Alternative method of generating gamma rays with orbital angular momentum
OOU-HET-1088
New method of generating gammarays with orbital angular momentum
Minoru Tanaka ∗ and Noboru Sasao † Department of Physics, Graduate School of Science, Osaka University, Toyonaka,Osaka 560-0043, Japan Research Institute for Interdisciplinary Science, Okayama University, Okayama,700-8530, Japan
February 2, 2021
Abstract
We propose a new method of generating gamma rays with orbital angular mo-mentum (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 OAMare emitted in their deexcitation process. We examine the excitation cross sectionand deexcitation rate.
It is now well known that light (electro-magnetic wave) has angular momentum in ad-dition to linear one. The fact was pointed out by Poynting [1] as early as in 1909, andwas confirmed experimentally by Beth [2] in 1936, who observed the torque exerted ona birefringent plate as the polarization state of the transmitted light was changed. Theexperiment in fact proved the angular momentum associated with spin (the spin angularmomentum). Spin degree-of-freedom is now widely used in various research fields.Light also can have orbital angular momentum (OAM), but this property had notwell exploited until Allen et al. [3] discovered in 1992 that a Laguerre-Gaussian beamcan carry OAM in a well-defined manner. One remarkable feature of such beams is a ∗ Email: [email protected] † Email: [email protected] a r X i v : . [ h e p - ph ] F e b haracteristic helical phase front with phase singularity at the center. Its intensity dis-tribution exhibits an annular profile, in particular, a completely dark spot at the center.Based on this property, lights with OAM are often called “twisted photons” or “opticalvortices” in literatures. Having such a new degree-of-freedom, twisted photon beamsare recognized as an excellent platform for new science and a myriad of applications[4, 5, 6, 7]. They include fundamental physics concerning interactions between particlesand photons [8, 9], quantum optics [10, 11], micro manipulation of particles/materials[12, 13], microscopy and imaging [14, 15, 16], optical data transmission [17, 18, 19], de-tection of astronomical rotating object [20, 21], among others. Now this research fieldis expanding rapidly.Many methods are proposed and being actually used to generate light beams withOAM. In the visible region, the common method is use of fork holograms, spiral phaseplates [22], lens-based mode converters [23], and q-plates [24]. In the X-ray region, high-harmonic radiation from a helical undulator [25, 26, 27] and/or coherent emission fromspirally-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 Comptonbackward scattering [31, 32, 33, 34, 35, 36, 37]. In this energy region, such beams mayplay an indispensable role in investigating nuclear structure, spin puzzle of nucleons [38]and phenomena in astrophysics associated with rotation.In this article, we present a new method of generating high-energy gamma rays withOAM. The method utilizes partially-stripped ions (PSIs) as an energy converter; acceler-ated PSIs absorb and re-emit photons with OAM. When initial PSIs have Lorentz boostfactor of γ , then the energy of photons re-emitted in the backward direction is amplifiedby a factor of 4 γ . Compared with more traditional backward Compton scattering, theprocess has an advantage of having much bigger fundamental cross section; the Rayleighscattering cross section proportional to square of the resonant wavelength versus theThomson scattering cross section proportional to square of the classical electron radius.We note that a similar method has been already proposed for generating intense gammarays without OAM [39].This paper is organized as follows. In the next section, we present a basic theory ofabsorption of photons with OAM by ions. The goal in this section is to calculate theabsorption rate for hydrogen-like PSIs. Calculation is done based on the Dirac theorysince high Z ions are used as a target. Then emission rate of photons from excited PSIsis evaluated; here we are interested in the emission of multi-pole photons, in particularE2 photons. We present the results of our numerical calculations in section 3 and asummary in section 4. Throughout this paper, the natural unit system c = (cid:126) = 1 isused. In this section, we present the formulas that describe the absorption and emission oftwisted photon by a hydrogen-like ion. Before discussing the relevant rates, we summa-2 = | m |= | m |= - - - - - x ( o r y ) β = m = | m |= | m |= - - - - x ( o r y ) β = Figure 1: Angular distributions of the quadrupole radiation of | m | = 0 , β = 0. Right: β = 0 . γ i + I → I ∗ → I + γ f , where I is aboosted 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 ω i = E eg γ (1 + β ) (cid:18) E eg m I (cid:19) (cid:39) E eg γ , (1)where ω i is the angular frequency of the initial photon γ i (in the laboratory frame), β and γ = 1 / (cid:112) − β are the boost factors of the initial ion. An approximate formula forthe case of γ (cid:29) β (cid:39)
1) and E eg /m I (cid:28) I ∗ → I + γ f , is given by ω f = E eg (1 + E eg / m I ) γ + ω i /m I − ( γβ − ω i /m I ) cos θ f (cid:39) E eg γ (1 − β cos θ f ) , (2)where θ f is the polar angle of the emitted photon momentum with respect to the directionof the ion boost, and an approximate formula for ω i , E eg (cid:28) m I is also shown. In theevents of backward scattering, namely θ f = 0, ω f becomes maximal and ω max f (cid:39) γE eg , (3)for γ (cid:29)
1. Thus the energy up-conversion factor is ω max f /ω i (cid:39) γ .The emission process is classified by its multipole nature. Denoting the projection ofangular momentum onto the quantization axis by m , a quadrupole or higher multipoleradiation of | m | (cid:54) = 1 possesses an OAM and a beam of twisted photons is obtained ifthe parent ions are boosted. In the numerical illustration in Sec. 3, we consider thedeexcitation from the 3 d / state of hydrogen-like ions to the ground state, which isdominated by the electric quadrupole (E2) radiation. In Fig. 1, we illustrate the angular3istributions of the quadrupole radiation in the rest frame of the ion (left) and thelaboratory frame in which the ion is accelerated as β = 0 . The distributionof | m | = 2 vanishes on the z axis, which indicates the phase singularity along the beamaxis and exhibits the nature of twisted photons. The Bessel beam is an instance of light beams with OAM. In this work, we suppose thata Bessel beam is irradiated to excite an ion to a state that is able to emit a twistedphoton in its deexcitation process.A Bessel beam propagating along the z axis is represented by the following superpo-sition of plane waves [31, 32]: A µm γ κ γ λ ( t , r ) := (cid:90) a m γ κ γ ( k T ) A µ k λ ( t , r ) d k T (2 π ) , (4)where a m γ κ γ ( k T ) := ( − i ) m γ e im γ ϕ k (cid:115) πκ γ δ ( | k T | − κ γ ) (5)is the weight of the superposition for a given photon transverse wave vector k T . Theplane wave field of wave vector k and helicity λ is given by A µ k λ ( t , r ) := 1 √ ω (cid:15) µλ ( k ) e − i ( ωt − k · r ) , ω = | k | , (6)where (cid:15) µλ ( k ) is the polarization vector. The wave vector of plane wave is parameterizedas 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 , 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 axisof the Bessel beam. We denote the impact parameter of the Bessel beam by b takingthe the position of the ionic nucleus as the origin of the coordinate system. Then, theelectromagnetic field at the electronic position r is given by Eq. (4) with [40] A µ k λ ( t , r ) := 1 √ ω (cid:15) µλ ( k ) e − i ( ωt − k · r − k T · b ) . (7) 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 keVfor heavy ions. In the realistic kinematics of up-conversion in Sec. 3, we employ β (cid:39)
1. The rather small β here ischosen for the visibility of the figure. H I = e α · A , (8)where e = | e | is the unit charge and α represents the Dirac matrices. The absorptionamplitude is given by a matrix element of V = H I e iωt . It is convenient to express the twisted photon amplitude as a superposition of planewave amplitudes like the twisted photon field itself. Then, the absorption amplitude iswritten as [41] M (tw) fi = (cid:104) f | V (tw) | i (cid:105) = (cid:90) d k T (2 π ) a m γ κ γ ( k T ) e − i k T · b M (pl) fi ( θ k , ϕ k ) (9)where M (pl) fi ( θ k , ϕ k ) := (cid:104) f | V (pl) | i (cid:105) is the plane wave amplitude in which the directionof the photon wave vector is specified by θ k and ϕ k with respect to the ionic spinquantization axis (taken to be the z axis). Such an inclined plane wave amplitude isexpressed as M (pl) fi ( θ k , ϕ k ) = e − i ( m f − m i ) ϕ k (cid:88) m (cid:48) f m (cid:48) i d j f m f m (cid:48) f ( θ k ) d j i m i m (cid:48) i ( θ k ) M m (cid:48) f m (cid:48) i , (10)where d jmm (cid:48) ( θ ) is Wigner’s d-function and M m (cid:48) f m (cid:48) i represents the ordinary plane waveamplitude of θ k = ϕ k = 0. Substituting Eq. (10) into Eq. (9), one obtains M (tw) fi = ( − i ) m γ + m i − m f e i ( m γ + m i − m f ) ϕ b (cid:114) κ γ π J m γ + m i − m f ( κ γ b ) M (pl) fi ( θ k , M m f m i ( b ) , (11)where ϕ b denotes the azimuthal angle of b . Here, we consider hydrogen-like ions in the Dirac theory. (See e.g. Ref. [42].) The wavefunction is given by ψ nκm ( r ) = G nκ ( r ) r Ω (cid:96) A jm ( θ, ϕ ) i F nκ ( r ) r Ω (cid:96) B jm ( θ, ϕ ) , (12)where κ = ∓ ( j + 1 / (cid:96) A = j ∓ / (cid:96) B = j ± / (cid:96)jm ( θ, ϕ ) denotes thespinor spherical harmonics. The wave function is normalized as (cid:82) d r | ψ nκm ( r ) | = (cid:82) dr { G nκ ( r ) + F nκ ( r ) } = 1. The electronic state of an ion may be denoted as | n, κ, m (cid:105) .5he plane wave matrix element in the right-hand side of Eq. (10) is expressed as M m f m i = (cid:104) n f , κ f , m f | V (pl) | n i , κ i , m i (cid:105) . (13)Using the Dirac wave function in Eq. (12) and the plane wave field in Eq. (6), we findthat M m f m i = − ie (cid:114) ω ( − j i + m f (cid:113) (2 j f + 1)(2 j i + 1) δ m f ,m i + λ × (cid:88) j,(cid:96) ( − j i (cid:96) √ (cid:96) + 1 C j − m f + m i j f − m f ,j i m i C j − m f + m i (cid:96) − m f + m i + λ, − λ ( − (cid:96) Af (cid:113) (2 (cid:96) Af + 1)(2 (cid:96) Bi + 1) C (cid:96) (cid:96) Af ,(cid:96) Bi (cid:96) Af j f / (cid:96) Bi j i / (cid:96) j I (cid:96)GF − ( − (cid:96) Bf (cid:113) (2 (cid:96) Bf + 1)(2 (cid:96) Ai + 1) C (cid:96) (cid:96) Bf ,(cid:96) Ai (cid:96) Bf j f / (cid:96) Ai j i / (cid:96) j I (cid:96)F G , (14)where C j m j m ,j m is the Clebsch-Gordan coefficient and a b cd e fg h i (15)represents the 9 j symbol, e.g. [43]. The radial integrals are defined by I (cid:96)GF := (cid:90) dr j (cid:96) ( kr ) G f ( r ) F i ( r ) , I (cid:96)F G := (cid:90) dr j (cid:96) ( kr ) F f ( r ) G i ( r ) , (16)where j (cid:96) ( kr ) denotes the spherical Bessel function and k = | k | is the wave number ofthe plane wave photon. The absorption rate of a twisted photon is proportional to the squared amplitude inEq. (11) |M m f m i ( b ) | = ( κ γ / π ) | M (pl) m f m i ( θ k , | J m γ + m i − m f ( κ γ b ) . (17)In experiments where the ion beam is not sufficiently collimated (as virtually all exper-iments we envisage), it is legitimate to average over the impact parameter b . Providedthat the (effective) beam radius R is large enough as κ γ R (cid:29)
1, (fat beam approxima-tion), one obtains 1 πR (cid:90) |M m f m i ( b ) | d b (cid:39) π R | M (pl) m f m i ( θ k , | , (18)6here we have used the asymptotic form of the Bessel function, J m ( z ) (cid:39) (cid:112) /πz cos( z − π/ − mπ/ m γ dependence disappears in this approximation.The photon number flux of the Bessel beam is given by f ( R ) = cos θ k /π R in the fatbeam approximation [32]. With this flux, we obtain the absorption cross section for agiven set of initial and final magnetic quantum numbers as σ ( ω ) = 2cos θ k Γ f / ω + E i − E f ) + Γ f / | M (pl) m f m i ( θ k , | , (19)where E i ( f ) is the energy of the initial (final) state and Γ f denotes the natural width ofthe final state. The photons emitted in multipole radiations of j ≥ H I = − eγ µ A µ in which the electromagnetic field is oneof the multipole fields introduced below describes the multipole emission. The emissionrate of | i (cid:105) → | f (cid:105) + γ is given by dw = 2 π | V fi | δ ( E i − E f − ω ) dω , w = 2 π | V fi | , (20)where V fi = (cid:104) f | H I e − iωt | i (cid:105) is the emission matrix element. We work in the rest frame ofthe ion as in the evaluation of the absorption cross section.We do not employ the long wavelength approximation. The scale of transition wave-lengths is 1 / ( Zα ) m e , and the size of an ion is ∼ /Zαm e . In the case Zα (cid:28)
1, thelong wavelength approximation is legitimate. For heavy ions of Zα = O (1), the longwavelength approximation is questionable. In the Coulomb gauge, the multipole vector potentials are written in terms of the vectorspherical harmonics, Y (cid:96)jm ( n ). We note thatˆ J Y (cid:96)jm = j ( j + 1) Y (cid:96)jm , ˆ J z Y (cid:96)jm = m Y (cid:96)jm , (21)ˆ L Y (cid:96)jm = (cid:96) ( (cid:96) + 1) Y (cid:96)jm , ˆ S Y (cid:96)jm = 2 Y (cid:96)jm . (22)The vector spherical harmonics may be expressed as Y (cid:96)jm ( n ) = (cid:80) m,σ C jm(cid:96)m, σ Y (cid:96)m ( n ) e σ ,where e σ denotes the covariant spherical basis vectors, e ± = ∓ ( e x ± i e y ) / √ e = e z [43].The electric multipole field of angular frequency ω and wave vector k is given by A ωjm ( k ) = 2( π/ω ) / δ ( | k | − ω ) Y ( e ) jm (ˆ k ) , (23) We ignore the width of excitation laser and other broadening effects for simplicity. The Gauss units in Ref. [44] and the Heaviside-Lorentz rationalized units employed here are relatedby α = e = e / (4 π ) and e G A G = e HL A HL . Y ( e ) jm (ˆ k ) := 1 (cid:112) j ( j + 1) ∇ ˆ k Y jm (ˆ k ) = (cid:115) j + 12 j + 1 Y j − jm + (cid:115) j j + 1 Y j +1 jm . (24)The magnetic multipole field is written as A ωjm ( k ) = 2( π/ω ) / δ ( | k | − ω ) Y ( m ) jm (ˆ k ) , (25)where Y ( m ) jm (ˆ k ) := ˆ k × Y ( e ) jm (ˆ k ) = i Y jjm . (26)We note that (cid:90) d Ω ˆ k Y ( e ) ∗ jm (ˆ k ) · Y ( e ) j (cid:48) m (cid:48) (ˆ k ) = (cid:90) d Ω ˆ k Y ( m ) ∗ jm (ˆ k ) · Y ( m ) j (cid:48) m (cid:48) (ˆ k ) = δ jj (cid:48) δ mm (cid:48) . (27)The parities of the electric and magnetic multipole fields are opposite. We assign( − j to the electric multipole field in Eq. (23) and ( − j +1 to the magnetic multipolefield in Eq. (25). For instance, the electric (magnetic) dipole field is parity odd (even).In the coordinate space, the field is given by A ωjm ( r ) = (cid:90) dk (2 π ) A ωjm ( k ) e i k · r . (28) Since the electric multipole field contains two components of orbital angular momentum, (cid:96) = j ±
1, as seen in Eq. (24), we rearrange them by a gauge transformation [44]. Agauge transformation results in the following vector and scalar potentials: A ωjm ( k ) = 2( π/ω ) / δ ( | k | − ω ) (cid:110) Y ( e ) jm (ˆ k ) + C ˆ k Y jm (ˆ k ) (cid:111) , (29) φ ωjm ( k ) = 2( π/ω ) / δ ( | k | − ω ) CY jm (ˆ k ) , (30)where C is a gauge parameter. Choosing C = − (cid:112) ( j + 1) /j , we obtain A ( e ) ωjm ( k ) = (cid:112) (2 j + 1) /j π/ω ) / δ ( | k | − ω ) Y j +1 jm (ˆ k ) , (31) φ ( e ) ωjm ( k ) = − (cid:112) ( j + 1) /j π/ω ) / δ ( | k | − ω ) Y jm (ˆ k ) . (32)In the long wavelength approximation, φ ( e ) ωjm dominantly contributes and A ( e ) ωjm ( k ) doessubdominantly. 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 isgiven by V ( φ ) fi = − e (cid:90) d rψ † f ( r ) ψ i ( r ) (cid:90) d k (2 π ) φ ( e ) ∗ ωjm ( k ) e − i k · r (33)= e (cid:115) j + 1 j √ ω π / (cid:90) d rψ † f ( r ) ψ i ( r ) (cid:90) d Ω k e − i k · r Y ∗ jm (ˆ k ) , (34)8here ψ i ( f ) ( r ) represents the initial (final) wave function given in Eq. (12). The angularintegral over Ω k is performed using e i k · r = 4 π ∞ (cid:88) (cid:96) =0 m = (cid:96) (cid:88) m = − (cid:96) i (cid:96) j (cid:96) ( kr ) Y ∗ (cid:96)m (ˆ k ) Y (cid:96)m ( ˆ r ) , (35)and one obtains V ( φ ) fi = e ( − i ) j (cid:115) j + 1 j (cid:114) ωπ (cid:90) d rψ † f ( r ) ψ i ( r ) j j ( kr ) Y ∗ jm ( ˆ r ) . (36)Substituting the Dirac wave function in Eq. (12) and integrating over the remainingangular variables, we obtain V ( φ ) fi = e π √ ω i j ( − j f + m f + j i +1 / (cid:115) ( j + 1)(2 j f + 1)(2 j i + 1) j (2 j + 1) C j − m f + m i j f − m f ,j i m i × (cid:20)(cid:26) (cid:96) Af (cid:96) Ai jj i j f / (cid:27) (cid:113) (2 (cid:96) Af + 1)(2 (cid:96) Ai + 1) C j (cid:96) Af ,(cid:96) Ai I jGG + (cid:26) (cid:96) Bf (cid:96) Bi jj i j f / (cid:27) (cid:113) (2 (cid:96) Bf + 1)(2 (cid:96) Bi + 1) C j (cid:96) Bf ,(cid:96) Bi I jF F (cid:21) , (37)where (cid:26) a b cd e f (cid:27) (38)represents the 6 j symbol and the radial integrals are given by I jGG := (cid:90) dr j (cid:96) ( kr ) G f ( r ) G i ( r ) , I jF F := (cid:90) dr j (cid:96) ( kr ) F f ( r ) F i ( r ) . (39)As for the contribution of the vector potential in Eq. (31), the k integration results in V ( A ) fi = e ( − i ) j +1 (cid:115) j + 1 j (cid:114) ωπ (cid:90) d rj j +1 ( kr ) ψ † f ( r ) α ψ i ( r ) · Y j +1 ∗ jm ( ˆ r ) . (40)Integrating over r , we find V ( A ) fi = e π √ ω i j ( − j i + m f (cid:115) j + 1)(2 j f + 1)(2 j i + 1) j C j − m f + m i j f − m f ,j i m i × (cid:96) Af j f / (cid:96) Bi j i / j + 1 j ( − (cid:96) Af (cid:113) (2 (cid:96) Af + 1)(2 (cid:96) Bi + 1) C j +10 (cid:96) Af ,(cid:96) Bi I j +1 GF − (cid:96) Bf j f / (cid:96) Ai j i / j + 1 j ( − (cid:96) Bf (cid:113) (2 (cid:96) Bf + 1)(2 (cid:96) Ai + 1) C j +10 (cid:96) Bf ,(cid:96) Ai I j +1 F G . (41)The total amplitude is V fi = V ( φ ) fi + V ( A ) fi . 9 .3.3 Magnetic multipole radiations in the Dirac theory The magnetic multipole radiation is described by the vector potential in Eq. (25). Theemission matrix element is evaluated in the similar manner as V ( A ) fi above and we obtain V fi = e π √ ω i j ( − j i + m f (cid:113) j f + 1)(2 j i + 1) C j − m f + m i j f − m f ,j i m i × (cid:96) Af j f / (cid:96) Bi j i / j j ( − (cid:96) Af (cid:113) (2 (cid:96) Af + 1)(2 (cid:96) Bi + 1) C j (cid:96) Af ,(cid:96) Bi I jGF − (cid:96) Bf j f / (cid:96) Ai j i / j j ( − (cid:96) Bf (cid:113) (2 (cid:96) Bf + 1)(2 (cid:96) Ai + 1) C j (cid:96) Bf ,(cid:96) Ai I jF G . (42) As an illustration, we consider the excitation of H-like ions from the ground state 1s / to the excited state of 3d / by the Bessel beam and the successive deexcitation back tothe ground state by the E2 emission. If the magnetic quantum number of the groundstate differs from that of the exited state by two (or larger), this process is not possiblewith the plane wave beam nor the dipole emission. The resonant excitation rate is described by the absorption cross section in Eq. (19)with ω = E eg = E (3d / ) − E (1s / ) and presented in Fig. 2 as a function of the pitchangle θ k . We choose m i = 1 / m f = 5 / λ = 1.The target ion is H-like Pb ( Z = 82) in the left panel. The level splitting is E eg =91 . ω i = 10 eV, and this implies γ = 4 . × and ω max f = 0 .
834 GeV. The width of the excited state is dominated bythe major E1 rate, Γ f (cid:39) Γ(3d / → / ) = 1 .
89 eV.In the right panel, the ion is H-like Ne ( Z = 10) and we take ω i = 1 eV. The relevantparameters in this case are γ = 606, ω max f = 1 .
47 MeV and Γ f (cid:39) . × − eV. Themuch larger cross section of Ne than Pb is due to the narrow width of the 3d / state.The pitch angle θ k can be O (1) in the laboratory frame. While, in the rest frameof the ion, it is O (1 /γ ) because the transverse momentum k T of twisted photons isinvariant under the Lorentz boost. We have chosen the ranges of the horizontal axis inFig. 2 following this observation and the difference of ω i ’s for the two species of ion. Theabsorption cross section is proportional to θ k for | θ k | (cid:28) θ k = 0 corresponds to a plane wave, and the cross section vanishes since the process of | m f − m i | > .0 0.1 0.2 0.3 0.4020406080 θ k × σ [ m b ] Pb, 1s / → / θ k × σ [ b ] Ne, 1s / → / Figure 2: Absorption cross section of 1s / , m =1 / → / , m =5 / . The photon helicity is λ = 1. The target ion is Pb (Ne) in the left (right) panel with ω i = 10(1) eV.We note the difference in the units of the vertical axes. E1E21 2 5 10 20 5010 - - - - Γ [ e V ] - - E f r a c t i on Figure 3: Left: E2 rate compared to E1. Right: ratio of E2 to E1.
The E2 transition rate from 3d / to 1s / is given by Eqs. (20), (37) and (41) with j = 2.In the left panel of Fig. 3, we present the E2 transition rate of H-like ion as a functionof Z . The dominant E1 rate from 3d / to 2p / is also shown for comparison. The ratioof the E2 rate to the E1 rate, which gives an approximate branching fraction to emit anE2 photon, is given in the right panel of Fig. 3. We observe that the E2 rate dependson Z as Z in good precision, while the E1 is approximately proportional to Z , so thatheavier ions exhibit larger branching fractions of emitting a twisted photon.For H-like Pb, we find Γ(3d / → / ) = 8 . × − eV, so that the branchingfraction of the E2 transition is Br(3d / → / ) (cid:39) . × − . As for H-like Ne,Γ(3d / → / ) = 3 . × − eV, and Br(3d / → / ) (cid:39) . × − . In the excitation process in Sec. 3.1, the final state of m f = 3 /
2, in addition to m f = 5 / / ( m = 3 /
2) state deexcites11o the ground state (1s / ) by the E2 emission of m = 1. Its emission pattern is shownin the right panel of Fig. 1 (the red dashed line) and exhibits no phase singularity onthe z axis. So that, the photon in this process is not twisted and could be a source ofbackgrounds.It turns out that the m f = 3 / σ m f =5 / /σ m f =3 / (cid:39) × − (3 × − ) for H-like Pb (Ne) in the case of θ = 1 /γ . This isbecause the process of | m f − m i | = 1 is possible with the plane wave and the cross sectiondoes not vanish even if θ k = 0. We note that the above ratios are O ( θ k ) = O (1 /γ ). 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 actualexperimental configurations, in particular accelerators and incident lasers. Consideringcurrent technologies, we expect reasonable flux useful to a variety of physics. As tothe backgrounds, we expect two major backgrounds in this method as discussed in theprevious section: one is due to the process from d-states to p-states and the other from m f = 3 / B T in the productionregion. 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. Notethat B T is amplified by the Lorentz boost factor γ when seen by the ions. Also notethat it is necessary to rotate the spin from transverse to longitudinal. Designing actualspin rotation systems requires more detailed studies, which is underway currently.In summary, we have proposed a new method of generating gamma rays with OAM.It exploits accelerated partially-stripped ions as an energy upconverter. Relativisticcalculations are performed to calculate the excitation cross section and deexcitation ratefor hydrogen-like ions, and their properties including flux and possible backgrounds arediscussed. For example, assuming 10 W laser power with 1 mm focusing area, 10 bunch/s with 10 H-likeions/bunch and a 10 m-long interaction section, we expect an order of 10 excitations per second to m f = 5 / Z = 10. cknowledgments The work of MT is supported in part by JSPS KAKENHI Grant Numbers JP 16H03993and 18K03621. The work of NS is supported in part by JSPS KAKENHI Grant NumberJP 16H02136.