form of Eˉ in Eq. (H.49) for an x-polarized wave traveling in the +zˆ direction, the overall sign of the irreducible representation for Bˉ must be chosen to ensure that the energy flow along +zˆ is positive. This requires that the Poynting vector Sˉ=EˉHˉ points along +zˆ. For this to be the case, Bˉ must have the same sign and form as the transverse components of the axial vector in Eq. (J.16):

SˉE+εˆ+Eεˆ+iB+εˆiBεˆ+.

Simplifying the cross products by using the results

εˆ±εˆ±=0,εˆ±εˆ=εˆεˆ±=±izˆ,

one can easily verify that Sˉ+zˆ.

Equation (J.16) displays the tensorial form suitable for representing axial vectors such as magnetic dipole moments and magnetic fields. With this result in hand, the magnetic field interaction Hamiltonian which is the scalar product of two axial vectors can readily be obtained. First, note that

(J.17)
AˉaxialBˉaxial=±iA+εˆ+iAεˆ++A0εˆ0iB+εˆ+iBεˆ++B0εˆ0.

The ± sign on the right-hand side reflects the sign ambiguity mentioned previously. Since different considerations may determine the overall sign of each axial vector comprising the product, the product itself may in principle be positive or negative. Next, Eq. (J.17) can be simplified by evaluating the scalar products of the basis vectors.

(J.18)
εˆ±εˆ±=εˆ0εˆ±=0

(J.19)
εˆ±εˆ=1

(J.20)
εˆ0εˆ0=1

Upon substitution of Eqs. (J.18)–(J.20) into Eq. (J.17) one finds

(J.21)
AˉaxialBˉaxial=A+B+AB++A0B0+c.c.A0B0+c.c..

In electric dipole interactions of light with isolated atoms, the induced dipole is parallel to and in the same direction as the electric field. However, magnetic dipoles induced by light are anti-parallel to the magnetic flux density Bˉ. Hence explicit forms for the optical magnetic field and its induced dipole are

(J.22)
Bˉ=12(iB+εˆiBεˆ+)ei(ωtkz)+c.c.,

(J.23)
μˉ(m)=12(iμ+(m)εˆ+iμ(m)εˆ+)ei(ωtkz)+c.c..