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Alternative method of generating gamma rays with orbital angular momentum

We propose a new method of generating gamma rays with orbital angular momentum (OAM). Accelerated partially-stripped ions are used as an energy up-converter. Irradiating an optical laser beam with OAM on ultrarelativistic ions, they are excited to a state of large angular momentum. Gamma rays with OAM are emitted in their deexcitation process. We examine the excitation cross section and deexcitation rate.


Introduction
It is now well known that light (electro-magnetic wave) has angular momentum in addition to linear one. The fact was pointed out by Poynting [1] as early as in 1909, and was confirmed experimentally by Beth [2] in 1936, who observed the torque exerted on a birefringent plate as the polarization state of the transmitted light was changed. The experiment in fact proved the angular momentum associated with spin (the spin angular momentum). Spin degree-of-freedom is now widely used in various research fields.
Light also can have orbital angular momentum (OAM), but this property had not well exploited until Allen et al. [3] discovered in 1992 that a Laguerre-Gaussian beam can carry OAM in a well-defined manner. One remarkable feature of such beams is a characteristic helical phase front with phase singularity at the center. Its intensity distribution exhibits an annular profile, in particular, a completely dark spot at the center.
Many methods are proposed and being actually used to generate light beams with OAM. In the visible region, the common method is use of fork holograms, spiral phase plates [22], lens-based mode converters [23], and q-plates [24]. In the X-ray region, highharmonic radiation from a helical undulator [25,26,27] and/or coherent emission from spirally-bunched electrons produced by combination of a laser and undulator [28,29,30] seem very promising methods. Less well established is generation of twisted gamma rays. Almost all methods proposed so far utilize up-conversion of wavelength by Compton backward scattering [31,32,33,34,35,36,37]. In this energy region, such beams may play an indispensable role in investigating nuclear structure, spin puzzle of nucleons [38] and phenomena in astrophysics associated with rotation.
In this article, we study an alternative method of generating high-energy gamma rays with OAM. The method utilizes partially-stripped ions (PSIs) as an energy converter; accelerated PSIs absorb and re-emit photons with OAM. This interesting possibility is suggested in Ref. [39] in connection with a proposal of Gamma Factory [40,41]. When initial PSIs have Lorentz boost factor of γ, then the energy of photons re-emitted in the backward direction is amplified by a factor of 4γ 2 . Compared with more traditional backward Compton scattering, the process has an advantage of having much bigger fundamental cross section; the Rayleigh scattering cross section proportional to square of the resonant wavelength versus the Thomson scattering cross section proportional to square of the classical electron radius. This paper is organized as follows. In the next section, we present a basic theory of absorption of photons with OAM by ions. The goal in this section is to calculate the absorption rate for hydrogen-like PSIs. Calculation is done based on the Dirac theory since high Z ions are used as a target. Then emission rate of photons from excited PSIs is evaluated; here we are interested in the emission of multipole photons, in particular E2 photons. We present the results of our numerical calculations in section 3 and a summary in section 4. Throughout this paper, the natural unit system c = = 1 is used.

Theory and formulas
In this section, we present the formulas that describe the absorption and emission of twisted photon by a hydrogen-like ion. Before discussing the relevant rates, we summarize the kinematics of the energy up-conversion.
We consider the resonant Rayleigh scattering γ i + I → I * → I + γ f , where I is a boosted ion in its ground state and I * represents an excited ion. The energy splitting of I and I * is denoted by E eg . Assuming the head-on collision, the resonant condition of the excitation process, γ i + I → I * , is expressed as where ω i is the angular frequency of the initial photon γ i (in the laboratory frame), β and γ = 1/ 1 − β 2 are the boost factors of the initial ion. An approximate formula for the case of γ 1(β 1) and E eg /m I 1 is also shown. The energy of the photon in the emission process, I * → I + γ f , is given by where θ f is the polar angle of the emitted photon momentum with respect to the direction of the ion boost, and an approximate formula for ω i , E eg m I is also shown. In the events of backward scattering, namely θ f = 0, ω f becomes maximal and for γ 1. Thus the energy up-conversion factor is ω max f /ω i 4γ 2 . The emission process is classified by its multipole nature. Denoting the projection of angular momentum onto the quantization axis by m, a quadrupole or higher multipole radiation of |m| = 1 possesses an OAM and a beam of twisted photons is obtained if the parent ions are boosted. In particular, beams of |m| ≥ 2 are nontrivial in the sense that one can access target states of larger magnetic quantum numbers with such beams. It is impossible to make a transition of |∆m| ≥ 2 with a plane wave in single-photon absorption. 1 In the numerical illustration in Sec. 3, we consider the deexcitation from the 3d 5/2 state of hydrogen-like ions to the ground state, which is dominated by the electric quadrupole (E2) radiation. In Fig. 1, we illustrate the angular distributions of the quadrupole radiation in (a) the rest frame of the ion and (b) the laboratory frame in which the ion is accelerated as β = 0.6. 2 The distribution of |m| = 2 vanishes on the z axis, which indicates the phase singularity along the beam axis and exhibits the nature of twisted photons.

Bessel beam
The Bessel beam is an instance of light beams with OAM. In this work, we suppose that a Bessel beam is irradiated to excite an ion to a state that is able to emit a twisted photon in its deexcitation process.
A Bessel beam propagating along the z axis is represented by the following superposition of plane waves [31,32]: where is the weight of the superposition for a given photon transverse wave vector k T . The plane-wave field of wave vector k and helicity λ is given by where µ λ (k) is the polarization vector. The wave vector of plane wave is parameterized as k = k(sin θ k cos ϕ k , sin θ k sin ϕ k , cos θ k ), so that k · x = k z z + k T · x T , k z = k cos θ k and k T = k sin θ k (cos ϕ k , sin ϕ k , 0). The polar angle θ k is called the pitch angle. We note that m γ represents the eigenvalue of J z , the z component of the total angular momentum, and κ γ = |k T | = |k| sin θ k .
When a Bessel beam is irradiated on an ion, the target ion is not always on the axis of the Bessel beam. We denote the impact parameter of the Bessel beam by b taking the position of the ionic nucleus as the origin of the coordinate system. Then, the electromagnetic field at the electronic position r is given by Eq. (4) with [42]

Absorption of twisted photons by hydrogen-like heavy ions
In the following, we evaluate the absorption cross section in the rest frame of the ion. In this frame the resonant energy of the incident photon is E eg , which is of order keV for heavy ions. The interaction of the electron with the radiation field in the Dirac theory is given by where e = |e| is the unit charge and α represents the Dirac matrices. The absorption amplitude is given by a matrix element of V = H I e iωt .

Amplitude superposition
It is convenient to express the twisted photon amplitude as a superposition of plane wave amplitudes like the twisted photon field itself. Then, the absorption amplitude is written as [43] M |i is the plane wave amplitude in which the direction of the photon wave vector is specified by θ k and ϕ k with respect to the ionic spin quantization axis (taken to be the z axis). Such an inclined plane wave amplitude is expressed as where d j mm (θ) is Wigner's d-function and M m f m i represents the ordinary plane wave amplitude of θ k = ϕ k = 0. Substituting Eq. (10) into Eq. (9), one obtains where ϕ b denotes the azimuthal angle of b.

Plane wave amplitude in the Dirac theory
Here, we consider hydrogen-like ions in the Dirac theory. (See e.g. Ref. [44].) The wave function is given by where κ = ∓(j + 1/2), A = j ∓ 1/2, B = j ± 1/2 and Ω jm (θ, ϕ) denotes the spinor spherical harmonics. The wave function is normalized as d 3 r |ψ nκm (r)| 2 = dr {G 2 nκ (r) + F 2 nκ (r)} = 1. The electronic state of an ion may be denoted as |n, κ, m . The plane wave matrix element in the right-hand side of Eq. (10) is expressed as Using the Dirac wave function in Eq. (12) and the plane wave field in Eq. (6), we find that where C j 3 m 3 j 1 m 1 ,j 2 m 2 is the Clebsch-Gordan coefficient and represents the 9j symbol, e.g. [45]. The radial integrals are defined by where j (kr) denotes the spherical Bessel function and k = |k| is the wave number of the plane wave photon.

Absorption cross section
The absorption rate of a twisted photon is proportional to the squared amplitude in Eq.
In experiments where the ion beam is not sufficiently collimated (as virtually all experiments we envisage), it is legitimate to average over the impact parameter b. Provided that the (effective) beam radius R is large enough as κ γ R 1, (fat beam approximation), one obtains where we have used the asymptotic form of the Bessel function, J m (z) 2/πz cos(z − π/4 − mπ/2). We note that the m γ dependence disappears in this approximation.
The photon number flux of the Bessel beam is given by f (R) = cos θ k /π 2 R in the fat beam approximation [32]. With this flux, we obtain the absorption cross section for a given set of initial and final magnetic quantum numbers as where E i(f ) is the energy of the initial (final) state and Γ f denotes the natural width of the final state. 3

Emission of twisted photons by hydrogen-like heavy ions
The photons emitted in multipole radiations of j ≥ 2 are twisted in the sense that they have nonzero orbital angular momentum. Accordingly, we consider the multipole radiations of hydrogen-like heavy ions. The interaction hamiltonian H I = −eγ µ A µ in which the electromagnetic field is one of the multipole fields introduced below describes the multipole emission. The emission rate of |i → |f + γ is given by where V f i = f |H I e −iωt |i is the emission matrix element. We work in the rest frame of the ion as in the evaluation of the absorption cross section. We do not employ the long wavelength approximation. The scale of transition wavelengths is 1/(Zα) 2 m e , and the size of an ion is ∼ 1/Zαm e . In the case Zα 1, the long wavelength approximation is legitimate. For heavy ions of Zα = O(1), the long wavelength approximation is questionable.

Multipole fields
In the Coulomb gauge, the multipole vector potentials are written in terms of the vector spherical harmonics, Y jm (n). We note that The vector spherical harmonics may be expressed as Y jm (n) = m,σ C jm m,1σ Y m (n)e σ , where e σ denotes the covariant spherical basis vectors, e ±1 = ∓(e x ± ie y )/ √ 2 and e 0 = e z [45].
The electric multipole field of angular frequency ω and wave vector k is given by 4 where The magnetic multipole field is written as where Y (m) We note that The parities of the electric and magnetic multipole fields are opposite. We assign (−1) j to the electric multipole field in Eq. (23) and (−1) j+1 to the magnetic multipole field in Eq. (25). For instance, the electric (magnetic) dipole field is parity odd (even).
In the coordinate space, the field is given by 5

Electric multipole radiations in the Dirac theory
Since the electric multipole field contains two components of orbital angular momentum, = j ± 1, as seen in Eq. (24), we rearrange them by a gauge transformation [46]. A gauge transformation results in the following vector and scalar potentials: where C is a gauge parameter. Choosing C = − (j + 1)/j, we obtain 4 The Gauss units in Ref. [46] and the Heaviside-Lorentz rationalized units employed here are related by α = e 2 G = e 2 HL /(4π) and e G A G = e HL A HL . 5 The time dependence of e −iωt is implicit.
In the long wavelength approximation, φ (e) ωjm dominantly contributes and A (e) ωjm (k) does subdominantly. As we mentioned above, we consider both contributions for heavy ions.
The contribution of the scalar potential in Eq. (32) to the emission matrix element is given by where ψ i(f ) (r) represents the initial (final) wave function given in Eq. (12). The angular integral over Ω k is performed using and one obtains Substituting the Dirac wave function in Eq. (12) and integrating over the remaining angular variables, we obtain where a b c d e f (38) represents the 6j symbol and the radial integrals are given by As for the contribution of the vector potential in Eq. (31), the k integration results in Integrating over r, we find The total amplitude is

Magnetic multipole radiations in the Dirac theory
The magnetic multipole radiation is described by the vector potential in Eq. (25). The emission matrix element is evaluated in the similar manner as V (A) f i above and we obtain

Numerical results
As an illustration, we consider the excitation of H-like ions from the ground state 1s 1/2 to the excited state of 3d 5/2 by the Bessel beam and the successive deexcitation back to the ground state by the E2 emission. If the magnetic quantum number of the ground state differs from that of the exited state by two (or larger), this process is not possible with the plane-wave beam nor the dipole emission. For instance, one may consider 2p 3/2 instead of 3d 5/2 . The excitation to the state of |m| = 3/2 is possible with the plane-wave beam. This is an E1 transition and its rate is significant. However, the subsequent deexcitation from 2p 3/2 (m = ±3/2) to 1s 1/2 (m = ∓1/2), which generates a twisted gamma ray of |m| = 2, is the magnetic quadrupole (M2) transition. It turns out that, even for Pb, this M2 rate is suppressed by a factor of ∼ 10 3 compared to the dominant E1 emission of an untwisted photon. This E1 photon becomes a serious background because its energy is the same as the M2 photon.

Excitation rate with twisted photons
The resonant excitation rate is described by the absorption cross section in Eq. (19) with ω = E eg = E(3d 5/2 ) − E(1s 1/2 ) and presented in Fig. 2 as a function of the pitch angle θ k . We choose m i = 1/2, m f = 5/2 and λ = 1. The target ion is H-like Pb (Z = 82) in Fig. 2(a). The level splitting is E eg = 91.3 keV. The energy of the excitation laser is taken as ω i = 10 eV, and this implies γ = 4.57 × 10 3 and ω max f = 0.834 GeV. The width of the excited state is dominated by the major E1 rate, Γ f Γ(3d 5/2 → 2p 3/2 ) = 1.89 eV.
In Fig. 2(b), the ion is H-like Ne (Z = 10) and we take ω i = 1 eV. The relevant parameters in this case are γ = 606, ω max f = 1.47 MeV and Γ f 4.26 × 10 −4 eV. The much larger cross section of Ne than Pb is due to the narrow width of the 3d 5/2 state.
The pitch angle θ k can be O(1) in the laboratory frame. While, in the rest frame of the ion, it is O(1/γ) because the transverse momentum k T of twisted photons is invariant under the Lorentz boost. We have chosen the ranges of the horizontal axis in Fig. 2 following this observation and the difference of ω i 's for the two species of ion. The absorption cross section is proportional to θ 2 k for |θ k | 1 as seen in Fig. 2. The case of θ k = 0 corresponds to a plane wave, and the cross section vanishes since the process of |m f − m i | > 1 is impossible with the plane wave as mentioned above.

Characteristics of photons generated by the method
In the excitation process in Sec. 3.1, the final state of m f = 3/2, in addition to m f = 5/2, is also possible with the same Bessel beam. Then, the 3d 5/2 (m = 3/2) state deexcites to the ground state (1s 1/2 ) by the E2 emission of m = 1. Its emission pattern is shown in Fig. 1(b) (the red dashed line) and exhibits no phase singularity on the z axis. So that, the photon in this process is not twisted and could be a source of backgrounds.
It turns out that the m f = 3/2 process dominates the absorption cross section as σ m f =5/2 /σ m f =3/2 6 × 10 −8 (3 × 10 −6 ) for H-like Pb (Ne) in the case of θ = 1/γ. This is because the process of |m f −m i | = 1 is possible with the plane wave and the cross section does not vanish even if θ k = 0. We note that the above ratios are O(θ 2 k ) = O(1/γ 2 ).

Summary
We first point out two important aspects of the method we proposed in this article: one is its final flux of gamma rays with OAM and the other is associated backgrounds. As shown in Fig. 2, the absorption cross sections are in a range of a few tens of mb (for Z = 82) to b (for Z = 10). The achievable flux depends heavily upon actual experimental configurations, in particular accelerators and incident lasers. Considering current technologies, we expect reasonable flux useful to a variety of physics. 7 As to the backgrounds, we expect two major backgrounds in this method as discussed in the previous section: one is due to the process from d-states to p-states and the other from m f = 3/2 of the d-states to s-states. The former type of the backgrounds has a different energy from that of the signal, i.e. the gamma rays with OAM. Whether they are tolerable or not depends upon details of particular experiments, for example detectors employed or reactions in interests may or may not be sensitive to such backgrounds. In general, however, it is desirable to reduce them; from this view point, we believe it important to consider other ion types than the hydrogen-like ions. On the other hand, the second type of the backgrounds has the same energy with the signal. In this case, it is highly recommended to eliminate or reduce these backgrounds. One way to eliminate such backgrounds is to introduce a transverse magnetic field B T in the production region. The Zeeman effect splits each magnetic quantum number m f , one of which (i.e. m f = 5/2) can be selected by choosing the frequency of the excitation photons. Note that B T is amplified by the Lorentz boost factor γ when seen by the ions. Also note that it is necessary to rotate the spin from transverse to longitudinal. Designing actual spin rotation systems requires more detailed studies, which is underway currently.
In summary, we have studied an alternative method of generating gamma rays with OAM. It exploits accelerated partially-stripped ions as an energy upconverter. Relativistic calculations are performed to calculate the excitation cross section and deexcitation rate for hydrogen-like ions, and their properties including flux and possible backgrounds are discussed.