Correlated two-photon emission by transitions of Dirac-Volkov states in intense laser fields: QED predictions

In an intense laser field, an electron may decay by emitting a pair of photons. The two photons emitted during the process, which can be interpreted as a laser-dressed double Compton scattering, remain entangled in a quantifiable way: namely, the so-called concurrence of the photon polarizations gives a gauge-invariant measure of the correlation of the hard gamma rays. We calculate the differential rate and concurrence for a backscattering setup of the electron and photon beam, employing Volkov states and propagators for the electron lines, thus accounting nonperturbatively for the electron-laser interaction. The nonperturbative results are shown to differ significantly compared to those obtained from the usual double Compton scattering.

Figure 1

The six Feynman diagrams contributing to (ordinary) double Compton scattering (initial electron four-momentum $p_i$, final four-momentum $p_f$). The electron line is denoted by a single customary fermion line. The frequencies of the two emitted photons with wave four-vectors $k_b$ and $k_c$ may be different and thus may be their wavelengths; this is explicitly indicated in the panel.

Figure 2

Feynman diagram for the two-photon decay of a Dirac–Volkov state. The electron line is dressed by the laser field and denoted by a zigzag laser photon superimposed on the fermion line.

Figure 3

Polarization vectors according to Eq. (24) for fixed, small $\theta \approx 0$, i.e., for a photon propagating in the positive $z$ direction. For example, we have at $\psi =\pi /2$ the two vectors ${ϵ}^1=\left(0,0,\text{cos}\theta ,-\text{sin}\theta \right)\approx \left(0,0,1,0\right)$ and ${ϵ}^2=\left(0,-1,0,0\right)$.

Figure 4

Clarification of the index $s$. Shown above is one of the contributing Feynman diagrams in the perturbative picture, where the laser photons are inserted one by one. The net number of laser mode absorbed photons in this case is $n=1$. The propagator momentum is $p=q_i-k_b+s\varkappa$, so that $s$ counts the net number of absorbed photons before emitting photon $k_c$, i.e., the momentum at the position of the label “$p$.” For the above diagram, $s=0$. Although $n$ must be positive for a net two-photon emission process, $s$ may be negative, and to get the total amplitude for fixed $n$, one should sum all diagrams of this kind with $s$ ranging from $-\infty$ to $+\infty$.

Figure 5

(Color online) Panel (a) shows ${\omega }_b^{\text{res}1}$ [Eq. (38)] and ${\omega }_b^{\text{res}2}$ [Eq. (39)], as a function of ${\theta }_b$. Recall that we write ${\mathit{k}}_b={\omega }_b\left(\text{sin}{\theta }_b\text{sin}{\psi }_b,\text{sin}{\theta }_b\text{cos}{\psi }_b,\text{cos}{\theta }_b\right)$, i.e., ${\theta }_b$ is the angle between ${\mathit{k}}_b$ and ${\mathit{p}}_i$. The parameters employed are $\omega =2.5\text{eV}$, $\xi =1$, $E_i={10}^{3}m$, ${\psi }_b={\psi }_c=0$, and ${\theta }_c={10}^{-3}$. In panel (b), the resonance position of the first harmonic at ${\theta }_b={10}^{-3}$ is plotted as a function of $n$. For ${\omega }_b^{\text{res}1}$, the first harmonic (the resonance at lowest possible ${\omega }_b$) means $s=1$, and since for ${\omega }_b^{\text{res}2}$ the value of $n-s$ tells us the order of the resonance, we have set $s=n-1$ for this curve. In fact, for large values of $n$, the resonance ${\omega }_b^{\text{res}2}$ with $s=n-1$ shifts down to low photon energies, so that there will be resonances for any photon energy ${\omega }_b>0$. However, these higher-order resonances will be suppressed by a large-order Bessel function, and effectively, one can say that the higher-order resonances will not contribute provided $\xi$ is not too large $\left(\sim 1\right)$.

Figure 6

(Color online) We illustrate the resonances of the fully differential rate (42) as a function of ${\omega }_b$. The parameters used are $\xi =1$, $E_i={10}^{3}m$, ${ϵ}_b={ϵ}_b^1$, ${ϵ}_c={ϵ}_c^1$, ${\theta }_c=0.002$, ${\psi }_b={\psi }_c=0$. In panels (a) and (b), a fixed value of ${\theta }_b=0.001$ is used, corresponding to a cut along the dashed line in (c). In panel (c), the color coding indicates the value of the decadic logarithm ${\text{log}}_{10}\frac{d\dot{W}}{d{\omega }_{b}d{\Omega }_{b}d{\Omega }_c}$, where the argument of the logarithm is measured in units of ${\text{s}}^{-1}{\text{sr}}^{-2}{\text{MeV}}^{-1}$. The regularization method employed is given by a finite laser pulse duration as in Eq. (44), with $\tau ={10}^4/\omega$. In the inset of panel (b), both method (43) [Method 1] and method (44) [Method 2] are shown for comparison. For ${\omega }_b<2\text{MeV}$, the application of the two methods yields numerically indistinguishable curves. The first Compton harmonic [the lowest bright curve in panel (c)] is broken at ${\theta }_b\approx 1.2\times {10}^{-3}$, which can be understood as the point where $\mathit{a}\cdot {\mathit{ϵ}}_b=0$ in the rest frame of the electron [in this frame, and in the gauge ${ϵ}_b=\left(0,{ϵ}_b\right)$, the Thomson cross section is proportional to ${\left|\mathit{a}\cdot {\mathit{ϵ}}_b\right|}^2$].

Figure 7

(Color online) Contour plot of the differential rate for different final photon polarizations. The color coding indicates the value of the differential rate $\frac{d\dot{W}}{d{\omega }_{b}d{\Omega }_{b}d{\Omega }_c}$, on linear scale, in units of ${\text{s}}^{-1}{\text{sr}}^{-2}{\text{MeV}}^{-1}$. The upper two rows [(a)–(f)] display the result of the nonperturbative formula, and the lower two rows [(g)–(l)] show the perturbative results. The parameters used are $\xi =1$, $\omega =2.5\text{eV}$, $E_i={10}^{3}m$, ${\omega }_b=1\text{MeV}$, and ${\theta }_b=2{\theta }_c={10}^{-3}$. In this case, ${\omega }_c$ is fixed by the scattering geometry but still depends on the number $n$ of exchanged photons; we present the differential rate summed over all $n$ and thus suppose that the energy ${\omega }_c$ is unobserved. For the polarizations, we have linear laser polarization in (a)–(c) and (g)–(i) (first and third row), and circular laser polarization in (d)–(f) and (j)–(l) (second and fourth row). In (a), (d), (g), and (j) (first column) we have ${ϵ}_b={ϵ}_b^1$, ${ϵ}_c={ϵ}_c^1$, where we recall that the polarization vectors are defined in Eq. (24). In (b), (e), (h), and (k) (middle column) we have ${ϵ}_b={ϵ}_b^1$, ${ϵ}_c={ϵ}_c^2$, and panels (c), (f), (i) and (l) (right column) show the differential rate summed over all possible polarizations of the final photons.

Figure 8

(Color online) Contour plot of the differential rate as a function of the polar angles ${\theta }_b$ and ${\theta }_c$. The color coding indicates the value of the decadic logarithm ${\text{log}}_{10}\frac{d\dot{W}}{d{\omega }_{b}d{\Omega }_{b}d{\Omega }_c}$, with the differential rate given in units of ${\text{s}}^{-1}{\text{sr}}^{-2}{\text{MeV}}^{-1}$. As in Fig. 7, the upper two rows [(a)–(f)] show nonperturbative, and the lower two rows [(g)–(l)] perturbative results, respectively. The parameters used are $\xi =1$, $\omega =2.5\text{eV}$, $E_i={10}^{3}m$, ${\omega }_b=1\text{MeV}$, and ${\psi }_b=0$, ${\psi }_c=\pi$. For the polarizations, we have linear laser polarization in (a)–(c) and (g)–(i) (first and third row), and circular laser polarization in (d)–(e) and (j)–(l) (second and fourth row). In (a), (d), (g), and (j) (left column) we have ${ϵ}_b={ϵ}_b^1$, ${ϵ}_c={ϵ}_c^1$, In (b), (e), (h), and (k) (middle column) we have ${ϵ}_b={ϵ}_b^1$, ${ϵ}_c={ϵ}_c^2$, and panels (c), (f), (i) and (l) (right column) show the differential rate summed over the final photon polarizations. Note that from Fig. 3, we have ${ϵ}_b^1\cdot {ϵ}_c^1\approx 1$ and ${ϵ}_b^1\cdot {ϵ}_c^2\approx 0$ here.

Figure 9

(Color online) Demonstration of the multiphoton character of the pair-creation rate, for linear and circular polarization of the laser field. In (a), ${\psi }_b=0$, ${\psi }_c=\pi$, and the final polarization is fixed to ${ϵ}_b={ϵ}_b^1$, ${ϵ}_c={ϵ}_c^1$, while in (b) we have ${\psi }_b=\pi /2$, ${\psi }_c=3\pi /2$, and ${ϵ}_b={ϵ}_b^1$, ${ϵ}_c={ϵ}_c^2$. Otherwise the parameters are the same as in Fig. 7.

Figure 10

(Color online) The integrated rate (50), defined in Eq. (50) for $\omega =2.5\text{eV}$ and $E_i={10}^{3}m$. The figure compares the nonperturbative formula for linear laser polarization [Eq. (42)], for circular laser polarization [Eq. (31)], and the perturbative expression [Eq. (48)].

Figure 11

(Color online) Contour plot of concurrence as a function of the azimuth angles ${\psi }_b$ and ${\psi }_c$. The color coding indicates the value of the concurrence $C\left({\rho }_f\right)$ [see Eq. (60)]. Panel (a) shows linear laser polarization, and (b) circular laser polarization, both calculated with the nonperturbative expressions for the amplitude. The results of the perturbative formula are displayed in panel (c) [linear laser polarization] and (d) [circular laser polarization]. The values of $E_i$, $\omega$, $\xi$, and ${\theta }_{b,c}$ are the same as in Fig. 7.

Figure 12

(Color online) Contour plot of concurrence as a function of the polar angles ${\theta }_b$ and ${\theta }_c$. The color coding indicates the value of the concurrence $C\left({\rho }_f\right)$ [see Eq. (60)]. Panel (a): linear laser polarization, nonperturbative, (b): circular laser polarization, nonperturbative, (c): linear laser polarization, perturbative, (d): circular laser polarization, perturbative. Laser parameters as well as the initial electron energy $E_i$ and the photon azimuth angles ${\psi }_{b,c}$ are the same as in Fig. 8.