Short-time emission of higher-angular-momentum photons by atomic transitions

The short-time regime of spontaneous light emission by few-electron ions is examined in detail, with a specific emphasis on the angular momentum of the emitted light. It is found that, in general, photons carrying a higher angular momentum are emitted with important probabilities, at short times, in transitions that are not of the electric dipole type. The probability of emission of such photons is found to be parametrically non-negligible in this time regime, and even numerically dominant for some cases. It is also found that, in all time regimes, the emission of electric 2 n+1-pole fields is typically numerically dominant over the emission of magnetic 2 n -pole fields by many orders of magnitude. These results refine and deepen our understanding of the emission of angular momentum-carrying light by simple atomic systems.


Introduction
Recent decades have seen increased interest in the orbital angular momentum (OAM) of light [1][2][3][4], with many differents points of emphasis, including the topological properties of light carrying such angular momentum [5][6][7], conversion from spin to orbital angular momenta [8,9], the relation of angular momenta to the symmetries of relativistic spacetime [10,11], generation of twisted electromagnetic beams [12,13] including at very high frequency [14], propagation of twisted light in media [15][16][17], sorting of light according to OAM [18,19], and corresponding applications in telecommunications [20,21] and quantum information [22,23]. Others have examined the interaction of twisted light with atomic systems * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. [24][25][26][27][28][29]. In this work, we reverse the latter situation, and study in detail the angular momentum properties of light emitted by simple atomic systems through the process of spontaneous emission, and exhibit non-trivial results in the regime of emission that precedes the establishment of Fermi's golden rule. This regime could be accessed experimentally with the help of the stimulated Raman adiabatic passage (STIRAP) technique, as discussed in reference [30]. Through dephasing measurements, this process can indeed 'freeze' the dynamics in the short-time regime [31].
In a recent work [30], one of us show (with co-workers) that, for atomic electronic transitions that are not of the electricdipole (E1) type, a strong enhancement of the decay with respect to the golden rule prediction is to be expected up to times that verify ω 0 t 1 (a condition which is necessary to ensure the validity of the two-level and rotating-wave approximations made here), with ω 0 the resonant frequency of the transition. In the present work, we highlight, and elaborate on a result that we had partially derived in passing but not focused upon. Namely, for non-E1 transitions involving no s states, in the time-regime described just above, photons of different angular momenta are emitted with intensities that only differ through non-trivial numerical pre-factors. This is in stark contrast with the usual formulation of the selection rules of atomic physics, which, in their oft-used, approximated version, state that only photons with an angular momentum J that satisfies J = |l e − l g | can be emitted for electric transitions, and also J = |l e − l g | + 1 if J = |l e − l g | corresponds to a magnetic transition. Here l e and l g are the OAM of the excited (with quantum numbers n e , l e and m e ) and ground states (with quantum numbers n g , l g and m g ) considered. In the specific case where l e = l g , these approximate selection rules are amended to J = 1, 2, where J = 1 corresponds to the magnetic dipole channel and J = 2 to the electric quadrupole channel, typically considered to have comparable decay rates [32,33]. As can be anticipated from the rules of addition of angular momenta, and as was confirmed by the work of Moses [34], the exact selection rules show that all angular momenta in the range |l e − l g | J l e + l g for electric transitions, and |l e − l g | J l e + l g − 1 for magnetic transitions, are allowed for emitted photons. However, it is expected that spontaneous emission is very clearly dominated by photons with J = |l e − l g |, and also J = |l e − l g | + 1 if J = |l e − l g | corresponds to a magnetic transition (or J = 1 and J = 2 in the case where l e = l g ), as enunciated by the approximate selection rules.
In this work, we show that this hierarchy of angular momenta for emitted photons is not generally valid at short times; and that, in Fermi's golden rule regime that takes over at longer times, the emission of electric 2 n+1 -pole fields is numerically (albeit not parametrically) dominant over the emission of magnetic 2 n -pole fields by many orders of magnitude. For non-E1 transitions, there exists a realistically accessible shorttime regime in which the emission rates of photons of all allowed momenta have the same parametric dependence on the characteristic frequencies of the system, that is, the resonant transition frequency, the cutoff frequency of the atom-field coupling, and the inverse of the measurement time. However, dramatic differences in numerical pre-factors mean that these emission rates are typically not commensurable. The angular momentum of photons emitted during spontaneous emission could be accessed experimentally by filtering the emitted light according to its OAM, for instance as done in references [18,19,35].
The rest of the manuscript is organised as follows. In section 2, the basic equations of the dynamics of atom-light interaction are presented in the two-level approximation. In section 3, the expressions for the rates of emission of photons of specific angular momenta are derived, both in Fermi's golden rule regime and in the short-time regime that precedes it. Numerical results for specific transitions and ions are presented in section 4, and conclusions are drawn in section 5.

Spontaneous emission in the two-level approximation
We consider a two-level atom in free space, consisting of a ground state |g and an excited state |e separated by the Bohr energy ω 0 , and prepared in |e at t = 0. The state of the electromagnetic field at t = 0 is the vacuum state |0 . In what follows, we monitor the emission rates of photons carrying all allowed angular momenta J. In the framework of the rotating wave approximation, the state of the system at time t reads The single-photon state |1 JMλω is labelled by its angular momentum J, its angular momentum projection number M, its helicity λ and its frequency ω, with the corresponding wave function a (vector) spherical wave [34]. The orthogonality relation is Schrödinger's equations of motion for the probability amplitudes read [36] c e (t) = i where the atom-field couplings are given by with the interaction Hamiltonian The matrix elements (3) have been calculated exactly in nonrelativistic hydrogen-like atoms (the relativistic corrections, which we do not consider here, bring contributions of relative order (Zα) 2 [37]). The calculation was initiated by Moses [34] and completed by Seke [38] (see appendix A for the explicit expressions of the matrix elements). In the following, we will examine the emission rates of photons of specific angular momenta J. Note that these rates are in general time-dependent [30], which might be counterintuitive. As is well known, in Fermi's golden rule regime, the emissions rates are constant. Emission rates, both in this golden rule regime, and in the short-time regime that precedes it, will now be investigated in more detail.

General expressions
We write the atomic states as |g = |n g , l g , m g and |e = |n e , l e , m e where each atomic state is described by three discrete quantum numbers n i , l i and m i which are respectively, as indicated above, the principal, angular momentum and magnetic quantum numbers. It is wellknown that the transition which we consider here can only cause the emission of photons with angular momenta J that verify |l e − l g | J l e + l g , and that J = |l e − l g |, and also J = |l e − l g | + 1 if J = |l e − l g | corresponds to a magnetic transition, is/are the dominant emission channel(s), as expected by following the approximate selection rules. We will show that there is a certain time regime for which emission of photons of all allowed angular momenta takes place at parametrically similar rates, with the only difference a non-trivial numerical pre-factor. For that purpose, we will use time-dependent perturbation theory, which yields [30,39] the emission rate [see equation (5)] of photons with angular momentum J as The spectral profile F t of the atom at time t is given by The form factor for the coupling of the atom to photons of angular momentum J is given by [30,34,38] (see also appendix A) where η J = 1 + 2J for magnetic transitions, and η J = −1 + 2J for electric transitions; μ = 2 n g + n e − 1 ; the D Jr are dimensionless constants involving the Clebsch-Gordan coefficients of the transition under consideration; and ω X ω 0 is the non-relativistic cutoff frequency with a 0 the Bohr radius and Z the atomic number, and α 1/137 the fine-structure constant of electrodynamics. Finally, the index at which the sum is terminated is N J = 2 n e + n g − 4 − J − l e − l g − with = 0 for electric transitions and = 1 for magnetic transitions. In the present work we focus on transitions for which neither l g nor l e is equal to zero, namely, in which no s states are involved. In this case, it is easily seen that several values of J are always allowed. For a given pair of initial (excited) and final (ground) states, both electric and magnetic transitions are generally allowed. As explained in the appendix A, for given initial and final states, in general, several photon angular momenta J are allowed. If l e + l g + J is even, then the corresponding photon is said to be emitted through the electric channel, while, if l e + l g + J is odd, it is emitted through the magnetic channel. The transition matrix elements are calculated from the vector potentialÂ x, t = 0 at the position of the electron and, for electric transitions, only the poloidal part of the vector potential contributes, while, for magnetic transitions, only the toroidal part contributes (see appendix A). Parameters of the atom-field coupling form factor given by equation (8) for the transitions 4F-2P, 5F-4F and 3P-2P. Here J refers to the allowed photon angular momenta, η and N J and μ to the corresponding parameters in the form factor (see definitions after equations (8) and (9)), while ω 0 is the resonant transition frequency and ω X the cutoff frequency of the form factor. The ratio ω X /ω 0 is given here as a function of the (effective) nuclear charge, while the values given for all four other parameters are valid for all systems within the single-electron treatment.
The parameters J, η, N J , μ and ω X corresponding to the transitions 4F-2P, 5F-4F and 3P-2P, to be investigated in detail in section 4, are given in table 1. These transitions are normally considered electric quadrupole, magnetic dipole and magnetic dipole transitions, respectively, because the (J = 2, = 0), (J = 1, = 1), and (J = 1, = 1) channels, respectively, correspond to the lowest allowed J in each case. However, they are accompanied by the emission of photons of higher J (see table 1). As we will now see, this is not in contradiction with the approximate selection rule J = |l e − l g | (or J = 1 and J = 2 in the case where l e = l g ), except potentially in the short-time regime of emission.

Photon emission in Fermi's regime
For sufficiently long times, it is known that it is permissible to where we used the hierarchy ω 0 /ω X 1 in the last step. That step is valid unless, of course, D J0 vanishes, which happens for instance for electric dipole transitions ( = 0, J = 1) between levels sharing the same principal quantum number [30,38]. However, we do not focus on this special case here. Explicit results for the Fermi decay rates of various transitions are given in tables 2 and 3, and compared with those compiled in the NIST Atomic Spectra Database [40]. We chose, in all cases, to implement the single-electron results in a simple effective nuclear charge framework, for which the valence electron is screened by the core electrons, yielding Z eff = Z − (N e − 1), with N e the number of electrons present in the ion. The good agreement with the NIST database confirms the approximate validity of the single-electron approximation for two-and three-electron systems (such as Be 2+ or F 3+ ). For systems with more electrons, such as Ti 11+ and Ca + , the results obtained  Table 3. Decay rates in the Fermi regime for the transitions 4F-2P, 5F-4F and 3P-2P, as calculated in the present work, and given in the NIST database [40]. Discrepancies by a factor of 10 5 for the 4F-2P transitions are almost certainly due to a mistake in the NIST database: in reference [44], from which the NIST values are taken, the quadrupole oscillator strengths are given in units of 10 −5 , which was probably overlooked. Discrepancies by a factor of 10 for the 3P-2P transitions are probably due to a systematic mistake in the NIST database, as we also found a result almost exactly ten times smaller than the database one for He and for Li + . are within one order of magnitude of the literature data. A more thorough treatment of screening would surely improve our results, but, while effective charges for neutral atoms were investigated in detail in reference [41], we are not aware of any detailed study of effective charges for ions in the literature. The simpler Slater rules [42], improve only moderately, at best, the agreement of our results with the NIST database, therefore, we opted not to pursue the Slater approach. Ultimately, of course, many-electron codes such as Grasp2K [43] yield the most reliable results for this type of problem, but our simpler approach highlights trends and patterns that could then be checked with such methods. Now, we focus on transitions for which several J are allowed. In the Fermi regime, we can calculate the ratio of emitted photons of two different angular momenta J and J as Since the cutoff frequency of the transition is much larger than the resonant frequency: ω 0 /ω X 1 [see equation (9)], it is clear from equation (11) that the more intense emission of the two should be expected to be that of the photons with the lower η J . If the lowest J allowed by the matrix elements, J min , is such that l e + l g + J min is even, then the lowest η J allowed is −1 + 2J min , which means that the electric channel of lowest allowed multipolar order dominates in the Fermi regime (also see references [32,33]). If l e + l g + J min is odd, then both J min and J min + 1 correspond to the same η = 1 + 2J min , which means that the magnetic channel of the lowest allowed multipolar order, and the electric channel of the next multipolar order, are expected to dominate in the Fermi regime. We find, however, that the non trivial pre-factors D J0 are such that the electric 2 n+1 -pole channels dominate the emission over the magnetic 2 n -pole channels by many orders of magnitude (here n = 1 stands for dipole, n = 2 for quadrupole, n = 3 for octupole, etc). For instance, for the 5F-4F transition in Ti 11+ , the calculated magnetic dipole decay rate is 26 orders of magnitude smaller than the electric quadrupole decay rate given in table 3, and for the 3P-2P transition in Be 2+ , it is smaller by 13 orders of magnitude.
Despite these hierarchies, we emphasise that all allowed J contribute to the decay/emission dynamics. As we will now see, photons with all allowed angular momenta are potentially important contributors to the dynamics at short times.

Photon emission before the onset of Fermi's regime
As we have previously found (See equation (C2) of reference [30]), there is a time regime 1/ω 0 t 1/ (2πω 0 ) ω X /ω 0 η J −1 for which all terms in the sum over r in (8) have a contribution to the modified emission rate that has an identical parametrical dependence on the characteristic frequencies ω 0 , ω X and 1/t of the system. For arbitrary J, the photon emission rate is given by with θ the Heaviside step function. This expression is easily seen to reduce to the golden rule value (10) for sufficiently large times 1/ω 0 1/ (2πω 0 ) ω X /ω 0 t. For shorter times, on the other hand, two different cases appear: for electric dipole fields, η J = 1, and the condition ω 0 t 1 guarantees that the first (Fermi) term on the rhs of (12) dominates over the sum over r at all realistic times. For all other multipolar orders for electric fields, and for all multipolar orders of magnetic fields, though, there is a time regime at which the sum over r dominates. Namely, in this regime, the emission rate of photons of a specific angular momentum J is We write η min the smallest value of η allowed in the |e → |g transition, and we focus on transitions for which the electric dipole channel does not contribute: η min = 1. In this case, for all allowed J, equation (13) is valid in a certain time regime. More specifically, for 1/ω 0 t 1/ (2πω 0 ) ω X /ω 0 η min −1 , the ratio of photon emission rates between two arbitrary allowed angular momenta is Here we have sums starting at owing to the Heaviside step function in equation (13). At least parametrically, the ratio (14) is of order unity, as it does not feature the characteristic frequencies of the system, namely, the atomic resonant frequency ω 0 , the cutoff frequency ω X , and the inverse observation time 1/t. Although, as soon as t ∼ 1/ (2πω 0 ) ω X /ω 0 η min −1 , photons with η min start dominating (at least parametrically) the emission, before this Fermi regime takes over, it is therefore seen through equation (14) that there exists a regime in which all allowed angular momenta contribute equally to the decay at the parametric level, with the only difference determined by non-trivial numerical pre-factors. In the next section, we turn to the calculation of these pre-factors, in order to describe emission at short times in terms of the photon angular momentum, in more detail.

Detailed study of a few specific transitions
Here we inspect three transitions in detail: 4F-2P, 5F-4F, and 3P-2P. The corresponding parameters were given in table 1.  Tables 4, 5, and 6 show the branching ratios of emission rates Γ J of photons of different angular momenta J, in the short-time regime of emission, defined by t 1/ (2πω 0 ) ω X /ω 0 η min −1 .
Numerical values for the upper limit of this time regime are also given in tables 4, 5, and 6. The results indicated are obtained by averaging over initial magnetic quantum numbers m e , and summing over final magnetic quantum numbers m g [45]. The branching ratios can be computed using equation (14), with the expressions for the D coefficients obtained through the expressions for the matrix elements given in appendix A. Our results are illustrated by figure 1, where the evolution of emission rates for photons of all allowed angular Table 5. 5F-4F transition. Branching ratios of emission rates Γ J of photons of different angular momenta J, in the short-time regime of emission. Γ 1 : magnetic dipole, Γ 2 : electric quadrupole, Γ 3 : magnetic octupole, Γ 4 : electric hexadecapole, Γ 5 : magnetic dotriacontapole, Γ 6 : electric tetrahexacontapole. The total rate is the sum Γ tot = Γ J=1 + Γ J=2 + Γ J=3 + Γ J=4 + Γ J=5 + Γ J=6 . The rates were obtained by averaging over initial magnetic quantum numbers m e , and summing over final magnetic quantum numbers m g . In the lower table, we give the numerical value of the end of the short-time regime [see paragraph before equation (12)].  ), it appears that the emission rate of the magnetic octupole channel is entirely negligible even in the short-time regime. On the other hand, the emission rate of the electric hexadecapole channel is not always negligible. Indeed, that channel contributes almost 10% of the total emission rate in the short-time regime.
For the 5F-4F transition (table 5), the results are striking. Indeed, the electric quadrupole channel dominates over the magnetic dipole one by many orders of magnitude. As it turns out, even the electric hexadecapole and tetrahexacontapole (64-pole) channels dominate over the magnetic dipole channel, and are seen to contribute approximately 10% and 15% of the total emission rate in the short-time regime.
For the 3P-2P transition (table 6), the results are similar to the preceding case. Indeed, the emission rate of the electric quadrupole channel dominates over that of the magnetic dipole channel by three orders of magnitude.
For all three transitions of interest, we also represent in figure 2, the evolution of the average angular momentum J of the photon, which can be shown quite straightforwardly, from the general expressions established in section 2, to be given as a function of time by The obtained curves illustrate the non-negligible presence, before the establishment of the Fermi regime, of photons of angular momenta different from that of the dominant emission channel. In particular for the 4F-2P and 5F-4F transitions, which are both dominated by the electric quadrupole (J = 2) in the Fermi regime, large deviations from J (t) = 2 are observed in the short-time regime, which represents a striking illustration of our results. It can be envisaged to access the short-time regimes that interest us experimentally, for instance, with the help of the STIRAP technique mentioned at the beginning of the present work. Through dephasing measurements, this process can 'freeze' the dynamics in the short-time regime [31]. In a different approach, the Zeno dynamics of a driven two-level transition was recently observed [46]. Then, the OAM of the emitted light could be analyzed, for instance with techniques discussed in references [18,19,35].

Conclusion
We have studied spontaneous emission by few-electron ions at short times with a focus on the angular momentum of the emitted photons. Our results show the emergence of a pattern whereby, for a given atomic transition, the pre-factors corresponding to the electric 2 n+1 -pole emission channels are typically numerically dominant over those of the emission of magnetic 2 n -pole channels by many orders of magnitude. This is true both in the short-time regime of emission t 1/ (2πω 0 ) ω X /ω 0 η min −1 , where the photon emission rates are given by equation (13) (only valid for non-E1 transitions), and in the Fermi regime, where they are given by equation (10). Indeed, in the short-time regime, for non-E1 transitions, photons of all allowed angular momenta are emitted with rates that have an identical parametric dependence, and, in the Fermi regime, we established through equation (11) that the emission rates for electric 2 n+1 -pole and magnetic 2 n -pole channels have the same parametric dependence on the characteristic frequencies of the system. However, in both cases, non-trivial numerical pre-factors cause the electric 2 n+1pole channels to dominate the emission process over the magnetic 2 n -pole channels by many orders of magnitude. These results complete, refine, and, to some extent, contradict the standard paradigm in which electric 2 n+1 -pole and magnetic 2 n -pole couplings are considered to be mutually commensurate [32,33]. Moreover, we have also established that, for some transitions, electric 2 n+2 -pole channels can compete with electric 2 n -pole channels in the spontaneous emission process at short times, as evidenced for instance in the case of the 4F-2P and 5F-4F transitions (see tables 4 and 6, as well as figure 2). This causes significant deviations, in the short-time regime, from the average value of the angular momentum of emitted photons that would be expected from the approximate selection rules.
Our results motivate a re-examination of the relative strengths of electric 2 n+1 -pole and magnetic 2 n -pole channels in atomic transitions. Given the time-symmetry between emission and absorption, our work also motivates further study into the problem of the interaction between OAM-carrying light and simple atomic systems, with potential relevance for quantum information applications. It would also be relevant to try, in a more detailed way, to extend this study to many-electron systems, for which non-E1 transitions are more easily accessible [30,47]. In this case, the D Jr coefficients in the form factor (8) have to be computed numerically.