Modulation effect in multiphoton pair production

We investigate the electron-positron pair production process in an oscillating field with modulated amplitude in quantum kinetic formalism. By comparing the number density in field with and without modulation, we find that the pair production rate can be enhanced by several orders when the photon energy just reach the threshold with the help of shifted frequency due to modulation. We also detect the same effect in a pulse train with subcycle structure. We demonstrate that the frequency threshold can be lowered by frequency of pulse-train due to modulation effect. We also find that the momentum distribution for $N$-pulse train can reach $N^2$ times the single pulse at the maximum value and the number density as a function of pulse number follows the power laws with index $1.6$ when the modulation effect is maximized.


I. INTRODUCTION
The electron-positron pair production from vacuum under the strong external field is one of remarkable predictions of the quantum electrodynamics (QED). After pioneering works of Sauter [1], Heisenberg and Euler [2], and Schwinger [3] a large number of investigations are dedicated to study on vacuum pair production through employing different methods such as proper time technique [4][5][6], Wentzel-Kramers-Brillouin (WKB) approximation [7], worldline instanton technique [8,9], quantum field theoretical simulation [10,11] as well as the quantum kinetic method [12][13][14][15]. The experimental verification of vacuum pair production is still remaining unavailable so far due to Schwinger threshold of external electric field E cr = m 2 c 3 /e = 1.32 × 10 18 V/m (m and −e denote mass and charge of electron, respectively.), which is too high to achieve in the laboratory at present.
With the rapid development of laser technology in recent years, it is being expectable that the field strength of experimental facility may be more approaching to the Schwinger threshold in the near future, for example, the European extreme-light-infrastructure (ELI) program is now advancing [17]. Motivated by this, various schemes have been proposed to support the upcoming possible experiments in recent years. A strongly enhanced pair production rate was presented in dynamically assisted Schwinger mechanism where a rapid oscillating electric field is superimposed onto a slowly varying one [18][19][20]. The study on time-domain multi-slit interference effect in alternating sign N-pulse electric field [21] showed that the maximum of central longitudinal momentum can reach N 2 times the single pulse value. The pair production in a short pulse with subcycle structure was studied [22] to present momentum spectrum extremely sensitive to subcycle dynamics. The accurate study on pair production in a pulsed electric field with subcycle structure [23] detected the signature of effective mass of electrons and positrons in the given strong electric field, where the effective mass is a little higher than real mass so that the frequency threshold rises a little correspondingly. These findings and schemes advance greatly the topic of pair production in strong field, more details can be seen in [24][25][26][27][28][29][30].
In present letter we introduce a scheme where the amplitude of a high frequency oscillating electric field is modulated. Amplitude modulating change the dynamics of oscillating field in the following two aspects. On the one hand the frequency of oscillating field is shifted up by modulation frequency, which may have a positive influence on the production rate, on the other hand the field strength is decreased to suppress the production rate in multiphoton regime. Therefore it is desirable to show overall influence of the amplitude modulation on pair production process. In this paper, we study the production rate and momentum distribution for different parameters in a high frequency oscillating field with amplitude modulated by a sinusoidal signal. The study is to highlight how the modulation effect influences the pair production. Furthermore, we investigate how the modulation effect can lower the frequency threshold in a more realistic field configurationpulse train with subcycle structure. We employ quantum Vlasov equation in the kinetic formalism and all quantities are working on the natural units ( = c = 1) in this paper.
The paper is organized as follows. In Sec.II we introduce the quantum Vlasov equation for a completeness. In Sec.III we get the numerical results and necessary theoretical analysis. In the last section we provide a brief conclusion.

II. THEORETICAL FORMALISM BASED ON QUANTUM VLASOV EQUATION
The spectral information of created particles in an external field is encoded in the distribution function f (p, t). The equation of motion for f (p, t) can be derived from canonical quantization with fully quantized spinor and the electromagnetic field as a classical background [14]. We are only interested in subcritical field strength regime E ≪ E cr where created particle density is so low that the collision effect and self consistent field current due to created particles can be neglected. Since the achievable spatial focusing scale is orders of magnitude larger than the Compton wavelength of electrons we ignore any spatial dependence, and we also ignore the magnetic field. With these simplifications, the quantum Vlasov equation for f (p, t) reads: where f (p, t) accounts for both spin directions due to absence of magnetic fields. Here, q(p, t) = eE(t)ε ⊥ /ω 2 (p, t) and Θ(p, t ′ , t) = t t ′ ω(p, τ)dτ with quantities as the electron/positron momentum p = (p ⊥ , p ), transverse energy squared ε 2 ⊥ = m 2 e + p 2 ⊥ , the total energy squared ω 2 (p, t) = ε 2 ⊥ + p 2 , and the longitudinal momentum p = P 3 − eA(t). This equation may be expressed as a linear, first order, ordinary differential equation system (ODEs) [15] for the convenience of numerical treatment:ḟ The term g(p, t), i.e. the integral part of Eq.(1) constitutes an important contribution to the source of pair production, where the quantum statistics character is represented by the term due to the Pauli exclusive principle. The term w(p, t) denotes a countering term to pair production, which is associated to the pair annihilation in pair creation process to some extent. The last one of ODEs means that the more pairs are created, the more pairs are annihilated probably in pair creation process. Note that the studied system has a typical non-Markovian character.
By numerically solving this ODEs with the initial conditions f (p, we can obtain spectral information f (p, t) of the created particles for any given spatially homogenous, time-dependent electric field. The number density n(t) of created particles can be obtain easily from f (p, t): where the factor 2 comes from the degeneracy of electrons. It is need to remind that the particle interpretation of f (p, t) is invalid in the presence of external field, it can be considered as the distribution function for real particles only at asymptotic times t = ±∞.

III. NUMERICAL RESULTS
In order to reveal the physical mechanism briefly as well as for the simplicity of numerical calculations, we just consider the problem in the one-dimensional momentum space for created electron-positron pair.

A. Pair production in a high frequency oscillating field with amplitude modulated by a sinusoidal signal
The spatially homogenous, time-dependent electric field in a given direction can be expressed . The transverse momentum of created particles which is perpendicular to electric field is fixed as p ⊥ = 0 as mentioned above. We introduce a high frequency oscillating field (carrier) with frequency ω c and amplitude E 0 modulated by a sinusoidal signal with modulation frequency ω m (ω m ≪ ω c ), where 0 ≤ M ≤ 1 denotes modulation degree (M = 0 for no modulation and M = 1 for full modulation) as displayed in Fig.1.
It indicates that modulating amplitude results in frequency shift, from which one can expect a possible positive influence on the pair production rate. On the other hand the average strength (power) of electric field is suppressed by a factor of due to modulation, which may have a negative influence on the pair production rate. To investigate the overall influence of amplitude modulation on the pair production rate, we compare produced number density with and without modulation for full frequency space. We find that the pair production rate can be enhanced due to modulation for carrier frequency near the frequency threshold of multiphoton process. We discuss the results corresponding to three-photon regime here, without losing generality. In Fig.2 we display the increasing factor n 1 /n 0 as a function of modulation frequency ω m for carrier frequencies ω c = 0.64m and ω c = 0.65m with field strength

B. The modulation effect among pulses in a pulse train
Now we consider a N-pulse train with subcycle structure where the field is a superposition of a serial pulses as E(t) = N n=1 E n (t) as displayed in Fig.5 when N = 3. Here the n th pulse E n (t) can be represented as: where E 0 denotes field strength of each pulse, ω c denotes the carrier frequency, τ denotes the pulse duration and T m denotes the time delay of pulse train and also it is associated to the peak positions of n th pulse field as nT m . Note that we fix the τ for each pulse so that the modulation by Gaussian We display the produced number density as a function of carrier frequency in a pulse train with subcycle structure for one pulse (N = 1) and ten pulses (N = 10) in Fig.6 with parameters The curve for one pulse (amplified by 10 times for convenience) is peaked at the ordinary frequency threshold ω c = 0.67m for the three-photon process as one may expect. However, the curve for ten pulse is peaked at several places with the same distance. We suppose this field can also be composed of three single oscillating fields with frequencies ω ± = ω c ± ω m and ω c analogous to the case in the last subsection. The peak at ω c = 0.668m is corresponding to the ordinary frequency threshold of three-photon process, with the relation 3ω c = 2m * (we take the effective mass m * = 1.002m in this subsection), and the peak at ω c = 0.649m is corresponding to 2ω c + ω + = 2ω + + ω − = 2m * , where the carrier frequency is lower than threshold frequency by 0.019m while the number density is about still of 80% of that in case of threshold frequency. These peaks at ω c = 0.631m, ω c = 0.612m and ω c = 0.687m are also corresponding to ω c + 2ω + = 2m * , 3ω + = 2m * and 2ω c + ω − = 2ω − + ω + = 2m * , respectively. The difference in peak heights can be explained by different strength of each frequency component as we expanded in last subsection.
This result indicates that the frequency threshold can be lowered by a shifted frequency due to modulation among pulses in a pulse train.
The produced number density as a function of pulse number N for ω c = 0.631m and ω m = 0.056m is displayed in Fig.7. As a comparison, the results for alternative modulation frequencies of ω m = 0.046m and ω m = 0.066m are shown also. Other parameters are the same as in Fig.6. The number densities for three values of ω m are the same when N = 1. This is not surprising since that for a single pulse the modulation among pulses is absent and the different T m just shift the position of peak field. However, as the pulse number increases, the number density is greatly increased for three sets of ω m but the increasing for ω m = 0.056m is most remarkable because the ω c + 2ω + = 2m * holds in this case in comparisons with either ω m = 0.046m or ω m = 0.066m.
Thus it represents a typical significance of modulation effect for the pair production in multiphoton regime. Especially the number density as a function of pulse number follows the power laws with index 1.6 for ω m = 0.056 as represented with the dashed line, which is very similar to the result in the case of interference effect with dynamically assisted Schwinger mechanism [27]. In the other word the power law presented in publication before and here can be explained briefly as a result of multiphoton matching condition of effective frequency for pair production. Obviously this effective frequency is constituted by the original field carrier frequency and the modulation effect among the pulses which is associated to the delay time of pulse train. multi-slit interference effect [21]. The small oscillations between the main peaks for the cases of N = 5 and N = 10 can be regard as a result of interference between pulses, but not very strong as in [21].

IV. CONCLUSION
The electron-positron pair production process in the high frequency oscillating field with amplitude modulation is investigated, where both of the problems that the field amplitude is modulated by a simple sinusoidal signal and field structure is modulated by a time-delay pulse train are considered, respectively.
Our main findings and conclusions are the follows: • The number density with and without modulation is compared for different carrier and modulation frequencies for the case of sinusoidal modulation. It is demonstrated that the pair production rate can be significantly enhanced due to modulation when the gap between threshold frequency and carrier frequency is just supplemented by the modulation frequency. The number density depends non-monotonically on the modulation degree. The significant change of the order of multiphoton process due to modulation is represented by the momentum distribution.
• The modulation effect in a pulse train with subcycle structure is also investigated. It is found that the frequency threshold can be lowered due to the modulation effect among pulses so that the carrier frequency need not just reach the ordinary threshold ω cr = 2m * /n to trigger n th order multiphoton process. It was also demonstrated that number density as a function of pulse number is fitted by the power laws with index 1.6 and the momentum distribution in maximum for Npulse can reach N 2 times the single pulse value when the modulation effect is maximal, i.e. that is matching the multiphoton condition of pair production.
The present study indicates that there is an effective frequency which is higher than laser frequency in a field configuration where the laser strength is periodically changing. It is reasonable to expect further investigations on the same effect in other scheme such as combined pulse or bi-frequent pulse train in future work. Certainly the modulation effect on pair production in full momentum space of created pairs is also worthy to study.