Elsevier

Annals of Physics

Volume 333, June 2013, Pages 104-126
Annals of Physics

Vacuum pair-production in a classical electric field and an electromagnetic wave

https://doi.org/10.1016/j.aop.2013.02.012Get rights and content

Highlights

  • The electron–positron pair-production rate in an electric field and an electromagnetic wave.

  • The pair-production rate enhanced by the amplitude of the electromagnetic wave.

  • Its application for the superposition of the static electric field and laser beams.

  • Medium effects on the pair-production rate.

  • The enhancement of the pair-production rate by the self-focusing property of laser beams.

Abstract

Using semiclassical WKB-methods, we calculate the rate of electron–positron pair-production from the vacuum in the presence of two external fields, a strong (space- or time-dependent) classical field and a monochromatic electromagnetic wave. We discuss the possible medium effects on the rate in the presence of thermal electrons, bosons, and neutral plasma of electrons and protons at a given temperature and chemical potential. Using our rate formula, we calculate the rate enhancement due to a laser beam, and discuss the possibility that a significant enhancement may appear in a plasma of electrons and protons with self-focusing properties.

Introduction

The creation of electron–positron pairs from the vacuum by an external uniform electric field in space–time was first studied by Sauter [1] as a quantum tunneling process. Heisenberg and Euler [2] extended his result by calculating an effective Lagrangian from the Dirac theory for electrons in a constant electromagnetic field. A more elegant reformulation was given by Schwinger [3] based on Quantum Electrodynamics (QED), where the result is obtained from a one-loop calculation of the electron field in a constant electromagnetic field yielding an effective action. A detailed review and relevant references can be found in Refs. [4], [5].

The rate of pair-production may be split into an exponential and a pre-exponential factor. The exponent is determined by the classical trajectory of the tunneling particle in imaginary time which has the smallest action. It plays the same role as the activation energy in a Boltzmann factor with a “temperature” ħ. The pre-exponential factor is determined by the quantum fluctuations of the path around that trajectory. At the semiclassical level, the latter is obtained from the functional determinant of the quadratic fluctuations. It can be calculated in closed form only for a few classical paths [6]. An efficient technique for doing this is based on the WKB wave functions, another on solving the Heisenberg equations of motion for the position operator in the external field [6]. If the electric field depends only on time, both exponential and pre-exponential factors were approximately computed by Brezin and Itzykson by applying Schwinger’s method to a purely periodic field E(t)=E0cosω0t [7]. The result was generalized by Popov with a first-quantized calculation in Ref. [8] to a general time-dependent field E(t). An alternative approach to the same problems was more recently employed using the so-called worldline formalism [9], sometimes called the “string-inspired formalism”. This formalism is closely related to Feynman’s orbital view of the propagators of quantum fields. The functional determinant of the electron field in Schwinger’s approach is calculated as a relativistic path integral over all fluctuating orbits of an electron in the external field as described in the textbook [6]. In the path integral formalism the tunneling problem has a standard formulation and the pre-exponential factor is calculated via an orbital fluctuation determinant for whose calculation simple formulas have been developed in Ref. [6]. These formulas were evaluated by Dunne and Schubert [10] and Dunne et al. [11] for various field configurations, such as the single-pulse field with a temporal Sauter shape 1/cosh2ωt.

In our previous article [12], we have derived a general expression for the pair-production rate in nonuniform electric fields E(z) pointing in the z-direction and varying only along this direction. A simple variable change in all formulas has led to results for electric fields depending on time rather than space.

The relevant critical field strength which creates a pair over two Compton wavelengths 2λC=2ħ/mec in two Compton times 2τC=2ħ/mec2 sets in Ecme2c3/eħ=1.3×1016V/cm, and field intensity Ic=Ec24.3×1029W/cm2. For electric fields EEc, the pair-production rate is exponentially reduced by a factor exp(πEc/E). In the laboratory, the electric field intensity Ic is, unfortunately, extremely difficult to reach in the laboratory [13], [14]. Motivated by these difficulties, people have studied possibilities of a dynamical enhancement of the pair-production rate by time-dependent oscillating or pulse electric fields [15], [16].

One possibility is to consider the superposition of a strong but slow field pulse and a weak but fast field pulse, which can lead to a significant enhancement of the pair-production rate [15], [17]. Another is a catalysis mechanism of pair production that has been studied in Ref. [16]. The setup is a superposition of a plane-wave X-ray probe beam with a strongly focused optical laser pulse. Namely, the optical laser pulse beams are focused onto a spot to yield a strong stationary electric field E, and the X-ray laser propagates through the focusing spot of optical laser beams. Since the X-ray laser wavelength (frequency) is much smaller (larger) than the optical one, namely the size of the focusing spot, the electric field created by focusing two optical laser beams can be approximated by a constant classical electric field in space and time. In that spot, a large number of coherent photons (X-ray laser) collide with virtual pairs of the vacuum in a strong classical electric field (optical intense pulse), and in consequence the pair-production rate must be enhanced.

The semiclassical WKB-approximation approach is an important method to study strong field QED [18] and electron–positron pair production [19], [12]. In this article, we continue and extend our semiclassical WKB-approach [12], by calculating the enhanced pair-production rate in the superposition of a strong (space- or time-dependent) classical field and an electromagnetic plane wave. In Sections 2 Semiclassical description of pair production, 3 Time-dependent electric fields, we present a general expression for the rate with a general enhancement factor. In Section 4 we apply this general expression to two cases; (i) a constant electric field in the finite spatial region which drops sharply to zero at the boundary; (ii) a softened version of this, where the production takes place in a Sauter electric step field. In Section 4.3, we extend our general formalism by calculating the enhancement factor in the presence of coherent laser photons and thermal photons at a finite temperature. In Sections 5.1 The presence of electrons with temperature and chemical potential, 5.2 The presence of bosons with temperature and chemical potential, we discuss the Pauli-suppression and Bose-enhancement of the pair-production rate in the presence of thermal electrons and bosons at a given temperature and chemical potential. Finally, in Section 5.3, we discuss the possibility that the pair-production rate can be greatly enhanced by the self-focusing phenomenon of laser beam propagating in the plasma of electrons and protons.

Section snippets

Semiclassical description of pair production

The phenomenon of pair production in an external electric field can be understood, in the historic first-quantized Dirac picture, as a quantum-mechanical tunneling process of electrons from the negative-energy Dirac sea to the positive energy conduction band [20], [21]. The electric field bends the positive and negative-energy levels of the Hamiltonian, leading to a level-crossing and a tunneling of the electrons in the negative-energy band to the positive-energy band. Let the field vector E(z)

Time-dependent electric fields

The above semiclassical considerations can be applied with little change to the different physical situation in which the electric field along the z-direction depends only on time rather than z. Instead of the time t itself we shall prefer working with the zeroth length coordinate x0=ct, as usual in relativistic calculations. As an intermediate step consider for a moment a vector potential Aμ=(A0(z),0,0,Az(x0)), with the electric field E=zA0(z)0Az(x0),x0ct. The associated Klein–Gordon

Applications

The most striking feature of the final formulas of the vacuum pair-production rate (42) is an exponential factor exp+12a2H(0,E) containing the fine structure constant α and the squared amplitude of the monochromatic electromagnetic field (see Eqs. (5), (39)). The enhancement of the vacuum pair-production rate due to monochromatic electromagnetic fields is mainly caused by this exponential factor. The term Erf(ϑ) in Eqs. (42), (43) is negligible. In this section, we apply formulas (42) to two

Medium effects on the vacuum pair-production rate

In order for the vacuum pair-production to occur, static electric fields must be near to the critical value (1) and the laser-field parameter (71) must not be much smaller than one. When strong static fields and laser fields (a pulse) enter a medium, one expects that the strength of fields will be damped, and the equilibrium in the medium will be altered, due to complex nonlinear interactions between these strong fields and the charged particles in the medium. In addition to the vacuum

Summary and remarks

In Ref. [12], we studied the process of electron–positron pair production from the vacuum as a quantum tunneling phenomenon, we derived in semiclassical approximation the general rate formula (42) with 12a2=0. This consists of a Sauter-like tunneling exponential, and a pre-exponential factor, and are applicable to any system where the field strength points mainly in one direction and varies only along this direction. In this article, we generalize these formulas to the presence of a

Acknowledgments

The authors thank R. Ruffini for discussions on the issue of pair-production phenomenon and its application in astrophysical scenario. One of the authors, S.-S. Xue thanks J. Rafelski for discussions on the self-focusing phenomenon.

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