PHYSICAL REVIEW D 97, 096004 (2018) Sauter-Schwinger pair creation dynamically assisted by a plane wave Greger Torgrimsson,1,2,3 Christian Schneider,1 and Ralf Schützhold1 1Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, 47057 Duisburg, Germany 2Theoretisch-Physikalisches Institut, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany 3Helmholtz Institute Jena, Fröbelstieg 3, 07743 Jena, Germany (Received 11 January 2018; published 7 May 2018) We study electron-positron pair creation by a strong and constant electric field superimposed with a weaker transversal plane wave which is incident perpendicularly (or under some angle). Comparing the fully nonperturbative approach based on the world-line instanton method with a perturbative expansion into powers of the strength of the weaker plane wave, we find good agreement—provided that the latter is carried out to sufficiently high orders. As usual for the dynamically assisted Sauter-Schwinger effect, the additional plane wave induces an exponential enhancement of the pair-creation probability if the combined Keldysh parameter exceeds a certain threshold. DOI: 10.1103/PhysRevD.97.096004 I. INTRODUCTION is strongly enhanced by adding a weaker time-dependent field to the strong field E. So far, most studies of this As one of the most striking consequences of quantum enhancement mechanism have been devoted to purely field theory, extreme external conditions can tear apart time-dependent fields [15]. quantum vacuum fluctuations and thereby create real As a step towards a more realistic field configuration, we particles. Already in 1939, Schrödinger predicted that consider a propagating plane wave superimposed to the the rapid expansion of the Universe could induce such a constant field E in the following. Plane waves propagating process [1]. As another example, the strong gravitational parallel to the constant electric field were already considered field around a black hole can tear apart quantum vacuum in [17–20], for example. It was found that such transversal fluctuations leading to Hawking radiation, i.e., black hole planewaves do not enhance the pair creation probability [18]. evaporation [2,3]. In analogy to the gravitational force, a Further, for longitudinal parallel waves, E ðt þ zÞ, the pair strong electric field can have a similar effect and create z creation probability is given by the locally constant field electron-positron pairs out of the quantum vacuum—the approximation [21–23], which implies that the enhancement Sauter-Schwinger effect [4–7]. For a constant electric field is comparably small. Both results can be understood by E, the pair creation probability (per unit time and volume) considering a Lorentz boost along the direction of the strong scales as (ℏ ¼ c ¼ 1) field which leaves the strong field invariant but reduces the m2 Peþe− ∼ exp −π ; ð1Þ frequency of the plane wave, see also [24]. In the transversal qE case, the field strength of the plane wave is reduced as well where q and m are the elementary charge and the mass of while the longitudinal wave retains its field strength. the positron/electron, respectively. In contrast to the parallel scenarios discussed in [18–23], Unfortunately, this fundamental prediction of quantum we consider the case of a transversal plane wave propa- field theory has not been directly verified experimentally yet gating perpendicular to the strong field E because the required field strength is very large. This E e ε Ω − e motivates the quest for ways to enhance the pair-creation ðt; xÞ¼E z þ E cosð ½t xÞ z; ð2Þ probability or, equivalently, to lower the required field corresponding to the vector potential (in temporal strength. One option is the dynamically assisted Sauter- gauge) A ðt; xÞ¼Et þ εE sinðΩ½t − xÞ=Ω. Schwinger effect [8–14], where the pair-creation probability z This scenario has several advantages: since such a transversal wave cannot create electron-positron pairs on its own (due to a similar Lorentz boost argument as above), Published by the American Physical Society under the terms of pair creation can only occur in cooperation with the strong the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to field E, which retains the nonperturbative character of this the author(s) and the published article’s title, journal citation, effect. Furthermore, the above profile (2) represents a and DOI. Funded by SCOAP3. vacuum solution to the Maxwell equations and could be 2470-0010=2018=97(9)=096004(9) 096004-1 Published by the American Physical Society TORGRIMSSON, SCHNEIDER, and SCHÜTZHOLD PHYS. REV. D 97, 096004 (2018) a reasonable approximation for an experimental setup the exponentials in PN grow according to Eq. (4).Asa where E represents the focus of an optical laser while the result, there could be a dominant order Nà which yields the plane wave is generated by an x-ray free-electron laser maximum contribution to the sum (3). In order to study this (XFEL). Finally, we found that this scenario (2) yields the question, we approximately treat N as a continuous variable N maximum enhancement of the pair creation probability. and apply the saddle point method to the term ε PN, i.e., Other profiles, polarizations and propagation directions will 2 2 pffiffiffiffiffiffiffiffiffiffiffiffi be discussed below in Sec. III and Appendix A. Note that this d m⊥ 2 −Njlnεj− ðarccosΣ − Σ 1 − Σ Þ ¼ 0: ð6Þ profile (2) was already considered in [25] using first-order dN qE perturbation theory in ε, whereas we are going to consider higher orders as well as a fully nonperturbative approach. This yields the dominant order Nà as solution of the transcendental equation II. PERTURBATIVE APPROACH Ω 4 γ 1 − Nà m qE ε crit 4 arctan Ω ¼ Ω jln j¼ γ ; ð7Þ At first, we employ a perturbative expansion of the total m Nà m pair creation probability Peþe− in powers of the relative strength ε of the plane wave, which is supposed to be small where we have introduced the (combined) Keldysh param- γ Ω γ ε ε ≪ 1 eter ¼ m =ðqEÞ and its threshold crit ¼jln j. We only X∞ obtain real solutions Nà if the right-hand side is less than N γ γ Peþe− ¼ ε PN; ð3Þ unity, i.e., if exceeds the threshold crit. At the threshold, 0 γ γ 0 N¼ ¼ crit, we find Nà ¼ which implies the original Schwinger result (1).Forγ>γ and Ω ≪ m, however, the where the contributions PN can be derived via the world-line crit formalism, for an introduction see [26,27] and references dominant order Nà can be quite large (which justifies γ 3γ therein. The zeroth order N ¼ 0 reproduces the original the continuum approximation). For example, for ¼ crit, ∼ 2 ε Sauter-Schwinger effect in Eq. (1), and odd orders vanish in the dominant order Nà scales as Nà m =ðqEj ln jÞ which this situation (but not always [28]). can be a large number for electric fields E well below the 2 The lowest-order term N ¼ 2 corresponding to the one- Schwinger limit ES ¼ m =q. γ γ ≫ 1 photon contribution has already been calculated in [25]. In the limit = crit , we may approximate the solution Deriving the exponential dependence for the higher-order of the transcendental equation (7) via terms, it turns out that the exponent for two photons rffiffiffiffiffiffiffiffiffiffi 4m γ (N ¼ 4) with frequency Ω is the same as that for a single γ ≫ γ ≈ NÃð critÞ Ω 3γ ; ð8Þ photon (N ¼ 2) with twice the frequency 2Ω, and so on for crit more photons (see Appendix A). which will also be a large number unless the frequency Ω Consequently, we find far exceeds the electron mass m. Inserting this approximate 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi m⊥ 2 N P ∼ exp − ðarccos Σ − Σ 1 − Σ Þ ; ð4Þ solution for the dominant order à back into the exponent N qE (4), we find Σ rffiffiffiffiffiffiffi where the function of in the exponentp isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi already known 2 γ m crit 2 2 − ∼ −8 from [14]. The effective mass m⊥ ¼ m þðNΩ=4Þ Peþe exp : ð9Þ qE 3γ reflects momentum conservation in x-direction, where the momentum NΩ=2 of the N=2 photons has to be In contrast to the dynamically assisted Sauter-Schwinger transferred to the electron-positron pair. As a result, the effect with a purely time-dependent field, we see that the effective mass m⊥ is higher than the original mass m, and exponent still crucially depends on the strong field E, hence the pair creation probability is lower than in the case which demonstrates the nonperturbative character of Σ of a purely time-dependent field. Finally, describes the this effect even for γ=γ ≫ 1. As mentioned in the 2 crit relative contribution of the energy of the N= photons in Introduction, this is a consequence of the fact that a plane 2 comparison to the effective mass gap m⊥ wave alone cannot create electron-positron pairs out of the Ω 2 Ω vacuum. Σ N = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN ¼ ¼ 2 2 : ð5Þ 2m⊥ 4 m þðNΩ=4Þ B. Improved approximation Ω↓0 Σ↓0 ↓ In the limit of , i.e., , where m⊥ m, we recover In the following, we try to improve the accuracy of the Eq. (1), as expected. approximation outlined in the previous section. The above estimate of the leading order Nà was based on the A. Dominant order competition between the factor εN and the exponent (4). Inspecting the terms in the sum (3) we find that the However, the prefactor in front of this exponent will also prefactors εN decrease as N increases (due to ε ≪ 1) while depend on N and thereby slightly modify the scaling 096004-2 SAUTER-SCHWINGER PAIR CREATION DYNAMICALLY … PHYS.