1.School of Science, China University of Mining and Technology - Beijing, Beijing 100083, China 2.State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology-Beijing, Beijing 100083, China 3.Key laboratory of Terahertz Optoelectronics, Ministry of Education, Capital Normal University, Beijing 100048, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 11974419, 11605286) and the National Key R&D Program of China (Grant No. 2018YFA0404802)
Received Date:10 December 2020
Accepted Date:27 June 2021
Available Online:09 September 2021
Published Online:05 December 2021
Abstract:Within an oscillating field with high frequency, electron-positron pairs can be generated from vacuum as the result of multi-photon transition process. In this paper, through the computational quantum field theory and the split operator technique, we use a numerical method to solve the spatiotemporally dependent Dirac equation, the result of which enables us to discuss the process of creating electron-positron pair under a time-dependent and spatially localized external field. By monitoring the total number and the energy distribution of created pairs, the effect of the field width on the creating electron-positron pair is discussed.For a wide width, the symmetric transition of single photon transition is dominant, because the momentum of the transition particle is approximately conserved due to a gradually varying space. For an oscillating field with frequency that exceeds the threshold $ 2mc^2$, the energy of a single-photon is sufficient to cross the energy gap between the positive energy continuum and the negative energy continuum. As a result, the electron-positron pairs will be generated continuously, where a transition with symmetric energy has the maximum probability. Meanwhile, higher-order photon transition also arises, especially for three-photon transition with one photon transition completely inside the negative energy continuum. To observe the effect of this photon, we artificially cut the negative energy at a specific value. Accordingly, in the energy distribution of the created pairs, the peak corresponding to three-photon transition disappears, which indicates that the photon inside the negative energy continuum is indispensable in a three-photon transition process. For a narrow field width where the conservation of the momentum breaks down, the production corresponding to the asymmetric transition becomes obvious. In the energy distribution, the peaks representing two-photon transition and three-photon transition become wide and are split into two small peaks. For the three-photon transition, if we cut the negative energy at a specific value, it affects only the peak with lower energy, which indicates a different transition mode of the case corresponding to a wide field. Furthermore, in a narrow field the transition probability of double-photon transition greatly increases, even to a similar order of magnitude of the single photon transition. Apart from transitions with energy equal to integer multiple of the frequency of the photon appearing with asymmetric patterns, there also exists transitions with other energy. The multi-photon transition process of the particles for a narrow field width is more complicated than for a wide field width. Keywords:pair creation/ strong laser field/ computational quantum field theory
3.外场宽度对多光子跃迁过程的影响本文对多光子跃迁过程进行深入讨论, 并研究不同外场宽度对粒子产生过程的影响. 首先运用第2节介绍的方法, 根据(3)式计算出粒子的产生总数. 在多光子跃迁效应中, 由于外场振荡频率ω直接对应光子的能量, 因此粒子的产生数与ω有着密切的关系. 图1给出了$ t = 0.002 $时刻, 外场频率在0—$ 4 c^2 $范围内的粒子产生总数N, 其中外场宽度$ W = 5/c $. 这里为了更加清晰地观察多光子跃迁效应, 所取场强为$ V_1 = 5.5 c^2 $. 图 1 粒子产生数随外场频率变化关系图. 其中$t=0.002$, 外场强度$V_1=5.5 c^2$, 宽度$W=5/c$ Figure1. Total number of created pairs for different frequencies of the external field. Here $t=0.002$, the field intensity is $V_1=5.5 c^2$ and width is $W=5/c$.
图1中出现多个峰值. 其中最大的峰在频率$ \omega > 2 c^2 $后开始出现, 对应的是单光子过程. 当一个光子的能量大于能带间隙$ 2 c^2 $(在原子单位制中$ m = 1 $), 则可实现正负能态的交叠, 持续产生正负电子对. 该结论与文献[20]中粒子产生数随频率变化规律一致. 为了研究多光子跃迁效应的影响, 本文中加强了外场强度, 因此还可以看到其他峰值. 比如, 当振荡场的频率$ \omega > c^2 $, 双光子过程就可实现正负能态的交叠, 因此在$ \omega > c^2 $的位置也可以观察到一个小峰. 另外, 图中$ \omega > 0.67 c^2 $处的小峰则与三光子过程相对应, 这与文献[21]中粒子产生数随频率变化规律一致. 需要注意的是, 对于$ \omega > 2 c^2 $的情况, 双光子或者三光子跃迁仍然可以发生, 只是相较单光子而言产量较小. 为进一步理解多光子跃迁过程, 图2以对称跃迁为例, 给出多光子跃迁过程的示意图. 图 2 多光子跃迁过程示意图, 外场频率为$\omega=2.5 c^2$ Figure2. Schematic diagram of the multiphoton transition, where the frequency of the field is $\omega=2.5 c^2$.
当外场宽度较大时, 多光子跃迁过程将呈现出对称的形式. 这是因为, 当场宽W明显大于康普顿波长$\lambda_{\rm c} = 1/c$时, 在正负粒子产生过程中, 整个系统可被认为是接近空间均匀的. 因此相应地, 能够发生跃迁的过程须满足动量近似守恒. 在产生粒子的能量分布上, 则会出现较明显的与单、双、三光子过程分别对应的主峰. 当然, 由于这里使用的是空间局域化的外场, 并不是完全均匀的, 因此主峰存在一定宽度. 也就是说, 跃迁过程虽然以对称跃迁为主, 但动量不守恒的非对称跃迁仍然可以发生. 而由于动量守恒的对称跃迁过程最易发生, 则概率最大, 与能量分布的峰值相对应. 根据第2节介绍的理论, 将(3)式在正能态和负能态上求和可得出产生粒子的总数, 而若只对负能态求和, 则可得出产生粒子在正能态上的分布. 再通过将动量换算成能量, 则可得出粒子产生量在能量上的概率分布, 如图3所示. 图 3 外场宽度$W=5/c$时粒子产生量在能量上的概率分布图, 外场频率和强度分别为$\omega=2.5 c^2$, $V_1=8.5 c^2$ Figure3. Energy distribution of the created particles for a wide field width $W=5/c$, where the frequency and intensity of the field are $\omega=2.5 c^2$ and $V_1=8.5 c^2$, respectively.
文献[20]所取外场均为宽场, 为进一步讨论多光子跃迁过程, 本节选取较小的外场场宽$ W = 1/c $进行讨论. 由于高阶多光子跃迁过程概率较小, 为便于观察, 选择较强外场强度$ V_1 = 8.5 c^2 $. 当场宽接近康普顿波长时, 空间的变化更加剧烈, 则粒子在跃迁时不受动量守恒的限制, 对称跃迁的机制将被打破. 图4(a)给出了场宽为$ W = 1/c $时粒子产量在能量上的概率分布, 其他参数与图3相同. 可以看到, 相较于宽场的情况, 粒子的能量分布发生了明显的变化. 由于对称跃迁的机制被打破, 图3(a)中的三峰结构已经完全消失, 而分裂成了多个小峰. 这里仍然用竖直虚线标出了各种数量的光子对称跃迁的能量. 仔细观察可以发现, 首先, 单光子跃迁仍然在对称跃迁处存在一个主峰, 而与三光子跃迁过程对应的峰变得不明显. 其次, 双光子过程以对称跃迁为中心分裂成了两个对称的小峰, 图中A峰和B峰的能量分别为$ 1.85 c^2 $和$ 3.15 c^2 $,与能量为$ 2.5 c^2 $的双光子跃迁相对应. 另外, 在高能部分还出现了C峰和D峰, 能量分别为$ 4.35 c^2 $和$ 5.65 c^2 $. 图 4 外场宽度$W=1/c$时粒子产生量在能量上的概率分布图, 外场频率和强度分别为$\omega=2.5 c^2$, $V_1=8.5 c^2$ Figure4. Energy distribution of the created particles for a narrow field width $W=1/c$, here the frequency and intensity of the field are $\omega=2.5 c^2$ and $V_1=8.5 c^2$,?respectively.
图 1 粒子产生数随外场频率变化关系图. 其中
图 2 多光子跃迁过程示意图, 外场频率为
图 3 外场宽度
图 4 外场宽度
图 5 场宽较小时多光子跃迁过程示意图
图 6 多光子跃迁过程概率在正负能量上的分布图 (a)场宽
