The Schwinger mechanism with perturbative electric fields
Hidetoshi TAYA (Fudan University; h_taya@fudan.edu.cn)

Ref: HT, PRD 99, 056006 (2019); X.-G. Huang, HT, 1904.08200

1. INTRODUCTION
1-1. The Schwinger mechanism

What happens if one adds a weak fast (perturbative)
time-depending electric field onto the slow strong field?

gap ~2𝜔𝒑Known facts in high energy community since 2008~
E field

Dynamically assisted Schwinger mechanism

tunneling[Schutzhold, Gies, Dunne 2008]✔ a QED analog of electrical breakdown ofsemi-conductors (Landau-Zener transition)✔ phenomenological applicationse.g.) heavy ion collisions, early Universe, laser✔ very well formulated for a static limit 𝑬 = 𝐜𝐨𝐧𝐬𝐭.
𝑉exp[−32𝜋(∓)
𝑛𝒑,𝑠 =

Q1:

pert. 𝛾✔ tunneling length ~2𝜔𝒑 is
reduced by a perturbative
kick by the fast weak E-fieldtunneling✔ big enhancement✔ not verified in experiments yet(hopefully in the near future)𝜋(𝑚2 +𝒑2⊥ )]𝑒𝐸[Schwinger 1951]

Q2:

What happens if the electric field
becomes time-dependent?

Franz-Keldysh effect

exp w/ Silicon[Franz 1958, Keldysh 1958][Yacoby 1966]

difference(■ー■)photoabsorption rate

Particles are spontaneously created from the vacuum
in a strong slow (non-perturbative) electric field.

vacuum ＝ Dirac sea

Known facts in cond-mat community since 1958~

1-2. Time-dep. by a weak fast perturbation

w/ E-field
w/o E-field(photon energy)-(gap energy)✔ the same setup in semi-conductor and measures
photo-absorption rate w/ & w/o strong E-field✔ a long history (> 60 years) in both theory
and experiment in cond-mat.
✔ not only enhancement, but also oscillation

Is it OK to understand the dynamically assisted Schwinger mechanism as an analog of the Franz-Keldysh
effect in QED? If yes, can we see the oscillation? How does it modify the momentum spectrum?

by deriving an analytical formula for the particle production from a strong slow E-field superimposed by a weak
fast E-field based on the perturbation theory in the Furry picture

AIM: Answer Q1 and Q2
2. THEORYഥ non-perturbatively, but 𝓔 perturbatively✔ Treat 𝑬✔ Evaluate diagrams up to first order in 𝓔
[Furry 1951]
ഥ (Furry picture):
with a propagator fully dressed by 𝑬

✔ QED in the presence of a static strong E-field
superimposed by a weak time-depending E-fieldഥ + 𝓔 𝑡 where |𝑬ഥ | ≫ |𝓔|𝑬 𝑡 =𝑬2

ഥ∥𝓔 𝑡✔ Assume 𝑬(for arbitrary direction, see [Huang, HT, 1904.08200] )✔ Assume spatial homogeneity(∓)
𝑛±𝒑,𝑠 = ෍ න 𝑑 3 𝒑′𝑠′+

𝓔(0)out†✔ Neglect backreaction (E-field is external)

ഥ𝑬

ഥ
𝑬=+ഥ𝑬

ഥ 𝑬
ഥ 𝑬
ഥ
𝑬+

+⋯

(0)in/outis the dressed propagator, and ± 𝜓𝒑𝑠is sol. of Dirac eq.

～(0)in/outҧ
0 = 𝑖𝜕
−𝑒𝐴−𝑚𝜓
± 𝒑,𝑠//2= ෍ න 𝑑 3 𝒑′ න 𝑑 3 𝒙± 𝜓𝒑,𝑠where～～～

Compute the production number analytically for～～2-2. Perturbation theory in the Furry picture～

2-1. Setup(0)in(0)out(0)in4𝑥 𝜓
ത
/
𝜓−න𝑑𝒜
𝜓
± 𝒑′ ,𝑠 ′± 𝒑𝑠± 𝒑′ ,𝑠 ′𝑠′with boundary conditions (plane wave at 𝑡 → ±∞)(0)in/out
lim ± 𝜓𝒑,𝑠
∝ exp[∓𝑖𝜔𝒑 𝑡 + 𝑖𝒑 ⋅ 𝒙]𝑡→−∞/+∞

2-3. Analytical formulaഥ ≪ 𝟏: ■ dominates✔ static limit 𝝎Τ 𝒆𝑬

The number of produced particles is given by2 +𝒑2 )𝑉𝜋(𝑚(∓)⊥𝑛𝒑,𝑠 =exp−2𝜋 3𝑒𝐸ത

1 𝑚2 + 𝒑2⊥ ∞ℰሚ 𝜔𝑖 𝜔2 + 4𝜔𝑝∥
𝑖 𝑚2 + 𝒑2⊥
𝑖 𝜔21 +න 𝑑𝜔
exp −𝐹1 1 −; 2;12 𝑒𝐸ത
𝐸ҧ
4𝑒𝐸ҧ
2 𝑒𝐸ത
2 𝑒𝐸ത
0usual Schwinger formula2

correction by perturbative 𝓔𝑉(∓)
𝑛𝒑,𝑠 ∼exp −32𝜋𝜋(𝑚2 +𝒑2⊥ )𝑒𝐸ത

1+222𝜋 𝑚 + 𝒑⊥ ℰ𝑒𝐸ത2𝐸ത𝑉𝜋(𝑚2 +𝒑2⊥ )∼exp −
32𝜋𝑒(𝐸ത + ℰ)ഥ ≫ 𝟏: ■ dominates✔ perturbative limit 𝝎Τ 𝒆𝑬
2 + 𝒑2 |ℰሚ 2𝜔 |2𝑉1𝑚𝒑(∓)⊥𝑛𝒑,𝑠 ∼2𝜋 3 4 𝜔𝒑2
𝜔𝒑2

3. RESULTS(∓)
3-3. Momentum distribution 𝑛𝒑,𝑠3-1. Setup

FK effect also modifies the✔ A demonstration for ℰ 𝑡 = ℰ0 cos Ω𝑡 with ℰ0 = 0.01𝐸ത

momentum distribution✔ FK effect (dyn. assisted Schwinger(∓)(∓)
33-2. Total production number 𝑁𝒑,𝑠 = ∫ 𝑑 𝒑 𝑛𝒑,𝑠mech.) enhances the production
Exactly the same behavior asfor low ΩFK effect in cond-mat !!

✔ FK effect oscillates the distribution
for large Ωnumerical✔ FK effect equally occurs✔ The peak position is lowered

in QED with strong slow

✔ an excellent agreement with thenumerical result obtained by directly
solving the Dirac eq. w/o any approx

E-field + weak fast E-field

3-3. Momentum distribution✔ dyn. assisted Schwinger mech.

3-4. Interpretation

can be understood as a low-Ω

Not only tunneling, but also reflection

behavior of FK effectis important✔ Tunneling✔ an oscillating behavior appears➡ wavefunc. get inside of the gap

for large Ω (FK oscillation)➡ enhancement in low Ω➡ non-negligible (~10% effect)(dyn. assisted Schwinger mech.)✔ Reflection✔ a sharp peak in the difference➡ interference b/w wavefunc.

e.g.) modulation spectroscopy

heading to and reflected by the gap

in cond-mat

2

prob. density ~ |wave function|ഥ)z (direction of 𝑬➡ oscillation & peak shift

4. SUMMARY
What I didI

studiedspontaneousparticle

production from the vacuum (theSchwinger

mechanism)

in

the

presence of a slow strong electricfield superimposed by a fast weakelectric field

What we obtained/learned

What to do✔ Analytical formula for the production in strong slow E-field superimposed by✔ Application to phenomenologye.g.) particle production in glasma

a weak fast E-field is obtained based on the pert. theory in the Furry picture✔ A QED analog of the Franz-Keldysh effect occurs, which✔ Generalization: spatial inhomogeneity;(i) enhances the production below the threshold (dyn. ass. Schwinger mech.);inclusion of magnetic fields; higher order(ii) shifts the perturbative peak to a lower frequency;

calculations, regularization of IR divergence...

(iii) results in an oscillating pattern above the threshold✔ Not only quantum tunneling, but also reflection plays an important role

✔ Apply ideas of cond-mat to the Schwinger
mechanism