Superadiabatic particle number in Schwinger and de Sitter particle production

We consider the time evolution of the adiabatic particle number in both time-dependent electric fields and in de Sitter spaces, and define a superadiabatic particle number in which the (divergent) adiabatic expansion is truncated at optimal order. In this superadiabatic basis, the particle number evolves smoothly in time, according to Berry’s universal adiabatic smoothing of the Stokes phenomenon. This superadiabatic basis also illustrates clearly the quantum interference effects associated with particle production, in particular, for sequences of time-dependent electric field pulses, and in eternal de Sitter space where there is constructive interference in even dimensions and destructive interference in odd dimensions.

Figure 1

The adiabatic particle number ${\tilde{\mathcal{N}}}_k(t)$ in (10), defined with respect to two different adiabatic bases, for the Schwinger effect in a time-dependent electric field with longitudinal vector potential $A_{\parallel }=-\frac{E}{a}\tanh (at)$, corresponding to a single electric field pulse, $E(t)=E{\text{sech}}^2(at)$. The pulse parameters for this plot are $E=0.25,a=0.1,k_{\perp }=0,k_{\parallel }=0$, all in units with $m=1$. The blue curves correspond to the basis $W_k(t)={\omega }_k(t)$ and $V_k(t)=0$, with associated evolution equations (15), while the red curves correspond to the basis $W_k(t)={\omega }_k(t)$ and $V_k(t)=-\frac{{\dot{\omega }}_k(t)}{{\omega }_k(t)}$, with associated evolution equations (16). The right-hand figure shows the late-time evolution (notice the different vertical scale). Notice that at intermediate times there is a large difference in the time evolution profile of ${\tilde{\mathcal{N}}}_k(t)$, both in the scale and form of the oscillations, while the final asymptotic value of the particle number is the same in the two bases.

Figure 2

The adiabatic particle number ${\tilde{\mathcal{N}}}_k(t)$ in (10), defined with respect to two different adiabatic bases, for particle creation in four-dimensional de Sitter space with conformal coupling. The blue curves correspond to the basis $W_k(t)={\omega }_k(t)$ and $V_k(t)=0$, with associated evolution equations (15), while the red curves correspond to the basis $W_k(t)={\omega }_k(t)$ and $V_k(t)=-\frac{{\dot{\omega }}_k(t)}{{\omega }_k(t)}$, with associated evolution equations (16). The de Sitter space has physical parameters: $H=0.5,k=25$, in units with mass scale $m=1$. The right-hand figure shows the late-time evolution (notice the very different vertical scale). Notice that at intermediate times there is a very large difference in the time evolution profile of ${\tilde{\mathcal{N}}}_k(t)$, both in the scale and form of the oscillations, while the final asymptotic value of the particle number is the same in the two bases.

Figure 3

For a single complex conjugate pair of turning points, $(t_c,t_c^{*})$ joined by a Stokes line (dashed blue line), we define $s_c$ as the time at which the Stokes line crosses the real axis. Particle production associated with this pair $(t_c,t_c^{*})$ corresponds to the jump across the Stokes line at $s_c$.

Figure 4

Time evolution of the adiabatic particle number for the first 6 orders of the adiabatic expansion, for Schwinger particle production in a constant electric field of magnitude $E=0.25$, with transverse momentum $k_{\perp }=0$ and longitudinal momentum $k_{\parallel }=0$, in units with $m=1$. The numerical results, from integrating (24), (25) are plotted in solid-blue lines, and Berry’s universal form (32) is plotted as a red-dashed curve in each subplot. Note the different vertical scales. The final asymptotic value of the particle number, at future infinity, is the same for all orders of truncation. At intermediate times there are large oscillations in the particle number, which become much smaller as the optimal order ($j=3$) is reached, and then grow again rapidly beyond this optimal order of truncation. Such behavior is characteristic of asymptotic expansions, where the order of truncation depends on the size of the expansion parameter, and going beyond this optimal order typically yields increasingly worse results.

Figure 5

Time evolution of the adiabatic particle number for the first 6 orders of the adiabatic expansion, for Schwinger particle production in a time-dependent single-pulse electric field $E(t)=E{\text{sech}}^2(at)$, with magnitude $E=0.25$, $a=0.1$, transverse momentum $k_{\perp }=0$, and longitudinal momentum $k_{\parallel }=0$, in units with $m=1$. The numerical results, from integrating (24), (25), are plotted in solid-blue lines, and Berry’s universal form (32) is plotted as a red-dashed curve in each subplot. Note the different vertical scales. The final asymptotic value of the particle number, at future infinity, is the same for all orders of truncation. At intermediate times there are large oscillations in the particle number, which become much smaller as the optimal order ($j=3$) is reached, and then grow again rapidly beyond this optimal order of truncation. Such behavior is characteristic of asymptotic expansions, where the order of truncation depends on the size of the expansion parameter, and going beyond this optimal order typically yields increasingly worse results.

Figure 6

Illustration of the universal error-function form (32) of the time evolution of the adiabatic particle number in the superadiabatic basis. These plots show the time evolution of the optimally truncated adiabatic particle number (solid blue curves) for the Schwinger effect, with the single-pulse time-dependent electric field with vector potential $A_{\parallel }=-\frac{E}{a}\tanh (at)$. The four plots correspond to different choices of field parameters $E$ and $a$, and also different longitudinal momenta $k_{\parallel }$ of the produced particles, in units with $m=1$. In order to show all plots on the same scale, we have normalized the adiabatic particle number by its final value $N_k\equiv {\tilde{\mathcal{N}}}_k(t=+\infty )$. In each plot the order $j$ of the optimal truncation is indicated by the superscript $(j)$ on ${\tilde{\mathcal{N}}}_k^{(j)}(t)$. The red dashed curves show the universal error function form in (32), while the green dot-dashed curves show the universal form with the approximate expression (34) for the function ${\sigma }_k(t)$. For the parameter values used in these plots, $F_k^{(0)}$ takes values 7.416, 2.967, 6.050, and 6.956, respectively, consistent with the estimate (35) for the optimal truncation order.

Figure 7

Time evolution of the adiabatic particle number for the first 6 orders of the adiabatic expansion, for Schwinger particle production in a time-dependent double-pulse electric field (37), with parameters $E=0.25$, $a=0.1$, $b=50$, transverse momentum $k_{\perp }=0$, and longitudinal momentum $k_{\parallel }=2.51555$, in units with $m=1$. The longitudinal momentum is chosen to correspond to constructive interference [68, 69]. The final asymptotic value of the particle number, at future infinity, is the same for all orders of truncation. Note that the final asymptotic value of the particle number is 4 times that of the intermediate plateau, which is the $n^2$ enhancement factor for coherent constructive quantum interference [69]. At intermediate times there are large oscillations in the particle number, which become much smaller as the optimal order ($j=3$) is reached, and then grow again rapidly beyond this optimal order of truncation. Such behavior is characteristic of asymptotic expansions, where the order of truncation depends on the size of the expansion parameter, and going beyond this optimal order typically yields increasingly worse results.

Figure 8

As in Fig. 7, but with parameters $E=0.25$, $a=0.1$, $b=50$, transverse momentum $k_{\perp }=0$, and longitudinal momentum $k_{\parallel }=2.49887$, in units with $m=1$. The longitudinal momentum is chosen to correspond to destructive interference [68, 69]. The final asymptotic value of the particle number, at future infinity, vanishes for each order of truncation. This vanishing of the final asymptotic value of the particle number is characteristic of coherent destructive quantum interference [69].

Figure 9

Time evolution of the adiabatic particle number for a sequence of two alternating-sign pulses, of the form (37), with parameters: $E=0.5$, $a=0.2$, $b=2.5$, transverse momentum $k_{\perp }=0$, and longitudinal momentum $k_{\parallel }=1.85$, in units with $m=1$. Here the pulses are much closer together than in Figure 7, and we observe in the first plot that at leading order of the adiabatic approximation it is more difficult to resolve the situation into two pulses with coherent constructive interference between the produced particles. For these parameters the optimal order is $j=2$, and we see clearly the two-step coherent constructive interference with plateaux in the ratio $1:4$.

Figure 10

As in Figure 9, but with longitudinal momentum $k_{\parallel }=1.22175$ Here the pulses are much closer together than in Figure 8, and we observe in the first plot that at leading order of the adiabatic approximation it is more difficult to resolve the situation into two pulses with coherent destructive interference between the produced particles. The optimal order is $j=2$, and we see clearly the two-step coherent destructive interference with vanishing asymptotic particle number.

Figure 11

Time evolution of the adiabatic particle number for a sequence of two alternating-sign pulses, of the form (39), with parameters $E=0.25$, $a=0.1$, $b=50$, transverse momentum $p_{\perp }=0$, and longitudinal momentum $k_{\parallel }=0.08336$, in units with $m=1$. The optimally truncated order is $j=3$, and the ratios of the plateaux are $1:3.99:8.9$, very close to the expected $1:4:9$ for coherent constructive interference of three pulses.

Figure 12

Time evolution of the superadiabatic particle number, normalized relative to the final value at future infinity, showing the coherent nature of the interference. The first plot is for a two-pulse field (37), with parameters $E=0.25$, $a=0.1$, $b=50$, transverse momentum $k_{\perp }=0$, and longitudinal momentum $k_{\parallel }=2.515556$, in units with $m=1$; the second is for a three-pulse field (39), with parameters $E=0.25$, $a=0.1$, $b=50$, $k_{\perp }=0$, and $k_{\parallel }=0.08336$, in units with $m=1$. The optimally truncated order is $j=3$ in each case, and the ratios of the plateaux follow $1:4$ for two pulses and $1:4:9$ for three pulses.

Figure 13

The longitudinal momentum spectrum $N_k$ as a function of the produced particle’s longitudinal momentum, for the cases of one, two, and three electric field pulses, of the forms (36), (37), and (39), respectively. The distribution is of the $n$-slit interference form, with the envelope being $n^2$ times the envelope for one pulse. When integrated over $k_{\parallel }$, the total particle number is just $n$ times that for a single pulse, but the modes are redistributed into the $n$-slit form, with constructive enhancement in some modes and destructive interference in other modes.

Figure 14

The approximate superadiabatic particle number (45) plotted as a function of time and the produced particle’s longitudinal momentum $k_{\parallel }=k$, for Schwinger pair production due to three alternating-sign pulses of the form (39) with parameters $E=0.5$, $a=0.25$, $b=7.5$, and transverse momentum $k_{\perp }=0$, in units with $m=1$. Note that in the time region $-15\ll t\ll 0$, between the first and second pulses, the momentum distribution corresponds to the asymptotic momentum distribution for a single pulse, while in the time region $0\ll t\ll 15$, between the second and third pulses, the momentum distribution corresponds to the asymptotic momentum distribution for a sequence of two alternating sign pulses; finally, for $t\gg 15$, after the third pulse, the momentum distribution corresponds to the asymptotic momentum distribution for a sequence of three alternating sign pulses. This can also be seen in the accompanying Fig. 15, which shows cross sections of the momentum distributions at times in between the pulses. Notice that the time evolution of the superadiabatic particle number critically depends on the longitudinal momentum: For certain $k$ there is enhancement due to constructive interference, while for other $k$ the interference is destructive.

Figure 15

As in (14), but plotted over specific time ranges, ending with cross sections at $t=-7.5$, $t=+7.5$, and $t=+30$, respectively, to illustrate the time evolution of the longitudinal momentum spectrum (red filled) of the produced particles during the time periods between the particle creation events due to the pulses at $t=0,\pm 15$. The highlighted longitudinal momentum spectrum of each subplot follows the $n$-slit interference pattern (44), which can be compared to the asymptotic momentum spectra shown in Fig. 13, for one, two, and three pulses, respectively.

Figure 16

Time evolution of the adiabatic particle number for the first 6 orders of the adiabatic expansion, for particle production in four-dimensional de Sitter space (3), (4), with parameters $k=25$ and $H=0.5$, in units with $m=1$. We use conformal coupling so that $\xi =\frac{1}{6}$ in $d=4$. Note the similarity to the case of a two-alternating-sign-pulse electric field in the Schwinger effect, with constructive interference, as in Fig. 7. The final asymptotic value of the particle number, at future infinity, is the same for all orders of truncation. Note that the final asymptotic value of the particle number is 4 $(=2^2)$ times that of the intermediate plateau, which is the $n^2$ enhancement factor for coherent constructive quantum interference and is consistent with the analysis of [31, 32]. At intermediate times there are large oscillations in the particle number, which become much smaller as the optimal order ($j=3$) is reached, and then grow again rapidly beyond this optimal order of truncation. Such behavior is characteristic of asymptotic expansions, where the order of truncation depends on the size of the expansion parameter, and going beyond this optimal order typically yields increasingly worse results.

Figure 17

Time evolution of the adiabatic particle number for the first 6 orders of the adiabatic expansion, for particle production in three-dimensional de Sitter space (3), (4), with parameters $k=25$ and $H=0.5$, in units with $m=1$. We use conformal coupling so that $\xi =\frac{1}{8}$ in $d=3$. Note the similarity with the case of a two-alternating-sign-pulse electric field in the Schwinger effect, with destructive interference, as in Fig. 8. The final asymptotic value of the particle number, at future infinity, is the same for all orders of truncation. The final asymptotic value of the particle number vanishes, due to coherent destructive quantum interference, consistent with the analysis of [71]. At intermediate times there are large oscillations in the particle number, which become much smaller as the optimal order ($j=3$) is reached, and then grow again rapidly beyond this optimal order of truncation. Such behavior is characteristic of asymptotic expansions, where the order of truncation depends on the size of the expansion parameter, and going beyond this optimal order typically yields increasingly worse results.