Franz-Keldysh effectin strong-field QED
Hidetoshi TAYA(Fudan University)
Based on [HT, PRD 99, 056006 (2019)][X.-G. Huang, HT, 1904.08200 (to appear in PRD)]@LPHYS2019 2019/7/9

Brief summary
What
Spontaneous particle production from the vacuum (Schwinger mech.)by a strong slow E-field superimposed by a weak fast E-field

HowDerive an analytical formula for the production based on theperturbation theory in the Furry picture

ResultA QED analog of Franz-Keldysh effect occurs
 a generalization of the dynamically assisted Schwinger mechanism non-trivial oscillating pattern in the production

INTRODUCTION⇓
THEORY⇓RESULTS⇓SUMMARYSchwinger mechanism
Particles are spontaneously produced from the vacuum inthe presence of a strong slow electric fieldgap ~2𝜔𝜔𝒑𝒑
vacuum ＝ Dirac sea

E-field
tunneling QED analog of electrical breakdown of a semi-conductor in cond-mat(Landau-Zener transition)

Schwinger mechanism
Particles are spontaneously produced from the vacuum inthe presence of a strong slow electric fieldgap ~2𝜔𝜔𝒑𝒑

E-field
tunnelingvacuum ＝ Dirac sea

 QED analog of electrical breakdown of a semi-conductor in cond-mat(Landau-Zener transition)- very well formulated for a static 𝑬𝑬 = 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜. (but not well for time-dep. 𝐸𝐸)-d3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )=exp[−]
d𝒑𝒑32𝜋𝜋 3𝑒𝑒𝑒𝑒[Schwinger 1951]

Schwinger mechanism
Particles are spontaneously produced from the vacuum inthe presence of a strong slow electric fieldgap ~2𝜔𝜔𝒑𝒑

E-field
tunnelingvacuum ＝ Dirac sea

 QED analog of electrical breakdown of a semi-conductor in cond-mat(Landau-Zener transition)- very well formulated for a static 𝑬𝑬 = 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜. (but not well for time-dep. 𝐸𝐸)
d3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )=exp[−]
d𝒑𝒑32𝜋𝜋 3𝑒𝑒𝑒𝑒[Schwinger 1951]- attainable E-field is too weak compared electron’s mass 𝑒𝑒𝐸𝐸 ≪ 𝑚𝑚𝑒𝑒2cf) Guinness world record = HERCULES laser = 𝑒𝑒𝑒𝑒 ∼1022 W⁄cm2 ≪1028 W⁄cm2 ∼ 𝑚𝑚𝑒𝑒2

Schwinger mechanism
Particles are spontaneously produced from the vacuum inthe presence of a strong slow electric fieldgap ~2𝜔𝜔𝒑𝒑

E-field
tunnelingvacuum ＝ Dirac sea

 QED analog of electrical breakdown of a semi-conductor in cond-mat(Landau-Zener transition)- very well formulated for a static 𝑬𝑬 = 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜. (but not well for time-dep. 𝐸𝐸)
d3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )=exp[−]
d𝒑𝒑32𝜋𝜋 3𝑒𝑒𝑒𝑒[Schwinger 1951]- attainable E-field is too weak compared electron’s mass 𝑒𝑒𝐸𝐸 ≪ 𝑚𝑚𝑒𝑒2cf) Guinness world record = HERCULES laser = 𝑒𝑒𝑒𝑒 ∼1022 W⁄cm2 ≪Q1: Can we enhance the production rate ?
Q2: What does happen if E becomes time-dep. ?

1028 W⁄cm2 ∼ 𝑚𝑚𝑒𝑒2

Enhancement by weak fast E-field[Schutzhold, Gies, Dunne 2008]

Dynamically assisted Schwinger mechanism= Schwinger mechanism is dynamically enhanced by superimposing
� -fielda weak fast time-depending 𝓔𝓔-field on top of a strong slow 𝑬𝑬
weak fast 𝓔𝓔tunneling perturbative kick by 𝓔𝓔 reduces the tunneling length ➡ big enhancement

Enhancement by weak fast E-field[Schutzhold, Gies, Dunne 2008]

Dynamically assisted Schwinger mechanism= Schwinger mechanism is dynamically enhanced by superimposing
� -fielda weak fast time-depending 𝓔𝓔-field on top of a strong slow 𝑬𝑬weak fast 𝓔𝓔tunneling perturbative kick by 𝓔𝓔 reduces the tunneling length ➡ big enhancement- usually studied w/i semi-classical methods (e.g. worldline, WKB), which are� ≪ 𝑚𝑚2 , not-so-fast 𝓔𝓔 s.t. 𝜔𝜔 ≪ 𝑚𝑚)
valid if sufficiently adiabatic (i.e., subcrit. 𝒆𝒆𝑬𝑬-Enhancement by weak fast E-field[Schutzhold, Gies, Dunne 2008]

Dynamically assisted Schwinger mechanism= Schwinger mechanism is dynamically enhanced by superimposing
� -fielda weak fast time-depending 𝓔𝓔-field on top of a strong slow 𝑬𝑬weak fast 𝓔𝓔tunneling perturbative kick by 𝓔𝓔 reduces the tunneling length ➡ big enhancement- usually studied w/i semi-classical methods (e.g. worldline, WKB), which are� ≪ 𝑚𝑚2 , not-so-fast 𝓔𝓔 s.t. 𝜔𝜔 ≪ 𝑚𝑚)
valid if sufficiently adiabatic (i.e., subcrit. 𝒆𝒆𝑬𝑬- not observed in exp. yet, but may be observed in the very near futureEnhancement by weak fast E-field[Schutzhold, Gies, Dunne 2008]

Dynamically assisted Schwinger mechanism= Schwinger mechanism is dynamically enhanced by superimposing
� -fielda weak fast time-depending 𝓔𝓔-field on top of a strong slow 𝑬𝑬
weak fast 𝓔𝓔tunneling perturbative kick by 𝓔𝓔 reduces the tunneling length ➡ big enhancement- usually studied w/i semi-classical methods (e.g. worldline, WKB), which are� ≪ 𝑚𝑚2 , not-so-fast 𝓔𝓔 s.t. 𝜔𝜔 ≪ 𝑚𝑚)
valid if sufficiently adiabatic (i.e., subcrit. 𝒆𝒆𝑬𝑬- not observed in exp. yet, but may be observed in the very near future

Deepening our understanding of dyn. ass. Schwinger mech. is important✔ for the up-coming laser experiments (e.g. ELI, HiPER)✔ for going beyond the Schwinger pair production for a static E-fieldIdea: Franz-Keldysh effect in cond-mat
A very similar situation has been studied both theoretically andexperimentally more than 60 years in cond-mat !!!
� -field and a weak fast time-depending 𝓔𝓔-field Apply a strong slow 𝑬𝑬(or a dynamical photon) onto a semi-conductor
 Measure photo-absorption rate (instead of e-h pair production number)Idea: Franz-Keldysh effect in cond-mat
A very similar situation has been studied both theoretically andexperimentally more than 60 years in cond-mat !!!
� -field and a weak fast time-depending 𝓔𝓔-field Apply a strong slow 𝑬𝑬(or a dynamical photon) onto a semi-conductor
 Measure photo-absorption rate (instead of e-h pair production number)

TheoryExp. (w/ Si)[Yacoby 1966]

difference(■ー■)� -fieldw/ 𝑬𝑬
� -field
w/o 𝑬𝑬

difference

photoabsorption rate

[Franz 1958] [Keldysh 1958](photon energy) - (gap energy)

 abs. rate is enhanced below the threshold (looks like dyn. ass. Schwinger mech. (?))
 not only enhancement, but also oscillation above threshold (FK oscillation)

Idea: Franz-Keldysh effect in cond-mat
A very similar situation has been studied both theoretically andexperimentally more than 60 years in cond-mat !!!
� -field and a weak fast time-depending 𝓔𝓔-field Apply a strong slow 𝑬𝑬(or a dynamical photon) onto a semi-conductor
 Measure photo-absorption rate (instead of e-h pair production number)

TheoryExp. (w/ Si)[Yacoby 1966]

difference(■ー■)� -fieldw/ 𝑬𝑬
� -fieldw/o 𝑬𝑬

difference

photoabsorption rate

[Franz 1958] [Keldysh 1958](photon energy) - (gap energy)

 abs. rate is enhanced below the threshold (looks like dyn. ass. Schwinger mech. (?))
 not only enhancement, but also oscillation above threshold (FK oscillation)

Q: Is this really similar? Does FK effect equally occur for QED?

Aim of this study
Analytically show if FK effect occurs in QED clarify if dyn. ass. Schwinger mech. can be understood as a part of FK effect clarify if FK oscillation above the threshold 𝜔𝜔 > 2𝑚𝑚 occurs in QED clarify how FK effect modifies the momentum spectrum

Aim of this study
Analytically show if FK effect occurs in QED clarify if dyn. ass. Schwinger mech. can be understood as a part of FK effect clarify if FK oscillation above the threshold 𝜔𝜔 > 2𝑚𝑚 occurs in QED clarify how FK effect modifies the momentum spectrum

valid only when
adiabatic 𝝎𝝎 ≪ 𝒎𝒎Semi-classical methods(e.g. worldline, WKB)Aim of this studyAnalytically show if FK effect occurs in QED clarify if dyn. ass. Schwinger mech. can be understood as a part of FK effect clarify if FK oscillation above the threshold 𝜔𝜔 > 2𝑚𝑚 occurs in QEDOK clarify how FK effect modifies the momentum spectrum

valid only whenadiabatic 𝝎𝝎 ≪ 𝒎𝒎

valid as long as�
𝓔𝓔 is weak 𝒆𝒆𝓔𝓔 ≪ 𝒎𝒎𝟐𝟐 , 𝒆𝒆𝑬𝑬

[Furry 1951][Fradkin et al. 1991][Torgrimsson et al. 2017]

Semi-classical methods Perturbation theory in(e.g. worldline, WKB)
Furry pictureINTRODUCTION⇓
THEORY⇓RESULTS⇓
SUMMARY

Perturbation theory in Furry picture (1/2)
Compute d3 𝑁𝑁/d𝒑𝒑3 =†𝑎𝑎𝒑𝒑,𝑠𝑠 𝑎𝑎𝒑𝒑,𝑠𝑠non-perturbatively w.r.t.
� , but perturbatively w.r.t. weak fast 𝓔𝓔
strong slow 𝑬𝑬Perturbation theory in Furry picture (1/2)
Compute d3 𝑁𝑁/d𝒑𝒑3 =†𝑎𝑎𝒑𝒑,𝑠𝑠 𝑎𝑎𝒑𝒑,𝑠𝑠

non-perturbatively w.r.t.
� , but perturbatively w.r.t. weak fast 𝓔𝓔
strong slow 𝑬𝑬

STEP 1 Separate the total E-field 𝑬𝑬 into strong 𝑬𝑬� and weak 𝓔𝓔
� + 𝓔𝓔 with 𝑬𝑬� ≫ 𝓔𝓔𝑬𝑬 = 𝑬𝑬Perturbation theory in Furry picture (1/2)
Compute d3 𝑁𝑁/d𝒑𝒑3 =†𝑎𝑎𝒑𝒑,𝑠𝑠 𝑎𝑎𝒑𝒑,𝑠𝑠

non-perturbatively w.r.t.
� , but perturbatively w.r.t. weak fast 𝓔𝓔strong slow 𝑬𝑬

STEP 1 Separate the total E-field 𝑬𝑬 into strong 𝑬𝑬� and weak 𝓔𝓔
� + 𝓔𝓔 with 𝑬𝑬� ≫ 𝓔𝓔𝑬𝑬 = 𝑬𝑬

STEP 2 Solve the Dirac eq. non-perturbatively w.r.t 𝑬𝑬� , but perturbatively w.r.t. 𝓔𝓔
�/ − 𝑚𝑚 𝜓𝜓� = 𝒆𝒆𝓐𝓐/ 𝜓𝜓�𝑖𝑖𝜕𝜕/ − 𝒆𝒆𝑨𝑨∞𝜓𝜓� 𝑥𝑥 = 𝜓𝜓� (0) 𝑥𝑥 + ∫−∞ d𝑦𝑦 4 𝑆𝑆 𝑥𝑥, 𝑦𝑦 𝒆𝒆𝒆𝒆/ (𝑦𝑦)𝜓𝜓� 0 𝑦𝑦 + 𝑂𝑂(|𝒆𝒆𝓐𝓐|2 )� asHere, 𝜓𝜓� (0) and 𝑆𝑆 are non-perturbatively dressed by 𝑬𝑬
�/ − 𝑚𝑚 𝜓𝜓� (0) = 0𝑖𝑖𝜕𝜕/ − 𝒆𝒆𝑨𝑨
�/ − 𝑚𝑚 𝑆𝑆(𝑥𝑥, 𝑦𝑦) = 𝛿𝛿 4 (𝑥𝑥 − 𝑦𝑦)𝑖𝑖𝜕𝜕/ − 𝒆𝒆𝑨𝑨

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

𝑎𝑎�𝒑𝒑,𝑠𝑠

in/out†𝑏𝑏�−𝒑𝒑,𝑠𝑠≡lim𝑡𝑡→−∞⁄+∞

� d3 𝒙𝒙

in/out, 𝑏𝑏�𝑝𝑝,𝑠𝑠(𝑢𝑢𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙 )†(𝑣𝑣𝒑𝒑,𝑠𝑠e+𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙 )†from 𝜓𝜓��𝜓𝜓(𝑥𝑥)Perturbation theory in Furry picture (2/2)
STEP 3in/outCompute in/out annihilation operators 𝑎𝑎�𝑝𝑝,𝑠𝑠
in/out𝑎𝑎�𝒑𝒑,𝑠𝑠

in/out†𝑏𝑏�−𝒑𝒑,𝑠𝑠≡lim𝑡𝑡→−∞⁄+∞

� d3 𝒙𝒙

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

inout
inout
𝑜𝑜�𝒑𝒑,𝑠𝑠, 𝑜𝑜�𝒑𝒑,𝑠𝑠
are inequivalent 𝑜𝑜�𝒑𝒑,𝑠𝑠≠ 𝑜𝑜�𝒑𝒑,𝑠𝑠
and related with each otherby a Bogoliubov transformation
out𝑎𝑎�𝒑𝒑,𝑠𝑠𝛼𝛼𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠3 𝒑𝒑′
�d=�∗out†−𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠𝑏𝑏�−𝒑𝒑,𝑠𝑠
𝑠𝑠𝑠where(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,𝑠𝑠

�/ − 𝑚𝑚𝑖𝑖𝜕𝜕/ − 𝒆𝒆𝑨𝑨

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

� w/ different B.C.
are sol. of the Dirac eq. dressed by 𝒆𝒆𝑨𝑨
0 in/out=0± 𝜓𝜓𝒑𝒑,𝑠𝑠w/lim𝑡𝑡→−∞⁄+∞

0 in/out+ 𝜓𝜓𝒑𝒑,𝑠𝑠
0 in/out− 𝜓𝜓𝒑𝒑,𝑠𝑠=𝑢𝑢𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙𝑣𝑣𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙

Perturbation theory in Furry picture (2/2)
STEP 3in/outCompute in/out annihilation operators 𝑎𝑎�𝑝𝑝,𝑠𝑠in/out𝑎𝑎�𝒑𝒑,𝑠𝑠in/out†𝑏𝑏�−𝒑𝒑,𝑠𝑠≡lim𝑡𝑡→−∞⁄+∞

� d3 𝒙𝒙

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

inout
inout
𝑜𝑜�𝒑𝒑,𝑠𝑠, 𝑜𝑜�𝒑𝒑,𝑠𝑠
are inequivalent 𝑜𝑜�𝒑𝒑,𝑠𝑠≠ 𝑜𝑜�𝒑𝒑,𝑠𝑠
and related with each otherby a Bogoliubov transformationout
𝑎𝑎�𝒑𝒑,𝑠𝑠𝛼𝛼𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠
3 𝒑𝒑′
�d=
�∗out†−𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠𝑏𝑏�−𝒑𝒑,𝑠𝑠
𝑠𝑠𝑠where(0)out†

𝛼𝛼𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠 = � d3 𝒙𝒙 + 𝜓𝜓𝒑𝒑,𝑠𝑠(0)out†

𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠 = � d3 𝒙𝒙 − 𝜓𝜓𝒑𝒑,𝑠𝑠0 in/out

Here, ± 𝜓𝜓𝒑𝒑,𝑠𝑠

STEP 4

�/ − 𝑚𝑚𝑖𝑖𝜕𝜕/ − 𝒆𝒆𝑨𝑨𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠∗𝛼𝛼𝒑𝒑,𝑠𝑠;𝒑𝒑′,𝑠𝑠𝑠in𝑎𝑎�𝒑𝒑′,𝑠𝑠′in†𝑏𝑏�−𝒑𝒑′,𝑠𝑠′(0)in(0)in4� (0)out 𝒆𝒆𝓐𝓐
/ + 𝜓𝜓𝒑𝒑′,𝑠𝑠′ + 𝑂𝑂(|𝒆𝒆𝒆𝒆|2 )+ 𝜓𝜓𝒑𝒑′,𝑠𝑠′ − 𝑖𝑖 � 𝑑𝑑 𝑥𝑥 + 𝜓𝜓𝒑𝒑,𝑠𝑠(0)in(0)in4� (0)out 𝒆𝒆𝓐𝓐
/ + 𝜓𝜓𝒑𝒑′,𝑠𝑠′ + 𝑂𝑂(|𝒆𝒆𝒆𝒆|2 )+ 𝜓𝜓𝒑𝒑′,𝑠𝑠′ − 𝑖𝑖 � 𝑑𝑑 𝑥𝑥 − 𝜓𝜓𝒑𝒑,𝑠𝑠

� w/ different B.C.
are sol. of the Dirac eq. dressed by 𝒆𝒆𝑨𝑨
0 in/out=0± 𝜓𝜓𝒑𝒑,𝑠𝑠w/lim𝑡𝑡→−∞⁄+∞

0 in/out+ 𝜓𝜓𝒑𝒑,𝑠𝑠
0 in/out− 𝜓𝜓𝒑𝒑,𝑠𝑠

Compute the in-vacuum expectation value of # operator=𝑢𝑢𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙
𝑣𝑣𝒑𝒑,𝑠𝑠 e−𝑖𝑖𝜔𝜔𝒑𝒑 𝑡𝑡 e𝑖𝑖𝒑𝒑⋅𝒙𝒙

d3 𝑁𝑁𝑒𝑒2 c.f.) evaluation based on WKBout† out
3≡ vac; in 𝑎𝑎𝒑𝒑,𝑠𝑠 𝑎𝑎𝒑𝒑,𝑠𝑠 vac; in = � � d 𝒑𝒑′ 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠′(valid for adiabatic case)d𝒑𝒑3[Torgrimsson et al. 2017]𝑠𝑠𝑠Formula� is sufficiently slow (i.e., static) and spatially uniformAssume 𝑬𝑬➨ analytical sol. of Dirac eq. ± 𝜓𝜓𝒑𝒑,𝑠𝑠is known➨ one can evaluate 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠 ′ analytically !!! [HT 2019]
0 in/outFormula� is sufficiently slow (i.e., static) and spatially uniform
Assume 𝑬𝑬➨ analytical sol. of Dirac eq. ± 𝜓𝜓𝒑𝒑,𝑠𝑠is known➨ one can evaluate 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠 ′ analytically !!! [HT 2019]
0 in/out

𝑑𝑑 3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )=exp −
𝑑𝑑𝒑𝒑32𝜋𝜋 3𝑒𝑒𝐸𝐸�

1 𝑚𝑚2 + 𝒑𝒑2⊥ ∞ℰ̃ 𝜔𝜔
𝑖𝑖 𝜔𝜔2 + 4𝜔𝜔𝑝𝑝∥
� 𝑑𝑑𝑑𝑑× 1 +exp −�2 𝑒𝑒𝐸𝐸�
4𝑒𝑒𝐸𝐸�
𝐸𝐸
01 𝐹𝐹1𝑖𝑖 𝑚𝑚2 + 𝒑𝒑2⊥
𝑖𝑖 𝜔𝜔21−; 2;2 𝑒𝑒𝐸𝐸�
2 𝑒𝑒𝐸𝐸�NOTE: 𝐸𝐸� ∥ ℰ and ℰ = ℰ(t) (i.e., spatially uniform) are assumed;
for generalization, see [Huang, HT 2019], [Itakura, HT (in prep)]2Formula� is sufficiently slow (i.e., static) and spatially uniform
Assume 𝑬𝑬➨ analytical sol. of Dirac eq. ± 𝜓𝜓𝒑𝒑,𝑠𝑠
is known➨ one can evaluate 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠 ′ analytically !!! [HT 2019]
0 in/out

𝑑𝑑 3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )=exp −𝑑𝑑𝒑𝒑32𝜋𝜋 3𝑒𝑒𝐸𝐸�

1 𝑚𝑚2 + 𝒑𝒑2⊥ ∞ℰ̃ 𝜔𝜔𝑖𝑖 𝜔𝜔2 + 4𝜔𝜔𝑝𝑝∥
� 𝑑𝑑𝑑𝑑× 1 +exp −�
2 𝑒𝑒𝐸𝐸�
4𝑒𝑒𝐸𝐸�
𝐸𝐸
01 𝐹𝐹1𝑖𝑖 𝑚𝑚2 + 𝒑𝒑2⊥
𝑖𝑖 𝜔𝜔21−; 2;2 𝑒𝑒𝐸𝐸�
2 𝑒𝑒𝐸𝐸�NOTE: 𝐸𝐸� ∥ ℰ and ℰ = ℰ(t) (i.e., spatially uniform) are assumed;
for generalization, see [Huang, HT 2019], [Itakura, HT (in prep)]�usual Schwinger formula by static 𝑬𝑬2

Formula� is sufficiently slow (i.e., static) and spatially uniform
Assume 𝑬𝑬➨ analytical sol. of Dirac eq. ± 𝜓𝜓𝒑𝒑,𝑠𝑠is known➨ one can evaluate 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠 ′ analytically !!! [HT 2019]
0 in/out

𝑑𝑑 3 𝑁𝑁𝑒𝑒
𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )=exp −
𝑑𝑑𝒑𝒑32𝜋𝜋 3𝑒𝑒𝐸𝐸�

1 𝑚𝑚2 + 𝒑𝒑2⊥ ∞ℰ̃ 𝜔𝜔
𝑖𝑖 𝜔𝜔2 + 4𝜔𝜔𝑝𝑝∥
� 𝑑𝑑𝑑𝑑× 1 +exp −�
2 𝑒𝑒𝐸𝐸�
4𝑒𝑒𝐸𝐸�
𝐸𝐸
01 𝐹𝐹1𝑖𝑖 𝑚𝑚2 + 𝒑𝒑2⊥
𝑖𝑖 𝜔𝜔21−; 2;2 𝑒𝑒𝐸𝐸�
2 𝑒𝑒𝐸𝐸�NOTE: 𝐸𝐸� ∥ ℰ and ℰ = ℰ(t) (i.e., spatially uniform) are assumed;
for generalization, see [Huang, HT 2019], [Itakura, HT (in prep)]�usual Schwinger formula by static 𝑬𝑬

2“perturbative assistance” by 𝓔𝓔

Formula� is sufficiently slow (i.e., static) and spatially uniform
Assume 𝑬𝑬➨ analytical sol. of Dirac eq. ± 𝜓𝜓𝒑𝒑,𝑠𝑠is known➨ one can evaluate 𝛽𝛽𝒑𝒑,𝑠𝑠;𝒑𝒑′ ,𝑠𝑠 ′ analytically !!! [HT 2019]
0 in/out

𝑑𝑑 3 𝑁𝑁𝑒𝑒
𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )=exp −
𝑑𝑑𝒑𝒑32𝜋𝜋 3𝑒𝑒𝐸𝐸�

1 𝑚𝑚2 + 𝒑𝒑2⊥ ∞ℰ̃ 𝜔𝜔
𝑖𝑖 𝜔𝜔2 + 4𝜔𝜔𝑝𝑝∥
� 𝑑𝑑𝑑𝑑× 1 +exp −�2 𝑒𝑒𝐸𝐸�
4𝑒𝑒𝐸𝐸�
𝐸𝐸
01 𝐹𝐹1𝑖𝑖 𝑚𝑚2 + 𝒑𝒑2⊥
𝑖𝑖 𝜔𝜔21−; 2;2 𝑒𝑒𝐸𝐸�
2 𝑒𝑒𝐸𝐸�NOTE: 𝐸𝐸� ∥ ℰ and ℰ = ℰ(t) (i.e., spatially uniform) are assumed;
for generalization, see [Huang, HT 2019], [Itakura, HT (in prep)]�usual Schwinger formula by static 𝑬𝑬“perturbative assistance” by 𝓔𝓔� ≪ 𝟏𝟏: ■ dominates ➡ usual Schwinger pair production
 slow limit 𝝎𝝎⁄ 𝒆𝒆𝑬𝑬
𝑑𝑑3 𝑁𝑁𝑒𝑒𝑉𝑉𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )𝜋𝜋 𝑚𝑚2 + 𝒑𝒑2⊥ ℰ∼
exp −1+𝑑𝑑𝒑𝒑32𝜋𝜋 32 𝑒𝑒 𝐸𝐸� 𝐸𝐸�
𝑒𝑒𝐸𝐸�

2𝑉𝑉
𝜋𝜋(𝑚𝑚2 +𝒑𝒑2⊥ )∼exp −2𝜋𝜋 3𝑒𝑒(𝐸𝐸� + ℰ)� ≫ 𝟏𝟏: ■ dominates ➡ perturbative production
 fast limit 𝝎𝝎⁄ 𝒆𝒆𝑬𝑬
𝑑𝑑3 𝑁𝑁𝑒𝑒
𝑉𝑉 1 𝑚𝑚2 + 𝒑𝒑2⊥ |𝑒𝑒 ℰ̃ 2𝜔𝜔𝒑𝒑 |2∼
𝑑𝑑𝒑𝒑32𝜋𝜋 3 4 𝜔𝜔𝒑𝒑2𝜔𝜔𝒑𝒑22

INTRODUCTION⇓
THEORY⇓RESULTS⇓SUMMARY

Total production number N
Total prod. #𝑁𝑁𝑒𝑒 / 𝑒𝑒 𝐸𝐸� 2 𝑉𝑉𝑉𝑉 For a monochromatic E-field 𝓔𝓔 = 𝓔𝓔𝟎𝟎 cos 𝜔𝜔𝜔𝜔For𝑚𝑚= 2.5�𝑒𝑒𝐸𝐸

Difference (■－■)
pert(𝑁𝑁𝑒𝑒 − 𝑁𝑁𝑒𝑒)/ 𝑒𝑒𝐸𝐸� 2 𝑉𝑉𝑉𝑉

perturbative�)(prod. w/o 𝑬𝑬
our formula�)(prod. w/ 𝑬𝑬

frequency

Schwinger

frequency 𝜔𝜔/ 𝑒𝑒 𝐸𝐸�(𝜔𝜔 − 2𝑚𝑚)/ 𝑒𝑒 𝐸𝐸�Total production number N
Total prod. #𝑁𝑁𝑒𝑒 / 𝑒𝑒 𝐸𝐸� 2 𝑉𝑉𝑉𝑉 For a monochromatic E-field 𝓔𝓔 = 𝓔𝓔𝟎𝟎 cos 𝜔𝜔𝜔𝜔
For𝑚𝑚

= 2.5
�

𝑒𝑒𝐸𝐸

Difference (■－■)
pert(𝑁𝑁𝑒𝑒 − 𝑁𝑁𝑒𝑒)/ 𝑒𝑒𝐸𝐸� 2 𝑉𝑉𝑉𝑉

perturbative�)(prod. w/o 𝑬𝑬
our formula�)(prod. w/ 𝑬𝑬

frequency

Schwinger

frequency 𝜔𝜔/ 𝑒𝑒 𝐸𝐸�(𝜔𝜔 − 2𝑚𝑚)/ 𝑒𝑒 𝐸𝐸�Total production number N
Total prod. #𝑁𝑁𝑒𝑒 / 𝑒𝑒 𝐸𝐸� 2 𝑉𝑉𝑉𝑉 For a monochromatic E-field 𝓔𝓔 = 𝓔𝓔𝟎𝟎 cos 𝜔𝜔𝜔𝜔For𝑚𝑚= 2.5
�

𝑒𝑒𝐸𝐸Difference (■－■)
pert(𝑁𝑁𝑒𝑒 − 𝑁𝑁𝑒𝑒)/ 𝑒𝑒𝐸𝐸� 2 𝑉𝑉𝑉𝑉

perturbative�)(prod. w/o 𝑬𝑬
our formula�)(prod. w/ 𝑬𝑬

frequency

Schwinger

frequency 𝜔𝜔/ 𝑒𝑒 𝐸𝐸�(𝜔𝜔 − 2𝑚𝑚)/ 𝑒𝑒 𝐸𝐸� exactly the same as cond-mat ➨ FK effect does occur in QED!

- production is enhanced below the threshold (dyn. ass. Schwinger mech.)- not only enhancement, but also oscillation above threshold (~ FK oscillation)- very sharp peak at the threshold in the number differencec.f.) modulation
spectroscopy

Interpretation of the oscillation✖ ✖
probability density

 quantum tunneling ➡ dynamically assisted Schwinger mechanism
 quantum reflection ➡ FK oscillation- non-uniform prob. dist. due to interference b/w in-coming and reflected waves- production occurs most efficiently at the maxima

Momentum distribution d3 𝑁𝑁𝑒𝑒 ⁄d𝒑𝒑𝟑𝟑
momentum dist.
2𝜋𝜋 3 d3 𝑁𝑁𝑒𝑒
𝑉𝑉 d𝒑𝒑𝟑𝟑

For𝑚𝑚𝑒𝑒𝐸𝐸�= 2.5

frequency 𝜔𝜔/ 𝑒𝑒 𝐸𝐸� FK effect: enhancement below threshold & oscillation above threshold the location of the perturbative peak is modified due to reflection
 excellent agreement b/w our analytical formula and numerical results

INTRODUCTION⇓
THEORY⇓RESULTS⇓SUMMARY

Summary I discussed spontaneous particle production from thevacuum (Schwinger mechanism) in the presence of astrong slow E-field superimposed by a weak fast E-field I derived an analytical formula for the production number
based on the perturbation theory in the Furry picture� ≫ 𝓔𝓔- reproduces the numerical results very well as long as 𝑬𝑬

 I claimed that a QED analog of Franz-Keldysh effect occurs:- enhancement below the threshold (dyn. ass. Schwinger mech.)- oscillation above the threshold (FK oscillation)- the location of the perturbative peak in d3 𝑁𝑁𝑒𝑒 ⁄d𝒑𝒑𝟑𝟑 is modified

 Not only quantum tunneling, but also reflection is important