PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

Mutual assistance between the Schwinger mechanism and the dynamical Casimir effect

Hidetoshi Taya *

Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, Chinaand Research and Education Center for Natural Sciences, Keio University 4-1-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8521, Japan

(Received 1 April 2020; accepted 22 May 2020; published 16 June 2020)

We study massive charged particle production from the vacuum confined between two vibrating plates in thepresence of a strong electric field. We analytically derive a formula for the production number based on theperturbation theory in the Furry picture, and show that the Schwinger mechanism by the strong electric fieldand the dynamical Casimir effect by the vibration assist with each other to dramatically enhance the productionnumber by orders of the magnitude.

DOI: 10.1103/PhysRevResearch.2.023346

I. INTRODUCTION

According to the quantum field theory, our vacuum is byno means a simple empty space, but should be regarded as asort of matter, in which virtual particles are ceaselessly createdand annihilated. Our vacuum, therefore, exhibits nontrivialresponses when exposed to external forces/fields just as ordi-nary matters do. In particular, if the external forces/fields arestrong, the virtual particles in the vacuum may acquire largeenergy through the interaction with the forces/fields. If theacquired energy is sufficiently large, i.e., larger than the massscale of the particles, the virtual particles become on-shell realparticles. Therefore, our vacuum is no longer a stable state inthe presence of strong forces/fields and decays spontaneouslyby producing particles.

An example of such a particle production mechanism isthe Schwinger mechanism, which is driven by a strong slowelectric field (for review, see, e.g., Refs. [1–3]). The originalidea of the Schwinger mechanism was first proposed by Sauterin 1931 [4] as a resolution to the Klein paradox in the barrierscattering problem of a Dirac particle [5]. The idea was thata strong slow electric field induces a level crossing betweenpositive and negative energy states. The level crossing, then,causes quantum tunneling from a negative energy state toa positive one, which can be interpreted as pair productionof a particle and an antiparticle. This production processis genuinely nonperturbative since it is driven by quantumtunneling unless the electric field is fast [6–8]. Sauter’s ideawas, then, sophisticated by Heisenberg and Euler by derivingthe one-loop effective action in the presence of a constantand homogeneous strong electromagnetic field [9] and bySchwinger by fully formulating Sauter’s idea and Heisen-berg and Euler’s calculation within the quantum field theory[10]. Later, the Schwinger mechanism for Dirac particles was

*h_taya@keio.jp

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.

generalized to scalar particles [11], vector particles [12], andparticles with arbitrary spin [13], which revealed that theSchwinger mechanism is insensitive to spin or boson/fermionstatistics provided that there is no magnetic field and thatthe electric field does not change its direction. Despite itslong history, the Schwinger mechanism is still a hot researchtopic in theoretical physics and has been developing con-tinuously. To name a few, examples include the following:study of finite size effects [14–22]; inclusion of back-reactioneffects [23–31]; proposal of avalanchelike particle productionmechanism (QED cascade) [32–36]; developments of newtheoretical techniques (e.g., the world-line instanton method[37–39], classical statistical method [40,41], and the pertur-bation theory in the Furry picture [42–47]); and phenomeno-logical applications (e.g., the early stage dynamics of ultrarel-ativistic heavy-ion collisions [23,24,48–56], magnetogenesisin the early universe [57–62], and spin-current generation inspintronics [63]).

From an experimental side, the Schwinger mechanismhas not been observed yet. Since the lightest charged par-ticle is electron, it requires extremely strong electric fieldeEcr ≡ m2

e ∼√

1029 W/cm2 for the Schwinger mechanismto be manifest. The available electric field strength in thecurrent laboratory experiments is, however, far below thiscritical value. In fact, eE ∼

√1022 W/cm2 is the present

world record established by the highest-intensity focusedlaser at HERCULES [64]. The forthcoming laser experimentsat, e.g., the Extreme Light Infrastructure (ELI) and at theExawatt Center for Extreme Light Studies (XCELS), mayreach eE ∼

√1024−25 W/cm2, which is still below the crit-

ical field strength eEcr by several orders of the magnitude.Therefore, much attention has been paid recently to how toenhance the Schwinger mechanism to make it experimentallyobservable. One of the most promising ideas is the so-calleddynamically assisted Schwinger mechanism [65–69] (see alsoRefs. [70–72]), which proposed that a superposition of afast weak electromagnetic field can enhance the Schwingermechanism by several orders of the magnitude. Recently,Refs. [42–47] showed that the enhancement can be capturedvery well within the perturbation theory in the Furry picture[73–75] and clarified that the parametric (or perturbative)

2643-1564/2020/2(2)/023346(14) 023346-1 Published by the American Physical Society

HIDETOSHI TAYA PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

particle production by the superimposed field is the essenceof the enhancement. This observation implies that a similarenhancement mechanism may occur not only with a fast weakelectromagnetic field but also with other kinds of forces/fieldsinvolving parametric particle production.

Another example of the particle production from the vac-uum is the dynamical Casimir effect (for review, see, e.g.,Refs. [76,77]), which is driven by vibrating plates (or, moregenerally, variations of geometries in time, e.g., accelerationof a single mirror [78] and rotation of an object [79,80]).The dynamical Casimir effect, named by Yablonovitch [81]and Schwinger [82], was first proposed by Moore in 1970[83]. It is a parametric effect, which occurs when the typicalfrequency of the vibration matches the mass gap (or theeigenfrequency of cavity modes inside the boundaries) andcan essentially be studied with perturbative approaches aslong as the amplitude of the vibration is sufficiently small[84–87]. Notice that these features are quite distinct fromthe Schwinger mechanism, which is not a parametric effectand hence does not exhibit any threshold behaviors. Also, anonperturbative treatment of a strong electric field is essentialto describe the Schwinger mechanism. The original idea ofthe dynamical Casimir effect proposed by Moore was formassless photon production, but was naturally generalized tomassive particle production, e.g., in Refs. [88–92].

The direct experimental observation of the dynamicalCasimir effect is still lacking, which requires a very fastvibration of plates, e.g., ω/2π ∼ 10 GHz to produce cavityphotons (which could be lowered by nonperturbative effects[93]) and much larger frequency for massive particles likeelectron ω/2π ∼ 2me/2π ∼ 100 EHz. Such a fast vibrationis not experimentally available at the present, although a closevalue ω/2π ∼ 6 GHz was achieved recently in Ref. [94]. Onthe other hand, there already exists an indirect observation ofthe dynamical Casimir effect utilizing some effective motioninstead of real vibration of plates. For example, Ref. [95]modulated electrical length of a coplanar transmission line bymodulating inductance of a superconducting quantum inter-ference device (SQUID) connected to the transmission line[96,97]. This effectively realized ω/2π ∼ 11 GHz vibration,which enabled us to observe the cavity photon production bythe dynamical Casimir effect for the first time. It is, however,still very difficult even with the indirect methods to observethe dynamical Casimir effect for massive particles. Hence, anew mechanism to lower the threshold frequency is highlydemanded for future experiments.

The purpose of this paper is to propose a new mechanismto enhance the Schwinger mechanism and the dynamicalCasimir effect for a massive charged particle. Namely, weconsider a charged massive scalar field confined inside oftwo vibrating plates and apply a strong external electric field.With this setup, we show that the dynamical Casimir effectby the vibrating plates assists the Schwinger mechanism bythe strong electric field and vice versa. In particular, weshow that the assistance effectively reduces the mass gap, andthe particle production is dramatically enhanced by ordersof the magnitude. This may be understood as an analog ofthe dynamically assisted Schwinger mechanism [65–69], inwhich the combination of the parametric particle productionmechanism and the nonperturbative Schwinger mechanism

enhances the particle production [42–47]. Our theory is basedon the perturbation theory in the Furry picture [42–47,73–75],in which the interaction with the electric field is treatednonperturbatively but that with the vibration perturbatively.The perturbation theory in the Furry picture is very success-ful in the analytical description of the dynamically assistedSchwinger mechanism [42–47] and, as we shall show inthis paper, can smoothly describe the interplay between thenonperturbative Schwinger mechanism and the parametric dy-namical Casimir effect, which is not feasible within adiabaticapproaches such as the world-line instanton method [37–39].Note that our formalism can naturally be extended to, e.g.,fermionic case and particles with arbitrary spin, which wouldbe discussed in another publication.

This paper is organized as follows: In Sec. II, we derivea general formula for the particle production number in thepresence of both vibrating plates and a strong electric fieldbased on the perturbation theory in the Furry picture. InSec. III, we discuss quantitative features of the productionnumber by considering a constant and homogeneous electricfield configuration, for which the general formula can beevaluated exactly. We show that the dynamical Casimir effectby the vibrating plates assists the Schwinger mechanism bythe strong electric field and vice versa, which results inan enhancement of the production number by orders of themagnitude. Section IV is devoted to summary and discussion.

II. GENERAL FORMULA

We derive a production number formula for a massivecharged scalar field in the presence of both vibrating platesand a strong electric field based on the perturbation theoryin the Furry picture [42–47,73–75]. We first explain ourphysical setup in Sec. II A, and write down the correspondingfield equation in Sec. II B. In Sec. II C, we solve the fieldequation perturbatively in terms of the vibration while thenonperturbative interaction with the strong electric field isfully taken into account by using a Green function dressed bythe strong electric field (the perturbation theory in the Furrypicture). Then, we canonically quantize the field operator atasymptotic states, which allows us to directly compute thevacuum expectation value of the number operator to derivethe production number formula.

A. Setup

We consider a massive charged scalar field φ with mass mand electric charge e confined between two vibrating plateswith (infinitely large) transverse area S⊥ put at z = z0, z0 +L(t ), and apply a strong electric field in the transverse direc-tion (see Fig. 1),

Aμ ≡ (0, A⊥, 0) =(

0,−∫ t

dt E⊥, 0

), (1)

where we adopted the temporal gauge A0 = 0. One can alwaysgauge out the longitudinal component A3 = 0 because there isno longitudinal electric field.

We assume (i) that the plates are vibrating just slightly, i.e.,the amplitude of the vibration is small (but the frequency canbe large) and (ii) that the plates adiabatically stop moving at

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MUTUAL ASSISTANCE BETWEEN THE SCHWINGER … PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

FIG. 1. A schematic picture of our system: a strong electric fieldE⊥ applied between two vibrating plates with transverse area S⊥ anddistance L(t ) = L0 + l (t ).

the infinite future and past. Namely, we separate L(t ) into atime-independent part L0 and a time-dependent part l (t ) as

L(t ) ≡ L0 + l (t ), (2)

and assume (i) smallness of the amplitude,∣∣∣∣ l

L0

∣∣∣∣ � 1, (3)

and (ii) adiabaticity,

limt→±∞ l = 0. (4)

These assumptions may be reasonable because (i) it is ex-perimentally difficult to realize a vibration with large am-plitude if its frequency is large, and (ii) it is impossibleto vibrate the plates forever. Theoretically, as we describebelow, we use (i) to justify the lowest order perturbation interms of the vibration l and (ii) to define asymptotic states ina well-defined manner. Note that Eq. (2) does not necessarilymean L0 is large, i.e., only the ratio l/L0 is assumed to besmall and the size of L0 is arbitrary.

B. Field equation and boundary condition

The field operator φ obeys the Klein-Gordon equation,

0 = [∂2

t − (∂⊥ − ieA⊥)2 − ∂2z + m2]φ. (5)

As the field φ cannot go outside of the two plates, φ shouldsatisfy a boundary condition,

0 = φ(z0) = φ(z0 + L(t )). (6)

The time-depending boundary condition (6) is quite in-convenient in solving the Klein-Gordon equation (5), and tosee how the vibrating plates affect the time evolution of thesystem. To circumvent these inconveniences, we change the

coordinates as(tz

)→(

τ (t, z)ξ (t, z)

)≡(

t(z − z0)/L(t )

). (7)

Then, the Klein-Gordon equation (5) and the boundary condi-tion (6) can be re-expressed as

0 =[(

∂

∂τ− L

Lξ

∂

∂ξ

)2

−(∂⊥−ieA⊥)2 −(

1

L

∂

∂ξ

)2

+ m2

]φ,

(8)

and

0 = φ(ξ = 0) = φ(ξ = 1). (9)

Notice that L explicitly enters in the new field equation (8),which effectively describes the interaction between particlesand the vibrating plates. Through the interaction, the vibratingplates can supply energy to the vacuum if L �= 0, and thus par-ticles can be excited from the vacuum if the supplied energyis larger than the mass gap ∼m (i.e., the dynamical Casimireffect). Mathematically speaking, the time-translational in-variance in Eq. (8) is explicitly broken by L, so that the pos-itive and negative frequency modes (particle and antiparticlemodes) ∝ e∓iωpt are mixed with each other during the timeevolution. As shown below, the vacuum expectation value ofthe number operator can have nonzero value because of thismixing.

C. Production number within the perturbation theory in theFurry picture

We evaluate the production number of particles and an-tiparticles from the vacuum in the presence of the vibrationand strong electric field: We first solve the field equation (8)within the perturbation theory in the Furry picture (i.e.,a perturbative calculation with a dressed propagator/wavefunction) [42–47,73–75]. As we assumed that the plates arevibrating just slightly, we are justified to expand the field φ

perturbatively in terms of the small displacement l . On theother hand, the electric field E⊥ is assumed to be strong,so that its interaction with the field φ should be treatednonperturbatively. That is, a free propagator/wave function isinappropriate to perform the perturbative calculation in termsof the vibration, and one has to use a dressed propagator/wavefunction which is fully dressed by the strong electric field.After obtaining a solution of the field equation (8) within theperturbation theory in the Furry picture, we canonically quan-tize the field operator to define annihilation operators at theasymptotic future and past. Then, we can explicitly evaluatethe in-vacuum expectation value of the number operator att = ∞ to get a production number formula up to the lowestorder in the displacement l , while the interaction with thestrong electric field is fully taken into account.

We solve the field equation (8) within the perturbationtheory in the Furry picture. To this end, we first expand Eq. (8)in terms of the small displacement l � L0. We find[

∂2

∂τ 2− (∂⊥ − ieA⊥)2 −

(1

L0

∂

∂ξ

)2

+ m2

]φ = V φ, (10)

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HIDETOSHI TAYA PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

where

V ≡(

l

L0+ 2

l

L0

∂

∂τ

)ξ

∂

∂ξ− 2

l

L0

(1

L0

∂

∂ξ

)2

+ O((l/L0)2)

(11)

describes effects of the vibration. Note that L−10 ∂ξ gives the

longitudinal momentum and is totally independent of thesmallness of l . Also, note that V should be adiabaticallyswitched off at the infinite future and past,

limτ→±∞V = 0, (12)

because of the adiabaticity of l (4). Then, by using the Greenfunction technique, one can write down a formal solution ofEq. (10) as

φ(x) =√

Zφin(x) +∫

d4y GR(x, y)V (y)φ(y)

=√

Z

[φin(x) +

∫d4y GR(x, y)V (y)φin(y)

+ O((l/L0)2)

], (13)

where Z is the field renormalization constant and GR is theretarded Green function satisfying

δ4(x − x′) =[

∂2

∂τ 2− (∂⊥ − ieA⊥)2

−(

1

L0

∂

∂ξ

)2

+ m2

]GR(x, x′)

0 = GR(τ < τ ′). (14)

Notice that the Green function GR is fully dressed bythe strong electric field E⊥. In Eq. (13), we required theLehmann-Symanzik-Zimmermann (LSZ) asymptotic condi-tion [98] onto φ as1

limτ→±∞[φ(x) −

√Zφas(x)] = 0, (15)

where φas (as = in/out) is a solution of the field equationwithout V (i.e., without the perturbative vibration) because Vis vanishing at the infinite future and past [see Eq. (12)].

In order to define the notion of a particle from the solutionof the field equation (13), we express the asymptotic fieldoperator φas in terms of a mode integral as

φas(x) = 2π

L0

∞∑n=1

∫d2 p⊥

× [+φas

p⊥,n(x)aasp⊥,n + −φas

p⊥,n(x)bas†−p⊥,n

], (16)

1Strictly speaking, the equality in Eq. (15) should be interpreted asa weak equality. It is not a strong equality among operators, but aweak equality holds only after operators are sandwiched by states. Inthis work, we are interested only in expectation values of the numberoperator, so that the difference between a strong and weak equalityis not important.

where ±φasp⊥,n are the positive and negative frequency mode

functions at the corresponding asymptotic times which aredefined as two independent solutions of[

∂2

∂τ 2− (∂⊥ − ieA⊥)2 − 1

L20

∂2

∂ξ 2+ m2

]±φas

p⊥,n = 0, (17)

with a boundary condition (i.e., a plane wave at the infinitefuture and past),

limτ→−∞/+∞ ±φin/out

p⊥,n = exp[∓i∫ τ dτ ωp⊥−eA⊥,n]√2ωp⊥−eA⊥,n

× eip⊥·x⊥

2π

√L0

πsin(nπξ ), (18)

where ωp⊥,n (n = 1, 2, . . .) is the on-shell energy,

ωp⊥,n ≡√

m2 + p2⊥ +

(nπ

L0

)2

. (19)

Here, we can safely identify the positive and negative fre-quency mode functions at the asymptotic times by the planewaves thanks to the adiabatic assumption (12): The vibrationadiabatically goes off, and the time-translational symmetryis restored at the asymptotic times. Therefore, the planewaves, which are the eigenfunctions of the time-translationaloperation −i∂τ , becomes a good basis for the mode expan-sion at the asymptotic times. It should be stressed here that±φin

p⊥,n �= ±φoutp⊥,n if L or A⊥ is time dependent because their

time dependence mixes the positive and negative frequencymodes. Only if L and A⊥ are time independent, can one have±φin

p⊥,n = ±φoutp⊥,n. Note that the plane waves contain a sine

function and the on-shell energy is quantized by the labeln = 1, 2, . . . ∈ N because of the boundary condition (9). Wealso note that the mode functions ±φas

p⊥,n are normalized as

(±φas

p⊥,n|±φasp′⊥,n′) = ± L0

2πδn,n′δ2(p⊥ − p′

⊥)(∓φas

p⊥,n|±φasp′⊥,n′) = 0, (20)

with (φ1|φ2) being the Klein-Gordon inner product defined as

(φ1|φ2) ≡ i∫

d3x φ∗1

↔∂ τφ2. (21)

Imposing the standard canonical commutation relationsonto φas, one can quantize aas

p⊥,n, basp⊥,n to obtain annihilation

operators at the corresponding asymptotic times. From thenormalization condition (20), the commutation relation for theannihilation operators read

L0

2πδn,n′δ2(p⊥ − p′

⊥) = [aas

p⊥,n, aas†p′

⊥,n′] = [

basp⊥,n, bas†

p′⊥,n′]

(others) = 0, (22)

which, as usual, can be interpreted that aasp⊥,n and bas

p⊥,n destroya particle and an antiparticle with quantum number n, p⊥ atthe corresponding asymptotic times, respectively.

The annihilation operators at in and out states are notindependent with each other but are related with each otherby a Bogoliubov transformation. To see this, we use Eq. (13)

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MUTUAL ASSISTANCE BETWEEN THE SCHWINGER … PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

and

GR(x, x′) = iθ (τ − τ ′)[φas(x), φas†(x′)] = iθ (τ − τ ′)2π

L0

∞∑n=1

∫d2 p⊥

[+φas

p⊥,n(x)+φas∗p⊥,n(x′) − −φas

p⊥,n(x)−φas∗p⊥,n(x′)

]. (23)

Then, we find (ain/out

p⊥,n

bin/out†−p⊥,n

)=( (

+φin/outp⊥,n

∣∣φin/out)

−(−φin/outp⊥,n

∣∣φin/out))

= limτ→−∞/+∞

1√Z

( (+φ

in/outp⊥,n

∣∣φ)−(−φ

in/outp⊥,n

∣∣φ))

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

(+φout

p⊥,n

∣∣φin)

+i∫ +∞

−∞d4y+φout∗

p⊥,n(y)V (y)φin(y) + O((l/L0)2)

−(−φoutp⊥,n

∣∣φin)

+ i∫ +∞

−∞d4y−φout∗

p⊥,n(y)V (y)φin(y) + O((l/L0)2)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (24)

Therefore,

aoutp⊥,n = 2π

L0

∑n′

∫d2 p′

⊥

[{(+φout

p⊥,n

∣∣+φinp′⊥,n′)+ i

∫d4y+φout∗

p⊥,nV +φinp′⊥,n′ + O((l/L0)2)

}ain

p′⊥,n′

+{(

+φoutp⊥,n

∣∣−φinp′⊥,n′)+ i

∫d4y+φout∗

p⊥,nV −φinp′⊥,n′ + O((l/L0)2)

}bin†

−p′⊥,n′

], (25a)

bout†−p⊥,n = 2π

L0

∑n′

∫d2 p′

⊥

[{−(−φout

p⊥,n

∣∣+φinp′⊥,n′)+ i

∫d4y−φout∗

p⊥,nV +φinp′⊥,n′ + O

((l/L0)2

)}ain

p′⊥,n′

+{−(−φout

p⊥,n

∣∣−φinp′⊥,n′)+ i

∫d4y−φout∗

p⊥,nV −φinp′⊥,n′ + O((l/L0)2)

}bin†

−p′⊥,n′

]. (25b)

An important point here is that the annihilation operators at out-state aoutp⊥,n, bout

p⊥,n contain the creation operators at in-state

ain†p⊥,n, bin†

p⊥,n if the inner products and the matrix elements are nonvanishing. It is easy to show that the matrix elements can be

nonvanishing only when ±φinp⊥,n �= ±φout

p⊥,n. This implies that the annihilation operators at out-state aoutp⊥,n, bout

p⊥,n cannot annihilatethe in-vacuum state if ±φin

p⊥,n �= ±φoutp⊥,n and, in turn, implies that the in-vacuum expectation value of the number operator at

out-state becomes nonvanishing.Now, we are ready to compute the total production number of particles N and antiparticles N by directly evaluating the

in-vacuum expectation value of the number operator at out-state t = τ → +∞. The result is

(−)N ≡ S⊥L0 × 2π

L0

∞∑n=1

∫d2 p⊥

(−)n p⊥,n, (26)

where S⊥L0 is the volume of the system, and

np⊥,n ≡ 1

S⊥L0

〈vac; in|aout†p⊥,naout

p⊥,n|vac; in〉〈vac; in|vac; in〉 = 1

S⊥L0

2π

L0

∞∑n′=1

∫d2 p′

⊥

∣∣∣∣(+φoutp⊥,n

∣∣−φinp′⊥,n′)+ i

∫d4y+φout∗

p⊥,nV −φinp′⊥,n′ + O((l/L0)2)

∣∣∣∣2

,

(27a)

np⊥,n ≡ 1

S⊥L0

〈vac; in|bout†p⊥,nbout

p⊥,n|vac; in〉〈vac; in|vac; in〉 = 1

S⊥L0

2π

L0

∞∑n′=1

∫d2 p′

⊥

∣∣∣∣−(−φoutp⊥,n

∣∣+φinp′⊥,n′)+ i

∫d4y−φout∗

p⊥,nV +φinp′⊥,n′ + O((l/L0)2)

∣∣∣∣2

,

(27b)

are the distributions of particles and antiparticles for eachmode p⊥, n, respectively. The in-vacuum state at t = τ →−∞, which we write |vac; in〉, is defined as

0 = ainp⊥,n |vac; in〉 = bin

p⊥,n |vac; in〉 for any p⊥, n. (28)

Note that n′, p′⊥ in Eqs. (27a) and (27b) can be interpreted as

the quantum number of the pair produced particle. The matrixelement in Eqs. (27a) and (27b) always has off-diagonal com-ponents n �= n′ because the vibration gives finite longitudinalmomentum to the produced particles.

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HIDETOSHI TAYA PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

The production number formula (26) smoothly connectsthe nonperturbative Schwinger mechanism and the parametricdynamical Casimir effect: In the limit of no vibration V → 0,the second term in Eqs. (27a) and (27b) vanishes. Then,we only have the first term, which describes the particleproduction by an electric field. This is, by definition, theSchwinger mechanism. On the other hand, the second termincludes the effect of the vibration. In the absence of an elec-tric field E⊥ → 0 (or, equivalently, V becomes much greaterthan E⊥), our formalism reduces to the standard perturbationtheory without E⊥, in which the first term vanishes and onlythe second term survives. Then, the particle production isdriven by the vibration, which is nothing but the dynami-cal Casimir effect. For general E⊥ and/or V , our formula(26) deviates from the standard formula for the Schwingermechanism or the dynamical Casimir effect because (i) wehave an interference between the first and second terms and(ii) our mode function ±φas

p⊥,n is fully dressed by the electricfield E⊥ and hence the second term is strongly affectedby e and E⊥ in a nonlinear manner. Note that the aboveargument suggests that the typical frequency of the vibrationcan control the interplay between the Schwinger mechanismand the dynamical Casimir effect. This is because the strengthof the potential V (11) increases with the frequency, withwhich one can control the relative strength between E⊥and V . We explicitly confirm this expectation in the nextsection.

III. CONSTANT AND HOMOGENEOUS ELECTRIC FIELD

We discuss quantitative aspects of the particle productionbased on the general formula obtained in Sec. II. As a demon-stration, let us assume that the electric field E⊥ is sufficientlyslow such that it is well approximated by a constant andhomogeneous electric field, i.e.,

E⊥(x) = Eex with eE > 0. (29)

An advantage of this field configuration is that the field equa-tion (17) is analytically solvable, with which one can exactlyevaluate the production number (26). Also, the Schwingermechanism for the constant and homogeneous electric fieldconfiguration is very well understood [10], while that for aspacetime-dependent electric field configuration is less under-stood. It is, therefore, good to consider the well-understoodfield configuration, so that we can better understand physicsof the dynamical assistance by the dynamical Casimir ef-fect to the Schwinger mechanism and vice versa on a clearfooting.

The rest of this section is organized as follows: In Sec.III A, we present the analytical expression for the productionnumber (26) using the analytical solution of the field equation(17). In Sec. III B, we analytically discuss how the particleproduction in the presence of both a strong electric field andvibrating plates is related to the standard Schwinger mecha-nism and the dynamical Casimir effect based on the analyticalexpression for the production number obtained in Sec. III A.We also show that the interplay between the two productionmechanisms can be controlled by the typical frequency of thevibration. In Sec. III C, we numerically discuss the dynamicalassistance between the two production mechanisms and show

that it dramatically enhances the particle production numbercompared to what the standard Schwinger mechanism or thedynamical Casimir effect naively expects.

A. Evaluation of the formula (26)

Let us evaluate the production number formula (26) for theconstant and homogeneous electric field configuration (29).For this field configuration, one can analytically solve themode equation (17) as(

+φasp⊥,n(x)

−φasp⊥,n(x)

)=(

asp⊥,n(τ )

as∗p⊥,n(τ )

)eip⊥·x⊥

2π

√L0

πsin(nπξ ), (30)

where

inp⊥,n = e−πap⊥ ,n/4

(2eE )1/4

×[

D−iap⊥ ,n−1/2

(−eiπ/4

√2

eE(eEτ + px )

)]∗,

outp⊥,n = e−πap⊥ ,n/4

(2eE )1/4

× D−iap⊥ ,n−1/2

(eiπ/4

√2

eE(eEτ + px )

). (31)

Here, Dν (z) is the parabolic cylinder function and

ap⊥,n ≡ m2 + p2y + (

nπL0

)2

2eE. (32)

Using the solution (30), one can exactly evaluate the innerproduct and the matrix element in Eqs. (27a) and (27b). Wefind (

+φoutp⊥,n

∣∣−φinp′⊥,n′)

= −[(+φoutp⊥,n

∣∣−φinp′⊥,n′)]∗

= δ2(p⊥ − p′⊥) × L0

2πδn,n′ × (+i)e−πap⊥ ,n , (33)

and

i∫

d4y±φout∗p⊥,nV ∓φin

p′⊥,n′

= δ2(p⊥ − p′⊥) × L0

2π× i

nπ√eEL0

n′π√eEL0

(−1)n+n′

×∫ ∞

0dω

l (ω)

L0ei ω2−4pxω

4eE

(ω√2eE

)i(ap⊥ ,n−ap⊥ ,n′ )

× 1F1

(1

2+ iap⊥,n; 1 + i(ap⊥,n − ap⊥,n′ ); −i

ω2

2eE

),

(34)

where

l (ω) ≡∫ +∞

−∞dt e−iωt l (t ), (35)

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MUTUAL ASSISTANCE BETWEEN THE SCHWINGER … PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

and O((l/L0)2) term is discarded. Putting Eqs. (33) and (34)into Eqs. (27a) and (27b), we arrive at

np⊥,n = n−p⊥,n

= 1

(2π )3

∞∑n′=1

e−π (ap⊥ ,n+ap⊥ ,n′ )

∣∣∣∣δn,n′ + nπ√eEL0

n′π√eEL0

× (−1)n+n′∫ ∞

0dω

l (ω)

L0ei ω2−4pxω

4eE

(ω√2eE

)i(ap⊥ ,n−ap⊥ ,n′ )

× 1F1

(1

2+ iap⊥,n; 1 + i(ap⊥,n − ap⊥,n′ ); −i

ω2

2eE

)∣∣∣∣2

.

(36)

The total production number (26) can be evaluated byintegrating Eq. (36). By noting

∫d px = eET with T being

the whole time interval [55,99], we find

N = N = S⊥L0T

(2π )3m4∑

n

∑n′

2π√

eEL0

∣∣∣∣eE

m2

∣∣∣∣2

e−π m2

eE

×[δn,n′ × e

−π ( nπ√eEL0

)2

{1 + 2π

T

∣∣∣∣ nπ√eEL0

∣∣∣∣2 l (0)

L0

}

+ 2π

T

1√eE

∣∣∣∣ nπ√eEL0

∣∣∣∣2∣∣∣∣ n′π√

eEL0

∣∣∣∣2

e−π n2+n′2

2

∣∣∣ π√eEL0

∣∣∣2

×∫

d py

∫ ∞

0dω e−π

p2y

eE | l (ω)

L0|2

×∣∣∣∣∣1F1

(1

2+ i

2

{m2 + p2

y

eE+∣∣∣∣ nπ√

eEL0

∣∣∣∣2}

; 1

+ in2 − n′2

2

∣∣∣∣ π√eEL0

∣∣∣∣2

; −iω2

2eE

)∣∣∣∣∣2⎤⎦. (37)

B. The interplay between the Schwinger mechanism and thedynamical Casimir effect

Let us discuss the basic features of the production num-ber formula (37). Namely, we analytically discuss how theformula (37) is related to the Schwinger mechanism andthe dynamical Casimir effect and how the interplay betweenthe two production mechanisms can be described in terms ofthe typical frequency of the vibration.

1. Slow vibration: the Schwinger mechanism

For a slow vibration, the plates cannot supply large en-ergy to the vacuum. Therefore, the main energy source forthe particle production should come from the electric field.This implies that the particle production is dominated by theSchwinger mechanism, for which the production number isexponentially suppressed by the mass gap.

To see this, let � be the typical frequency of the vibrationand assume � ∼ 0. This is equivalent to assuming

l ∼ l0 ⇔ l ∼ 2πδ(ω) × l0. (38)

Then, one can simplify the distribution(−)n p⊥,n (36) as

np⊥,n = n−p⊥,n,

�→0−−→ 1

(2π )3

[1 + 2π

(nπ√eEL0

)2 l0L0

+ O((l0/L0)2)

]

× exp

[−π

m2 + p2y + (nπ/L0)2

eE

]

= 1

(2π )3exp

[−π

m2 + p2y + (nπ/L)2

eE

]

× [1 + O((l0/L0)2)], (39)

where

L = L0 + l0 (40)

is the total size of the system (2) in the slow limit (38). Up toO((l0/L0)2), Eq. (39) reproduces the well-known Schwingerformula [10] for a system with size L, in which the longitudi-nal momentum pz is quantized as pz → nπ/L. Therefore, theparticle production is, indeed, dominated by the Schwingermechanism for a slow vibration. It should be noted that thefiniteness of the system always makes the mass gap ωp⊥,n

larger by ∼nπ/L and hence the particle production is alwayssuppressed if the frequency is small, i.e., if the production isdominated by the Schwinger mechanism.

The total production number(−)N can be evaluated by

integrating Eq. (39). By neglecting terms of the order ofO((l0/L0)2), we find

(−)N

�→0−−→ S⊥LT

(2π )3m4

∣∣∣∣eE

m2

∣∣∣∣2

e−π m2

eE

× π√eEL

[−1 + ϑ3(0, e−π ( π√eEL

)2

)]

≡(−)N (Sch;L→∞) × F

(π√eEL

)

≡(−)N (Sch), (41)

where ϑ3 is the Jacobi elliptic function of the third kind, and(−)N (Sch;L→∞) is the Schwinger formula for the total productionnumber with infinite system size L → ∞ [10],

(−)N (Sch;L→∞) ≡ S⊥L

∫d3 p

(2π )3exp

[−π

m2 + p2x + p2

z

eE

]

= S⊥LT

(2π )3m4

∣∣∣∣eE

m2

∣∣∣∣2

e−π m2

eE . (42)

The factor F � 1 in Eq. (41) accounts for the finite size effect,which suppresses the Schwinger mechanism as

F

(π√eEL

)→{

1 − π√eEL

for π√eEL

� 1

2 π√eEL

e−π ( π√eEL

)2

for π√eEL

� 1. (43)

Namely, the finite size effect is not important for largeL � 1/

√eE , while it gives an exponential suppression

for small L � 1/√

eE for which the mass gap ωp =

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HIDETOSHI TAYA PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

√m2 + p2

y + (nπ/L)2 becomes larger than the electric fieldstrength no matter how small m is. Note that the finitetransverse size effect to the Schwinger mechanism was pre-viously discussed in Refs. [16–22] (with various theoreticalapproaches and boundary conditions), whose results are con-sistent with ours.

2. Fast vibration: the dynamical Casimir effect

If the vibration is fast enough, the plates are able to supplylarge energy to the vacuum. Then, the particle productionshould be driven by the vibrating plates rather than by theelectric field. Therefore, the dynamical Casimir effect shoulddominate the production, for which the production numberis just suppressed by powers of the mass gap and is freefrom the strong exponential suppression unlike the Schwingermechanism.

To see this, let us assume that the vibration is dominatedby a high-frequency mode � → ∞, for which the distribution(−)n p⊥,n (36) reads

np⊥,n = n−p⊥,n

�→∞−−−→ 1

(2π )3

∑n′

∣∣∣∣δn,n′ exp

[−π

m2 + p2y + (nπ/L0)2

eE

]

+ (−1)n+n′

√ωp⊥,nωp⊥,n′

nπ

L0

n′πL0

l∗(ωp⊥,n + ωp⊥,n′ )

L0

∣∣∣∣2

∼ 1

(2π )3

∑n′

1

ωp⊥,nωp⊥,n′

×(

nπ

L0

)2(n′πL0

)2∣∣∣∣∣ l (ωp⊥,n + ωp⊥,n′ )

L0

∣∣∣∣∣2

. (44)

Therefore, we obtain

lim�→∞

(−)N ∼ S⊥T

(2π )3

∑n

∑n′

∫d py

1

ωp⊥,nωp⊥,n′

×(

nπ

L0

)2(n′πL0

)2∣∣∣∣∣ l (ωp⊥,n + ωp⊥,n′ )

L0

∣∣∣∣∣2

≡(−)N (Cas). (45)

Equations (44) and (45) are independent of the electric fieldE and are dependent only on the vibration l , and reproducethe known formulas for the dynamical Casimir effect [87].Therefore, the particle production is, indeed, dominated by thedynamical Casimir effect for a fast vibration. Note that, to getthe second line of Eq. (44), we neglected the first term, whichcomes from the inner product in Eqs. (27a) and (27b), becauseit is exponentially suppressed by the mass gap ωp⊥,n. In otherwords, the contribution from the first term (i.e., the Schwingermechanism) is subleading compared to the second term (i.e.,the dynamical Casimir effect), which is just suppressed bypowers of the mass gap ωp⊥,n. For a very strong electricfield comparable to the mass gap eE � ω2

p⊥,n, the first term(i.e., the Schwinger mechanism) is free from the exponentialsuppression, for which case one may not neglect it. Note

FIG. 2. � dependence of the total production number N (37) forthe monochromatic vibration (46). As a comparison, the productionnumber for the Schwinger mechanism with finite L (41) and withinfinite L → ∞ (42), and for the dynamical Casimir effect (45)are shown in blue, dashed cyan, and red lines, respectively. Theparameters are fixed as eE/m2 = 0.1, mL0 = 30, and ml0 = 0.3.

also that the argument of l is ωp⊥,n + ωp⊥,n′ , which appearsbecause the vibration must supply energy ωp⊥,n + ωp⊥,n′ tocreate a pair of particles with energy ωp⊥,n and ωp⊥,n′ . Thisimplies that the (lowest-order parametric) dynamical Casimireffect never occurs for vibrations whose typical frequency �

is below ωp⊥,n + ωp⊥,n′ 2.

C. The dynamical assistance

At intermediate frequencies (i.e., the vibration is neitherfast or slow), both the Schwinger mechanism and the dynami-cal Casimir effect become important, and they assist with eachother to enhance the particle production.

We demonstrate how the assistance occurs by explicitlyevaluating the total production number N (37) by consideringa monochromatic vibration as an example:

l = l0 sin �t, (46)

for which the Fourier component l reads

l = l0 × iπ [δ(ω + �) − δ(ω − �)]. (47)

1. � dependence

Figure 2 shows the result for the total production numberN (37) as a function of the frequency �. As a demonstration,we considered a subcritical electric field eE/m2 = 0.1, suf-ficiently large system size mL0 = 30, and a small vibrationml0 = 0.3 (i.e., l0/L0 = 1/100).

2Note that our discussion here is based on the first-order pertur-bation theory with respect to the vibration, which means that thevibration can interact with particles only once. If one considershigher nth-order perturbations, i.e., multiple interactions between thevibration and particles, the threshold frequency can be lowered as� = (ωp⊥,n + ωp⊥,n′ )/n. Such higher order perturbations are, how-ever, suppressed by (l0/L0 )n and hence can safely be neglected.

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MUTUAL ASSISTANCE BETWEEN THE SCHWINGER … PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

Figure 2 clearly shows the interplay between theSchwinger mechanism and the dynamical Casimir effect thatwe discussed in Sec. III B. Namely, the Schwinger mecha-nism (the dynamical Casimir effect) dominates the particleproduction when the frequency is small � � m,

√eE , 1/L0

(large � � m,√

eE , 1/L0), for which the production numberis strongly (weakly) suppressed by an exponential (powers) ofthe mass gap ωp⊥,n. Notice that the dynamical Casimir effectoccurs only above |�| � 2

√m2 + (π/L0)2 > 2m because of

the energy conservation as we discussed below Eq. (45).This point may become clearer if one carries out the py

integration in Eq. (45), which is analytically doable for themonochromatic vibration (46) as

(−)N (Cas) = S⊥L0T

(2π )3m4 × 2π3

∣∣∣∣eE

m2

∣∣∣∣2∣∣∣∣ l0

L0

∣∣∣∣2

×∑

n

∑n′

∣∣∣∣nπ

L0

∣∣∣∣2∣∣∣∣n′π

L0

∣∣∣∣2 1

L0|�|

× �

⎛⎝|�| −

√m2 +

∣∣∣∣nπ

L0

∣∣∣∣2

−√

m2 +∣∣∣∣n′π

L0

∣∣∣∣2⎞⎠.

(48)

Hence, if the production is dominated by the dynamicalCasimir effect, the n, n′th modes can contribute to the produc-tion only when |�| �

√m2 + (nπ/L0)2 +

√m2 + (n′π/L0)2

is satisfied. This is in contrast to the production by theSchwinger mechanism, which does not have such a sharpthreshold behavior for the modes n, n′ because an electric fieldcan mix up different energy states.

At intermediate frequencies (cf. �/m ∼ 0.5 in Fig. 2),the production number N is dramatically enhanced by ordersof the magnitude compared to the naive expectations of theSchwinger mechanism and the dynamical Casimir effect. Thisis the dynamical assistance effect between the two productionmechanisms. Note that the size of the enhancement changesquadratically with l and also is dependent on the system sizeL0 and the electric field strength E , which will be discussedin Secs. III C 2 and III C 3, respectively. One may interpretthis assistance in two different ways (although the physics isthe same) depending on which mechanism one compares withthe result: (i) Since the production number in the presenceof the vibrating plates is much more abundant than the naiveSchwinger formula without vibrating plates, one may saythat the Schwinger mechanism is enhanced by the dynamicalCasimir effect. This may be understood as an analog of thedynamically assisted Schwinger mechanism not by a super-position of an additional electromagnetic field [65–69] but bythe vibrating plates. Physically, the energy supply from thevibrating plates reduces the mass gap, and hence it becomeseasier for the quantum tunneling by a strong electric field (i.e.,the Schwinger mechanism) to occur. (ii) The particle produc-tion occurs even below the threshold frequency for the low-est mode |�| �

√m2 + (nπ/L0)2 +

√m2 + (n′π/L0)2. Since

the naive (lowest-order parametric) dynamical Casimir effectalone cannot create particles if the frequency is below thethreshold, one may say that the Schwinger mechanism assiststhe dynamical Casimir effect to lower the threshold frequency.

FIG. 3. The contribution from the n, n′th modes to the totalproduction number N (37) for the monochromatic vibration (46).Different colors distinguish the maximum value of the summationn, n′ � nmax = 0, 1, . . . , 20, and the black curve is for nmax = 25.The parameters are the same as Fig. 2, i.e., eE/m2 = 0.1, mL0 = 30,and ml0 = 0.3.

Physically, the energy supplied by the electric field reduces themass gap, which assists the perturbative pair production bythe vibrating plates (i.e., the dynamical Casimir effect). Thismay be understood as an analog of the Franz-Keldysh effect[42,100–103], in which photoabsorption rate in the presenceof a strong electric field becomes finite even below the gapenergy.

Figure 3 shows how the n, n′th modes contribute to theproduction number N . The lowest mode n, n′ = 1 has thesmallest mass gap, so that it gives the largest contributionto the production number. In particular, the lowest modedominates the production if the system size is very smallL0 � |�|−1, |eE |−1/2, m−1 (cf. the lowest Landau approxi-mation in a strong magnetic field), for which the mass gapof the higher modes becomes very large and their produc-tion is strongly suppressed. With increasing � (i.e., injectingmore energy to the vacuum), higher n, n′th modes begin tocontribute at around the naive threshold frequency |�| ∼√

m2 + (nπ/L0)2 +√

m2 + (n′π/L0)2. Notice that there is nosharp threshold behavior as the dynamical Casimir effectexpects (48) and the particle production has a tail below thethreshold. This is nothing but the dynamical assistance by theSchwinger mechanism to the dynamical Casimir effect thatwe discussed in the last paragraph. Also, note that the produc-tion number for each mode exhibits an oscillating behavior asa function of � above the threshold. This can be understoodas an analog of the Franz-Keldysh oscillation [42,102,103],which occurs because of the quantum reflection process (adual process of the quantum tunneling) in the presence of astrong electric field. If L0 is large such that higher n, n′ modescan contribute to the total production number N , it is hardto see the oscillation behavior in N because the oscillation ateach mode cancels with each other after the n, n′ summation.Inversely, the oscillating behavior in N may survive if L0

is small such that only a few modes can contribute to theproduction number N .

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HIDETOSHI TAYA PHYSICAL REVIEW RESEARCH 2, 023346 (2020)

FIG. 4. L0 dependence of the total production number N (37) forthe monochromatic vibration (46). As a comparison, the productionnumber for the Schwinger mechanism with finite L (41) and withinfinite L → ∞ (42) are shown in blue and dashed cyan, respectively.The parameters are fixed as eE/m2 = 0.1, �/m = 1, and ml0 = 0.3.

2. L0 dependence

Figure 4 shows the system size L0 dependence of thetotal production number N (37). As a demonstration, weconsidered a subcritical electric field eE/m2 = 0.1 and an in-termediate frequency �/m = 1. We fixed the amplitude of thevibration as ml0 = 0.3, and therefore the ratio l0/L0 decreasesas increasing L0. Note that the naive dynamical Casimireffect expects no particle production N(Cas) = 0 [Eq. (45)] for�/m = 1.

For small L0, the particle production is strongly suppressedbecause of the finite size effect. Although the dynamicalassistance enhances the production by orders of the magnitudecompared to the naive Schwinger formula with finite L0,the finite size effect is so large that the production numberbecomes much smaller than the naive Schwinger formulawith infinite L0 → ∞. Indeed, one may analytically take theL0 → 0 limit of Eq. (37) to find

limL0→0

(−)N

= S⊥L0T

(2π )3m4 × 2

π√eEL0

∣∣∣∣eE

m2

∣∣∣∣2

e−π m2

eE e−π ( π√

eEL0)2

×[

1 + 2π

T

∣∣∣∣ π√eEL0

∣∣∣∣2 l (0)

L0

+2π

T

∣∣∣∣ π√eEL0

∣∣∣∣3 ∫ ∞

0

dω

2π

√eE

ωe

2 ω√eE

π√eEL0

∣∣∣∣ l (ω)

L0

∣∣∣∣2].

(49)

It is evident that the production number is suppressed expo-nentially by L0 due to the overall factor exp[−π (π/

√eEL0)2],

i.e., the finite size effect strongly suppresses the Schwingermechanism as well as the dynamical assistance. Also, thethird term (i.e., the dynamical assistance) is exponentiallylarge ∝ exp[2(ω/

√eE )(π/

√eEL0)] compared to the first and

second terms (i.e., the Schwinger mechanism with finite L0).This implies that the relative magnitude of the dynamicalassistance increases with an exponential of L−1

0 .

FIG. 5. E dependence of the total production number N (37) forthe monochromatic vibration (46). As a comparison, the productionnumber for the Schwinger mechanism with finite L (41) and withinfinite L → ∞ (42) are shown in blue and dashed cyan, respectively.The parameters are fixed as �/m = 1, mL0 = 30, and ml0 = 0.3.

For large L0, the production number asymptotes the naiveSchwinger formula with infinite L0 → ∞, i.e., no dynamicalassistance, since l0/L0 → 0. Indeed, in the limit of L0 → ∞,Eq. (37) behaves as

limL0→∞

(−)N

= S⊥L0T

(2π )3m4 ×

∣∣∣∣eE

m2

∣∣∣∣2

e−π m2

eE

⎡⎣1 + 1

L0Tl (0)

+ 1

L0T

1

eE

∫ +∞

−∞d pzd p′

zd py

∫ ∞

0dω |l (ω)|2

× p2z

eE

p′z2

eEe−π

p2y+ p2

z +p′z2

2eE

∣∣∣∣∣1F1

(1

2+ i

2

m2 + p2y + p2

z

eE; 1

+ ip2

z − p′z2

2eE; −i

ω2

2eE

)∣∣∣∣∣2⎤⎦, (50)

where we used

nπ

L0→ pz,

2π

L0

∑n

→∫ +∞

−∞d pz. (51)

Therefore, the dynamical assistance decays slowly ∝L−10 for

large L0 if l0 is fixed. Note that, if one fixes the ratio l0/L0

instead of fixing l0, the production number linearly increaseswith L0. This is simply because the dynamical assistancebecomes stronger with L0 since l0 increases (i.e., the vibrationbecomes stronger) with L0.

At intermediate L0, the dynamical assistance overwhelmsthe finite size effect, and the production number becomesmore abundant than the naive Schwinger formula for infiniteL0 → ∞.

3. E dependence

Figure 5 shows the electric-field strength E dependenceof the total production number N (37). As a demonstration,we considered sufficiently large system size mL0 = 30, and

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a small vibration ml0 = 0.3 (i.e., l0/L0 = 1/100). We alsofixed the frequency as �/m = 1, for which the naive dynam-ical Casimir effect expects no particle production N(Cas) = 0[Eq. (45)].

For a supercritical electric field exceeding the mass gap,the particle production is dominated by the Schwinger mech-anism, and the dynamical assistance becomes unimportant.This is because the Schwinger mechanism becomes freefrom the strong exponential suppression for a supercriticalfield, while the dynamical assistance is always suppressedby powers of the mass gap independent of the electric fieldstrength.

For a subcritical electric field strength below the massgap, the dynamical assistance becomes important, and theparticle production is enhanced by orders of the magnitudecompared to the naive Schwinger mechanism. This is be-cause the Schwinger mechanism is strongly suppressed byan exponential of the mass gap, while the dynamical as-sistance is suppressed just weakly by powers of the massgap.

IV. SUMMARY AND DISCUSSION

We have discussed massive charged particle productionfrom the vacuum in the presence of vibrating plates and astrong electric field. Based on the perturbation theory in theFurry picture, we have analytically derived a formula for theparticle production number, and have shown that (i) the for-mula smoothly describes the interplay between the Schwingermechanism by the strong electric field and the dynamicalCasimir effect by the vibrating plates, and the interplay iscontrolled by the typical frequency of the vibration; (ii) atintermediate frequency, the Schwinger mechanism and the dy-namical Casimir effect assist with each other to dramaticallyenhance the particle production by orders of the magnitude;(iii) the dynamical assistance can be greater than the finite sizeeffect, which gives an exponential suppression on the produc-tion number; and (iv) the dynamical assistance becomes moreimportant for smaller system size and/or subcritical electricfield strength.

Our results suggest a novel method to enhance theSchwinger mechanism by introduction of vibrating plates.The vibration could be realized either mechanically or effec-tively, just as the usual experimental setups for the dynamicalCasimir effect. This could be interesting to the current intenselaser experiments, in which the available field strength isstill far below the critical strength eEcr and hence somemechanism to enhance the Schwinger mechanism is highlydemanded. Our results indicate that the enhancement becomeslarger for weaker electric field strength below the critical one,which is exactly the relevant parameter regime at the currentexperiments. Note that one may combine the usual dynami-cally assisted Schwinger mechanism (i.e., superimposition ofa fast electromagnetic field) [65–69] to further enhance theproduction number.

One may interpret the above application in an oppositemanner: One may use a strong electric field to enhance thedynamical Casimir effect. Although the dynamical Casimireffect is usually applied for photon production, the sameproduction mechanism, in principle, can be applied to massive

particles (e.g., electron) as well. However, it is very difficultwithin the current experimental technologies to achieve a veryfast oscillation with frequency comparable to the mass scaleof, e.g., electron, so that such massive particle productionis not feasible in laboratory experiments at the present. Ourresults suggest that one may lower the threshold frequencyfor the dynamical Casimir effect by applying a strong electricfield if the massive particle is charged. In other words, onemay produce massive particles via the dynamical Casimireffect even with a slow vibration under the assistance by astrong electric field.

Another interesting application is ultrarelativistic heavy-ion collision experiments operated at Relativistic Heavy IonCollider (RHIC) at Brookhaven National Laboratory and theLarge Hadron Collider (LHC) at the European Organizationfor Nuclear Research (CERN). In those experiments, expand-ing color flux tubes (sometimes called glasma) are producedjust after a collision of ions [104–108]. The speed of theexpansion is relativistically fast, so that the dynamical Casimireffect may take place [109] on top of the Schwinger mecha-nism by the chromo-electromagnetic field of the flux tubes.If this is the case, the dynamical Casimir effect as well asthe dynamical assistance between the two mechanisms couldleave some experimental traces in, e.g., hadron multiplicitiesand rapidity correlations.

It is also interesting to pursue a condensed-matter analogof our results. Electrical breakdown of materials (Landau-Zener transition [110–113]) is one of the possible examples:Electrical breakdown can be understood as an analog of theSchwinger mechanism. Usually, materials are assumed to bestatic in discussing electrical breakdown. Our results suggestthat addition of some mechanical perturbations onto materials(e.g., acoustic wave on the surface) and/or varying materialproperties (e.g., dielectric constant) in time may trigger elec-trical breakdown even below the naive threshold, which couldbe useful in designing, e.g., optical devices.

In the present paper, we have concentrated on the dynam-ical assistance between the Schwinger mechanism (electricfield) and the dynamical Casimir effect (vibrating plates).In principle, any kinds of external forces/fields can assistthe Schwinger mechanism and vice versa. In fact, no matterwhat the physics origin of the potential V (11) is, one getsthe same assistance effect for the Schwinger mechanismfor the same V . It is, therefore, interesting to pursue othercombinations of forces/fields. For example, one may con-sider the assistance between the Schwinger mechanism andgravitational fields, which could be important to understandthe magnetogenesis in the early universe [57–62]. Anotherexample is the axion production by axion condensate [114],which may take place at the early universe where the curvatureis large and hence the curvature effect could enhance the axionproduction and vice versa. We leave these topics as futurework.

ACKNOWLEDGMENTS

The author would like to thank Xu-Guang Huang forcollaboration at the early stage of this work, and YukawaInstitute for Theoretical Physics at Kyoto University (YITP)for its warm hospitality during a workshop “Quantum kinetic

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theories in magnetic and vortical fields (YITP-T-19-06),”where a part of this work was done. The author was supported

by National Natural Science Foundation in China (NSFC)under Grant No. 11847206.

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