Multipole Expansions in Radiation Theory of Quantum Systems

Multipole expansions in quantum radiation theory

M. Ya. Agre National University of “Kyiv-Mohyla Academy”, 04655 Kyiv, Ukraine E-mail: [email protected]

With the help of mathematical technique of irreducible tensors the multipole expansion for the probability amplitude of spontaneous radiation of a quantum system is derived. It is shown that the found series represents the total radiation amplitude in the form of the sum of radiation amplitudes of electric and magnetic 2l-pole (l = 1, 2, 3,…) photons. All information about the radiating system is contained in the coefficients of the series which are the irreducible tensors being determined by the current density of transition. The expansion can be used for solving different problems that arise in studying electromagnetic field-quantum system interaction both in long-wave approximation and outside its framework.

PACS number(s): 31.15.ag, 32.70.-n, 32.90.+a

I. INTRODUCTION aligned (i.e., spin-polarized) atoms in long-wave approximation1. The known multipole expansion for Multipole expansions in classical electrodynamics the probability amplitude of a quantum transition are the seriess for potentials or field strengths, where accompanied by radiation of photon with definite the coefficients are the irreducible tensors dependent polarization and direction of motion is based on on the charge or current distributions of the system that representation of function e expik  r, where e is is the source of the field. In every term of the multipole the vector of right(left)-hand circular polarization and k series the irreducible tensors have been convolved in is the wave vector (i.e., the momentum of the photon in scalars (for the scalar potential) or vectors (for the the units of  ), in the form of the series in the vector potential or field strength) with the irreducible spherical vectors or the fields of electric and magnetic tensors specifying the corresponding fields (called the multipoles (see, e.g., [1]). The coefficients of the series multipole fields [1]). L The expression for multipole expansion in are proportional to Wigner’s DM ,1,,0 -function, electrostatics can be found in any textbook on classical where  and  are the azimuthal and polar angles electrodynamics (see, e.g., [2]). In that case the specifying the direction of the vector k. In the irreducible tensors specifying the source are the well interesting for us case of multipole radiation by spin- known electric multipole moments of the charge polarized atoms this expansion led to cumbersome system. The structure of multipole expansion in formulas that essentially hampered the study. That is magnetostatics is more complicated owing to the vector why we were forced to seek for other form of multipole nature of the magnetic field potential. And no wonder expansion. As a result, the multipole series had been that one can only find the first term of the expansion in derived which is highly efficient, in our opinion, in the textbook which is determined by the field of solution of different problems of quantum radiation magnetic dipole and contains the magnetic (magnetic theory both in long-wave approximation and outside its dipole) moment of the the current system as the framework. coefficient. The total multipole expansion for magnetic Note that analyzing the long-wave approximation in field of a current system containing magnetic multipole quantum radiation theory the authors of known moments has been recently derived by us [3]. textbooks on quantum electrodynamics [6,7] do at all The different forms of multipole expansions in without the multipole expansion for the amplitude of classical radiation theory are known [1,4]. Note that the photon radiation with definite momentum and expansion for the potential of radiation field given by polarization. On the first stage they limit themselves to Dubovik and Cheshkov [4] is considerably more derive the total radiation probability of electric 2l-pole convenient for applications. It has allowed to (El) or magnetic 2l-pole (Ml) photons. Considering accomplish the correct analysis of long-wave further the multipole radiation of given direction and approximation in the radiation theory and to take into polarization the authors [6] actually construct in account the contribution of toroid multipole moments artificial way the corresponding term of the multipole arising in higher orders of the approximation [4]. expansion, where the Wigner’s D-function arises again. Multipole expansion has also to work effectively in In such way one cannot furthermore take into account the quantum radiation theory, especially under the possible interference of Ml and El', l' l 1 , consideration of transitions between the states with definite values of angular momentum of the radiating system when the angular momentum selection rules take place. We needed such expansion in theoretical 1 The results of the study have been presented at ECAMP11 study of multipole radiation of light by oriented and [5]. 1

amplitudes which have, generally speaking, the same  order in the long-wave approximation. gl (x)  J l1 2 (x) , (4) The paper is organized as follows. In Section II the 2x mathematical technique for deriving multipole expansion in quantum radiation theory is stated and the J (x) is Bessel function and Ylm is spherical multipole series for the radiation amplitude is found. function. Substituting this expansion in equation (2) Several equivalent forms of the multipole series are gives the following series for I : given in Sec. III. The possible applications of the fi derived expansion are discussed in the final section of the paper. l  ˆ I fi  4   i  gl (kr)j fi (r)Ylm (rˆ)drYlm (k) . (5) l, m

II. MULTIPOLE EXPANSION FOR THE To derive the multipole expansion we take AMPLITUDE OF SPONTANEOUS advantage of mathematical technique of angular RADIATION momentum algebra (i.e., the technique of irreducible tensors which is based on the theory of irreducible The probability of spontaneous radiation per unit representation of the rotation group) and separate out in time of a photon with frequency , unit polarization the series (5) and respectively in the expansion of the vector e and wave vector k in spatial angle element d radiation amplitude (1), (3) the irreducible tensors is determined by the well known expression (see, e.g., being determined by the current density of transition. [6]) We use further a number of standard definitions and designations of the technique [9] which we give here  2 for the convenience. The spherical components dWfi  V fi d , a , m  0, 1 , of arbitrary vector a forming the first 2c3 m rank irreducible tensor are expressed through its Cartesian components as follows: where

1  a0  az , a1   ax  iay . V fi  e   j fi (r)exp ik  rdr . (1) 2

The radiating system passes with that from the initial One can compose of two irreducible tensors Alm and state i into the final state f . The value V fi (1), Bl'm' having respectively the ranks l and l' the where j fi is the current density vector of the transition, irreducible tensors (the tensor products of the we shall call the probability amplitude of spontaneous irreducible tensors) of the ranks radiation and shall seek the multipole expansion just for it. Although the explicit structure of the current L  l  l', l  l' 1,..., l  l' density vector is not used in the derivation of multipole expansion, the corresponding formulas are given in after the rule Appendix A to complete the statement. Let us introduce the vector LM Al  Bl'LM  Clml'm' Alm Bl'm' , (6) m, m' I  j (r)expik rdr (2) fi  fi LM and write the radiation amplitude (1) in the form of where Clml'm' is the Clebsh-Gordan coefficient. If scalar product: l  l' , the scalar (the zero rank irreducible tensor) is also composed. The scalar is proportional to the scalar product of two tensors that will be designated by V  e I . (3) fi fi parentheses:

Let us also use the known expansion [8] for the m exponential function Al  Bl   1 Al,mBlm m l  l  1 2l 1Al  Bl 00 . (7) exp ik r  4 il g (kr)Y (rˆ)Y  (kˆ) ,     l lm lm l0ml In particular, the zero rank tensor composed of two

vectors is proportional to the ordinary scalar product of where rˆ  r r and kˆ  k k are the unit vectors, the vectors:

1 we find as a result the following series for the a  b00   a  b . (8) 3 probability amplitude of spontaneous radiation V fi (3):

With the help of the technique analogous of that,  l 2L 1 (l) ˆ  used by us in derivation of multipole expansion in V fi  1 AL  Yl (k) e . (11) 3 1 magnetostatics [3], one can separate out the irreducible l0 L tensors dependent on the current density of transition in the expression (5). To do it we note that the spherical Here the irreducible tensors components of the vector I fi (5) contain the A(l)  4  il g (kr)Y (rˆ)  j (r) dr (12) construction of the following form: LM   l l fi LM

j 1m'Y (rˆ)Y (kˆ) . (9) are introduced which are determined by the current  fi m l,m' lm' m' density of transition and thus specify the radiating quantum system. Note also that the index L in the internal sum (11) satisfies the triangle condition Here j  is the spherical component of the current fi m l 1  L  l 1 (i.e., L  l, l 1 if l  0 and L 1 density vector of transition and the identity l  0  m if ) in accordance with angular momenta addition Ylm  1 Yl,m for the spherical function has been rule. taken into consideration. Taking also into account the The irreducible tensors A(l) (12) can also be definition of the scalar product of two irreducible LM tensors (7), one can write the expression (9) in the form expressed in terms of spherical vectors of the tensor product in the designations (6), Y (rˆ)  C LM Y (rˆ)e (13) LlM  lm11m20 lm1 m2 l ˆ m1m2 1 2l 1j fi Yl (rˆ) Yl (k0 , 1m

[1,7]. Here the unit vectors e , m  0,1, are and it is sufficient to change by standard way [9] the m coupling scheme of angular momenta in this expression determined in terms of unit vectors ex , e y and ez of to achieve our object. In given case the tensor product Cartesian coordinate system: is simple, thus one can do without 6j-symbols changing the coupling scheme. Really, with the help of relation, 1 which is inverse to (6), we find that e0  ez , e1   ex  ie y . 2 j  Y  (rˆ)  1m'j  Y (rˆ) fi m lm' fi m l,m' Taking into account the expression a  a  e for m m  1m' C LM j Y (rˆ) . the spherical components of the vector a, we find the  1ml,m' fi l LM L, M irreducible tensor A(l) (12) in equivalent form LM Making further use of symmetry properties of Clebsh- Gordan coefficients leading to the identity (l) l ˆ ALM  4i  gl (kr)YLlM (r) j fi (r)dr . (14)

2L 1 The last step to derive the multipole expansion C LM  1L1m' C1m , 1ml,m' 3 LMlm' consists in changing the coupling scheme of angular momenta in the scalar products which enter the series one reduces the expression (9) to the form (11). Making use for that the formula (B2) of Appendix B, we derive from (11) the multipole expansion of the radiation amplitude (1): j Y  (rˆ)Y (kˆ )  fi m  lm' lm' m'  l1 V  A(l)  e Y (kˆ) . (15) L1 2L 1 ˆ fi   L  l L   1 j fi Yl (rˆ) Yl (k) . (10) l0L l1 3 L 1m L

Let us also give the multipole series for the value, Substituting the identity (10) in (5) and taking into complex conjugate of the radiation amplitude (1): consideration the relation

 l1L Vfi  e jif (r)expik rdr , j fi Yl (rˆ)  1 Yl (rˆ)  j fi  ,  LM LM

 where the relation j fi  jif (see Appendix A) has been ˆ 3 ˆ Y1m (k)  km . taken into consideration. Comparing this expression 4 with (1) shows that the multipole expansion for V  fi Therefore, taking the identity (8) into consideration, we can be found from (15) by substitutions of find that ˆ  ˆ e, k and jif for e , k and j fi respectively.  ˆ 1  ˆ ˆ l ˆ e Y1(k)   e k  0 Considering also that Ylm(k)  1 Ylm(k) , one 00 4 derives the multipole expansion in the following form: in accordance with the transversality condition for the  l1 radiation field. The pointed reason allows us to write  l (l) ˆ Vfi   1 BL e Yl (k)L , the expansion (15) in the following form: l0L l1  L1 (l) (l)  ˆ (18) where the irreducible tensors B are determined by Vfi   AL e Yl (k)L . LM L1 lL1 the equation (12) or (14) with the substitution of

jif for j fi (i.e., with the rearrangement of the indices The irreducible tensors composed of the polarization of the current density vector). vector and the spherical function in (18) (see definition (6)) can be expressed in terms of the spherical vectors (13): III. MULTIPOLE EXPANSION AND RADIATION OF El AND Ml-PHOTONS  ˆ l1L ˆ  e Yl (k)Lm  1 Yl (k) e Lm

l1L  ˆ It is known that the angular dependence of the photon  1 e  YLlm (k) . (19) wave function in the states of electric (El)-type (the angular momentum quantum number is equal to l and The scalars of three types enter the expansion (18) of parity is equal to 1l ) is determined in momentum the radiation amplitude. Let us introduce the following designation for one of them: representation by linear combination of two spherical vectors (13): (M ) (l)  ˆ Vl  Al e Yl (k)  l (E) l 1 (l)  ˆ Y (kˆ )  Y (kˆ )  Al e  Yll (k) , (20) lm 2l 1 l,l 1,m

l where  Y (kˆ ) , (16) 2l 1 l,l 1,m (l)  ˆ m (l)  ˆ Al e Yll (k) 1 Al,me Yllm(k), whereas for the states of magnetic (Ml)-type (the m angular momentum quantum number is equal to l and l1 and the identity (19) and the definition of scalar parity is equal to 1 ) the angular part of the product (7) are taken into account. The spherical vector photon wave function is determined by spherical vector entering the expression (M ) (20) is directly alone: Vl connected with the photon wave function in the state of (M) ˆ ˆ Ml-type, thus one can write using the designation (17) Ylm (k)  Yllm(k) (17) that

[6,7]. Let us prove that the series (15) found by us can V (M)   A(l)  e Y(M)(kˆ) . (21) be considered as an expansion of the amplitude of a l  l  l  photon radiation in direction kˆ with polarization The scalar product of the polarization vector and the vector e in the corresponding radiation amplitudes of spherical vector (17) in equation (21) is proportional to electric-type and magnetic-type photons. projection of the photon Ml-state onto the state with l 1 and L  0 Note first of all that the term with is definite polarization and momentum that allows of absent in the series (15). Really, the spherical function (M ) l ˆ interpreting Vl as the contribution of magnetic 2 - Y1m(k) is proportional to the spherical components of pole photon to the total radiation amplitude [6]. ˆ the unit vector k  k k , Let us further write the expansion of the radiation amplitude (18) in the form

 El and Ml-photons. Such interpretation gives the (M ) (E) Vfi  Vl Vl  (22) additional argumentation to the name “multipole l1 expansion of the radiation amplitude”: the expansion in the contributions of the photons of different and ascertain the physical meaning of the amplitude multipolarity. One can also prove that the irreducible (E) (l) Vl , tensor nature of the coefficients ALM (12), (14) of the multipole expansion leads to the angular momentum (E) (l1)  ˆ Vl  Al e Yl1(k)l  selection rule: the tensor is only nonzero (l1)  ˆ provided , where and  Al e Yl1(k)l  J f  Ji  L  Ji  J f J i J f

 A(l1) e  Y (kˆ ) are the angular momentum quantum numbers in the l l,l1 initial and final states of the radiating system. (l1)  ˆ Moreover, the parity selection rule also arises:  Al e  Yl,l1(k). (23) l1 Pf  1 Pi . The given selection rules just For that purpose, making use of (16) and the expansion correspond to the angular momentum and parity ˆ ˆ of the longitudinal vector kYlm(k) in the spherical conservation laws in the radiation of El or Ml-photons. vectors, Let us also note that in the long-wave approximation ka  1 (here a determines the linear dimensions of the radiating system) expanding the spherical Bessel ˆ ˆ l ˆ l 1 ˆ kYlm (k)  Yl,l1,m (k)  Yl,l1,m (k) function (4) in the Taylor series and confining 2l 1 2l 1 ourselves to the first term,

[7], we express the spherical vectors entering (23) in xl terms of (E) and the longitudinal vector: gl (x)  , Ylm 2l 1!!

l 1 (E) (l) Y (kˆ )  Y (kˆ ) we find from (12) or (14) that ALM has the order l,l1,m 2l 1 lm kal . Furthermore, in the first nonvanishing order of l ˆ ˆ (l)  kYlm (k) , the long-wave approximation proves to be 2l 1 AlM (24) proportional to well known in the theory of multipole ˆ l (E) ˆ radiation [6,7] magnetic 2l-pole moment of transition Yl,l1,m (k)  Ylm (k) 2l 1 (l1) and AlM  to electric one. l 1  kˆY (kˆ ) . Note finally that having eliminated the spherical 2l 1 lm ˆ vector Yl,l1,m(k) from (23) one can represent the

(E) Substituting (24) in (23) (the transversality condition radiation amplitude of El-photon Vl (25) entering for the radiation field has to be taken into account) the multipole expansion (22) in more simple form. It is leads to the equivalent expression for V (E) , (E) ˆ l sufficient for that to express the vector Ylm (k) from the first identity (24), to substitute it in (25) and to use again the transversality condition of the radiation field. (E)  l 1 (l1) V   A As a result we derive the equivalent expression for l  2l 1 l  (E) Vl : l    A(l1)  e Y(E) (kˆ )  , (25) l   l  (E)  ˆ 2l 1  Vl  Dl e  Yl,l1(k)   D e Y (kˆ )  , (26) which clearly shows that this term determines the l l-1 l contribution of the electric 2l-pole photon to the where radiation amplitude . V fi (M ) Thus, the series (22) found by us, where Vl is (l1) l (l1) DlM  AlM  AlM . (27) (E) l 1 determined by the equation (21) and V  by the l equation (25), really represents the total amplitude of As appears from the above, the tensor DlM (27) has photon radiation in the direction kˆ with the l1 l polarization e as the sum of the radiation amplitudes of the order ka and is proportional to the electric 2 - 5 pole moment of transition in the first nonvanishing where the terms in the parenthesis of the sum over l order of the long-wave approximation. The next term l (E) 0 have the order ka and V has the order ka . l1 1 in the expansion of DlM has the order ka and, as The term determines the amplitude of dipole it can be proved, is proportional to so-called toroid 2l- pole moment of transition, which is determined by the radiation, V (M ) is the amplitude of magnetic dipole expression for 2l-pole toroid moment of classical 1 radiation, (E) is the amplitude of quadrupole current j  j fi introduced by Dubovik and Cheshkov V2 [4]. radiation and so on. In the first nonvanishing order of the long-wave approximation the series will contain no more than two terms owing to the selection rules, and IV. CONCLUSIONS (M ) (E) therefore Vl and Vl1 amplitudes which have, generally speaking, the same order can only interfere. It In the present paper making use of mathematical has been shown by us [5] that in accordance with technique of irreducible tensors the effective, in our above-mentioned the interference of the amplitudes is opinion, multipole expansion for the probability manifested only provided the so-called state multipoles amplitude of spontaneous radiation (1) has been [10] specifying the orientation and alignment of the derived. The expansion is written down in the form of radiation atom are nonzero, i.e., the atom is spin- the series (15), (18) or (22). The last form (22) as the l polarized. sum of the radiation amplitudes of electric 2 -pole Note in conclusion that the derived here multipole (E) l expansion can prove to be useful in the study of photons Vl (26) and magnetic 2 -pole photons photoeffect outside the dipole approximation and also (M ) Vl (20) is the most convenient for different in the theory of multiphoton transitions. In the last case applications. the expressions of the form (1) enter as the structural The multipole series contains of course the infinite elements the so-called composite matrix elements [11]. number of terms, but in case of transitions between the We would like to hope that the multipole expansion states of the radiating system with definite values of will find an application for solving outlined here and angular momentum and parity the angular momentum other problems that arise in studying electromagnetic and parity selection rules keep only few terms in the field-quantum system interaction. series. Note that the interference terms of the form

ReV (M)V (E) , ReV (M )V (M ) and (E) (E)  l l'   l l'  ReVl Vl'  APPENDIX A: CURRENT DENSITY OF arising in the expression for the probability of TRANSITION radiation, which contain the products of the amplitudes of different multipolarities allowed by the selection In case of relativistic electron the expression for the rules, disappear in the total radiation probability. current density vector of transition in the radiation Really, after summation over two independent photon amplitude (1) has the following form (see [6,7]): polarizations the products of two different spherical vectors (see (20), (25) and (16)), integrated over all  j  ec ˆ  , (A1) directions of kˆ , turn into zero in consequence of fi f i orthogonality condition for the spherical vectors. The interference terms will only contribute to the angular where e is the electron charge, i, f is the wave distribution (and polarization) of the radiation provided function (the Dirac bispinor) of the initial and final that the initial state of the radiating system is not states, spherically symmetric, for example, the radiating atom is spin-polarized. For spherically symmetric initial state  the angular distribution of the radiation is also  0 ˆ  ˆ    , spherically symmetric and thus only differs by the  ˆ   0  multiplier 4 1 from the total radiation probability   in which the interference of the amplitude is not ˆ manifested.   are the Pauli spin matrices. For the nonrelativistic The found here multipole expansion was used by us electron the expression (A1) turns into following one in the theoretical study of peculiarities of multipole [7]: radiation of light by spin-polarized (i.e., oriented and aligned) atoms in long-wave ka  1 approximation ie   j fi   f i  if  [5]. To analyze the long-wave approximation it is 2m convenient to write down the series (22) in the form  ˆ  c f  i . (A2)  (E) (M ) (E) Vfi V1  Vl Vl1 , l1

 e  APPENDIX B: CHANGING THE COUPLING Here ˆ  ˆ is the intrinsic magnetic moment 2mc SCHEME IN THE SCALAR PRODUCT operator of the electron, m is the electron mass,  i, f One can change the coupling scheme in the scalar is the wave function (the spinor taking into product A  B   a without 6j-symbols. Let us consideration the electron spin) of the initial and final L l 1 states. write down the scalar product, taking into account the One can also derive the equation (A2) for the current definitions (6) and (7), in the form of the sum: density vector of transition in nonrelativistic approximation making use of the expression for the AL  Bl 1 a operator of nonrelativistic electron-electromagnetic  C1m A B a 1 m . (B1) field interaction in the form  Lm1lm2 Lm1 lm2 m   m,m1,m2 e e2  ˆ ˆ ˆ ˆ ˆ 2 ˆ ˆ (A3) Making further use of the symmetry property for the  p  A  A p 2 A   A  , 2mc 2mc Clebsh-Gordan coefficient, where pˆ  i is the momentum operator of electron 1m lm2 3 L,m1 CLm lm  1 C , and Aˆ is the vector potential of quantized 1 2 2L 1 1,mlm2 electromagnetic field [6]. In the first order of perturbation theory the quadratic in Aˆ term entering and substituting this identity in (B1), where (A3) does not contribute, and the known expression for m  m1  m2 , we find the sought relation: spontaneous radiation probability [6] (see Sec. II of the paper) is derived, where the current density vector of transition entering the probability amplitude (1) is l 3 AL  Bl  a 1 determined by the formula (A2). The expression (A2) 1 2L 1 for the current density vector of transition is obviously AL a  Bl  . (B2) generalized in case of N-electron system: L

N Note that the relation (B2) with L  l 1 turns into  ie   obvious identity for the scalar triple product of the j fi (r)    f ai  iaf  a1  2m vectors.  ˆ  ca  f ai  (r  ra )dr1dr2...drN .

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[1] M. E. Rose, Multipole Fields (New York: Wiley, [7] A. I. Akhiezer and V. B. Berestetskii, Quantum 1955). Electrodynamics (New York: Interco. Publ., 1965). [2] L. D. Landau and E. M. Lifshitz, Classical Theory of [8] L. D. Landau and E. M. Lifshitz, Quantum Mechanics Fields (Oxford: Butterworth-Heinemann, 1994). (Oxford: Pergamon Press, 1977). [3] M. Ya. Agre, Phys. Usp. 54, 167 (2011). [9] D. A. Varshalovich, A. N. Moskalev and V. K. [4] V. M. Dubovik and A. A. Cheshkov, Sov. J. Part. Nucl. Khersonskii, Quantum Theory of Angular Momentum 5, 318 (1974). (Singapore: World Sci. Publ., 1988). [5] M. Ya. Agre, in Proceedings of the 11th European [10] K. Blum, Density Matrix Theory and Applications Conference on Atoms, Molecules and Photons, Aarhus, (New York and London: Plenum Press, Second Denmark, 24th-28th June 2013, poster Mo-T5-2 Edition, 1996). (www.ecamp11.au.dk/poster-sessions/monday-poster- [11] N. L. Manakov, V. D. Ovsiannikov and L. P. Rapoport, session/). Phys. Rep. 141, 320 (1986). [6] V. B. Berestetskii, E. M. Lifshitz and L. P. Pitaevskii, Quantum Electrodynamics (Oxford: Pergamon Press, 1982).