The vacuum pair production in the external electric field in the frame of quantum electrodynamics (QED) was first proposed in 1931 by Sauter^[1] and also studied in 1936 by Heisenberg.^[2] In particular in 1951, Schwinger studied this problem systemically by employing the proper-time technique.^[3] He obtained the critical value of the constant external field to create the electron–positron (e⁻e⁺) pairs from the vacuum as , where m_e is the electron mass and −e is an elementary electron charge. This critical field strength is related to the laser intensity . The intensity is so high really, however, as the new experimental development in ultrahigh intensity laser techniques,^[4–7] it may be possible to get a subcritical laser field.^[8,9]

On the other hand, because of the proposed dynamically assisted Schwinger mechanism, which combines different pulse laser fields, the vacuum pair creation may be observed even in a much lower intensity laser field.^[10–15] So many ways can be used to reduce the requirement of laser field but enhance the pair production, for example, the combination of sinusoid/ cosine with exponential laser pulse,^[16–18] by using super-Gaussian instead of Gaussian pulse to widen pulse width,^[19,20] the usages of multi-slit interference due to the alternating sign N-pulse electric fields or/and polarized field,^[21–23] and so on.

Beside changing the shape or carrier phase,^[16,19] it is also expected and found that, through adding a small frequency chirping, it can influence the momentum distribution and then possibly increase the created e⁻e⁺ pair number density.^[24,25] In Ref. [24], different frequency chirps lead to the changing of the momentum spectra and it is explained by a turning point. In our previous work,^[25] the field time was divided into three parts and different frequency chirps are applied. Also in our work of Ref. [26] we have gained the enhanced electron–positron pair production of a vacuum in a strong laser pulse field by the possible irregular frequency variation. These few researches make this topic very interesting meanwhile leaving some problems still open. For example, whether the momentum spectra and pair number are dependent on the sign of frequency chirping? What phenomena occur if we change the one-color pulse with single frequency chirping to a two-color pulse with double frequencies chirping or if we change the sign of chirp parameter from positive (negative) to negative (positive) at the peak position of one-color laser pulse? How the two-color laser pulse field with different chirping affect the momentum spectra and the pair production rate?

In this paper, we will answer these questions. We have studied the one-color and two-color laser pulse fields by adding positive or negative frequency chirp parameters and analyze the effects of the frequency chirp parameters on the laser pulse shape, momentum spectrum and number density. It is found that a small frequency chirp can shift the momentum spectrum along the momentum axis. The positive and negative frequency chirp parameters play the same role in increasing the pair number density. The sign change of frequency chirp parameter at the moment leads the pulse shape and momentum spectrum to be symmetric. The case of first positive and then negative frequency chirp lead to the wider time lasting of pulse within the neighbor of peak field strength at than the opposite case of first negative and then positive chirp, therefore, the pair number density dominated by the tunneling mechanism is larger in the former case. On the other hand, the number density of produced pairs in the two-color pulse fields is also much higher than that in the one-color pulse field. Moreover, in the two-color case, the relation between the number density and the frequency ratio of two colors is not sensitive to the small frequency chirp field but sensitive to the large frequency chirp field.

For the completeness of the paper, we have to include the description about the basic ideas and formula of the quantum Vlasov equation while the content would be unavoidable repetition of other published papers. Here we outline it similar to Ref. [19].

The source term of pair production, s(p, t), obviously depends on the applied external field as well as the electron/positron kinetic property. From , where is the momentum distribution of the created pairs, we obtain the quantum Vlasov equation (QVE) in the following integro–differential equation form

(2)

Moreover when the integral part is represented by g(p, t) in Eq. (2), the equation can be expressed as a set of first order ordinary differential equations (ODEs)^[16]

(3)

(6)

We use the electrons quantities as the normalized ones, i.e., the length , the time , and the momentum . The field strength is normalized by E_cr. In our study some typical parameters are given as laser frequency or/and , the pulse length , and the field strength is or/and , and so on.

In this section, we will analyze the effect of the frequency chirp parameter on the pair production rate by discussing the laser pulse field, momentum spectrum and the number density.

3.1. One-color laser pulse field

The one-color laser pulse field is given by

(8)

where , , , , and . The frequency chirp parameters are b₁ and b₂ for low frequency field E₁(t) and high frequency field E₂(t), respectively, and in general , are required.

For some typical frequency chirp parameters b₁ and b₂ the momentum spectrum are depicted in Fig. 4. By comparing the blue dashed lines in Fig. 4(a) and Fig. 1(a) we found that the maximum value of the momentum spectrum in two-color chirp-free laser pulse is two orders of magnitude higher than that of the one-color chirp-free laser pulse.

Fig. 4.

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	Fig. 4. (color online) The same as in Fig. 1 except in two-color pulse laser field with different frequency parameters b1 and b2.

In the case of the two-color laser pulse, first we study the effect of frequency chirping with the same signs of b₁ and b₂. The momentum spectrum curve is depicted in Fig. 4(a) when , (black solid line) and (red dotted line). The same as Fig. 4(a) is shown in Fig. 4(b) except that the frequency parameters are increased as , (black solid line) and , (red dotted line). Some phenomena can be observed from Figs. 4(a) and 4(b). It is seen from Fig. 4(a) that, by adding a small frequency chirp to the two-color laser pulse, the momentum spectrum shifts along the longitudinal momentum axis, especially for the positive frequency chirping. In comparison of chirping with chirp-free the maximum value of the momentum spectrum does not increase remarkably but an oscillation structure with a small peak appears. In the comparison of Fig. 4(b) with Fig. 4(a), when the chirp increases 6 times, the maximum of the momentum spectrum increases 8 orders of magnitude.

Second let us examine the effect of frequency chirping with the opposite signs of b₁ and b₂ when , , and . The momentum spectrum are depicted in Fig. 4(c) and Fig. 4(d), where and , respectively. Similarly after adding a small frequency chirp to the two-color laser pulse, the momentum spectrum shifts along the longitudinal momentum axis and the maximum value has no obvious increase but the appearance of a small peak lacking an oscillation structure. On the other hand, when the chirp increases 6 times the maximum of the momentum spectrum increases 9 orders of magnitude. By the way in all cases of chirping the black solid line and the red dotted line have a momentum-reverse symmetry except a nonzero momentum value as the symmetry point.

When and are fixed, by changing the b₁ and b₂, we obtain how the number density curve depends on in Fig. 5. Since we keep so the horizonal axis can be characterized by a unified quantity . It can be seen from the figure that, although the shape or value of the momentum spectrum of the four fields are different, the number density of them are the same. It reveals the intrinsic symmetry again about the effect of chirping on the momentum spectrum and concludes that this time-reverse or/and momentum-reverse symmetry leads to the same pair production.

Fig. 5.

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	Fig. 5. (color online) Electron–positron number density versus the ratio of magnitude of frequency chirp to the original frequency, where , in two-color pulse laser field .

Now we will investigate which laser pulse chirping field contributes more to the pair production rate when changes. We can see the effect of single frequency chirping, for example, keep fixed but change b₁, or keep fixed but change b₂. Some typical results about how the number density curves depend on the ratio for different single frequency chirping are shown in Fig. 6. The concrete different values of b₁ and b₂ can be seen directly from the figure labels. As a comparison we plotted the line with a black square symbol as the number density curve related to the chirp-free two-color laser pulse field. It is found that, as the ratio increases the number density increases slowly and tends to reach a saturation value. Different from Fig. 2, in the two-color laser pulse here, the number density curve does not exhibit the oscillation behavior. On the other hand it is interesting to find that the number density curves are in coincidence almost. It can be concluded that in the two-color laser pulse field, if we add a small frequency chirp in either of or , the effect of them on the pair production are the same and the increasing of the number density are not obvious compared to that of the chirp-free one. The main reason for the number density increase is the increase of the original frequency ratio of the two-color laser pulse field.

Fig. 6.

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	Fig. 6. (color online) Electron–positron number density vs the original two-frequency ratio in two-color pulse laser field with a single small frequency chirp of either b1 or b2.

Finally, we keep fixed and change a single parameter b₂, or we change b₁ within a small value but keep changing the corresponding within a relatively large value. We depict the number density curves vs the ratio in Fig. 7 for different values of b₁ and b₂, which can be seen directly from the figure labels. In Fig. 7, the values of b₂ are different in the four groups of curves labeled 1, 2, 3, and 4 in the left side of the curves. For example, the marked 1 curves (except the chirp-free black solid line) correspond to and the marked 4 curves correspond to . From these four sets of curves it is found that when the frequency ratio is not large, the number density curves seem very sensitive to the frequency chirp increase. For example, in the case of , when the chirp parameter b₂ increases 6 times from marked 1 to 4 the number density increase from to , i.e., about 10 orders of magnitude. As the frequency ratio increases the differences between the four groups of number density curves are reduced, but there are still 5 orders of magnitude difference at . On the other hand two curves in each group have little difference. But for the frequency ratio , the difference of the two curves is evident because whether the small frequency chirp is added plays a role, especially in the case of group “2”.

Fig. 7.

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	Fig. 7. (color online) Electron–positron number density versus the original two-frequency ratio in two-color pulse laser field with a relative large chirping for b2 and the chirping of b1 is either absent or small still.

If we choose the number density curve of chirp-free, and , as a reference, it is found that the differences in the curves 1 are very small. A little difference exists when the ratio increases. It may be caused by the diversification of the frequency.^[29,30,32] All the curves in each group 1, 2, 3, 4 are almost the same except when . This is because as the frequency ratio increased the small frequency chirp in the field E₁(t) can be ignored. When the frequency chirp becomes larger, as the frequency chirp increased, the number density curves become less sensitive to the increasing of the frequency ratio , in the group 4 the number density curves even do not change almost. By comparing the values of the curves, we found that the large frequency chirp of high frequency weak field contributes more to increasing the number density. Certainly to meet the condition we do not add too large frequency chirp parameters about b₁ as well b₂. As the frequency ratio becomes larger the number density curve presents a small oscillation behavior, see the inset of Fig. 7, this may be caused by the larger frequency chirp.

By solving the quantum Vlasov equation, we investigated the effect of frequency chirping parameter on the electron–positron pair production in one- and two-color laser pulse fields. The main findings of the present study are listed in the following.

(i) A positive or negative frequency chirp parameter is added to the one-or two-color pulse field, then the momentum spectrum and the number density curve of the created pairs in this field is obtained. It is found that in either one- or two-color laser pulse field the small frequency chirp shifts the momentum spectrum along the momentum axis, especially for the positive chirping. It can expand the detection probability by spectrometer.

(ii) Adding frequency chirp to the one-color pulse field results in a smoothing number density curve and the widening of the peaks in the multiphoton pair production process. The positive and negative frequency chirp parameters play the same role in increasing the number density. It is a natural result of time-reverse symmetry.

(iii) The sign change of frequency chirp parameter at the moment leads pulse shape and momentum spectrum to be symmetric and the number density to be increased. The chirping of first positive and then negative has a relatively higher number density than that of first negative and then positive.

(iv) In the two-color pulse field, the number density is much higher than that in the one-color pulse field. The larger frequency chirped pulse field contributes more to increasing the pair production rate.

(v) In the two-color pulse field, the relation between the frequency ratio of two colors and the number density is not sensitive to the parameters of small frequency chirp added in either a low frequency field or a high frequency field but sensitive to the parameters of large frequency chirp added in a high frequency field.

(vi) When the frequency chirp parameter increases 6 times, the number density can increase 10 orders of magnitude. As the original frequency ratio becomes larger the number density curve presents a small oscillation behavior.

We believe that the results obtained here are helpful to deepening the understanding of the pair production by including the effect of frequency chirping, and in an alternative way one can expect to control the pair production through the appropriate frequency chirping.