Spin-Statistic Selection Rules for Multi-Equal-Photon Transitions in Atoms: Extension of the Landau-Yang Theorem to Multiphoton Systems
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Spin-Statistic Selection Rules for Multi-Equal-Photon Transitions in Atoms: Extension of theLandau-Yang Theorem to Multiphoton Systems
T. Zalialiutdinov , D. Solovyev , L. Labzowsky , and G. Plunien Department of Physics, St.Petersburg State University,Ulianovskaya 1, Petrodvorets, St.Petersburg 198504, Russia Petersburg Nuclear Physics Institute, 188300, Gatchina, St. Petersburg, Russia Institute f¨ur Theoretische Physik, Technische Universit¨at Dresden, Mommsenstrasse 13, D-10162, Dresden, Germany
We establish the existence of spin-statistic selection rules (SSSR) for multi-equal-photon transitions in atomicsystems. These selection rules are similar to those for systems of many equivalent electrons in atomic theory.The latter ones are the direct consequence of Pauli exclusion principle. In this sense the SSSR play the role ofthe exclusion principle for photons: they forbid some particular states for the photon systems. We establishedseveral SSSR for few-photon systems. 1) First rule (SSSR-1): two-equivalent photons involved in any atomictransition can have only even values of the total angular momentum J . This selection rule is an extension of theLandau-Yang theorem to the photons involved in atomic transitions. 2) second rule (SSSR-2): three equivalentdipole photons involved in any atomic transition can have only odd values of the total angular momentum J = 1 , . 3) third rule (SSSR-3): four equivalent dipole photons involved in any atomic transition can have onlyeven values of the total angular momentum J = 0 , , . We also suggest a method for a possible experimentaltest of these SSSR by means of laser experiments. I. INTRODUCTION
Landau-Yang theorem [1], [2], together with the Bose-Einstein condensation can be viewed as the most spectacular confirma-tions of Bose-Einstein statistics for integer-value-spin particles. Landau-Yang theorem forbids two photons to participate in anyprocess that would require them to be in a state with total angular momentum one. In the high-energy physics an evident exampleis the prohibition the two-photon decay for the neutral spin-one Z -boson. The same concerns also the annihilation decay oforthopositronium (also spin-1 state). However both this decays are also forbidden by charge-parity conservation law. Positro-nium presents a real neutral system (it coincides with itself after charge conjugation), therefore it possesses a definite chargeparity [3], connected with the total spin value S : parapositronium ( S = 0 ) is charge-positive and orthopositronium ( S = 1 ) ischarge negative. Since the charge parity of a system of N γ photons equals ( − N γ [4], parapositronium can not decay into anodd number of (not necessarily equivalent) photons and orthopositronium can not decay into an even number of photons. The Z -boson as a charge-parity-negative particle can not decay into an even number of photons.A similar situation exists in atomic physics. Already the early calculations of the two-photon decay of the singlet S ≡ (1 s s ) S and triplet S ≡ (1 s s ) S excited states of the He-like ions to the ground S ≡ (1 s ) S state revealed thecrucial difference in the photon frequency distributions in both cases [5]. The decay probability for the triplet case tends to zerowhen the frequencies of the emitted photons are equal (see Fig. 1). Later these conclusions were confirmed within the fullyrelativistic calculations (see, for example [6]).Unlike the positronium two-electron ions do not possess a definite charge parity. The neutral He atom also is not a real neutralparticle and also does not have a definite charge parity. Therefore, only the spin-statistic properties can be responsible for thisspecial selection rule. Its connection with the Landau-Yang theorem and hence with Bose-Einstein statistics was first emphasizedin [7] where an experimental limit for the violation of the Spin-Statistic Theorem (SST) was obtained. Recently this limit wasimproved in [8]. A confirmation of the Spin-Statistic-Selection Rules (SSSR) for two-equal-photon transitions in atomic physicswas obtained in [9] where it was demonstrated that the frequency distributions for the transition rates S → S + 2 γ with E , M , E photons correspond to the type Fig. 1 (right panel), while the distributions for the same transition with E M , E M (nonequivalent) photons belong to the type Fig. 1 (left panel). Finally in [10] the SSSR-suppressed two-photontransition induced either by the hyperfine interaction or by an external magnetic field were investigated. The hyperfine interactionchanges the value of the total angular momentum J e of electron state in an atom and in an external magnetic field the angularmomentum, in principle, is not conserved. Therefore in both cases the prohibition of the atomic transition J e = 1 → J e = 0 with two equivalent photons can be relaxed.We formulate the SSSR for the multi-equal-photon atomic transitions which present an extension of the Landau-Yang theoremas follows:1) SSSR-1: Two equivalent photons involved in any atomic transition can have only even values of the total angular momentum J ,2) SSSR-2: Three equivalent dipole photons involved in any atomic transition can have only odd values of the total angularmomentum J = 1 , ,3) SSSR-3: Four equivalent dipole photons involved in any atomic transition can have only even values of the total momentumvalues J = 0 , , .Note that SSSR-2, SSSR-3 do not hold, in general, for the photon multipolarity j > (see sections IV, V).The Landau-Yang theorem in case of atomic transitions corresponds to SSSR-1 for number of photons N γ = 2 and multipo-larity J = 1 involved in the transitions J e = 1 → J e = 0 or J e = 0 → J e = 1 via the emission or absorption of two photons.In principle, the original Landau-Yang theorem does not require that the two photons are equivalent; it is assumed only thatthey are not propagating in the same direction, so that one can choose the frame of reference where the center-of-inertia for twophotons is at rest. Moreover, originally Landau-Yang theorem was applied to the processes when the initial particle disappearsafter decay and is converted to a system of two photons. From the energy-momentum conservation law it follows that thesetwo photons should be collinear (opposite direction) and should have equal frequencies. The total angular momentum and spaceparity for these photons were defined in the center-of-inertia rest frame for the two photons. This frame coincides with the restframe for decaying particle which disappears after decay ( Z -boson, annihilation decay of orthopositronium). For introducingof the SSSR for two-photon atomic decays we employ a different picture (see section II). We define the equivalent photons asthe photons having the same frequency, angular momentum and parity in the rest frame of a decaying atom. In atomic processesthe decaying particle (atom) does not disappear in the decay process, therefore the rest frame of an atom does not coincide withthe center-of-inertia rest frame for two emitted photons. The difference in the definitions of the total angular momenta and spaceparity in both reference frames is important. The value of the photon orbital angular momentum and therefore the value of thephoton total angular momentum depends on the choice of the reference frame.For all these reasons the SSSR-1 which will be proven explicitly in section II for the two-photon decay does not fully coincidewith the results of the standard Landau-Yang theorem for the two-photon decay of disappearing particle [1]-[4]. In the standardformulation a two-photon system after decay of an initial (disappearing) particle can have any total angular momentum J , except J = 1 .This result was obtained in the center-of-inertia rest frame for two photons and is not immediately applicable to an arbitrarytwo-photon decays in He-like ions. According to the SSSR-1 no odd values for the total angular momentum are allowed. Thusour SSSR-1 in case of two equivalent photons is more restrictive than one could expect from the standard formulation of theLandau-Yang theorem.The standard formulation of Landau-Yang theorem can not be applied also to laser photons: these photons are collinear buthave the same direction and the center-of-inertia rest frame does not exist in this case. With our approach we can consider theabsorption of the laser photons in the rest frame of an absorbing atom or ion. The SSSR will work for the absorption transitionas well. Though the incident laser photons do not have certain total angular one-photon momentum in the rest frame of anabsorbing atom, the fixing of the photon frequency usually defines the initial and final atomic states in the process of absorption.The total electron angular momenta J e of the initial and final atomic states then define the total angular momentum for a photonwhich can be absorbed in this particular transition. In case of multiphoton absorption the total angular momenta of the absorbedphotons are defined by the vector coupling scheme.Apart from the general proof of SSSR-1 in case of two-photon decay we support this proof by the direct evaluation of thefrequency distribution of transition rate in section II for the transition J e = 3 → J e = 0 + 2 γ ( E , i.e. for the case when the2-photon system has a total angular momentum J = 3 . The transition rate tends to zero when the frequency of both photons areequal.In section III we search for the analogy between the values of a total angular momentum for the system of equivalent photons(allowed by the SST) and the values of a total angular momentum for the system of equivalent atomic electrons also allowedby SST, only with the Fermi-Dirac statistics. Section IV contains a detailed derivation of the 3-photon transition rate in Highly-Charged Ions (HCI) and the proof of SSSR-2 for the particular case of 3-photon transition J e = 2 → J e = 0 + 3 γ ( E .The results for the other 3-dipole-photon transitions are also included in this section. Section V contains the analytic proof ofthe SSSR-2 with equations for the Coefficients of Fractional Parentage traditionally used in the theory of atomic spectra forconstructing the wave functions for equivalent electrons. In section VI the analytic proof is given dor SSSR-3 and the particularexamples are provided for the 4-photon transitions which support this proof. Section VII contains discussion and outlook. Apossible experimental test of the results presented in this paper is briefly discussed. II. PROOF OF THE SSSR-1 FOR TWO-PHOTON TRANSITIONS
We describe a photon emitted or absorbed by an atom by wave functions ~A ( s ) ∗ ωjm ( ~k ) or ~A ( s ) ωjm ( ~k ) in momentum space, respec-tively [3]. Here ω is in the frequency, jm are the angular momentum and its projection and index s denotes the type of thephoton - electric ( s = E ) or magnetic ( s = M ) . The type of the photon together with j value determines the parity P of thephoton: P = ( − j +1 for s = E or P = ( − j for s = M ; the argument ~k denotes the photon momentum. We denote also thevector component of the photon wave function as (cid:16) ~A ( s ) ωjm (cid:17) i where the index i = 1 , , . For the real transverse E, M photonsthe index i takes only two values i = 1 , , while i = 3 corresponds to the longitudinal component which is absent for E, M photons. Each component of these wave functions is the eigenstate of the total one-photon angular momentum operator b ~j (cid:16) ~A ( s ) ωjm (cid:17) i = j ( j + 1) (cid:16) ~A ( s ) ωjm (cid:17) i , (1) b ~j z (cid:16) ~A ( s ) ωjm (cid:17) i = m (cid:16) ~A ( s ) ωjm (cid:17) i . (2)The two-photon wave function for two photons with the same frequency can be constructed as a symmetrized tensor product (cid:16) Φ s s ωJM ( ~k ~k ) (cid:17) i ,i = N X m m C JMj m j m (cid:20)(cid:16) ~A ( s ) ωj m ( ~k ) (cid:17) i (cid:16) ~A ( s ) ωj m ( ~k ) (cid:17) i + (cid:16) ~A ( s ) ωj m ( ~k ) (cid:17) i (cid:16) ~A ( s ) ωj m ( ~k ) (cid:17) i (cid:21) . (3)Here indices 1, 2 correspond to the two photons, JM are the total angular momentum for the two-photon system and itsprojection; one of the standard notations for the Clebsh-Gordan coefficient C JMj m j m is used. An explicit expression for thenormalization factor N is not necessary for our purposes. The components of the tensor wave function (3) are the eigenstates ofthe total two-photon angular momentum operator (cid:16)b ~j + b ~j (cid:17) (cid:16) Φ ( s s ) ωJM (cid:17) i i = J ( J + 1) (cid:16) Φ ( s s ) ωJM (cid:17) i i (4)and its projection (cid:16)b j z + b j z (cid:17) (cid:16) Φ ( s s ) ωJM (cid:17) i i = M (cid:16) Φ ( s s ) ωJM (cid:17) i i . (5)Bose-Einstein symmetry is already implemented in Eq. (3) via symmetrization. For the equivalent photons j = j = j and s = s = s . Then, changing the notations for the summation indices m ⇆ m in the second term in square brackets in Eq.(3) we can rewrite this equation like (cid:16) Φ s sωJM ( ~k ~k ) (cid:17) i ,i = N X m m (cid:16) ~A ( s ) ωjm ( ~k ) (cid:17) i (cid:16) ~A ( s ) ωjm ( ~k ) (cid:17) i (cid:2) C JMjm jm + C JMjm jm (cid:3) . (6)Employing the symmetry properties for the Clebsh-Gordan coefficients (with integer j , j ): C JMj m j m = ( − j + j + J C JMj m j m (7)we obtain finally an expression for the wave function of two equivalent photons (cid:16) Φ s sωJM ( ~k ~k ) (cid:17) i ,i = N (cid:2) − j + J (cid:3) X m m (cid:16) ~A ( s ) ωjm ( ~k ) (cid:17) i (cid:16) ~A ( s ) ωjm ( ~k ) (cid:17) i . (8)This wave function vanishes for the odd values of J . Thus, SSSR-1 prohibits all the odd values of J for two equivalent photonsand, consequently the corresponding transitions. According to Eq. (8) the two-photon wave function for two equivalent photonsalways is of even parity. We should note that description of the properties of a two-photon wave function, close to presentedhere can be found also in [11], as well as similar to our treatment of the Landau-Yang theorem in atomic processes.The SSSR-1 for S → S + 2 γ ( E two-equal-frequency-photon transition in He-like ions follows directly from Eq.(8). To illustrate SSSR-1 more directly and to support our analytical proof we performed also the evaluation of the transitionrate D ≡ (3 d s ) D → S + 2 γ ( E for He-like U ( Z = 92 ). This evaluation is fully similar to evaluation of S → S + 2 γ ( E transition rate (see capture to Fig. 1). Photon frequency distribution for the two-photon D → S + 2 γE (2) transition is given Fig. 2. The results in Fig. 2 demonstrate that SSSR-1 prohibits for two equivalent photons emitted in atomictransition to have a total angular momentum J = 3 . As it was explained in the Introduction, it does not contradict to the”standard” formulation of the Landau-Yang-theorem which allows this value for the two-photon decay of the particle providedthat this particle disappears in the process of the decay and the rest system for the center-of-inertia for two photons is employed. III. SSSR FOR EQUIVALENT ELECTRONS IN ATOMS: COMPARISON WITH THE SSSR FOR MULTI-EQUAL-PHOTONTRANSITIONS
There exists an analogy between the total electron momentum J e values allowed by the corresponding SSSR within jj -coupling scheme (in this case the spin-statistics is of the Fermi-Dirac type) and the total photon momentum values J allowedby Bose-Einstein SSSR. The one-electron Dirac wave functions in coordinate space we denote like ψ nj e l e m e ( ~r ) , where n is theprincipal quantum number, j e , m e are the electron total angular momentum and its projection, l e is the orbital angular momentumvalue which defines the parity of the state P = ( − l e , ~r is the space coordinate. In what follows we will omit the spinor indices.The two-electron atomic wave function (a 16-component spinor) having definite total angular momentum J e and its projection M e can be constructed as ψ ( l e l e ) n n J e M e ( ~r , ~r ) = N e X m e m e C J e M e j e m e j e m e (cid:2) ψ n j e l e m e ( ~r ) ψ n j e l e m e ( ~r ) − ψ n j e l e m e ( ~r ) ψ n j e l e m e ( ~r ) (cid:3) , (9)where N e is the normalization factor. The Fermi-Dirac statistics is implemented in Eq. (5) via the antisymmetrization. Theparity of the two-electron wave function is defined as P = ( − l e + l e . For the equivalent electrons (i.e. the electrons from theone nonclosed shell) n = n = n , j e = j e = j e , l e = l e = l e . Then, using the same procedure as in section II, we obtain ψ ( l e l e ) nnJ e M e ( ~r , ~r ) = N e (cid:2) − ( − j e + J e (cid:3) X m e m e C J e M e j e j e m e m e ψ nj e l e m e ( ~r ) ψ nj e l e m e ( ~r ) . (10)The parity of this function is always even as for photon wave function Eq. (8). From Eq. (10) it follows that the SSSR allows fortwo equivalent electrons also only even values of J e , exactly as for the equivalent photons. This happens because in the secondterm in square brackets in Eq. (10) we have an additional factor ( − unlike in Eq. (8), while the values of j e , unlike the valuesof j , are half-integer.In case of the number of equivalent electrons N e > there is no simple way to define the allowed values of the total angularmomentum J e . To determine these values one has to write down all sets of projections m e , . . . m e Ne which do not violate thePauli principle. Then for each set the total projection M e should be defined and all M e values should be distributed betweenpossible values of J e . This is a lengthy procedure (for example, for j e = 7 / and N e = 4 the number of such sets equals to 70)which can be only partly simplified with the use of group theory. The results can be found in books on atomic spectroscopy (forexample [12]) and are presented in Table 1. In general, all numbers in this Table follow from the Pauli principle. However, ifwe apply SSSR-1, SSSR-3, etc. for defining the allowed values of J e we will see that it will work up to j e = 7 / , N e = 4 . Itis violated by the presence of J e = 5 for j e = 7 / , N e = 4 . So the SSSR-3 for the equivalent electrons is limited by the value j e = 5 / . This should be compared with the limitation of the SSSR-2, SSSR-3 etc. by the photon multipolarity j = 1 .Finally, we should note that the similarity between the systems of equivalent photons and equivalents electrons could be mademore close with the use of the matrix form of Maxwell equations [13]. IV. SSSR-2 FOR THREE-DIPOLE PHOTON TRANSITIONS
For photon numbers N γ > the SSSR can be formulated only for dipole photons. For 3-dipole-photon transitions we cangive few examples which prove the validity of SSSR-2 for N γ = 3 . In this section we present a detailed derivation for thisproof. First, we present a computationally convenient fully relativistic form of a general expression for the 3-photon decay ratein H-like ion for arbitrary combination of electric and magnetic multipoles and in an arbitrary gauge for the electromagneticpotentials.The S -matrix element for the process i → f + 3 γ ( i and f denote the initial and final states of H-like ion respectively) reads[3], [4], [14] S (3) fi = ( − ie ) Z d x d x d x ψ f ( x ) γ µ A ∗ ( ~k ~e ) µ ( x ) S ( x , x ) γ µ A ∗ ( ~k ~e ) µ ( x ) S ( x , x ) γ µ A ∗ ( ~k ~e ) µ ( x ) ψ i ( x ) , (11) ψ n ( x ) = ψ n ( ~r ) e − iE n t , (12) ψ n ( ~r ) is the solution of the Dirac equation for the atomic electron, E n is the Dirac energy, ψ n = ψ + n γ is the Dirac conjugatedwave function, γ µ ≡ ( γ , ~γ ) are the Dirac matrices and x ≡ ( ~r, i t ) are the space-time coordinates. In this paper the Euclideanmetric with an imaginary fourth component is adopted. The photon wave function (electromagnetic field potential) is describedby A ( ~k,~e ) µ ( x ) = r πω e µ e ik µ x µ = A ( ~k,~e ) µ ( ~r ) e − iωt , (13)where k ≡ ( ~k, iω ) is the photon momentum 4-vector, ~k is the photon wave vector, ω = | ~k | is the photon frequency, e µ are thecomponents of the photon polarization 4-vector, ~e is the 3-dimensional polarization vector for real photons, A ( ~k,~e ) µ correspondsto the absorbed photon, A ∗ ( ~k,~e ) µ corresponds to the emitted photon, respectively.For the real transverse photons ~A ( x ) = r πω ~ee i ( ~k~r − ωt ) ≡ r πω ~A ~e,~k e − iωt . (14)The electron propagator for bound electrons we present in the form of the eigenmode decomposition with respect to one-electron eigenstates [3], [4] S ( x , x ) = 12 πi ∞ Z −∞ dωe iω ( t − t ) X n ψ n ( ~r ) ψ n ( ~r ) E n (1 − i
0) + ω , (15)where summation is runs over entire Dirac spectrum for atomic electron. Insertion of the expressions (12)-(15) into Eq. (11) andperforming the integrations over time and frequency variables yields S (3) fi = − πie δ ( E i − E f − ω − ω − ω ) X n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) fn ′ (cid:16) ~α ~A ∗ ~e ,~k (cid:17) n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) , (16)where ~α are the Dirac matrices, ( . . . ) kn denotes the matrix element with Dirac wave function ψ k , ψ n . The amplitude U of theprocess is related to the S-matrix via S fi = − πiδ ( E i − E f − ω − ω − ω ) U fi . (17)The probability (differential transition rate) of the process is defined as dW γi → f dω dω dω = 2 πδ ( E i − E f − ω − ω − ω ) (cid:12)(cid:12)(cid:12) U (3) fi (cid:12)(cid:12)(cid:12) . (18)We will be interested in the expression for the transition rate integrated over the directions ~ν = ~k/ | ~k | and summed over thephoton polarizations ~e of all the emitted photons. Then, taking into account all the permutations of photons and integrating over ω , we find dW i → f ( ω , ω ) dω dω = ω ω ω (2 π ) X ~e ,~e ,~e Z d~ν d~ν d~ν × (19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) fn ′ (cid:16) ~α ~A ∗ ~e ,~k (cid:17) n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) fn ′ (cid:16) ~α ~A ∗ ~e ,~k (cid:17) n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) fn ′ (cid:16) ~α ~A ∗ ~e ,~k (cid:17) n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) fn ′ (cid:16) ~α ~A ∗ ~e ,~k (cid:17) n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) fn ′ (cid:16) ~α ~A ∗ ~e ,~k (cid:17) n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) fm (cid:16) ~α ~A ∗ ~e ,~k (cid:17) n ′ n (cid:16) ~α ~A ∗ ~e ,~k (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where ω = E i − E f − ω − ω . The total transition rate can be defined as W i → f = 13! 12 j i + 1 X M i ,M f Z Z ω > dW i → f ( ω , ω ) . (20)Expanding the plane waves into spherical waves in Eq. (19) we go over to the description of photons by the total angularmomentum J , its projection M and parity (type of the photon). Then summation over polarizations and integration over photonemission angles in Eq. (19) yields dW i → f ( ω , ω ) dω dω = ω ω ω (2 π ) X λ λ λ X J J J X M M M (21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ′ n (cid:16) Q ( λ ) J M ω (cid:17) fn ′ (cid:16) Q ( λ ) J M ω (cid:17) n ′ n (cid:16) Q ( λ ) J M ω (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) Q ( λ ) J M ω (cid:17) fn ′ (cid:16) Q ( λ ) J M ω (cid:17) n ′ n (cid:16) Q ( λ ) J M ω (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) Q ( λ ) J M ω (cid:17) fn ′ (cid:16) Q ( λ ) J M ω (cid:17) n ′ n (cid:16) Q ( λ ) J M ω (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) Q ( λ ) J M ω (cid:17) fn ′ (cid:16) Q ( λ ) J M ω (cid:17) n ′ n (cid:16) Q ( λ ) J M ω (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) Q ( λ ) J M ω (cid:17) fn ′ (cid:16) Q ( λ ) J M ω (cid:17) n ′ n (cid:16) Q ( λ ) J M ω (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) + X n ′ n (cid:16) Q ( λ ) J M ω (cid:17) fn ′ (cid:16) Q ( λ ) J M ω (cid:17) n ′ n (cid:16) Q ( λ ) J M ω (cid:17) ni ( E n ′ − E f − ω )( E n − E f − ω − ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where we employ the reduction of the matrix elements to the radial integrals developed in [15], [16] D n α j α l α m α | Q ( λ ) J γ Mω | n β j β l β m β E = ( − j α − m α (cid:18) j α J γ j β − m α M γ m β (cid:19) × (22) ( − i ) J + λ − ( − j α − / (cid:18) π J + 1 (cid:19) / [(2 j α + 1)(2 j β + 1)] / (cid:18) j α J γ j β / − / (cid:19) M ( λ,J γ ) n α l α n β l β . Here J γ , M γ are the total angular momentum of the photon and its projection, λ characterizes the type of the photon: λ = 1 corresponds to electric and λ = 0 corresponds the magnetic photons, n α , j α , l α , m α is a standard set of one-electron Diracquantum numbers. The radial integrals M ( λ,J ) n α l α n β l β are defined as M (1 ,J ) n α l α n β l β = "(cid:18) JJ + 1 (cid:19) / (cid:2) ( κ α − κ β ) I + J +1 + ( J + 1) I − J +1 (cid:3) − (cid:18) J + 1 J (cid:19) / (cid:2) ( κ α − κ β ) I + J − − JI − J − (cid:3) (23) − G (cid:2) (2 J + 1) J J + ( κ α − κ β ) (cid:0) I + J +1 − I + J − (cid:1) − JI − J − + ( J + 1) I − J +1 (cid:3) ,M (0 ,J ) n α l α n β l β = 2 J + 1[ J ( J + 1)] / ( κ α + κ β ) I + J , (24) I ± J = ∞ Z ( g α f β ± f α g β ) j J (cid:16) ωrc (cid:17) dr, (25) J J = ∞ Z ( g α g β + f α f β ) j J (cid:16) ωrc (cid:17) dr, (26) g α and f α are the large and small components of the radial Dirac wave function as defined in [15], κ is the Dirac angular number, ω is the photon frequency, j J is the spherical Bessel function G is the gauge parameter for the electromagnetic potentials. In ourcalculations we employ the ”velocity” form of the matrix element containing the electromagnetic field potentials Eq. (23) andthe values G = 0 and q J +1 J [17]. Then we obtain what is traditionally called ”velocity” and ”length” gauges, respectively.The results can be further simplified by producing explicitly the summations over magnetic quantum numbers. For thispurpose we define the radial integral part for a particular combination of multipoles as S j n ′ j n ( i, j, k ) = X l n ′ ,l n X n,n ′ M ( λ i ,J γi ) f,n ′ ( ω i ) M ( λ j ,J γj ) n ′ ,n ( ω j ) M ( λ k ,J γk ) n,i ( ω k ) (cid:0) E n ′ j n ′ l n ′ − E n f j f l f − ω k − ω j (cid:1) (cid:0) E nj n l n − E n f j f l f − ω k (cid:1) × (27) ∆ j n ′ ,j n π l n ′ f ( i ) π l n n ′ ( j ) π l i n l ( k ) , where π lk ( t ) = (cid:26) if l k + l + J γ t + λ t = odd if l k + l + J γ t + λ t = even , (28) ∆ j n ,j m ( i, j, k ) = 4 π [ j f , j n ′ , j n , j i ] / (cid:2) J γ i , J γ j , J γ k (cid:3) / (cid:18) j f J γ i j n ′ / − / (cid:19) (cid:18) j n ′ J γ j j n / − / (cid:19) (cid:18) j n J γ k j i / − / (cid:19) Θ( i, j, k ) (29)and Θ( i, j, k ) = [ j n ′ , j n ] / X m n ′ ,m n ( − m i + m f +1 (cid:18) j f J γ i j n ′ − m f M γ i m n ′ (cid:19) (cid:18) j n ′ J γ j j n − m n ′ M γ j m n (cid:19) (cid:18) j n J γ k j i − m n M γ k m i (cid:19) . (30)Here indices i, j, k denotes the serial number of the photon and each one can take the values , or . The notation [ j, k, . . . ] means (2 j + l )(2 k + 1) . . . .The final expression for the decay rate for the one-electron ion is dW i → f ( ω , ω ) dω dω = ω ω ω (2 π ) X λ ,λ ,λ X J γ ,J γ ,J γ X M γ ,M γ ,M γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j n ,j n ′ S j n ′ j n (1 , ,
3) + (5 permutations ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (31)Summation over all projections appearing in the expression (31) can be easily performed numerically for each value of corre-sponding angular momenta.In the approximation of noninterecting electrons the two-electron wave function can be presented in the form Ψ J e M e ( ~r , ~r ) = 1 √ X m ,m C J e M e j m j m (cid:12)(cid:12)(cid:12)(cid:12) ψ n j l m ( ~r ) ψ n j l m ( ~r ) ψ n j l m ( ~r ) ψ n j l m ( ~r ) (cid:12)(cid:12)(cid:12)(cid:12) . (32)Then the two-electron matrix element can be reduces to the one-electron matrix element [18] D n α n α J α M α (cid:12)(cid:12)(cid:12) Q ( λ ) J γ Mω (cid:12)(cid:12)(cid:12) n β n β J β M β E = (33) ( − J α − M α + j α + j α [ J α J β ] / (cid:18) J α J γ J β − M α M M β (cid:19) (cid:26) J α J γ J β j α j j β (cid:27) D n α j α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( λ ) J γ ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n β j β E , D n α j α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( λ ) J γ ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n β j β E = ( − i ) J γ + λ − ( − j α − / (cid:18) π J γ + 1 (cid:19) / [ j α , j β ] / (cid:18) j α J γ j β / − / (cid:19) M ( λ,J γ ) αβ . (34)The final expression for the decay rate for two-electron ion is dW if dω dω = ω ω ω (2 π ) X λ ,λ ,λ X J γ ,J γ ,J γ X M γ ,M γ ,M γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X J n ′ ,J n X j n ′ ,j n S J n ′ J n j n ′ j n (1 , ,
3) + (5 permutations ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (35)where S J n ′ J n j n ′ j n ( i, j, k ) = X l n ′ ,l n X n l ,n ′ l M ( λ i ,J γi ) f,n ′ l ( ω i ) M ( λ j ,J γj ) n ′ l ,n l ( ω j ) M ( λ k ,J γk ) n l ,i ( ω k ) (cid:16) E n ′ l j n ′ l n ′ − E n f j f l f − ω k − ω j (cid:17) (cid:0) E n l j n l n − E n f j f l f − ω k (cid:1) × (36) ∆ J n ′ ,J n ,j n ′ ,j n ( i, j, k ) π l n ′ f ( i ) π l n n ′ l ( j ) π l i n l ( k ) , ∆ J n ′ ,J n ,j n ′ ,j n ( i, j, k ) = (4 π ) / (cid:2) j f , j n ′ , j n , j i (cid:3) / (cid:2) J γ i , J γ j , J γ k (cid:3) / (cid:18) j f J γ i j n ′ / − / (cid:19) (cid:18) j n ′ J γ j j n / − / (cid:19) (cid:18) j n J γ k j i / − / (cid:19) (37) × (cid:2) j n ′ , j n (cid:3) / [ J n ′ , J n ] / (cid:26) J f J γ i J n ′ j n ′ j j f (cid:27) (cid:26) J n ′ J γ j J n j n j j n ′ (cid:27) (cid:26) J n J γ k J i j i j j n (cid:27) Θ( i, j, k ) and Θ( i, j, k ) = [ J n ′ , J n ] / X M n ′ ,M n ( − M f + M n ′ + M n (cid:18) J f J γ i J n ′ − M f M γ i M n ′ (cid:19) (cid:18) J n ′ J γ j J n − M n ′ M γ j M n (cid:19) (cid:18) J n J γ k J i − M n M γ k M i (cid:19) . (38)For the intermediate states of He-like HCI in the approximation of noninteracting electrons we can use simple products of theone-electron wave functions not possessing the correct two-photon symmetry. This symmetry will be automatically restored inthe two-electron matrix elements with initial or final two-electron states with proper symmetry in Eq. (32).As an illustration of SSSR-2 for three equal photons we consider P → S + 3 γ ( E transition in the He-like HCIand demonstrate that the value J = 2 of the total angular momentum for the three equivalent dipole photons is prohibited. Inprinciple, this decay can proceed via several channels: 1) P → n S + γ ( E → n ′ P (cid:2) n ′ P (cid:3) +2 γ ( E → S +3 γ ( E ;this channel is prohibited since the transition n S + γ ( E → n ′ P (cid:2) n ′ P (cid:3) + 2 γ ( E is prohibited by SSSR-1, 2) P → n D + γ ( E → n ′ P (cid:2) n ′ P (cid:3) + 2 γ ( E → S + 3 γ ( E ; this channel is also prohibited by SSSR-1 since the transition n D → S + 2 γ ( E is prohibited by this rule, 3) P → n D (cid:2) n D (cid:3) + γ ( E → n ′ P (cid:2) n ′ P (cid:3) + 2 γ ( E → S .The states admixed by the spin-orbit interaction are placed in the square brackets, n, n ′ are arbitrary integer numbers. Thecontribution of the channel 3) does not turn to zero so evidently.However, using an exact expression (35) we find that for the three equal-frequency photons ( ω = ω = ω ) the totalexpression for the transition rate turns to zero. This can be traced analytically from Eq. (35), but we also illustrate it bynumerical calculations for Z = 92 (see Fig. 3). Note that in the channel 1) there is the resonance (the situation when the energydenominator turns to zero), in this case: P → S + γ ( E → S + 3 γ ( E . The presence of the cascade-producing state S in the sum over all the intermediate states in the transition amplitude Eq. (36) leads to the arrival of the high, but narrow”ridge” in the frequency distribution dW ( ω ,ω ) dω dω starting at the point ω = E (2 P ) − E (2 S ) on the ω axis and extendingparallel to the ω axis; alternatively, this ”ridge” can start from the point ω = E (2 P ) − E (2 S ) and extend parallel to the ω axis. The ”ridge” does not influence the existence of the SSSR-2: it does not correspond to the case of three equivalentphotons. This is a general conclusion for all the possible cascade transitions. In Fig. 3 one can see ”pit” in the frequencydistribution for the transition rate which arises around the point ( ω = ω = ω ) due to the SSSR-2. For comparison in Fig. 4we present a picture for the frequency distribution for transition P → S + 3 γ ( E also for Z = 92 , where instead of theSSSR-2-induced ”pit” there is a ”hump”, corresponding to the maximum in the two-dimensional distribution Fig. 1 (left panel).In this transition the total angular momentum J = 1 is not prohibited for 3 equivalent dipole photons.In the same way (using Eq. (35)) we prove SSSR-2 in case of transition D → S + 3 γ ( M . However, neither thevalue J = 2 for the total angular momentum of 3-photon system is forbidden for the transition D → S + 3 γ ( E , northe value J = 4 is forbidden for F → S + 3 γ ( M transition. The same concerns S → γ ( E transition where thevalue J = 0 is not forbidden for the 3-photon system. In all these cases the value J is fixed by the initial J e i and final J e f valuesfor the total electronic angular momentum. All these examples confirm our remark in the Introduction that the SSSR-2 works,in general, only for the dipole photons.In some cases, for example for transition S → S + 3 γ ( E the J = 0 value is excluded not by the SSSR-2 but formore trivial reasons: according to SSSR-1 the two E photons can have a total angular momentum , . Adding the angularmomentum of the third E photon to these values, it is impossible to receive the total angular momentum J = 0 for 3-photonsystem. With the three E photons the situation is different: according to SSSR-1 a two-photon system can have values of thetotal angular momentum equal to , , . Adding to these values the value for the angular momentum of the third E photonone can receive the value J = 0 for the total angular momentum of three photons. A direct check with Eq. (35) shows that thevalue J = 0 , is not forbidden in this case. V. GENERAL ANALYTIC PROOF OF THE SSSR-2
In this section we give a general proof of the SSSR-2 not connected with any particular transitions in atoms. For this purposewe will use the equations for the Coefficients of Fractional Parentage (CFP) usually employed in case of Fermi-Dirac statisticsfor the construction of the wave functions for the groups of the equivalent electrons (see, for example, [12]).We consider a system of 3 photons with equal frequencies ω , equal angular momenta j = j = j = j and of the same type: s = s = s = s . At first we will distinguish these quantum numbers and will put them equal to each other later. We choose onepossible coupling scheme for three photons where at first we couple the photons with indices i = 1 , and then add to them thethird photon with index i = 3 . The wave function for the pair of photons with i = 1 , we construct according to Eq. (8) and thenadd the third photon i = 3 using only the standard angular momentum coupling scheme and aiming to obtain the 3-photon wavefunction with the total angular momentum J and its projection M . We denote this wave function as Φ ωJM ( j j [ J ′ ] j ) , wherethe angular momentum J ′ in square brackets denotes one of the possible values for the intermediate total angular momentumfor two photons. The wave function Φ ωJM ( j j [ J ′ ] j ) is symmetrized with respect to the permutations of the photons , and therefore J ′ takes only even values according to SSSR-1. Then we decompose this wave functions via the wave functionscorresponding to another coupling scheme Φ ωJM ( j j [ J ′ ] j ) . The latter wave functions are not yet symmetrized with respectto the photon permutations , . The decomposition looks like Φ ωJM ( j j [ J ′ ] j ) = X J ′′ ( j j [ J ′′ ] j J | j j [ J ′ ] j J ) Φ ωJM ( j j [ J ′′ ] j ) , (39)where ( j j [ J ′′ ] j J | j j [ J ′ ] j J ) are the Racah coefficients connected with j -symbols via ( j j [ J ′′ ] j J | j j [ J ′ ] j J ) = ( − j + j + j + J p (2 J ′ + 1)(2 J ′′ + 1) (cid:26) j j J ′ j J J ′′ (cid:27) . (40)The summation in Eq. (39) extends over all the values J ′′ allowed by the vector coupling scheme for two angular momenta j , j . To ensure the symmetry of the wave function with respect to the permutation of the variables , , according to SSSR-1we have to retain only the even values of J ′′ in these expansion. For this purpose we replace the wave function Eq. (39) by thelinear combination Φ ωJM ( j j j ) = X J ′ ( j j [ J ′ ] j J } j j j J ) Φ ωJM ( j j [ J ′ ] j ) , (41)where ( j j [ J ′ ] j J } j j j J ) are the CFP and define the CFP from the requirement that all the terms with odd values of J ′′ inthe expansion Φ ωJM ( j j j ) = X J ′ ( j j [ J ′ ] j J } j j j J ) X J ′′ ( j j [ J ′′ ] j J | j j [ J ′ ] j J ) Φ ωJM ( j j [ J ′′ ] j ) (42)turn to zero. Since the presence of the odd J ′′ values in the wave function Eq. (39) contradicts to the requirement of its sym-metrization with respect to the permutation of variables , , the absence of these terms guarantees the corresponding symmetry.It is easy to check that any function of three variables , , , symmetric with respect to permutations , and , will be sym-metric also with respect to the permutation , . Thus the absence of the odd J ′′ values in the expansion Eq. (42) makes thewave function Φ ωJM ( j j j ) fully symmetric with respect to the permutations of the photon variables.The condition for the absence of the odd J ′′ values in the expansion Eq. (42) reduces to the system of equations X J ′ ( j j [ J ′ ] j J } j j j J ) ( j j [ J ′′ ] j J | j j [ J ′ ] j J ) = 0 (43)for all possible odd J ′′ values. The summation in Eq. (43) extends over all possible even J ′ values for the two-photon system.Setting now j = j = j = j we rewrite this equations as X J ′ ( jj [ J ′ ] jJ } jjjJ ) ( jj [ J ′′ ] jJ | jj [ J ′ ] jJ ) = 0 . (44)For the CFP in case of equivalent photons we will use shorthand notation ( jj [ J ′ ] jJ } jjjJ ) ≡ C jJ ′ J . (45)Then with Eq. (40) taken into account for the equivalent photons Eq. (44) reduces to X J ′ C jJ ′ J √ J ′ + 1 (cid:26) j j J ′ j J J ′′ (cid:27) = 0 . (46)For the dipole photons we have to set j = 1 and according to SSSR-1 J ′ = 0 , , J ′′ = 1 . This yields C J (cid:26) J (cid:27) + C J √ (cid:26) J (cid:27) = 0 . (47)In this case we have 1 equation for two coefficients C J and C J . In principle, normalization condition for the function Eq.(41) would give one more equation but it is not necessary to employ it for our purposes. The first j -symbol in Eq. (47) is equalto (cid:26) J (cid:27) = ( − J p J + 1) δ J . (48)0According to the angular momentum coupling rules the possible J values for 3 dipole photons are J = 1 , , . If we set J = 1 Eq. (47) always has a solution with nonzero values of C
10 1 , C
12 1 . However, setting J = 2 we arrive at the equality C
22 2 √ (cid:26) (cid:27) = 0 . (49)Since j -symbol in Eq. (49) is not equal to zero, it follows that G
12 2 = 0 . This means that for the J = 2 value of the totalangular momentum for three dipole photons in the wave function Eq. (42) the value J ′ = 2 for the two-photon subsystem shouldbe absent. This contradicts to the SSSR-1 and we conclude that the existence of the total angular momentum value J = 2 forthe three-photon system is inconsistent with the SSSR-1 for the two-photon system. Thus the value J = 2 is forbidden for thesystem of three equivalent dipole photons. It remains to consider the case J = 3 . For J = 3 Eq. (47) reads C
12 3 √ (cid:26) (cid:27) = 0 . (50)However, (cid:26) (cid:27) = 0 (51)and the coefficient C
12 3 is nonzero, but arbitrary. This does not contradict to the SSSR-1 and the vallue J = 3 for the totalangular momentum of the three equivalent dipole photon system is allowed. This proves the SSSR-2. VI. SSSR-3 FOR 4 DIPOLE PHOTON TRANSITIONS
It is important to check SSSR for the 4-photon transitions and especially for the higher J values since the fermionic analoguefor the SSSR is violated as we have seen in section III, for 4-electron systems and J e = 5 . Therefore, we give first the analyticproof of the SSSR-3 for 4 dipole photons.Proceeding in the same way as for the proof of SSSR-2, we consider two coupling schemes for constructing of the wavefunction of the 4-photon system. The first scheme corresponds to the wave function Φ ωJM ( j j [ J ] j j [ J ] JM ) , (52)where we first couple momenta j j to the intermediate momentum J , then couple momenta j j to another intermediatemomentum J and finally couple two intermediate momenta J and J to the final total angular momentum of the 4-photonsystem J with the projection M . We assume that the wave function Eq. (42) is symmetric with respect to the permutation of thevariables , and with respect to the permutation of variables , . Then according to the SSSR-1 the momenta J and J cantake only even values. Another coupling scheme will be representeed by the wave function Φ ωJM ( j j [ J ] j j [ J ] JM ) . (53)In this wave function we do not assume the symmetrization with respect to variables , and , and decompose the wavefunction Eq. (52) in terms of the wave functions Eq. (53): Φ ωJM ( j j [ J ] j j [ J ] JM ) = X J J ( j j [ J ] j j [ J ] J | j j [ J ] j j [ J ] J ) Φ ωJM ( j j [ J ] j j [ J ] JM ) , (54)where ( j j [ J ] j j [ J ] J | j j [ J ] j j [ J ] J ) are the Fano coefficients connected with j -symbols via ( j j [ J ] j j [ J ] J | j j [ J ] j j [ J ] J ) = (55) p (2 J + 1)(2 J + 1)(2 J + 1)(2 J + 1) j j J j j J J J J . In Eq. (54) in the expansion over J , J all the values of J J allowed by the angular momenta vector coupling arepresent. To symmetrize the wave function Eq. (54) with respect to the permutation of variables , and with respect to thepermutation , we have to replace it by the linear combination Φ ωJM ( j j j j ) = X J J ( j j [ J ] j j [ J ] J } j j j j J ) Φ ωJM ( j j [ J ] j j [ J ] JM ) = (56) X J J X J J ( j j [ J ] j j [ J ] J } j j j j J ) ( j j [ J ] j j [ J ] J } j j [ J ] j j [ J ] J ) × Φ ωJM ( j j [ J ] j j [ J ] JM ) , ( j j [ J ] j j [ J ] J } j j j j J ) are the CFP for the 4-particle systems (bosons). For symmetrization of the wave func-tion Eq. (56) with respect to the permutation of variables , and with respect to the permutation , it is necessary to requirethe terms with odd values of J , J to vanish in the summation over J , J in Eq. (56). This requirement leads to the systemof equations for the CFP: X J J ( j j [ J ] j j [ J ] J } j j j j J ) ( j j [ J ] j j [ J ] J | j j [ J ] j j [ J ] J ) = 0 , (57)where J , J take only the odd values and the summation is extended over the even values of J , J . When these equationsare satisfied we can consider the wave function Eq. (56) as symmetric with respect to the permutations of the variables , witheach other and the variables , with each other. If any function of 4 variables is symmetric with respect to the permutationswithin pairs (1 , , (3 , , (1 , and (2 , it is fully symmetric. Indeed, let us fix for example, variable 4 and consider thepermutations within a group of variables , , . If the function is symmetric with respect to the permutation , and to thepermutation , it will be symmetric also with respect to the permutation , as it follows from the symmetrization of thefunctions of the three variables. In this way we can prove the symmetry with respect to arbitrary pair of variables. Thus the wavefunction of the four photon constructed as described above will be fully symmetric, i.e. will obey the Bose-Einstein statistics.Remembering that we consider equivalent photons j = j = j = j = j and using the shorthand notations for the 4-particleCFP ( jj [ J ] jj [ J ] J } jjjjJ ) ≡ G jJ , J , J (58)we rewrite Eq. (57) as X J ,J G jJ , J , J p (2 J + 1)(2 J + 1) j j J j j J J J J = 0 . (59)Going over to the case of dipole photons ( j = 1) we should extend the summation in Eq. (59) over the values J = 0 , and J = 0 , . In the right-hand side of Eq. (59) we have to set J = 1 , J = 0 , or J = 0 , , J = 1 or J = J = 1 . Allthese cases should be excluded from the summation over J , J in Eq. (56). Eq. (59) now looks like X J ,J G J , J , J p (2 J + 1)(2 J + 1) J J J J J = 0 . (60)In general, for the fixed J value there are 5 equations for the 4 coefficients G , , J , G , , J , G , , J and G , , J . However,two of these equations for the case J = 1 , J = 0 , coincide with the other two equations for the case J = 0 , , J = 1 due to invariance of j -symbol in Eq. (59) under the permutation of the two first rows. Then the system of equations looks like G , , J J + G , , J J √ G , , J J √ G , , J J , (61) G , , J J + G , , J J √ G , , J J √ G , , J J , (62) G , , J J + G , , J J √ G , , J J √ G , , J J . (63)Eq. (61) corresponds to the case J = 1 , J = 0 , Eq. (62) corresponds to the case J = 1 , J = 2 and Eq. (63) correspondsto the case J = J = 1 .First, we analyze the system of equations Eq. (61)-Eq. (63) for J = 1 . Considering Eq. (61) we see that all the j -symbolsin the left-hand side of Eq. (61) but the last one are zero [19]. The last j -symbol is nonzero only for J = 1 : J = − √ (cid:26) (cid:27) δ J . (64)2This means that for J = 1 G , , = 0 and the value J = 1 for the total angular momentum of 4-photon system is inconsistentwith SSSR-1 which allows the value J = 2 for the two-electron system (condition G , , = 0 means that the value J = 2 is notallowed for the two-photon subsystems of the 4-photon system). All other possible values J = 0 , , , for the 4-dipole-photonsystem are not forbidden by Eq. (61). Continuing this analysis we find that in Eq. (62) for J = 0 all the j -symbols turn tozero, so this equation has a solution with arbitrary values of the CFP and the value J = 0 is not forbidden for the 4-equal-dipole-photon system. For J = 1 in Eq. (62) all the j -symbols but the last one are zero. Then G , , = 0 what contradicts to SSSR-1and the value J = 1 is forbidden. For J = 2 only the first j -symbol in Eq. (62) is zero, so that Eq. (62) allows for the nonzerosolution for CFPs and the value J = 2 is allowed. For J = 3 all the j -symbols but the last one are zero, so G , , = 0 whichcontradicts again to the SSSR-1 and the value J = 3 is forbidden. In the same way we will find that Eq. (63) forbids only thevalue J = 1 . Finally, for J = 4 all the j -symbols in Eq. (62) (as well as in Eqs. (61), (63)) are zero. Then the CFP with J = 4 are fully arbitrary and the value J = 4 is allowed for the 4-photon system. In total, our analysis demonstrates that for the4-equal-dipole-photon system the values of the total angular momentum of the system J = 0 , , are allowed and the values J = 1 , are forbidden.To support the general proof and using the same QED approach as for 3-photon systems we have checked the SSSR-3 for thetransitions S → S + 4 γ ( E and D → S + 4 γ ( E . In both cases transition rates turn to zero for equal frequencyphotons ω = ω = ω = ω . This can be traced analytically from equations similar to Eq. (35) and proves that the value J = 1 is forbidden for the equal-frequency photons in the first transition and J = 3 is forbidden in the second transition. Thus, SSSR-3holds for the 4-photon transitions. VII. CONCLUSIONS AND OUTLOOK
We have formulated the Spin-Statistic Selection Rules (SSSR) for the multiphoton atomic processes which originate from thefundamental properties of spin-1 particles, obeying the Bose-Einstein statistics. In a sense, these rules can be considered as anexclusion principle for bosons (photons) since they prohibit some states for the system of equivalent particles. This resemblanceis strengthened by comparison with the properties of the equivalent electrons in atomic physics.However the SSSR are formulated exclusively for the atomic processes, i.e. for the emission or absorption of photons byatomic systems (atoms, ions). The SSSR are related to the total angular momentum quantum numbers, the orbital angularmomenta of photons also being included. The latter ones depend on the choice of the frame of reference, in case of SSSR thischoice is the rest frame of an atom emitting or absorbing the photons.This makes the difference with the Landau-Yang theorem which states that two-photon system can not have a total angularmomentum equal to one. This statement is formulated in the rest frame for the center-of-inertia for two photons. In this referenceframe two photons are collinear (opposite directed) and have equal frequencies. The different choice of the reference framecompared to the SSSR leads to a different formulations when the higher multipolarities are involved , i.e. the orbital angularmomentum definition begins to be important. Otherwise the SSSR can be considered as an extension of the Landau-Yangtheorem to multiphoton systems in atomic processes.Our calculations were performed for the two-electron highly charged ions. That is, they were fully relativistic but with fullneglect of the interelectron interaction. We have chosen the He-like HCI because with the neglect of the interelectron interactionthe calculations are as simple as for the one-electron ions. On the other side the He-like ions have essentially more reach spectrumwhich allows to find many possibilities for the application of the SSSR. The SSSR, as based on the symmetry properties remainthe same independent on the inclusion or neglect of the interelectron interaction . Of course the numerical values for the transitionrates obtained in such an approximation will be far from accurate for the neutral helium, but will become more accurate for theHCI with high Z values. This is the reason why we have chosen uranium ( Z = 92 ) for our particular examples. The SSSRshould hold not only for two-electron atoms and ions but also for the multiphoton processes in any many-electron atoms or ions.Our choice of the two-electron HCI was made because in these systems the action of the SSSR becomes most transparent.An important question is in which kind of the experiments the influence of the SSSR on atomic processes can be observed.It is natural to use lasers for such observation. Let the laser light propagate through atomic vapour of an atom with suitabletransition frequency ω a between the pair of atomic levels. An advantage of the use of the laser source is that all the photons willhave the same frequency. If we divide this frequency by an integer number N γ and adjust the laser frequency ω l to this value, ω l = ω a /N γ , the number of photons N γ in the absorption process will be fixed. The value of the total angular momentum J forN-photon system can be fixed by choosing the appropriate values J e i and J e f for the initial (lower) and final (upper) levels inthe transition process. For example, if we choose J e i = 0 , J e f = 2 and N γ = 3 we will check the SSSR-2 for N γ = 3 , J = 2 .What we can not fix is the multipolarity of the transition, i.e. a total angular momentum j of every separate photon. A laser lightin the beam can be decomposed in all possible multipolarities. This means, for example that together with E E E transition,all transitions with the same total parity constructed with the higher multipoles, i.e. E M E , E E M etc. will be alwaysabsorbed. However the process with the photons of higher multipolarities are usually strongly suppressed in atoms. Due to thissuppression the E E E transition will be dominant. Measuring the absorption rate at the ω l = ω a /N γ frequency one canestablish the validity or non-validity of the particular SSSR.3AcknowledgmentsThe work was supported by RFBR (grants No. 12-02-31010 and No. 14-02-00188). T. Z., D. S. and L. L. acknowledge thesupport by St.-Petersburg State University with a research grant 11.38.227.2014. D. S. is grateful to the Max Planck Institute forthe Physics of Complex Systems for financial support and to the Dresden University of Technology for hospitality. [1] L.D. Landau, Dokl. Akad. Nauk SSSR , 207 (1948).[2] C. N. Yang, Phys. Rev. , 242 (1950).[3] V. B. Berestetskii, E. M. Lifshitz and L. P. Pitaevskii, Quantum Electrodynamics , Oxford, Pergamon, 1982.[4] A. I. Akhiezer and V. B. Berestetskii,
Quantum Electrodynamics , New York, Wiley 1965.[5] G. W. F. Drake, G. A. Victor and A. Dalgarno, Phys. Rev. , 25 (1969).[6] A. Derevianko and W. R. Johnson, Phys. Rev. A , 1288 (1997).[7] D. DeMille, D. Budker, N. Derr and E. Deveney, Phys. Rev. Lett. , 3978 (1999).[8] D. English, V. V. Yashchuk and D. Budker, Phys. Rev. Lett. , 253604 (2010).[9] R. W. Dunford, Phys. Rev. A , 062502 (2004).[10] M. G. Kozlov, D. English and D. Budker, Phys. Rev. A , 042504 (2009).[11] D. DeMille, D. Budker and E. Deveney, AIP Conf. Proc. , 227 (2000).[12] I. I. Sobelman, Theory of Atomic Spectra , Alpha Science International Limited, 2006.[13] P. J. Mohr, Annals of Physics , 607 (2010).[14] O. Yu. Andreev, L. N. Labzowsky, G. Plunien and D. A. Solovyev, Phys. Rep. , 135 (2008).[15] I. P. Grant. J. Phys. B: Atom. Molec. Phys. , 1458 (1974).[16] S. P. Goldman and G. W. F. Drake, Phys. Rev. A , 183 (1981).[17] L. Labzowsky, D. Solovyev, G. Plunien and G. Soff, Eur. Phys. J. D , 335 (2006).[18] J. Santos, P. Patte, F. Parente, P. Indelicato, Eur. Phys. J. D 13, 27 (2001).[19] D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, Quantum Theory of Angular Momentum , World Scientific, Singapore (1988)FIG. 1: Photon frequency distributions for the two-photon transitions (1 s s ) S → (1 s ) S + 2 γ ( E (left panel) and (1 s s ) S → (1 s ) S + 2 γ ( E (right panel) in He-like U (nuclear charge Z = 92 ). The transition rates dWdω in units s − is plotted versus the photonfrequency ω in units ω/ ∆ and ω/ ∆ . The values ∆ , ∆ correspond to the energy intervals ∆ = E (2 S ) − E (1 S ) and ∆ = E (2 S ) − E (1 S ) , respectively. The calculations are performed fully relativistically, with Dirac one-electron wave functions and relativisticexpressions for the electromagnetic vector potentials (photon wave functions). The Dirac wave functions for an extended nucleus describedby the Fermi distribution were employed. The interelectron interaction was fully neglected; the expected error is about /Z i.e. about for Z = 92 . The one-electron wave functions for the initial and final two-electron states were coupled to present S , S and S states, respectively. Summation over the full set of two-electron states taken as products of one-electron Dirac states was performed within theB-spline approach. The calculations were carried out in two relativistic ”forms”, corresponding to the nonrelativistic ”length” and ”velocity”forms [17]; the results coincide with 12 digits. For the ions with lower Z values and for the neutral He atom the accuracy for W valuesobtained from calculations with the neglect of interelectron interaction becomes poorer, but the difference between the frequency distributionsfor S → S + 2 γ ( E and S → S + 2 γ ( E transitions qualitatively remains the same. Ω ó ´ ´ ´ ´ ´ ´ Ω ó ´ ´ ´ ´ FIG. 2: Photon frequency distribution for the (1 s d ) D → (1 s ) S + 2 γ ( E two-photon transition. Notations, units etc. are the sameas in Fig. 1. ∆ denotes the energy difference ∆ = E (3 D ) − E (1 S ) . Calculation is performed for He-like U ion ( Z = 92 ). The totaltransition probability (integral value) is W γ (3 D − S ) = 3 . × s − , W γ (3 D − S ) = 3 . × s − in the relativistic”length” and ”velocity” forms respectively. ΩD FIG. 3: 3-dimensional plot for frequencies distribution of the transition rate P → S + 3 γ ( E in He-like uranium. On the verticalaxis the transition rate dWdω dω in s − is plotted; on the horizontal axes the photon frequencies are plotted in units ω / ∆ , ω / ∆ where ∆ denotes the energy difference ∆ = E (2 P ) − E (1 S ) . The lowest (zero) point is the point with coordinates ω / ∆ = ω / ∆ = 1 / atthe bottom of the ”pit” in the frequency distribution for the transition rate which arises due to SSSR-2. The calculations were done neglectingthe interelectron interaction.TABLE I: Allowed values of the total electron momentum J e for the systems of different numbers N e of equivalent electrons with angularmomenta j e within jj -coupling scheme. The results are given only for the even numbers N e , since for odd N e the J e values are half-integerand no analogy with photons can be traced. The numbers in parenthesis show how many times a particular value J e can arrive among theallowed values. The maximum number of equivalent electrons with particular value j e is j e + 1 . The allowed values of J e are the same forconfigurations with N e and j e + 1 − N e electrons, respectively. j e N e J e / / , / , , , / , , , ,
64 0 , , , , , FIG. 4: 3-dimensional plot for frequencies distribution of the transition rate P → S + 3 γ ( E in He-like uranium. All the detailsare the same as in Fig. 3. ∆ = E (2 P ) − E (1 S ) . In this case, there is no SSSR-2 induced ”pit” in the frequency distribution for thetransition rate. The calculations are performed under the same approximation as for Fig. 3. The total transition probability (integral value) is W γ (2 P − S ) = 13 . × s − , W γ (2 P − S ) = 13 . × s −1