Schwinger mechanism with
time-dependent electric fields
Hidetoshi TAYARIKEN iTHEMS
・Interplay between perturbative & non-perturbative particle production:[HT, Fujii, Itakura, PRD (2014)] [HT, Fujimori, Misumi, Nitta, Sakai, JHEP (2021)]
・Dynamically assisted Schwinger mechanism and Furry-picture perturbation theory:[HT, PRD (2019)] [Huang, HT, PRD (2019)]
・Review on strong-field QED:
[Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

What is the Schwinger mechanism?
[Sauter (1932)] [Heisenberg, Euler (1936)] [Schwinger (1951)]✔ Vacuum pair production occurs in the presence of strong E-fields
Intuitive picture for a slow electric field = quantum tunneling~ 2mpositive energy bandgap
Dirac sea

~ 2mslow & strong E~ 2m/eEtunnelingWhat is the Schwinger mechanism?
[Sauter (1932)] [Heisenberg, Euler (1936)] [Schwinger (1951)]✔ Vacuum pair production occurs in the presence of strong E-fields
Intuitive picture for a slow electric field = quantum tunneling~ 2mpositive energy bandgap
Dirac sea

~ 2mslow & strong E✔ For constant strong E-field 𝐸𝐸 𝑡𝑡, 𝒙𝒙~ 2m/eEtunneling(at least,= 𝐸𝐸, it’s understood well theoretically)

𝑒𝑒𝑒𝑒 2 𝑉𝑉𝑉𝑉
𝑚𝑚2
𝑁𝑁𝑒𝑒 ± =× exp −𝜋𝜋
∼ exp[−# × gap height × (gap length)]
2𝜋𝜋 3
𝑒𝑒𝑒𝑒

[Schwinger (1951)]

・ Non-perturbative ⇒ Interesting, since it is the unexplored region of QED (or QFT)
・ Vacuum process ⇒ Fundamental, since all the physical processes occur on top of vacuum
・ However, exponentially suppressed i.t.o. mass ⇒ Not confirmed by experiments yet…
cf.) Guinness world record:
HERCULES laser 𝑒𝑒𝑒𝑒~ 1022 W/cm2 ∼ 0.01 MeV 2 ≪ 𝑚𝑚𝑒𝑒2 ∼[Yanovsky et al (2008)]

1029 W⁄cm2 ∼ 0.5 MeV 2

Timeliness✔ Now is the best time to study the Schwinger mechanism !
Developments in experimental technologies ⇒ Novel strong-field sources

Timeliness✔ Now is the best time to study the Schwinger mechanism !
Developments in experimental technologies ⇒ Novel strong-field sourcesex.1) Intense lasers
Schwinger’s
limit

Current
world
record[Mourou, Tajima, Bulanov (2006)]

Timeliness✔ Now is the best time to study the Schwinger mechanism !
Developments in experimental technologies ⇒ Novel strong-field sourcesex.2) Heavy-ion collisionsex.1) Intense lasers
Schwinger’slimit・(ultra)peripheral collisions
𝑒𝑒𝑒𝑒, 𝑒𝑒𝑒𝑒 ∼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼∼ 𝛼𝛼𝛼𝛼𝛼𝛼 × 𝑚𝑚𝜋𝜋2𝑟𝑟 2[Skokov, Illarionov, Toneev (2009)][Deng, Huang (2012)]

Current
world
record[Mourou, Tajima, Bulanov (2006)]

・glasma (color flux tube)
𝑔𝑔𝐸𝐸color , 𝑔𝑔𝐵𝐵color ∼ 𝑄𝑄s2 ∼ 𝑂𝑂 1 GeV 2[Lappi, McLerran (2006)]

・Also in other geometries:- asymmetric coll. [Hirono, Hongo, Hirano (2014)] [Voronyuk, et al (2014)]- low-energy coll. [Review: Rafelski, et al (2014)] [Maltsev et al (2019)]

Timeliness✔ Now is the best time to study the Schwinger mechanism !
Developments in experimental technologies ⇒ Novel strong-field sourcesex.2) Heavy-ion collisionsex.1) Intense lasers
Schwinger’slimit・(ultra)peripheral collisions
𝑒𝑒𝑒𝑒, 𝑒𝑒𝑒𝑒 ∼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼∼ 𝛼𝛼𝛼𝛼𝛼𝛼 × 𝑚𝑚𝜋𝜋2𝑟𝑟 2[Skokov, Illarionov, Toneev (2009)][Deng, Huang (2012)]

Current
world
record[Mourou, Tajima, Bulanov (2006)]

・glasma (color flux tube)
𝑔𝑔𝐸𝐸color , 𝑔𝑔𝐵𝐵color ∼ 𝑄𝑄s2 ∼ 𝑂𝑂 1 GeV 2[Lappi, McLerran (2006)]

・Also in other geometries:- asymmetric coll. [Hirono, Hongo, Hirano (2014)] [Voronyuk, et al (2014)]- low-energy coll. [Review: Rafelski, et al (2014)] [Maltsev et al (2019)][Allor, Cohen, McGady (2008)][Solinas, Amoretti, Giazotto (2021)]ex.3) Cond-mat analogues: Graphene, Cold atom, Superconductor, Semiconductor, …
[Szpak, Shutzhold (2012)][Thesis by Linder; 1807.08050]

Timeliness✔ Now is the best time to study the Schwinger mechanism !
Developments in experimental technologies ⇒ Novel strong-field sourcesex.2) Heavy-ion collisionsex.1) Intense lasers
Schwinger’slimit・(ultra)peripheral collisions
𝑒𝑒𝑒𝑒, 𝑒𝑒𝑒𝑒 ∼𝛼𝛼𝛼𝛼𝛼𝛼𝛼𝛼∼ 𝛼𝛼𝛼𝛼𝛼𝛼 × 𝑚𝑚𝜋𝜋2𝑟𝑟 2[Skokov, Illarionov, Toneev (2009)][Deng, Huang (2012)]

Current
world
record[Mourou, Tajima, Bulanov (2006)]

・glasma (color flux tube)
𝑔𝑔𝐸𝐸color , 𝑔𝑔𝐵𝐵color ∼ 𝑄𝑄s2 ∼ 𝑂𝑂 1 GeV 2[Lappi, McLerran (2006)]

・Also in other geometries:- asymmetric coll. [Hirono, Hongo, Hirano (2014)] [Voronyuk, et al (2014)]- low-energy coll. [Review: Rafelski, et al (2014)] [Maltsev et al (2019)][Allor, Cohen, McGady (2008)][Solinas, Amoretti, Giazotto (2021)]ex.3) Cond-mat analogues: Graphene, Cold atom, Superconductor, Semiconductor, …
[Szpak, Shutzhold (2012)][Thesis by Linder; 1807.08050]✔ Schwinger’s result is insufficient for actual problems ⇒ Beyond Schwinger !
・ Inhomogeneous field (e.g., time- & space-dependence, more efficient field config., …)・ Realtime dynamics (e.g., backreaction, intermediate particle number, …)・ Higher-loop effects (e.g., radiation, mass shift, exponentiation conjecture, …)・ Other observables (e.g., spin, chirality, high-harmonic generation, …)・…

Today’s talk
Topic: the Schwinger mechanism with time-depending E-field
Part I: Interplay b/w non-pert. & pert. production mechanisms[HT, Fujii, Itakura, PRD (2014)] [HT, Fujimori, Misumi, Nitta, Sakai, JHEP (2021)]

Part II: Dynamically assisted Schwinger mechanism[HT, PRD (2019)] [Huang, HT, PRD (2019)]

The most important message
Perturbation theory in the Furry picture gives you a very powerful analytical
approach for the Schwinger mechanism with time-dependent E-fieldIntroduction⇓

Part I:

Interplay b/w non-pert. & pert. production mechanisms⇓

Part II:

Dynamically assisted Schwinger mechanism⇓

Summary

Background (1/2): Interplay b/w non-pert. & pert. mech.
✔ Consider time-dependent E-field, having strength 𝒆𝒆𝑬𝑬𝟎𝟎 and frequency 𝛀𝛀Background (1/2): Interplay b/w non-pert. & pert. mech.
✔ Consider time-dependent E-field, having strength 𝒆𝒆𝑬𝑬𝟎𝟎 and frequency 𝛀𝛀・ The tunneling picture should break down if the frequency Ω is large
Non-pert. tunneling 𝑁𝑁~ exp[#/𝑒𝑒𝐸𝐸0 ]~ 2𝑚𝑚~ 2𝑚𝑚/𝑒𝑒𝐸𝐸0

tunneling
Tunneling time Δ𝑡𝑡 ~

2𝑚𝑚
𝑒𝑒𝐸𝐸0

⇒ E-field should be slower than Δ𝑡𝑡⇒ Ω−1 ≿ Δ𝑡𝑡
⇒1≿

Δ𝑡𝑡
Ω𝑚𝑚
=
≡ 𝛾𝛾 (Keldysh parameter)
Ω−1
𝑒𝑒𝐸𝐸0

Background (1/2): Interplay b/w non-pert. & pert. mech.
✔ Consider time-dependent E-field, having strength 𝒆𝒆𝑬𝑬𝟎𝟎 and frequency 𝛀𝛀・ The tunneling picture should break down if the frequency Ω is large

Pert. multi-photon 𝑁𝑁~𝑒𝑒𝐸𝐸02𝑛𝑛

Non-pert. tunneling 𝑁𝑁~ exp[#/𝑒𝑒𝐸𝐸0 ]
~ 2𝑚𝑚

Ω
Ω
Ω

~ 2𝑚𝑚/𝑒𝑒𝐸𝐸0

tunneling
Tunneling time Δ𝑡𝑡 ~

2𝑚𝑚
𝑒𝑒𝐸𝐸0

⇒ E-field should be slower than Δ𝑡𝑡
⇒Ω

−1

⇒1≿

≿ Δ𝑡𝑡

Δ𝑡𝑡
Ω𝑚𝑚
=
≡ 𝛾𝛾 (Keldysh parameter)
Ω−1
𝑒𝑒𝐸𝐸0

~ 2m

Pair production when 𝑛𝑛Ω > 2𝑚𝑚

・ For large Ω, E-field may behave like a photon and interact incoherently (perturbatively)⇒ an analogue of photo-electric effect in materials

Background (2/2): Semi-classical analysis✔ The interplay is said to be “confirmed” by (but only by) semi-classical analysis

Background (2/2): Semi-classical analysis✔ The interplay is said to be “confirmed” by (but only by) semi-classical analysis
・ Trans-series expansion in terms of ℏ

𝑁𝑁 = � 𝑁𝑁𝑛𝑛,𝑚𝑚 ℏ𝑛𝑛 e−𝑚𝑚𝑚𝑚/ℏ = (𝑁𝑁0,1 + 𝑂𝑂(ℏ)) 𝐞𝐞−𝑺𝑺/ℏ + 𝑂𝑂(e−2𝑆𝑆/ℏ )𝑛𝑛,𝑚𝑚・ Valid in the slow limit ℏ ≪ 1 ⇔ Ω ≪ 1 :・ Steepest descent analysis [Brezin, Itzykson (1970)]
・ Imaginary-time method [Popov (1972)]
・ Divergent asymptotic series method [Berry (1989)]
・ Worldline instanton method [Dunne, Shubert (2005)]・ (exact) WKB [HT, Fujimori, Misumi, Nitta, Sakai (2020)]
・…

∵ iℏ𝜕𝜕𝑡𝑡 𝜓𝜓 = 𝐻𝐻 Ω𝑡𝑡 𝜓𝜓𝜏𝜏 ≡ 𝑡𝑡/ℏi𝜕𝜕𝜏𝜏 𝜓𝜓 = 𝐻𝐻 ℏΩ𝜏𝜏 𝜓𝜓Background (2/2): Semi-classical analysis✔ The interplay is said to be “confirmed” by (but only by) semi-classical analysis
・ Trans-series expansion in terms of ℏ

𝑁𝑁 = � 𝑁𝑁𝑛𝑛,𝑚𝑚 ℏ𝑛𝑛 e−𝑚𝑚𝑚𝑚/ℏ = (𝑁𝑁0,1 + 𝑂𝑂(ℏ)) 𝐞𝐞−𝑺𝑺/ℏ + 𝑂𝑂(e−2𝑆𝑆/ℏ )𝑛𝑛,𝑚𝑚・ Valid in the slow limit ℏ ≪ 1 ⇔ Ω ≪ 1 :・ Steepest descent analysis [Brezin, Itzykson (1970)]
・ Imaginary-time method [Popov (1972)]
・ Divergent asymptotic series method [Berry (1989)]
・ Worldline instanton method [Dunne, Shubert (2005)]・ (exact) WKB [HT, Fujimori, Misumi, Nitta, Sakai (2020)]
・…

∵ iℏ𝜕𝜕𝑡𝑡 𝜓𝜓 = 𝐻𝐻 Ω𝑡𝑡 𝜓𝜓𝜏𝜏 ≡ 𝑡𝑡/ℏi𝜕𝜕𝜏𝜏 𝜓𝜓 = 𝐻𝐻 ℏΩ𝜏𝜏 𝜓𝜓

・ Production number 𝑁𝑁 is controlled solely by the Keldysh parameter 𝛾𝛾 ≡𝑚𝑚2𝑆𝑆 = 𝜋𝜋
𝑔𝑔 𝛾𝛾
𝑒𝑒𝐸𝐸0

Example: 𝑒𝑒𝑒𝑒 𝑡𝑡 = 𝑒𝑒𝑒𝑒0 cos(Ω𝑡𝑡)

See, e.g., [Dunne, Gies, Schubert, Wang (2006)]
for an explicit expression for 𝑔𝑔(𝛾𝛾)

𝜋𝜋

𝑚𝑚2

𝑚𝑚2
−𝜋𝜋
+ 𝑂𝑂 𝛾𝛾 1 ⇒ 𝑁𝑁 ∝ e 𝑒𝑒𝐸𝐸0
𝑒𝑒𝐸𝐸0

𝑚𝑚Ω
𝑒𝑒𝐸𝐸0

2𝑚𝑚
𝑒𝑒𝐸𝐸0
ln 𝛾𝛾 2 + const. +𝑂𝑂 𝛾𝛾 −1 ⇒ 𝑁𝑁 ∝𝑚𝑚2
Ω

# of photons

pert. power-dep.
after exponentiation

2𝑚𝑚Ω[Keldysh (1965)](slow limit 𝛾𝛾 ≪ 1)
(fast limit 𝛾𝛾 ≫ 1)

Background (2/2): Semi-classical analysis✔ The interplay is said to be “confirmed” by (but only by) semi-classical analysis
・ Trans-series expansion in terms of ℏ

𝑁𝑁 = � 𝑁𝑁𝑛𝑛,𝑚𝑚 ℏ𝑛𝑛 e−𝑚𝑚𝑚𝑚/ℏ = (𝑁𝑁0,1 + 𝑂𝑂(ℏ)) 𝐞𝐞−𝑺𝑺/ℏ + 𝑂𝑂(e−2𝑆𝑆/ℏ )𝑛𝑛,𝑚𝑚・ Valid in the slow limit ℏ ≪ 1 ⇔ Ω ≪ 1 :・ Steepest descent analysis [Brezin, Itzykson (1970)]
・ Imaginary-time method [Popov (1972)]
・ Divergent asymptotic series method [Berry (1989)]
・ Worldline instanton method [Dunne, Shubert (2005)]・ (exact) WKB [HT, Fujimori, Misumi, Nitta, Sakai (2020)]
・…

∵ iℏ𝜕𝜕𝑡𝑡 𝜓𝜓 = 𝐻𝐻 Ω𝑡𝑡 𝜓𝜓𝜏𝜏 ≡ 𝑡𝑡/ℏi𝜕𝜕𝜏𝜏 𝜓𝜓 = 𝐻𝐻 ℏΩ𝜏𝜏 𝜓𝜓

・ Production number 𝑁𝑁 is controlled solely by the Keldysh parameter 𝛾𝛾 ≡𝑚𝑚2𝑆𝑆 = 𝜋𝜋
𝑔𝑔 𝛾𝛾
𝑒𝑒𝐸𝐸0

Example: 𝑒𝑒𝑒𝑒 𝑡𝑡 = 𝑒𝑒𝑒𝑒0 cos(Ω𝑡𝑡)

See, e.g., [Dunne, Gies, Schubert, Wang (2006)]
for an explicit expression for 𝑔𝑔(𝛾𝛾)

𝜋𝜋

𝑚𝑚2

𝑚𝑚2
−𝜋𝜋
+ 𝑂𝑂 𝛾𝛾 1 ⇒ 𝑁𝑁 ∝ e 𝑒𝑒𝐸𝐸0
𝑒𝑒𝐸𝐸0

𝑚𝑚Ω
𝑒𝑒𝐸𝐸0

2𝑚𝑚
𝑒𝑒𝐸𝐸0
ln 𝛾𝛾 2 + const. +𝑂𝑂 𝛾𝛾 −1 ⇒ 𝑁𝑁 ∝𝑚𝑚2
Ω

# of photons

pert. power-dep.
after exponentiation

2𝑚𝑚Ω[Keldysh (1965)](slow limit 𝛾𝛾 ≪ 1)
(fast limit 𝛾𝛾 ≫ 1)

✔ However, you should not be convinced with the semi-classical argument
・ Fast limit γ ≫ 1 is quite dangerous⇒ Q1: What is really happening at 𝛄𝛄 ≫ 𝟏𝟏 ?

・ Another dimensionless parameter should exist∵ 3 dimensionful parameters 𝑚𝑚, 𝑒𝑒𝐸𝐸0 , Ω

𝑒𝑒𝑒𝑒ex). Sauter pulsed E-field 𝑒𝑒𝑒𝑒 𝑡𝑡 = cosh20(Ω𝑡𝑡)

𝛾𝛾 < 1𝛾𝛾 > 1⇒ 2 dimensionless parameters ⇒ Q2: Why only 𝛄𝛄 ? The other has no role in the interplay ?

Idea: One-photon process✔ One-photon process is the key to answer Q1 & Q2 !

Idea: One-photon process✔ One-photon process is the key to answer Q1 & Q2 !
✔ Larger 𝛀𝛀, more perturbative ⇒ one-photon dominates eventually・because there is no time to interact many times, no matter how strong 𝑒𝑒𝑒𝑒0 is
𝑬𝑬

𝑬𝑬

𝑬𝑬

𝑬𝑬

𝑬𝑬

… … 𝑬𝑬……

𝑬𝑬

larger 𝛀𝛀 ≃ shorter E-fieldtime

multiple scatterings

・ analytical formula is available (for any field config. !)3

𝑁𝑁1−photon = � �d 𝒑𝒑 d 𝒑𝒑′
𝑠𝑠,𝑠𝑠𝑠

𝑬𝑬

𝑬𝑬

𝑬𝑬

𝑬𝑬

… … 𝑬𝑬
……

𝑬𝑬

time

finite # of scatterings

・ cover different region, as # of photons should be huge

3

𝑬𝑬

𝑬𝑬

𝑬𝑬𝒑𝒑, 𝑠𝑠

𝒑𝒑𝒑, 𝑠𝑠𝑠

2𝑚𝑚→ ∞ in the semi-classics
Ω

2
∞
𝑉𝑉
4𝑚𝑚2 14𝑚𝑚2=
� d𝜔𝜔 1 − 22+ 2
4𝜋𝜋 2 2𝑚𝑚𝜔𝜔 3𝜔𝜔𝑒𝑒𝐸𝐸� 𝜔𝜔

2

[HT, Fujii, Itakura (2014)] [QFT textbook by Itzykson, Zuber]

Result: Interplay b/w tunneling, multi-, one-photon𝒆𝒆𝒆𝒆✔ An explicit demonstration for Sauter pulsed E-field 𝒆𝒆𝒆𝒆 𝒕𝒕 = 𝐜𝐜𝐜𝐜𝐜𝐜𝐡𝐡𝟐𝟐𝟎𝟎(𝛀𝛀𝒕𝒕)

・ One-photon dominates for large 𝛀𝛀,where semi-classical methods fail,・ One-photon is more efficient𝛾𝛾 > 1𝜈𝜈 < 1than tunneling ; 𝑁𝑁 is the largest at Ω ∼ 2𝑚𝑚

Result: Interplay b/w tunneling, multi-, one-photon𝒆𝒆𝒆𝒆✔ An explicit demonstration for Sauter pulsed E-field 𝒆𝒆𝒆𝒆 𝒕𝒕 = 𝐜𝐜𝐜𝐜𝐜𝐜𝐡𝐡𝟐𝟐𝟎𝟎(𝛀𝛀𝒕𝒕)

・ One-photon dominates for large 𝛀𝛀,where semi-classical methods fail,・ One-photon is more efficient𝛾𝛾 > 1𝜈𝜈 < 1✔ Two dimensionless parameters 𝜸𝜸 =・ Sauter field is solvable, which can becompared w/ Schwinger & one-photon𝛾𝛾 ≫ 1, 𝜈𝜈𝜈𝜈 ≪ 1 ⇒ pert. one−photon𝛾𝛾 ≪ 1, 𝜈𝜈𝜈𝜈 ≫ 1 ⇒ non−pert. tunneling・ 𝜈𝜈 =(work done by E-field)
𝑒𝑒𝐸𝐸0 /Ω=Ω
(photon energy)= (# of photons involved)

than tunneling ; 𝑁𝑁 is the largest at Ω ∼ 2𝑚𝑚
𝒎𝒎𝛀𝛀
𝒆𝒆𝑬𝑬
, 𝝂𝝂 ≡ 𝛀𝛀𝟐𝟐𝟎𝟎 control the interplay
𝒆𝒆𝑬𝑬𝟎𝟎Message of Part I(1) The interplay b/w non-pert. & pert. production mechanisms is controlled
by 𝛾𝛾 =

𝑚𝑚Ω
𝑒𝑒𝑒𝑒
(Keldysh parameter) and also by 𝜈𝜈 = 2 (~ # of photons involved)𝑒𝑒𝑒𝑒Ω(2) Semi-classical methods (e.g., worldline) are dangerous for 𝛾𝛾 ≫ 1;
It breaks down for 𝜈𝜈 ≿ 1, where one-photon process dominates(3) One-photon pert. production is very efficient, compared to non-pert. tunneling⇒ This is the essence of the dynamically assisted Schwinger mech., as I show from now[HT, Fujii, Itakura, PRD (2014)][HT, Fujimori, Misumi, Nitta, Sakai, JHEP (2021)]

Introduction⇓

Part I:

Interplay b/w non-pert. & pert. production mechanisms⇓

Part II:

Dynamically assisted Schwinger mechanism⇓

Summary

Background: Dynamically assisted Schwinger mechanism✔ In Part I, I discussed E-field having a single frequency mode 𝛀𝛀✔ What if E-field is bi-frequent; superposition of slow strong + weak fast E-fields ?

Background: Dynamically assisted Schwinger mechanism✔ In Part I, I discussed E-field having a single frequency mode 𝛀𝛀✔ What if E-field is bi-frequent; superposition of slow strong + weak fast E-fields ?
・ Combinatorial effects of non-pert. & pert. mechs. ⇒ dynamically assisted Schwinger mech.

perturbative scattering

quantum tunneling[Dunne, Gies, Schutzhold (2008), (2009)][Piazza et al (2009)] [Monin, Voloshin (2010)]

𝑁𝑁 ∼ exp[−# × gap height × (gap length)] ⇒ Enhancement in production(even though the fast field is very weak)
reduced by pert.

Background: Dynamically assisted Schwinger mechanism✔ In Part I, I discussed E-field having a single frequency mode 𝛀𝛀✔ What if E-field is bi-frequent; superposition of slow strong + weak fast E-fields ?
・ Combinatorial effects of non-pert. & pert. mechs. ⇒ dynamically assisted Schwinger mech.

perturbative scattering

quantum tunneling[Dunne, Gies, Schutzhold (2008), (2009)][Piazza et al (2009)] [Monin, Voloshin (2010)]

𝑁𝑁 ∼ exp[−# × gap height × (gap length)] ⇒ Enhancement in production(even though the fast field is very weak)
reduced by pert.
・ Phenomenological importance:ex1) laser: available E-field is still weak ⇒ needs enhancement to observe the Schwinger mechex2) heavy-ion collisions: (Mini-)jets on top of glasma, Event generators (e.g., PYTHIA)

Background: Dynamically assisted Schwinger mechanism✔ In Part I, I discussed E-field having a single frequency mode 𝛀𝛀✔ What if E-field is bi-frequent; superposition of slow strong + weak fast E-fields ?
・ Combinatorial effects of non-pert. & pert. mechs. ⇒ dynamically assisted Schwinger mech.

perturbative scattering

quantum tunneling[Dunne, Gies, Schutzhold (2008), (2009)][Piazza et al (2009)] [Monin, Voloshin (2010)]

𝑁𝑁 ∼ exp[−# × gap height × (gap length)] ⇒ Enhancement in production(even though the fast field is very weak)
reduced by pert.
・ Phenomenological importance:ex1) laser: available E-field is still weak ⇒ needs enhancement to observe the Schwinger mechex2) heavy-ion collisions: (Mini-)jets on top of glasma, Event generators (e.g., PYTHIA)・ Typically, analyzed within semi-classical methods

BUT: NOT quantitatively good
Q: Better method ???

Exact
numerical result
Semi-classicalresult⇒ A: Perturbation theory in the Furry picture
Schwinger result
for const. E-fieldPerturbation theory in Furry picture
Idea
Split the field 𝐸𝐸 into strong 𝐸𝐸S and weak 𝐸𝐸w parts and treat

𝐸𝐸S and 𝐸𝐸w non-perturbatively and perturbatively, respectively.

cf. distorted Born wave approximation in quantum mechanics

Perturbation theory in Furry picture (1/3)
STEP 1

Split the total 𝑬𝑬 into strong slow 𝑬𝑬𝐒𝐒 & weak fast 𝓔𝓔𝐰𝐰

𝑬𝑬 = 𝑬𝑬𝐬𝐬 + 𝓔𝓔𝐰𝐰

Perturbation theory in Furry picture (1/3)
STEP 1

STEP 2

Split the total 𝑬𝑬 into strong slow 𝑬𝑬𝐒𝐒 & weak fast 𝓔𝓔𝐰𝐰

𝑬𝑬 = 𝑬𝑬𝐬𝐬 + 𝓔𝓔𝐰𝐰

Solve Dirac eq. non-pert. w.r.t. 𝑬𝑬𝐬𝐬 , but just pert. w.r.t. 𝓔𝓔𝐰𝐰
𝑖𝑖𝑖𝑖/ − 𝒆𝒆𝑨𝑨/𝐬𝐬 − 𝑚𝑚 𝜓𝜓� = 𝒆𝒆𝓐𝓐
/ 𝐰𝐰 𝜓𝜓�

∞
(0)
�
�
𝑥𝑥
=
𝜓𝜓
𝑥𝑥
+
𝜓𝜓
∫−∞ d𝑦𝑦 4 𝑆𝑆R 𝑥𝑥, 𝑦𝑦 𝒆𝒆𝓐𝓐⇒
/ 𝐰𝐰 (𝑦𝑦)𝜓𝜓� 0 𝑦𝑦 + 𝑂𝑂(|𝒆𝒆𝓐𝓐𝐰𝐰 |2 )Here, 𝜓𝜓� (0) and 𝑆𝑆R are non-perturbatively dressed by 𝑬𝑬𝐬𝐬 as𝑖𝑖𝑖𝑖/ − 𝒆𝒆𝑨𝑨/𝐬𝐬 − 𝑚𝑚 𝜓𝜓� (0) = 0

𝑖𝑖𝑖𝑖/ − 𝒆𝒆𝑨𝑨/ 𝐬𝐬 − 𝑚𝑚 𝑆𝑆R (𝑥𝑥, 𝑦𝑦) = 𝛿𝛿 4 (𝑥𝑥 − 𝑦𝑦)Perturbation theory in Furry picture (2/3)
STEP 3in/outCompute in/out annihilation operators 𝑎𝑎� 𝑝𝑝,𝑠𝑠
in/out𝑎𝑎�𝒑𝒑,𝑠𝑠

in/out†𝑏𝑏�−𝒑𝒑,𝑠𝑠≡lim

𝑡𝑡→−∞⁄+∞

� d3 𝒙𝒙(𝑢𝑢𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙 )†(𝑣𝑣𝒑𝒑,𝑠𝑠e+𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙 )†in/out, 𝑏𝑏�𝑝𝑝,𝑠𝑠�𝜓𝜓(𝑥𝑥)

from 𝜓𝜓�

Perturbation theory in Furry picture (2/3)
STEP 3in/outCompute in/out annihilation operators 𝑎𝑎� 𝑝𝑝,𝑠𝑠
in/out𝑎𝑎�𝒑𝒑,𝑠𝑠

in/out†𝑏𝑏�−𝒑𝒑,𝑠𝑠≡lim

𝑡𝑡→−∞⁄+∞

� d3 𝒙𝒙(𝑢𝑢𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙 )†(𝑣𝑣𝒑𝒑,𝑠𝑠e+𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙 )†in/out, 𝑏𝑏�𝑝𝑝,𝑠𝑠

from 𝜓𝜓��𝜓𝜓(𝑥𝑥)

in
out
in
out
, 𝑜𝑜�𝒑𝒑,𝑠𝑠
are inequivalent 𝑜𝑜�𝒑𝒑,𝑠𝑠≠ 𝑜𝑜�𝒑𝒑,𝑠𝑠
and related with each⇒ 𝑜𝑜�𝒑𝒑,𝑠𝑠

other by a Bogoliubov transformationout
𝑎𝑎�𝒑𝒑,𝑠𝑠𝛼𝛼𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠3
out† = � � d 𝒑𝒑′ −𝛽𝛽 ∗𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠𝑏𝑏�−𝒑𝒑,𝑠𝑠
𝑠𝑠𝑠where, up to 1st order in 𝒆𝒆𝓐𝓐𝐰𝐰 ,(0)out†𝛼𝛼𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠 = � d3 𝒙𝒙 + 𝜓𝜓𝒑𝒑,𝑠𝑠(0)out†𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠 = � d3 𝒙𝒙 − 𝜓𝜓𝒑𝒑,𝑠𝑠
0 in/out

Here, ± 𝜓𝜓𝒑𝒑,𝑠𝑠𝑖𝑖𝑖𝑖/ − 𝒆𝒆𝑨𝑨/𝐬𝐬 − 𝑚𝑚

𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠∗
𝛼𝛼𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠in𝑎𝑎�𝒑𝒑′,𝑠𝑠′in†𝑏𝑏�−𝒑𝒑′,𝑠𝑠′(0)in4
� 0 out 𝒆𝒆𝓐𝓐𝐰𝐰 + 𝜓𝜓 (0)in+ 𝑂𝑂(|𝒆𝒆𝓐𝓐𝐰𝐰 |2 )+ 𝜓𝜓𝒑𝒑′,𝑠𝑠′ − 𝑖𝑖 � 𝑑𝑑 𝑥𝑥 + 𝜓𝜓𝒑𝒑,𝑠𝑠
𝒑𝒑′ ,𝑠𝑠 ′

/

(0)in
4
� 0 out 𝒆𝒆𝓐𝓐𝐰𝐰 + 𝜓𝜓 (0)in+ 𝑂𝑂(|𝒆𝒆𝓐𝓐𝐰𝐰 |2 )+ 𝜓𝜓𝒑𝒑′,𝑠𝑠′ − 𝑖𝑖 � 𝑑𝑑 𝑥𝑥 − 𝜓𝜓𝒑𝒑,𝑠𝑠
𝒑𝒑′ ,𝑠𝑠 ′/are sol. of Dirac eq. dressed by 𝒆𝒆𝑨𝑨𝐬𝐬 w/ different B.C.
0 in/out=0± 𝜓𝜓𝒑𝒑,𝑠𝑠w/

lim

𝑡𝑡→−∞⁄+∞

0 in/out+ 𝜓𝜓𝒑𝒑,𝑠𝑠
0 in/out− 𝜓𝜓𝒑𝒑,𝑠𝑠=

𝑢𝑢𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙
𝑣𝑣𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙

Perturbation theory in Furry picture (3/3)
STEP 4

Evaluate the in-vacuum expectation value of # operator
d3 𝑁𝑁𝑒𝑒2out† out
3 𝒑𝒑′ 𝛽𝛽
′
′
�
d
≡
vac;
in
𝑎𝑎
𝑎𝑎
vac;
in
=
�
𝒑𝒑,𝑠𝑠
𝒑𝒑,𝑠𝑠
𝒑𝒑,𝑠𝑠;𝒑𝒑 ,𝑠𝑠
d𝒑𝒑3
𝑠𝑠𝑠

Perturbation theory in Furry picture (3/3)
STEP 4

Evaluate the in-vacuum expectation value of # operator
d3 𝑁𝑁𝑒𝑒2out† out
3 𝒑𝒑′ 𝛽𝛽
′
′
�
d
≡
vac;
in
𝑎𝑎
𝑎𝑎
vac;
in
=
�
𝒑𝒑,𝑠𝑠
𝒑𝒑,𝑠𝑠
𝒑𝒑,𝑠𝑠;𝒑𝒑 ,𝑠𝑠
d𝒑𝒑3
𝑠𝑠𝑠

Assume 𝑬𝑬𝐬𝐬 is so slow that it can be approximated as a static E-field
0 in/out⇒ analytical sol. of Dirac eq. ± 𝜓𝜓𝒑𝒑,𝑠𝑠⇒ one can evaluate 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠′ exactly !

is known

Perturbation theory in Furry picture (3/3)
STEP 4

Evaluate the in-vacuum expectation value of # operator
d3 𝑁𝑁𝑒𝑒2out† out
3 𝒑𝒑′ 𝛽𝛽
′
′
�
d
≡
vac;
in
𝑎𝑎
𝑎𝑎
vac;
in
=
�
𝒑𝒑,𝑠𝑠
𝒑𝒑,𝑠𝑠
𝒑𝒑,𝑠𝑠;𝒑𝒑 ,𝑠𝑠
d𝒑𝒑3
𝑠𝑠𝑠

Assume 𝑬𝑬𝐬𝐬 is so slow that it can be approximated as a static E-field
0 in/out⇒ analytical sol. of Dirac eq. ± 𝜓𝜓𝒑𝒑,𝑠𝑠

is known⇒ one can evaluate 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠′ exactly !

✔ Remarks

・ Directly computing VEV of # operator⇒ inclusive quantity that includes all the processes up to 1st order in 𝓔𝓔𝐟𝐟
out† out
vac; in 𝑎𝑎𝒑𝒑,𝑠𝑠
𝑎𝑎𝒑𝒑,𝑠𝑠 vac; in = � 𝑒𝑒𝒑𝒑,𝑠𝑠 𝑋𝑋; out|vac; in

=�
𝑋𝑋

2

X;out

=

𝑋𝑋

2

Schwinger

2

+

+

2

𝓔𝓔𝐟𝐟

[Baltz, McLerran (2001)]

2

+

2
𝓔𝓔𝐟𝐟

+

2
𝓔𝓔𝐟𝐟

+…

multiple
Schwinger
Breit-Wheeler Schwinger
Schwinger+annihilation+ absorption

・ Production number ≠ Vacuum decay rate
・ No approximation in evaluating 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠′

[Cohen McGady (2008)] [Fukushima, Gelis, Lappi (2009)]
within 0-th order WKB [Torgrimsson et al (2017)]

How good is the Furry picture ?

Result: Furry picture vs exact & semi-classical results✔ Case of 𝑬𝑬 = (𝟎𝟎, 𝟎𝟎, 𝑬𝑬𝐬𝐬 + 𝓔𝓔𝐰𝐰 𝐜𝐜𝐜𝐜𝐜𝐜 𝛀𝛀𝒕𝒕)

Semi-classical result Exact numerical result
Furry picture

Schwinger result
for const. E-field𝑒𝑒𝑒𝑒slow ⁄𝑚𝑚2 = 0.25,
𝐸𝐸fast /𝐸𝐸slow = 0.01,
𝒑𝒑 = 0✔ Excellent agreement b/w Furry picture and exact results !
・ Furry picture is super powerful
・ Physics lesson:
The dynamically assisted Schwinger mechanism= One-photon process under E-field ⇒ Perturbative effect is the essenceTechnically nice, but physically what are new ?
✔ Furry-picture perturbation theory revealed:(1) Dynamically assisted Schwinger mech. beyond
the semi-classical regime(2) Spin production from the vacuum(3) Effects spatial inhomogeneity

Result (1/3): Beyond the semi-classical regime✔ Case of 𝑬𝑬 = (𝟎𝟎, 𝟎𝟎, 𝑬𝑬𝐬𝐬 + 𝓔𝓔𝐰𝐰 𝐜𝐜𝐜𝐜𝐜𝐜 𝛀𝛀𝒕𝒕)
Semi-classical result

Exact numerical result
Furry picture

Schwinger result
for const. E-field𝑒𝑒𝑒𝑒slow ⁄𝑚𝑚2 = 0.25,
𝐸𝐸fast /𝐸𝐸slow = 0.01,
𝒑𝒑 = 0”enhancement” is just a one aspect of the dynamically assisted Schwinger mech.
・ shift of the one-photon peak Ω > 2𝑚𝑚 ⇒ increase of the effective mass in E-field・ non-trivial oscillations above the one-photon peak

⇐ related to the “band structure” of the QED vacuum (next slide)cf) Pondermotive energy
in AC E-field・ can be interpreted as an analogue of the Franz-Keldysh effect in semi-conductor(next next slide)Interpretation: “Band structure” of QED vacuum

××
energyPositive energy band

Dirac sea

Electron probability density
gap

E-field direction

・ Quantum tunneling ⇒ Enhancement
・ Quantum reflection ⇒ Oscillation⇒ The spectrum of the dynamically assisted Schwinger mech.
reflects the structure of the Dirac sea = QED vacuumAnalogue in semi-conductor: Franz-Keldysh effect✔ Enhancement & oscillation are analogous to Franz-Keldysh effectin semi-conductors[Franz (1958)] [Keldysh (1958)]⇐ Apply strong slow E-field & a photon (~ weak fast E-field) onto a semi-cond.,
and measure photo-absorption rate

photo-absorption rate

Semi-conductor✔ The observables are essentially the same:
photo-absorption rate ～ Im[1-loop action] ～ total # of particles = int. of spectrum

Analogue in semi-conductor: Franz-Keldysh effect✔ Enhancement & oscillation are analogous to Franz-Keldysh effectin semi-conductors[Franz (1958)] [Keldysh (1958)]⇐ Apply strong slow E-field & a photon (~ weak fast E-field) onto a semi-cond.,
and measure photo-absorption rate

photo-absorption rate

Semi-conductor

QED

E≠0
E=0

✔ The observables are essentially the same:
photo-absorption rate ～ Im[1-loop action] ～ total # of particles = int. of spectrum✔ The curves are the same !
⇒ the dynamically assisted Schwinger mech. (2008~) = Franz-Keldysh effect (1958~)

Technically nice, but physically what are new ?
✔ Furry-picture perturbation theory revealed:(1) Dynamically assisted Schwinger mech. beyond
the semi-classical regime(2) Spin production from the vacuum(3) Effects spatial inhomogeneity

Result (2/3): Spin production from the vacuum✔ The Schwinger mechanism is usually independent of spin
𝑁𝑁𝑒𝑒 ± =𝑒𝑒𝑒𝑒 2 𝑉𝑉𝑉𝑉
× exp
2𝜋𝜋 3

𝑚𝑚2
−𝜋𝜋
𝑒𝑒𝑒𝑒

(spin quantization axis ∝ E-field)∼ exp[−# × gap height × (gap length)]

・ linearly-polarized E-field does not couple to spin ⇒ gap is independent of spin・ true for the dynamically assisted Schwinger mechanism for 𝐸𝐸 = (0,0, 𝐸𝐸s + ℰw (𝑡𝑡))Result (2/3): Spin production from the vacuum✔ The Schwinger mechanism is usually independent of spin
𝑁𝑁𝑒𝑒 ± =𝑒𝑒𝑒𝑒 2 𝑉𝑉𝑉𝑉
× exp
2𝜋𝜋 3

𝑚𝑚2
−𝜋𝜋
𝑒𝑒𝑒𝑒

(spin quantization axis ∝ E-field)∼ exp[−# × gap height × (gap length)]

・ linearly-polarized E-field does not couple to spin ⇒ gap is independent of spin・ true for the dynamically assisted Schwinger mechanism for 𝐸𝐸 = (0,0, 𝐸𝐸s + ℰw (𝑡𝑡))✔ If E-field is circularly polarized (e.g., has transverse components 𝐸𝐸 = (ℰw 𝑡𝑡 , 0, 𝐸𝐸s )),the Schwinger mechanism can depend on spin・ intuitive reason: spin-orbit coupling ∝ 𝒔𝒔 ⋅ (𝒗𝒗 × 𝑬𝑬)[Foldy, Wouthuysen (1950)] [Tani (1951)]

Lab. frame

Rest frame

𝑬𝑬

𝑩𝑩 ~ 𝑬𝑬 × 𝒗𝒗

𝒗𝒗

boost

⇒ spin dependence
due to Zeeman splittingΔ(gap) ∼ 𝒔𝒔 ⋅ 𝑩𝑩 ∼ 𝒔𝒔 ⋅ (𝒗𝒗 × 𝑬𝑬)

Result (2/3): Spin production from the vacuum✔ The Schwinger mechanism is usually independent of spin
𝑁𝑁𝑒𝑒 ± =𝑒𝑒𝑒𝑒 2 𝑉𝑉𝑉𝑉
× exp
2𝜋𝜋 3

𝑚𝑚2
−𝜋𝜋
𝑒𝑒𝑒𝑒

(spin quantization axis ∝ E-field)∼ exp[−# × gap height × (gap length)]

・ linearly-polarized E-field does not couple to spin ⇒ gap is independent of spin・ true for the dynamically assisted Schwinger mechanism for 𝐸𝐸 = (0,0, 𝐸𝐸s + ℰw (𝑡𝑡))✔ If E-field is circularly polarized (e.g., has transverse components 𝐸𝐸 = (ℰw 𝑡𝑡 , 0, 𝐸𝐸s )),the Schwinger mechanism can depend on spin・ intuitive reason: spin-orbit coupling ∝ 𝒔𝒔 ⋅ (𝒗𝒗 × 𝑬𝑬)[Foldy, Wouthuysen (1950)] [Tani (1951)]

Lab. frame

Rest frame

𝑬𝑬

𝑩𝑩 ~ 𝑬𝑬 × 𝒗𝒗

𝒗𝒗

boost

⇒ spin dependence
due to Zeeman splittingΔ(gap) ∼ 𝒔𝒔 ⋅ 𝑩𝑩 ∼ 𝒔𝒔 ⋅ (𝒗𝒗 × 𝑬𝑬)✔ Technically, there was NO good analytical approach
𝑂𝑂(ℏ0 ) terms will be important (=geometrical effects)・ naïve semi-classical formula can be applied only to linearly-polarized case[Takayoshi, Wu, Oka, (2021)][Kitamura, Nagaosa, Morimoto, (2019)]⇒ the perturbation theory in the Furry picture can deal with this problem !

Result (2/3): Spin production from the vacuum✔ Non-linearly-polarized case1 d3 𝑁𝑁↑
momentum dist. of spin up
𝑉𝑉 d𝒑𝒑𝟑𝟑ℰw cos Ω𝑡𝑡 × cos 𝜃𝜃ℰ⊥
𝐸𝐸 = ℰw cos Ω𝑡𝑡 × sin 𝜃𝜃ℰ⊥
𝐸𝐸s

diff. b/w spin up and down

1 d3 𝑁𝑁↑1 d3 𝑁𝑁↓−𝑉𝑉 d𝒑𝒑𝟑𝟑
𝑉𝑉 d𝒑𝒑𝟑𝟑

For 𝑒𝑒𝐸𝐸slow ⁄𝑚𝑚2 = 0.4

・ Technical point:ℰfast ⁄𝐸𝐸slow = 0.025

Good agreement with numerics (though I haven’t plotted here)
・ Physics lessons:- Non-negligible (~10%) spin imbalance appears- Angular dependence due to spin-orbit coupling ∝ 𝒔𝒔 ⋅ (𝒑𝒑 × 𝑬𝑬)- No essential difference in enhancement & oscillation from the linearly-polarized case

Technically nice, but physically what are new ?
✔ Furry-picture perturbation theory revealed:(1) Dynamically assisted Schwinger mech. beyond
the semi-classical regime(2) Spin production from the vacuum(3) Effects spatial inhomogeneity

Result (3/3): Spatial inhomogeneity✔ What if E-field has spatial inhomogeneity 𝑬𝑬 𝑡𝑡 → 𝑬𝑬(𝑡𝑡, 𝒙𝒙) ?
・ No established analytical methods[Bulanov et al. (2004)]e.g.) locally-constant-field approximation: 𝑁𝑁 ∼ 𝑉𝑉 exp⇒ NOT good even without spatial inhomogeneity…

𝜋𝜋𝑚𝑚2
−
𝑒𝑒𝑒𝑒

3

→ ∫ d 𝒙𝒙 exp

・ Numerical simulation is also heavy: 1+0+3 dim. → 1+3+3 dim.
𝑡𝑡

𝒙𝒙

𝒑𝒑

𝜋𝜋𝑚𝑚2
−
𝑒𝑒𝑒𝑒(𝑡𝑡,𝑥𝑥)

Result (3/3): Spatial inhomogeneity✔ What if E-field has spatial inhomogeneity 𝑬𝑬 𝑡𝑡 → 𝑬𝑬(𝑡𝑡, 𝒙𝒙) ?
・ No established analytical methods[Bulanov et al. (2004)]e.g.) locally-constant-field approximation: 𝑁𝑁 ∼ 𝑉𝑉 exp⇒ NOT good even without spatial inhomogeneity…

𝜋𝜋𝑚𝑚2
−
𝑒𝑒𝑒𝑒

3

→ ∫ d 𝒙𝒙 exp

𝜋𝜋𝑚𝑚2−𝑒𝑒𝑒𝑒(𝑡𝑡,𝑥𝑥)

・ Numerical simulation is also heavy: 1+0+3 dim. → 1+3+3 dim.
𝑡𝑡

𝒙𝒙

𝒑𝒑

✔ Perturbation theory in the Furry picture can be applied if:
d3 𝑁𝑁𝑒𝑒2out† out
3 𝒑𝒑′ 𝛽𝛽
′
′
�
d
≡
vac;
in
𝑎𝑎
𝑎𝑎
vac;
in
=
�
𝒑𝒑,𝑠𝑠
𝒑𝒑,𝑠𝑠
𝒑𝒑,𝑠𝑠;𝒑𝒑 ,𝑠𝑠
d𝒑𝒑3
where

(0)out†𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠 = � d3 𝒙𝒙 − 𝜓𝜓𝒑𝒑,𝑠𝑠
(0)in

𝑠𝑠𝑠

(0)in
4
� 0 out 𝒆𝒆𝓐𝓐w + 𝜓𝜓 (0)in+ 𝑂𝑂(|𝒆𝒆𝓐𝓐𝐰𝐰 |2 )+ 𝜓𝜓𝒑𝒑′,𝑠𝑠′ − 𝑖𝑖 � 𝑑𝑑 𝑥𝑥 − 𝜓𝜓𝒑𝒑,𝑠𝑠
𝒑𝒑′ ,𝑠𝑠 ′(1) the wave function ± 𝜓𝜓𝒑𝒑′ ,𝑠𝑠′ under 𝒆𝒆𝑬𝑬s is known(2) 𝒆𝒆𝓐𝓐w is sufficiently weakResult (3/3): Spatial inhomogeneity✔ Example: 𝑬𝑬 = (𝟎𝟎, 𝟎𝟎, 𝐸𝐸s + ℰw sin(𝑄𝑄0 𝑡𝑡 − 𝑄𝑄3 𝑧𝑧))
Spectrum

d3 𝑁𝑁
d𝒑𝒑𝟑𝟑

[HT, unpublished](reduce to non-linear Breit-Wheeler
for 𝑄𝑄0 = 𝑄𝑄3 )Total number 𝑁𝑁

at 𝒑𝒑 = 𝟎𝟎

𝑒𝑒𝐸𝐸s
= 0.2,
𝑚𝑚2

𝑒𝑒ℰw= 0.001𝑚𝑚2

・ Essentially the same as the case without spatial inhomogeneitye.g.) enhancement below the threshold & oscillation above the threshold・ Role of spatial inhomogeneity= modify the thresholds by supplying momentum 𝑄𝑄0 ∼ 2𝑚𝑚 → 2 𝑚𝑚2 + 𝑄𝑄32

⇒ Time-dependence (i.e., energy injection) is the essence of the Schwinger mechanism
Spatial inhomogeneity may result in something interesting when the spatial inhomogeneity(1) is strong and/or (2) has transverse components, for which magnetic effects (e.g., spin/chiral stuffs) should appear

Message of Part II(1) The perturbation theory in the Furry picture is a very powerful analytical approachfor the Schwinger mechanism with (space)time-dependent E-field(2) The essence of the dynamically assisted Schwinger mech. is the pert. one-photon process(3) The dynamically assisted Schwinger mechanism has rich physics other than simply
enhancing the particle production (e.g., oscillation, mass shift, spin generation)[HT, PRD (2019)] [Huang, HT, PRD (2019)]

Introduction⇓

Part I:

Interplay b/w non-pert. & pert. production mechanisms⇓

Part II:

Dynamically assisted Schwinger mechanism⇓

SummarySummary

I discussed the Schwinger mechanism under time-depending E-field:
Part I: Interplay b/w non-pert. & pert. production mechanisms
・ The interplay is controlled by 𝛾𝛾 ≡

𝑚𝑚Ω
𝑒𝑒𝑒𝑒
(Keldysh parameter) and also by 𝜈𝜈 ≡ 2𝑒𝑒𝑒𝑒
Ω・ Semi-classical methods are invalid when 𝜈𝜈 ≿ 1, where one-photon process dominates
・ One-photon production is very efficient, compared to non-pert. tunnelingPart II: Dynamically assisted Schwinger mechanism
・ The perturbation theory in the Furry picture is a powerful analytical approachfor the Schwinger mechanism with (space)time-dependent E-field・ The essence of the dynamically assisted Schwinger mech. is the pert. one-photon process
・ The dynamically assisted Schwinger mechanism has rich physics other than simply
enhancing the particle production (e.g., oscillation, mass shift, spin generation)・Interplay between perturbative & non-perturbative particle production:[HT, Fujii, Itakura, PRD (2014)] [HT, Fujimori, Misumi, Nitta, Sakai, JHEP (2021)]
・Dynamically assisted Schwinger mechanism and Furry-picture perturbation theory:[HT, PRD (2019)] [Huang, HT, PRD (2019)]
・Review on strong-field QED:
[Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, 2203.00019]

BACKUPValidity of Furry-picture perturbation theory

Furry-picture formula for parallel 𝑬𝑬𝐬𝐬 || 𝓔𝓔𝐟𝐟

✔ Analytical formula for 𝑬𝑬 = (0,0, 𝐸𝐸s + ℰf (𝑡𝑡)), with arbitrary time-dep.

・ is applicable even for very fast ℰf and reproduces numerics very well (show later)
d3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )
=
exp −
d𝒑𝒑3
2𝜋𝜋 3
𝑒𝑒𝐸𝐸s

1 𝑚𝑚2 + 𝒑𝒑2⊥ ∞ℰ̃𝐟𝐟 𝜔𝜔
𝑖𝑖 𝜔𝜔2 + 4𝜔𝜔𝑝𝑝∥× 1 +� 𝑑𝑑𝑑𝑑exp −𝑒𝑒𝐸𝐸s
2 𝑒𝑒𝐸𝐸s
𝐸𝐸s
40

Schwinger mech. by slow 𝑬𝑬𝐬𝐬

1 𝐹𝐹1

𝑖𝑖 𝑚𝑚2 + 𝒑𝒑2⊥
𝑖𝑖 𝜔𝜔21−; 2;
2 𝑒𝑒𝐸𝐸s
2 𝑒𝑒𝐸𝐸s2

Dynamical assistance by fast 𝓔𝓔𝐟𝐟

・ describes interplay b/w Schwinger & one-photon process smoothly:- Slow limit 𝜔𝜔⁄ 𝑒𝑒𝐸𝐸s ≪ 1: ■ dominates ⇒ Schwinger formulad3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )𝜋𝜋 𝑚𝑚2 + 𝒑𝒑2⊥ ℰf
∼
exp −
1+
d𝒑𝒑3
2𝜋𝜋 3
𝑒𝑒𝐸𝐸s
2 𝑒𝑒𝐸𝐸s 𝐸𝐸s2𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )
∼
exp −
2𝜋𝜋 3
𝑒𝑒(𝐸𝐸s + ℰf )- Fast limit 𝜔𝜔⁄ 𝑒𝑒𝐸𝐸s ≫ 1: ■ dominates ⇒ one-photon process
d3 𝑁𝑁𝑒𝑒
𝑉𝑉 1 𝑚𝑚2 + 𝒑𝒑2⊥ |𝑒𝑒ℰ�f 2𝜔𝜔𝒑𝒑 |2∼d𝒑𝒑3
2𝜋𝜋 3 4 𝜔𝜔𝒑𝒑2
𝜔𝜔𝒑𝒑2

Furry-picture formula for general 𝑬𝑬𝐬𝐬 ∦ 𝓔𝓔𝐟𝐟

✔ Generalization of the analytical formula for 𝑬𝑬𝐬𝐬 || 𝓔𝓔𝐟𝐟 → 𝑬𝑬𝐬𝐬 ∦ 𝓔𝓔𝐟𝐟
d3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )
=
exp −
d𝒑𝒑3
2𝜋𝜋 3
𝑒𝑒𝐸𝐸s

×

Schwingermechanismby slow 𝑬𝑬𝐬𝐬∞

1 + � 𝑑𝑑𝑑𝑑 e
0∞+𝑖𝑖 � 𝑑𝑑𝑑𝑑 e0+𝑠𝑠 × 𝑖𝑖 �
+ �

−𝑖𝑖

𝜔𝜔𝑝𝑝∥
2 �
2
1
𝑚𝑚
+
𝒑𝒑
⊥ 𝓔𝓔f 𝜔𝜔𝑒𝑒𝐸𝐸s2

𝜔𝜔𝑝𝑝∥ �𝓔𝓔 𝜔𝜔𝑒𝑒𝐸𝐸s f

⋅ 𝑬𝑬s
2

e

−𝑖𝑖

𝜔𝜔2
4𝑒𝑒𝐸𝐸s1 𝐹𝐹1𝑖𝑖 𝑚𝑚2 + 𝒑𝒑2⊥
𝑖𝑖 𝜔𝜔21−; 2;
2 𝑒𝑒𝐸𝐸s
2 𝑒𝑒𝐸𝐸s

2
2
2
𝜔𝜔
⋅ 𝒑𝒑⊥
𝑖𝑖
𝑚𝑚
+
𝒑𝒑
𝑖𝑖
𝜔𝜔
−𝑖𝑖
⊥
Re e 4𝑒𝑒𝐸𝐸s 1 𝐹𝐹1 1 −; 1;
𝐸𝐸s 𝜔𝜔
2 𝑒𝑒𝐸𝐸s
2 𝑒𝑒𝐸𝐸s𝜔𝜔𝑝𝑝 �
−𝑖𝑖 ∥ (𝓔𝓔f 𝜔𝜔𝑑𝑑𝑑𝑑 e 𝑒𝑒𝐸𝐸s

∞

0

𝐸𝐸s2

𝑒𝑒𝐸𝐸s

𝑥𝑥
𝜔𝜔𝑝𝑝
−𝑖𝑖 ∥ 𝑚𝑚 ℰ̃f 𝜔𝜔
𝑑𝑑𝑑𝑑 e 𝑒𝑒𝐸𝐸s

∞

0

−𝑖𝑖

Dynamical assistance by fast 𝓔𝓔𝐟𝐟

𝜔𝜔

𝑦𝑦
𝜔𝜔2
+ 𝑖𝑖𝑖𝑖ℰ̃f 𝜔𝜔
𝑖𝑖 𝑚𝑚2 + 𝒑𝒑2⊥
𝑖𝑖 𝜔𝜔2
−𝑖𝑖
4𝑒𝑒𝐸𝐸
s
Im e
; 1;
1 𝐹𝐹1 1 −𝐸𝐸s
2 𝑒𝑒𝐸𝐸s
2 𝑒𝑒𝐸𝐸s・ becomes complicated (green = new terms), but the basic structure is the same・ spin-dependence appears even without magnetic fields [Takayoshi, Wu, Oka (2020)]∵ Dirac particle has a spin-orbit coupling 𝒔𝒔 ⋅ 𝒑𝒑 × 𝓔𝓔
[Foldy, Wouthuysen (1950)] [Tani (1951)]

2

2

𝜔𝜔
× 𝒑𝒑⊥ ) ⋅ 𝑬𝑬s
𝑖𝑖 𝑚𝑚2 + 𝒑𝒑2⊥
𝑖𝑖 𝜔𝜔2
−𝑖𝑖
4𝑒𝑒𝐸𝐸
s 𝐹𝐹
Im e
; 1;
1 1 1−
2 𝑒𝑒𝐸𝐸s
2 𝑒𝑒𝐸𝐸s
𝐸𝐸s2 𝜔𝜔

𝐸𝐸sℰf・ can be applied to rotating E-fields: 𝐸𝐸 = 𝐸𝐸0 cos Ω𝑡𝑡 , sin Ω𝑡𝑡 , 0 ∼ (𝐸𝐸0 , 𝐸𝐸0 Ω𝑡𝑡, 0)

Numerical studies: [Blinne, Strobel (2015)] [Strobel, Xue (2015)] [Woller, Bauke, Keitel (2015)] [Kohlfurst (2019)]

2