Interferences effects in polarized nonlinear Breit-Wheeler process

In the presence of a strong electromagnetic field, high-energy photons can be converted into an electron-positron pair via the nonlinear Breit-Wheeler process Ritus (1985); Baier et al. (1998); Burke et al. (1997); Bamber et al. (1999); Bulanov et al. (2013); Di Piazza (2016); Wan et al. (2020a); Blackburn and Marklund (2018); Blackburn and King (2022); Tang (2022); Meuren et al. (2016). This an appealing physical process as it allows for the conversion of light into particles of matter, and has practical implications for astrophysics, high-energy physics, and laser-plasma interactions Fedotov et al. (2023); Gonoskov et al. (2022). The E-144 experiment conducted at SLAC in the 1960s provided the first experimental evidence for the nonlinear Breit-Wheeler process (NBW) Burke et al. (1997); Bamber et al. (1999). Pair production involving multiphoton is observed by colliding a 46.6GeV electron beam with a terawatt laser with peak intensity of $\sim 10^{18}$W/cm². The polarized (linear) Breit-Wheeler process has been measured at the Relativistic Heavy Ion Collider, which reveals a large fourth-order angular modulation of the observed yields Adam et al. (2021). The advancement of high-power laser technology Danson et al. (2015); vul ; ELI has sparked interest in revisiting the E-144 experiment with lasers of higher intensity. The upcoming experiments at FACET II at SLAC Salgado et al. (2021) and LUXE at DESY Abramowicz et al. (2021) aim to explore NBW in the tunneling regime, where the production rate follows an exponential scaling law of $\exp(-\pi E_{\text{cr}}/E)$ Schwinger (1951). Here $E$ is the electric field strength and $E_{\text{cr}}=m^{2}/|e|$ the critical field of quantum electrodynamics (QED), with $e<0$ and $m$ being the electron charge and mass, respectively. However, the experiments will be initially conducted in the intermediate laser intensity regime, where the transition from multiphoton to tunneling regime is expected to occur. The separation of regimes is determined by the nonlinear parameter $a_{0}=|e|E/m\omega_{0}$, with $\omega_{0}$ being the laser carrier frequency.

In the tunneling regime of $a_{0}\gg 1$, the formation length of pair creation is much smaller than the spatial and temporal inhomogeneities of the laser field Baier et al. (1998); Ritus (1985); Wiedemann and Wiedemann (2003). The probability can be calculated under the locally constant field approximation (LCFA) Ritus (1985). Recently, various LCFA probabilities have been derived incorporating electron-positron spins and photon polarization Seipt et al. (2018); Seipt and King (2020); Chen et al. (2022); Li et al. (2019a, 2020a); Torgrimsson (2021), and has been applied to QED-simulations codes for studying polarization effects in strong-field QED processes Song et al. (2022); Zhuang et al. (2023); Li et al. (2019b, 2020a); Chen et al. (2019); Wan et al. (2020b); Dai et al. (2022); Li et al. (2020b, 2022); Chen et al. (2022); Seipt et al. (2019). Ultrarelativistic positrons (electrons) beam with high degree of polarization can be generated via NBW in asymmetric laser fields Chen et al. (2019), as a result of spin preferences parallel (antiparallel) to the instantaneous quantization axis during creation. However, the current or upcoming experiments are performed with intermediate values of laser intensity ($a_{0}\sim 1$), where the formation length of pair production $l_{f}\sim\lambda/a_{0}$ is comparable with the laser field inhomogeneities. In this case, interferences effects arising from the pair creation process may significantly modify the positrons polarization.

The interference effects has been investigated in several works Wistisen (2014); Meuren et al. (2016); Titov et al. (2013, 2016); Ilderton (2020); Akkermans and Dunne (2012); Jansen and Müller (2017); Granz et al. (2019); Ilderton et al. (2020). It has been reported that the interferences occurring at macroscoptically separated space-times points could induce interference structures in the momentum distribution of the the final particles Meuren et al. (2016), and interference between pair creation channels of different photon number combinations could enhance the yield of pairs in bichromatic fields Jansen and Müller (2016). Meanwhile, the carrier-envelope phase (CEP) effects are pronounced for a few-cycle pulses with moderate laser intensity Meuren et al. (2015), which induces shifts of spectra and angular distribution due to interferences Meuren et al. (2016); Jansen and Müller (2016); Titov et al. (2013, 2016). Moreover, the polarization effects in pair production has been investigated in the transition regime ($a_{0}\sim 1$) Jansen and Müller (2016); Krajewska and Kamiński (2012); Tang and King (2021). The oscillation pattern in angular spectra of positrons Krajewska and Kamiński (2012) and positron yield Wistisen (2020a); Gao and Tang (2022) can be enhanced by tuning photon polarization. The relevance of spin effects on NBW has been investigated by comparing the production of spinor and scalar particles Villalba-Chávez and Müller (2013); Jansen and Müller (2017). However, a comprehensive study of the positron polarization in this regime has yet to be conducted.

To improve the accuracy of simulation codes in the intermediate intensity regime, a method based on locally monochromatic approximation (LMA) has been developed taking into account interference effects at the scale of the laser wavelength Bamber et al. (1999); Heinzl et al. (2020); Hartin (2018). In contrast to LCFA, LMA is more suitable for exploring the transition region for planewave-like fields with slowly varying envelope. While for a short laser pulse, LMA would underestimate the pair production as it loses the high-frequency components induced by the finite-pulse envelope. In fact, these overlooked components have the potential to effectively enhance pair production Tang (2022). While for a short laser pulse, the fully spin-resolved pair-creation probability could be obtained using the semiclassical operator method of Baier et al. Baier et al. (1998); Baier and Katkov , which is applicable to non-plane-wave background. The probability associated with particular spin states and photon polarization can be calculated by coherently integrating over the electron trajectory. The numerical results obtained for plane waves exhibit good agreement with the predictions of the Volkov state approach Volkov (1935), implying that the semiclassical method is capable of describing the polarized Breit-Wheeler process in the intermediate intensity regime Wistisen (2020b, a); Nielsen et al. (2022).

In this paper, we investigate the NBW in a linearly polarized ultrastrong laser pulse using the semiclassical method beyond LCFA. It allowed us to identify interference effects in positrons polarization and angular distribution. We show that the positrons polarization exhibits unique features due to interferences that cannot be captured by LCFA. The spectra and angular distribution of final particles are depend sensitively on the spin states, leading to an angular dependent polarization of positrons. By cutting the positrons distributed at small angle region, one could obtain polarized positrons with polarization degree $\sim 25\%$. Moreover, the polarization degree is proportional to CEP, which is attractive for relativistic positron generation with controlled polarization and CEP measurement in strong laser fields.

Using the semiclassical method and the parameters given in Sec. II A, one could obtain the spectrum for different particles’ polarization states (see Appendix B). For a more intuitive perspective, we average over photon polarizations and sum over electron polarizations to specifically examine the spin-resolved probability of positron with respect to $\mathbf{\hat{y}}$, see Fig. 1 (a). It is evident that the probability for positron spin-down is higher than that for positron spin-up as a result of asymmetry of the short laser pulse, while the LCFA underestimates this difference in the probabilities for positron spin-up and spin-down. In the semiclassical method, interference peaks appear in the spectra [Fig.1 (a)]. The interferences lead to an enhancement of pair production probabilities and a more pronounced asymmetry between different polarization states of positrons. Counterintuitively, the probability for positron spin-down exceeds spin-up. This is because interference effects arising from pair production at negative half-cycles with $B_{y}<0$ are more significant than that at positive half-cycles, leading to a notable enhancement of pair production probability with negative positron spin, i.e., $\zeta^{+}_{f}<0$.

The disparities in probability give rise to distinct polarization characteristics across different approaches. For the LCFA approach, the polarization of positron decreases with the increase of positron energy, which coincides with that in a constant field with $B_{y}>0$. The average polarization of the positron beam is $3.8\%$. However, the polarization undergoes significant changes in the results from the semiclassical method, in which interferences are introduced [see Fig. 1 (b)]. Surprisingly, the average polarization becomes $-1.5\%$. The positron polarization $\zeta_{f}^{+}$ decreases with large oscillations at small position energy $\delta_{+}$, exhibits a negative dip around $\delta_{+}=0.5$ with $|\zeta_{f}^{+}|$ reaching to $\sim 10\%$, and finally diminishes to zero when $\delta_{+}>0.6$. The LCFA approach fails to capture the interference structures, resulting in an overestimation of polarization for low energy positrons. Conversely, it underestimates the polarization near the density peak. Therefore, for $a_{0}=1$, the LCFA performs poorly in accurately describing the polarization of the created particle.

For the LCFA approach, the positron polarization arises due to the asymmetry of the ultrashort laser pulse and the preference for pair production in spin-up positrons, as the LCFA probability is solely depend on the local quantum parameter $\chi_{\gamma}\propto a_{0}\omega$ and the positrons are mostly produced at the intense positive half-cycle II₊. Since the pair production is more probable for spin-up positron, the positrons produced at the half-cycle II₊ are more likely to be polarized with $\zeta_{f}^{+}>0$ [Fig. 2 (b)]. However, due to the cancellation of polarization effect between positive and negative half-cycles, the average polarization of positron in an ultrashort laser pulse is rather small ($\sim 3.8\%$).

When interferences are taken into account, the density of positrons with $\zeta_{f}^{+}<0$ gets larger than that of $\zeta_{f}^{+}>0$, indicating more pairs are produced at the negative half-cycles I_- and II_- than the positive peak II₊ as shown in Fig. 2. This is because interferences play an essential role in determining the spin-resolved pair production probabilities. The influences of interferences within each laser cycle and between macroscopically separated cycles are illustrated in Fig. 2 (c) and (d). It can be seen that the positrons with spin parallel with $\mathbf{\hat{y}}$ are mostly comes from pair production at the pulse peak II₊. The positive half-cycles I₊ and III₊ also contribute but with a much lower amplitude [Fig. 2 (c)]. Due to the interferences within each laser cycle, the pair production amplitude at I₊, II₊ and III₊ is all enhanced compared to the results obtained from the LCFA [Fig. 2 (e)]. Meanwhile, at $a_{0}\lesssim 1$, the coherence length is of the order of the period of the electron trajectory, which means that the process is formed along all multicycle dynamics, i.e., effectively the formation length is the whole trajectory length. In this case, the interferences between cycles also make an important role. When considering pair productions during all the positive half-cycles, interference structures appear in the spectrum.

In contrast, the spin-down positrons mainly come from pair production at negative half-cycles I_- and II_-, which contribute equally to positron spectrum. The interferences within each half-cycle I_- and II_- increase the corresponding probability compared with the LCFA results in Fig. 2 (e). More importantly, the constructive interferences arising from pair production occurring at I_- and II_- result in an enhancement of positrons density with $\zeta_{f}^{+}>0$ and $\delta_{+}\approx 0.5$ [Fig. 2 (d)]. With respect to $\mathbf{\hat{y}}$, the pair production is more probable for spin-down positron around $\delta_{+}\in(0.4,0.6)$. However, outside of this region, probability for spin-up dominates over spin-down. This is because the half-cycle II₊ is more intense than the negative half-cycles [Fig. 2 (a)], there is a higher number of photons $n$ involved in pair production during $\mathrm{II}_{+}$ compared to the other cycles. The width of the spectrum increases as the harmonic order $n$ increases Ivanov et al. (2005). The maximum of $n$ are determined by the field intensity where pairs are produced. The interference between different half-cycles only affects the amplitude of harmonics through an interference pattern, but it doesn’t change the harmonic order. Consequently, the spectrum for spin-down positrons exhibits a broader width compared to the negative half-cycles, leading to a positive polarization outside the range of $\delta_{+}\in(0.4,0.6)$. It is important to note that the observed interference effects, including interference structures in spectrum and negative polarization, predominantly arise from pair production events that occur at macroscopically separated formation regions. The interferences within each cycle, on the other hand, only result in an overall enhancement of the respective probability.

The interferences within each half-cycle always result in an increase in pair production probability compared with LCFA, see Fig. 3. This is because the laser field in a finite time domain has a wide spectrum. The high frequency components make a significant contribution to the pair production probability, which are ignored by LCFA. In contrast, the interferences between macroscopically separated cycles could be destructive or constructive determined by the phase differences. See the discussion in Ref. Ilderton (2019). In our case, the interferences patten of two adjacent negative (positive) half-cycles can be quantitatively interpreted by the interferences from two trajectory segments that have the same velocity and acceleration. In this case, the integral over the entire trajectory can be estimated with $P\propto|e^{i(\varepsilon_{-}/\varepsilon_{+})k_{\mu}x_{1-}^{\mu}}+e^{i(\varepsilon_{-}/\varepsilon_{+})k_{\mu}x_{2-}^{\mu}}|^{2}$, where $x_{1-}^{\mu}$ and $x_{2-}^{\mu}$ are four-coordinates for the two trajectory segments with a $2\pi$ difference. The destructive interferences occur when $n\pi=(\varepsilon_{-}/\varepsilon_{+})\omega\Delta t(1-\beta_{-}\cos\theta)\approx m^{2}\pi/\delta_{+}\left(1-\delta_{+}\right)\omega\omega_{0}$, resulting in the emergence of valleys in the spectrum at $\delta_{+}=n\omega\omega_{0}\pm\sqrt{-4m^{2}n\omega\omega_{0}+n^{2}\omega^{2}\omega_{0}^{2}}/2n\omega\omega_{0}$ Jackson (1999); Baier et al. (1998); Di Piazza et al. (2012). Similarly, constructive peaks emerge when $2n\pi=(\varepsilon_{-}/\varepsilon_{+})\omega\Delta t(1-\beta_{-}\cos\theta)$. Therefore, the probability of pair production is enhanced mainly due to the interference within each half-cycles, and interference between cycles results in destructive valleys and constructive peaks superimposed on the enhanced probability.

The angular distributions of the positrons in different spin states are illustrated in Fig. 4. For the semiclassical approach without LCFA, the distribution of positrons exhibits two symmetric circular regions around $\theta_{x}=0$. The separation is induced by longitudinal acceleration appeared in $J\propto\mathbf{\dot{v}}_{z}\sim v_{x}B_{y}$. At $\theta_{x}\approx 0$, the positron density decreases as $J$ approaches zero, and increases to a maximum at $\theta_{x}\approx\pm a_{0}/2\gamma\sim\pm 3\times 10^{-6}$rad. The separation between the two regions is larger for the positrons with spin-up than spin-down with respect to $\mathbf{\hat{y}}$ [Fig. 4 (a) and (c)]. This because the final momentum of positrons is determined by the laser intensity at creation, i.e. $p_{f}^{x}\approx eA(\varphi_{+})$ with $\varphi_{+}$ being the creation phase. The positrons with $\zeta^{+}_{f}>0$ are mostly generated in the high laser intensity region, consequently, they have higher transverse momentum $p_{x}$ compared to the positrons with $\zeta^{+}_{f}<0$. Therefore, the small angle region is dominated by the spin-down positrons created at negative half-cycles, while the large angle region is determined by the spin-up positrons created at positive half-cycles, leading to an angle dependent polarization [see Fig. 4 (e)]. By collecting the positrons at small angle regions, one could obtain polarized positrons with average polarization reaching to $\sim 26\%$. In contrast, the angular distribution obtained with LCFA approach exhibits an elliptical shape for different spin states [Fig. 4 (b) and (d)]. It is because positrons are most created at laser peaks with $p_{x}(\varphi_{+})\approx 0$ [Fig.2 (b)], and consequently have a small $\theta_{x}$ out of laser. Moreover, the interference ring structure, which is present in the semiclassical approach, disappears in the LCFA approach. The probability for spin-up always dominates over spin-down, which is more pronounced for low-energy positrons deflected to large angle region [Fig. 4 (f)].

For the polarized Breit-wheeler process in short laser pulse, it is possible to identify the CEP effects in the positron polarization. In the case of $\varphi_{0}=\pi/2$ [Fig. 5 (a)], the pair production probabilities for different spin states are equivalent, leading to a vanished polarization throughout the spectrum [Fig. 5 (b)]. This is because the ultrashort laser pulse becomes symmetric when $\varphi_{0}=\pi/2$. Polarization effects in different half-cycles cancel out due to the symmetry of the laser field. By collecting the small angle positrons, we obtain the scaling law of average polarization on the CEP of laser [Fig. 5 (c)]. The polarization increases monotonically from -25% to 25% when the CEP increases from 0 to $\pi$, and decreases from 25% to -25% when the CEP increases from $\pi$ to $2\pi$. The sensitive dependence of polarization on CEP provides an potential way of detecting strong laser’s CEP.

In this paper, we have investigated the polarization effects of the NBW using a short laser pulse in the intermediate intensity regime. By comparing the semiclassical approach beyond LCFA and the LCFA approach, we scrutinized the validity of the LCFA in predicting the polarization of created pairs. The positrons produced in a short pulse exhibit a net polarization as a result of the asymmetry of the background field. However, the polarization features obtained from different approaches can be completely different, even when considering the same asymmetric field. The LCFA approach demonstrates a positive polarization due to the dominant influence of the peak positive half-cycle, and the polarization degree decreases monotonically with the increase of positron energy. When taking interference effects into account, it is observed that the density of positrons with $\zeta^{+}<0$ surpasses that of positrons with $\zeta^{+}>0$ around $\delta_{+}=0.5$, leading to a negative polarization in the spectrum center. The variation of polarization feature is result from the interferences arising from separated formation lengths. The pair production occurring at negative half-cycles interferences constructively, leading to an enhancement of pair production for positrons with $\zeta^{+}<0$. Interferences structures appear in spectrum for spin-up positrons but without a substantial enhancement of probability. Consequently, a negative polarization emerges around $\delta_{+}=0.5$, which is contrast to the predictions of LCFA. Meanwhile, positrons presents an angular dependent polarization. The positrons deflected towards the small angle region exhibit a negative polarization of $-25\%$ for $\varphi_{0}=0$. This negative polarization increases as the carrier-envelope phase is increased before reaching $\varphi_{0}=\pi$. However, after $\varphi_{0}=\pi$, the trend is reversed and the polarization starts to decrease. The sensitive dependence of polarization on CEP provides an alternative way of generating and manipulating the positron polarization, as well as CEP measurement in strong laser fields.

In this section, we compared the semiclassical results with the full QED calculation derived from the standard S-matrix approach, see Eqs (20) and (21) in Ref. Tang (2022). The spin-resolved probabilities from the QED calculation display excellent agreement with the semiclassical approach, markedly differing from the results yielded by the LCFA [Fig. 6 (a)]. The polarizations obtained from different approaches also coincide with each other except for the low energy region of $\delta_{+}<0.6$ [Fig. 6 (b)]. The relative difference between the QED and Semi. results within the range of $\delta_{+}\in[0.4,0.6]$ is represented by $R_{i}=\left|\left(\frac{dP_{\text{QED}}^{i}}{d\delta_{+}}-\frac{dP_{\textrm{Semi.}}^{i}}{d\delta_{+}}\right)/\frac{dP_{\text{QED}}^{i}}{d\delta_{+}}\right|<1\%$ with $i\in\left\{\uparrow,\downarrow\right\}$. Due to the excellent agreement within this range, the average polarization remains consistent at approximately the level of -1.5% over the entire spectrum. The QED calculation is derived using the parameter $s=kp_{+}/kk^{\prime}$, while semiclassical method is derived using the parameter $\delta_{+}=\varepsilon_{+}/\omega$. In the case of large positron energy $\varepsilon_{+}$, these two parameters are equivalent, which allows for a comparison of results obtained using different approaches. However, in the low-energy regime, the approximation of $s=kp_{+}/kk^{\prime}\approx\delta_{+}$ may not hold, which could be responsible for the discrepancy between different approaches. Moreover, the discrepancy could be induced by the numerical error of semiclassical method. To accurately describe the spectrum in the low-energy region, it is necessary to extend the trajectory integral to infinity. However, this requires an infinite number of discrete points for numerical integration, which is not possible in practice. Therefore, a finite integration range could result in discrepancies in the results. This deviation can be considered inconsequential when evaluating the average polarization due to the negligible positron density in this region. Furthermore, the angular distribution of positrons obtained using the semiclassical approach closely matches that of the QED approach [Figs. 7 (a) and (b)], as well as the interferences rings exhibited in the polarization distribution [Fig. 7 (c)].

The semiclassical method allows one to obtain the spectrum for arbitrary particles’ polarization states, see Fig. 8. All curves are symmetric about $\delta_{+}$ = 0.5 except $\uparrow\uparrow\parallel$ and $\downarrow\downarrow\parallel$, and the curves for $\uparrow\uparrow\parallel$ and $\downarrow\downarrow\parallel$ are mirror images of each other about $\delta_{+}$ = 0.5. The probability is dominated by $dP_{\uparrow\downarrow\mathbf{e}_{\perp}}$, which corresponds to the scenario where the incoming photon is polarized perpendicularly to the laser polarization along the direction of magnetic field $\mathbf{\hat{y}}=(0,1,0)$, while the final electron has spin-up and the positron has spin-down with respect to $\mathbf{\hat{y}}$. In the case that the photon is initially polarized along magnetic field direction, the spin-resolved probabilities and polarization of created positron are roughly the same with the unpolarized case. On the other hand, when the photon has initial polarization along the electric field direction, the positron’s polarization is negligible, making it unsuitable for studying positron polarization.

Acknowledgement: We gratefully acknowledge helpful discussions with Karen Z. Hatsagortsyan. This work is supported by the National Natural Science Foundation of China (Grants No. 12074262 and Grant No.12104428), the National Key R&D Program of China (Grant No. 2021YFA1601700), and the Shanghai Rising-Star Program.