Pair production in inhomogeneous electric fields with phase modulation

The DHW formalism is a relativistic phase-space method that has been widely used to investigate vacuum pair creation within arbitrary electromagnetic fields. Since the complete derivation of DHW formalism has been obtained in Refs. Hebenstreit:2011wk ; Kohlfurst:2015zxi ; Ababekri:2019qiw , we only present key points of this method.

We start with the gauge-covariant density operator which is composed of two commutative Dirac field operators, i.e.,

where $r=(r_{1}+r_{2})/2$ denotes the center-of-mass coordinate and $s=s_{1}-s_{2}$ represents the relative coordinate. The Wilson line factor

can be used to guarantee the invariant of the density operator. The covariant Wigner operator is obtained via the Fourier transform of Eq. (1)

We can define the covariant Wigner function by taking the vacuum expectation value of Eq. (3) as

Being a Dirac-matrix valued quantity, the Wigner function can be expanded in terms of $16$ covariant Wigner coefficients

Since we are dealing with the $e^{-}e^{+}$ pair production, and we want to describe it as an initial value problem, the equal-time Wigner function can be obtained by taking the energy average of the covariant Wigner function

When the $e^{-}e^{+}$ pair production in time dependent field with spatial inhomogeneity of $x$-axis ($1+1$ dimensional time-space) is investigated, the complete DHW equations of motion can be reduced to the following form

with the pseudodifferential operator

The corresponding vacuum initial conditions are

where $\Omega$ represents the one-particle energy, which can be expressed as $\Omega=\sqrt{p_{x}^{2}+m^{2}}$. By subtracting these vacuum terms, the modified Wigner component can be written as

here $\mathbbm{w}_{k}$ is the Wigner component in Eqs. (7)-(10), and we define $\mathbbm{w}_{0}=\mathbbm{s}$, $\mathbbm{w}_{1}=\mathbbm{v}_{0}$, $\mathbbm{w}_{2}=\mathbbm{v}_{1}$ and $\mathbbm{w}_{3}=\mathbbm{p}$. The $\mathbbm{w}_{{k}\,\rm{vac}}$ denotes the corresponding vacuum initial condition in Eq. (12). The particle number density can be expressed as

We can obtain the particle number density of momentum space via integrating Eq. (14) with respect to $x$,

Consequently, the total particle yield of the whole phase space can be written as

Moreover, in order to extract the nontrivial effect of spatial scale $\lambda$, we calculate the reduced quantities $\bar{n}\left(p_{x},t\right)\equiv n\left(p_{x},t\right)/\lambda$ and $\bar{N}\left(t\rightarrow\infty\right)\equiv N\left(t\rightarrow\infty\right)/{\lambda}$.

To further discuss the momentum spectrum and the particle number of the created particles, the semiclassical WKB approximation method is introduced. The pair production from vacuum is similar to the over-the-barrier scattering problem in quantum mechanics, meanwhile, the physical picture can be reflected in the typical turning point structure. The approximate expression of the particle creation rate can be described as

with

and

where $t_{i}$ and $t_{j}$ are the solutions of equation $\omega_{\bf p}(t)=\sqrt{m^{2}+p_{\perp}^{2}+(p_{x}-eA(t))^{2}}=0$, which denote the different turning points. $\theta_{(i,j)}$ represents the phase accumulated between different pairs of turning points, which is also called interference term. It is noticed that the turning points closest to the real $t$ axis dominate the number of created particles, while the distances of turning points along the real-axis direction dominate the interference effect in the momentum spectrum. The detailed explanation of the turning point structure to the momentum spectrum and the particle number will be analyzed and discussed in Sec. V.

We investigate pair production in $1+1$ dimensional spatially inhomogeneous electric field with the sinusoidal phase modulation, where the field model can be described as Gong:2019sbw

where $E_{0}$ denotes the field strength, $\lambda$ is the spatial scale, $\tau$ is the pulse duration, $\omega$ represents the central frequency, $b$ and $\omega_{m}$ are the amplitude and frequency of the phase modulation, respectively. The nonzero modulated frequency $\omega_{m}$ and modulated amplitude $b$ lead to the time-dependent effective frequency $\omega_{\text{eff}}(t)=\omega+b\omega_{m}\cos(\omega_{m}t)$. In order to keep the modulation within a reasonable range, we set $|b\omega_{m}\cos(\omega_{m}t)|\leq\alpha\omega$ with $0<\alpha<1$. Because of $|b\omega_{m}\cos(\omega_{m}t)|_{max}=b\omega_{m}$, the inequality $b\omega_{m}\leq\alpha\omega$ can be derived. We can further obtain the relationship $b\leq\alpha\omega/\omega_{m}$, which also indicates that the upper and lower limits of the modulated amplitude can be obtained. Without losing the generality, we select the regime of $0\leq\alpha\leq 0.9$ and the modulated frequency $\omega_{m}\approx(1/5\sim 1/10)\omega$, meanwhile, it is noted that when we choose the maximum modulated amplitude, the minimum value of modulated frequency is considered, i.e., $\omega_{m}=1/10\omega$. For high frequency field, the field parameters are set to $E_{0}=0.3E_{\mathrm{cr}}$, $\omega=0.5$, $\tau=100$, therefore the corresponding maximum values of modulated amplitude and frequency can be selected as $b=0.9\omega/\omega_{m}=9$ and $\omega_{m}=1/5\omega=0.1$ , respectively. For low frequency field, we set $E_{0}=0.5E_{\mathrm{cr}}$, $\omega=0.1$, $\tau=25$, and the corresponding maximum values of modulated amplitude and frequency are chosen as $b=0.9\omega/\omega_{m}=9$ and $\omega_{m}=1/5\omega=0.02$, respectively.

The model in Eq. (18) may be viewed as a field generated in the antinode of the standing-wave mode. Since the particles are mainly produced in the $x$-axis direction of the electric field, $p_{\perp}=0$ can be assumed.

We first study the influence of the modulated amplitude and frequency on the momentum spectrum for various spatial scales, respectively.

In this subsection, we study the effect of modulation parameters on the reduced particle number for various spatial scales, in the different cases, such as modulating only in amplitude, modulating only in frequency and modulating in both amplitude and frequency.

Figures 3(a) and (b) show the reduced particle number dependence on spatial scales for various modulated amplitude and frequency, respectively. It can be seen that when the modulation parameters (either the modulated amplitude or frequency) are fixed, with increasing spatial scale, the reduced particle number is increased rapidly at small spatial scales, while it tends to be a constant at large spatial scales. Because at small spatial scales, the electric field strength becomes large with spatial scale, more particles are created in the whole range of the electric field. It is noted that for certain modulated frequency $\omega_{m}=0.07$ and $\omega_{m}=0.1$, the particle number is enhanced in the range of $1.6<\lambda<7$, while it is decreased in the range of $7<\lambda<9$, as shown in Fig. 3(b). It means that the reduced particle number is more sensitive to modulated frequency, this reason will be discussed separately in the following section.

When the spatial scale is fixed, the reduced particle number is enhanced significantly for various modulated amplitude and frequency. At large spatial scales, compared to the case of electric field without modulation, the particle number is increased rapidly at about one order of magnitude by either the large modulated amplitude or small modulated frequency, as shown in Fig. 3. At small spatial scales, the enhancement of the particle number for various modulated amplitude and frequency is different. With modulated amplitude, the particle number can be increased at about one order of magnitude, as shown in Fig. 3(a), while for modulated frequency, it is enhanced about $5$ times, as shown in Fig. 3(b). Therefore, it indicates that the change of modulated amplitude is beneficial for the pair production. Meanwhile, when spatial scale is small, for small modulated amplitude, the particle number is enhanced rapidly with spatial scale, while there is no significant increase for large modulated amplitude, as shown in Fig. 3(a). Since when modulated amplitude is large, the particle production process is dominated by multiphoton absorption that is less influenced by the spatial scale.

In order to study the effect of modulated frequency and amplitude on the reduced particle number more comprehensively, we present the contour plot as shown in Fig. 4. Due to limited computational resources, we select an intermediate spatial scale $\lambda=100$ to study. One can see that there are several different regions on the diagram, which are divided by three typical curves, where these curves represent $b=0.06\omega/\omega_{m}$, $b=0.18\omega/\omega_{m}$ and $b=0.36\omega/\omega_{m}$, respectively. It is found that in the modulation range of $b\leq 0.06\omega/\omega_{m}$, the reduced particle number is not sensitive to the modulation parameters, while it presents an obvious variation in the modulation range of $0.06\omega/\omega_{m}<b\leq 0.18\omega/\omega_{m}$. Since when $b\leq 0.06\omega/\omega_{m}$, the maximum value of effective frequency is $\omega+0.06\omega$, which has less distinction to the original center frequency. We also observe that when the modulation range is $0.18\omega/\omega_{m}<b<0.36\omega/\omega_{m}$, there is no pronounced change for the particle number, while it presents a significant variation in the range of $b\geq 0.36\omega/\omega_{m}$. Moreover, the significant maximum (wine dot) and minimum (white dot) values of the reduced particle number on the different regions can be obtained, as shown in Table 2. It is found that when the modulation range are $0.06\omega/\omega_{m}<b\leq 0.18\omega/\omega_{m}$ and $b\geq 0.36\omega/\omega_{m}$, the reduced particle number is enhanced significantly. Particularly, in the range of $b\geq 0.36\omega/\omega_{m}$, the maximum value of particle number is about $70$ times larger than that without modulation. Interestingly, we find that the same reduced particle number can be got through different set of modulation parameters.

To study whether there are optimal modulation parameters for $e^{-}e^{+}$ pair production in high frequency field with sinusoidal phase modulation, we further explore the variation of the reduced particle number in the following two cases at the spatial scale $\lambda=100$, as shown in Fig. 5(a). One case is that the electric field does not have any modulation, only the central frequency $\omega$. The other case is that the electric field has both central frequency and modulation, where the central frequency is $\omega=0.5$, and the modulation is that when the modulated frequency is fixed $\omega_{m}=0.01$, but the modulated amplitude $b$ changes. It is found that the trends of the results in the above two cases are almost identical. Since in the second case, we perform the Fourier transform of the time-dependent electric field and regard the frequency with the largest amplitude on the frequency spectrum as the original center frequency of the electric field without modulation, which makes the trends almost identical. Moreover, in the first case, it is found that the reduced particle number is extremely sensitive to the central frequency of the external field and presents an obvious nonlinear variation. In particular, when the central frequency is $\omega=0.69$, the particle number reaches the maximum value, i.e., $\bar{N}=0.1331$, and it is enhanced significantly by more than two orders of magnitude compared to the case without modulation, i.e., $\omega=0.5$. In the second case, we can also observe the sensitivity of the reduced particle number to the modulated amplitude and find that when modulated frequency $\omega_{m}=0.01$ and modulated amplitude $b=20.7$, the reduced particle number reaches the maximum value, i,e., $\bar{N}=0.1367$, which is almost the same as the maximum value in the first case.

Figure 5(b) shows the change in the ratio of the reduced particle number in the above two cases and the particle number under the electric field without modulation. In the first case, one can see that when the central frequency of the external field is $\omega=0.69$, the ratio reaches the maximum value, i.e., $\bar{N}(b=0)/\bar{N}(\omega=0.5,b=0)\approx 211$, while in the second case, when modulated frequency $\omega_{m}=0.01$ and modulated amplitude $b=20.7$, the maximum value of the ratio is $\bar{N}(\omega_{\text{m}}=0.01)/\bar{N}(\omega=0.5,b=0)\approx 217$. It indicates that in the two case, the reduced particle numbers are enhanced significantly about $200$ times compared to the case without modulation, i.e., $\omega=0.5$. Therefore, we can obtain that $\omega=0.69$ is the optimal value of the central frequency of the external field to get the largest reduced particle number, meanwhile, $\omega_{m}=0.01$ and $b=20.7$ are the optimal values of the modulated frequency and the modulated amplitude to get the largest reduced particle number.

We study the effect of the modulated amplitude and frequency on the momentum spectrum for various spatial scales, respectively.

In this subsection, we investigate the effect of modulated frequency and amplitude on the reduced particle number for different spatial scales, as shown in Fig. 8.

It can be seen from the Fig. 8 that when modulation parameters (either the modulated amplitude or frequency) are fixed, with spatial scale, the reduced particle number is enhanced significantly at small spatial scales, while it is almost unchangeable at large spatial scales. Interestingly, at small spatial scales, when the modulated frequency are $\omega_{m}=0.015$ and $\omega_{m}=0.02$, the reduced particle number shows a clear nonlinear variation, as shown in Fig. 8(b). When spatial scale is fixed, both large modulated amplitude and large modulated frequency are beneficial to the increase of reduced particle number at small spatial scales. Compared to the case of electric field without modulation, in the case of modulated amplitude, the particle number is enhanced about $9$ times, see Fig. 8(a), while in the case of modulated frequency, it is increased about $3$ times, see Fig. 8(b). At large spatial scales, the reduced particle number is insensitive to modulated amplitude, while for the modulation frequency, it is enhanced by a small amount.

According to the semiclassical WKB approach in the previous section, the interference effect of the momentum spectrum can be understood qualitatively by the location of turning points in the complex $t$ plane. Figure 9 shows the turning points distribution corresponding to the maximum peak values of the momentum spectrum for various modulated frequency. It can be seen from the Fig. 9(a), when $\omega_{m}=0$, there are two pairs of turning points closest to the real $t$ axis, while for small modulated frequency, we observed five pairs of turning points are closest to the real $t$ axis, see Fig. 9(b). Since the more pairs of turning point closest to the real $t$ axis means that the remarkable interference effect will appear in the momentum spectrum. Therefore, there is weak interference on the momentum spectrum in Fig. 1(a), while it presents an obvious interference effect in Fig. 2(a). With modulated frequency, one can see that the neighbouring distance of turning points is shorter and shorter, which depends on the modulated frequency, therefore, it leads to the more turning points near the real $t$ axis, as shown in Figs. 9(c) and (d). Correspondingly, the interference effect of the momentum spectra becomes more and more remarkable, as shown in Figs. 2(b) and (d). Compared with the case of Fig. 9(a), the turning points in Figs. 9(b), (c) and (d) present the closer distance to the real $t$ axis, which implies that the particles production rate in the modulating case is larger than that without modulation. Moreover, all the turning point structures in Fig. 9 are approximately symmetrical, which results in a symmetry of the corresponding momentum spectrum.

In order to see why the reduced particle number appears obvious nonlinear effects on spatial scale when $\omega_{m}=0.07$ and $\omega_{m}=0.1$ in Fig. 3(b), we employ the view point from the action of worldline instanton and the corresponding position distribution to make some discussions on this results, respectively.

It is known from Ref. Linder:2015 that the pair production rate can be written as $P_{e^{+}e^{-}}\sim e^{-S}$, where $S$ denotes the action of worldline instanton. In the time-dependent oscillation electric field $E(t)=E_{0}g(\omega t)$, the Keldysh adiabatic parameter is $\gamma_{\omega}=m\omega/eE_{0}$, where $\omega$ denotes the frequency of electric field. The well known result is that the corresponding $S_{\omega}$ decreases monotonically with $\gamma_{\omega}$ Gies:2005bz , therefore, it leads to a monotonically increasing of the pair production rate $P_{e^{+}e^{-}}$. In contrary, if the field is only space-dependent oscillation as $E(x)=E_{0}f(kx)$, the corresponding adiabatic parameter is expressed as $\gamma_{k}=mk/eE$, where $k$ represents wave number, and $S_{k}$ increases monotonically with $\gamma_{k}$ Gies:2005bz . Therefore, the $P_{e^{+}e^{-}}$ decreases monotonically. Now it is noted that our electric field model, Eq.(18), is composed by the space and time dependence, therefore, it should exhibit a complicated behavior due to the competition between the pair number increasing by the larger frequency and the pair number decreasing by the higher wave number.

Indeed, in Fig. 3(b), it can be seen that when $\omega_{m}=0.07$ or $\omega_{m}=0.1$, the particle number appears an obvious nonlinear variation with spatial scale $\lambda$, i.e., there is a transition point at $\lambda=7$. It is noticed that the left and right sides of the transition point correspond to $\lambda=5$ and $9$, respectively. According to the description of the space-dependent field above, the particle number should enhance monotonously as spatial scale $\lambda$ increases (the corresponding $k$ decreases), but it increases nonlinearly, which is caused by the competitive relationship between the effect of the temporal part of the field on the particle number and the effect of the spatial part on the particle number. In the case of $\omega_{m}=0.07$, when the spatial scale $\lambda$ changes from $5$ to $7$, the effect of the temporal part of the field should be greater than that of the spatial part, which leads to a net effect of the increasing of the particle number. However, when the spatial scale varies from $7$ to $9$, the situation is vice versa, i.e., the effect of the temporal part of the field is smaller than the spatial part, resulting in a net effect of the decreasing of the particle number. Therefore, the particle number presents an obvious nonlinear variation with spatial scale. The discussion about the case of $\omega_{m}=0.1$ is similar to that of $\omega_{m}=0.07$.

On the other hand, the result can be understood by the position distributions for $\omega_{m}=0.07$ and $0.1$ in Fig. 10. It can be seen from the Fig. 10(a), for $\omega_{m}=0.07$, the position distribution presents the largest peak and the broadest distribution at $\lambda=7$ compared with the case of $\lambda=5$ and $9$, which indicates that the more particles can be created in the position space. The phenomenon and discussion of Fig. 10(b) is similar to the case of Fig. 10(a). Moreover, for $\omega_{m}=0.07$, we can also read the maximum peak values of the momentum spectra at $\lambda=5$, $7$ and $9$, as shown in the upper part of Table 3. For $\lambda=7$, the corresponding maximum values are $0.0277$ and $0.0292$, which are larger than the case of $\lambda=5$ and $9$. Therefore, the corresponding reduced particle number is also the largest, i.e., $\bar{N}=0.0129$. When $\omega_{m}=0.1$, the maximum peak values of the momentum spectra at $\lambda=5$, $7$ and $9$ are shown in the lower part of Table 3, where the result is similar to the case of $\omega_{m}=0.07$.