Application of the exact WKB method
to particle production
and high-harmonics generation (HHG)
from the vacuum

Hidetoshi TAYA (RIKEN)[1] HT, Fujimori, Misumi, Nitta, Sakai, JHEP 03, 082 (2021) [2010.16080][2] HT, Hongo, Ikeda, 2105.12446

Problem: HHG from the vacuum
Pair production by E-field leads to emission of photons

E-field

Acceleration
photons

Dirac sea

VacuumPair production

Photon emission

Problem: HHG from the vacuum

Pair production by E-field leads to emission of photons

E-field

Acceleration
photons

Dirac sea

VacuumPair production

Photon emission✔ Photons may have high harmonics due to strong-field effects
Suppose E-field has a typical frequency Ω: 𝐸𝐸� 𝜔𝜔 ∝ 𝛿𝛿(Ω − 𝜔𝜔)・ 𝐸𝐸 is weak/fast (large 𝛾𝛾Keldysh ) ⇒ 𝜔𝜔d𝑁𝑁𝛾𝛾

� ⇒ peaked at 𝜔𝜔 = 𝟏𝟏 × Ω= (linear in 𝐸𝐸)

d𝜔𝜔
d𝑁𝑁𝛾𝛾

・ 𝐸𝐸 is strong/slow (small 𝛾𝛾Keldysh ) ⇒ 𝜔𝜔d𝜔𝜔� ⇒ 𝜔𝜔 = 𝒏𝒏 × Ω appear= (non − linear in 𝐸𝐸)

Problem: HHG from the vacuum

Pair production by E-field leads to emission of photons

E-field

Acceleration
photons

Dirac sea

VacuumPair production

Photon emission✔ Photons may have high harmonics due to strong-field effects
Suppose E-field has a typical frequency Ω: 𝐸𝐸� 𝜔𝜔 ∝ 𝛿𝛿(Ω − 𝜔𝜔)・ 𝐸𝐸 is weak/fast (large 𝛾𝛾Keldysh ) ⇒ 𝜔𝜔d𝑁𝑁𝛾𝛾

� ⇒ peaked at 𝜔𝜔 = 𝟏𝟏 × Ω= (linear in 𝐸𝐸)

d𝜔𝜔
d𝑁𝑁𝛾𝛾

・ 𝐸𝐸 is strong/slow (small 𝛾𝛾Keldysh ) ⇒ 𝜔𝜔d𝜔𝜔� ⇒ 𝜔𝜔 = 𝒏𝒏 × Ω appear= (non − linear in 𝐸𝐸)✔ Observation in semi-conductors (also in many other materials)
・ Key features: Odd harmonics 𝑛𝑛 = 1,3,5, … & Plateau structure

・ Microscopic mechanism of solid-state HHG is still under debate
and no established analytical methods(e.g., 3 step model of gas-HHG does not necessarily work)[Corkum (1993)][Ghimire et al., (2011)]

Problem: HHG from the vacuum

Pair production by E-field leads to emission of photons

E-field

Acceleration
photons

Dirac sea

VacuumPair production

Photon emission✔ Photons may have high harmonics due to strong-field effects
Suppose E-field has a typical frequency Ω: 𝐸𝐸� 𝜔𝜔 ∝ 𝛿𝛿(Ω − 𝜔𝜔)・ 𝐸𝐸 is weak/fast (large 𝛾𝛾Keldysh ) ⇒ 𝜔𝜔d𝑁𝑁𝛾𝛾

� ⇒ peaked at 𝜔𝜔 = 𝟏𝟏 × Ω= (linear in 𝐸𝐸)

d𝜔𝜔
d𝑁𝑁𝛾𝛾

・ 𝐸𝐸 is strong/slow (small 𝛾𝛾Keldysh ) ⇒ 𝜔𝜔

[Ghimire et al., (2011)]d𝜔𝜔� ⇒ 𝜔𝜔 = 𝒏𝒏 × Ω appear= (non − linear in 𝐸𝐸)✔ Observation in semi-conductors (also in many other materials)
・ Key features: Odd harmonics 𝑛𝑛 = 1,3,5, … & Plateau structure

・ Microscopic mechanism of solid-state HHG is still under debate
and no established analytical methods(e.g., 3 step model of gas-HHG does not necessarily work)[Corkum (1993)]✔ Natural to expect in QED, but not intensively discussed thus far
・ Previous works are based on the Euler-Heisenberg effective action⇒ valid only in the static limit Ω → 0 ⇒ Not satisfactory for Ω ≠ 0[Piazza, Hatsagortsyan, Keitel (2005)][Fedotov, Narozhny (2007)][Bohl, Ruhl, King (2015)]

What are needed for the formulation ?

① Need to know wavefunction 𝜓𝜓±∵ Photon spectra are given i.t.o. current, which is expressed i.t.o. wavefunctions 𝜓𝜓±
d3 𝒑𝒑
d𝑁𝑁
2
�
𝜇𝜇
𝐽𝐽 𝑡𝑡 ≔ 0; in 𝜓𝜓𝛾𝛾 𝜓𝜓� 0; in = �𝜓𝜓�−,𝒑𝒑 𝛾𝛾 𝜇𝜇 𝜓𝜓−,𝒑𝒑where𝜔𝜔∼ 𝜔𝜔𝐽𝐽̃ 𝜔𝜔
3
2𝜋𝜋
d𝜔𝜔
� 𝒙𝒙) = � d3 𝒑𝒑
𝜓𝜓(𝑡𝑡,

ei𝒑𝒑⋅𝒙𝒙(in)(in)†
𝜓𝜓 (𝑡𝑡)𝑎𝑎�𝒑𝒑 + 𝜓𝜓−,𝒑𝒑 (𝑡𝑡)𝑏𝑏�−𝒑𝒑
2𝜋𝜋 3/2 +,𝒑𝒑with B.C.

lim 𝜓𝜓±,𝒑𝒑 ∝ e∓i𝜔𝜔𝒑𝒑 𝑡𝑡 = plane waves

𝑡𝑡→−∞

Nice way to include ① & ② ??? ⇒ Exact WKB !!!

What are needed for the formulation ?

① Need to know wavefunction 𝜓𝜓±∵ Photon spectra are given i.t.o. current, which is expressed i.t.o. wavefunctions 𝜓𝜓±
d3 𝒑𝒑
d𝑁𝑁
2
�
𝜇𝜇
𝐽𝐽 𝑡𝑡 ≔ 0; in 𝜓𝜓𝛾𝛾 𝜓𝜓� 0; in = �𝜓𝜓�−,𝒑𝒑 𝛾𝛾 𝜇𝜇 𝜓𝜓−,𝒑𝒑where𝜔𝜔∼ 𝜔𝜔𝐽𝐽̃ 𝜔𝜔
3
2𝜋𝜋
d𝜔𝜔
� 𝒙𝒙) = � d3 𝒑𝒑
𝜓𝜓(𝑡𝑡,

ei𝒑𝒑⋅𝒙𝒙(in)(in)†
𝜓𝜓 (𝑡𝑡)𝑎𝑎�𝒑𝒑 + 𝜓𝜓−,𝒑𝒑 (𝑡𝑡)𝑏𝑏�−𝒑𝒑
2𝜋𝜋 3/2 +,𝒑𝒑with B.C.

lim 𝜓𝜓±,𝒑𝒑 ∝ e∓i𝜔𝜔𝒑𝒑 𝑡𝑡 = plane waves

𝑡𝑡→−∞② Need to include Stokes phenomenon of 𝜓𝜓±∵ Particle prod. is the origin of HHG and is interpreted as a Stokes phenomenon of ODEs・ In Math: Stokes phenomenon ~ Sudden mixing up of positive & negative frequency solutions
Ex.) Airy function: 0 = 𝜕𝜕𝑧𝑧2 − 𝑧𝑧 Ai 𝑧𝑧

3⁄2

Im[𝑧𝑧]

3⁄2

𝑐𝑐− ≠ 0, c+ ≠ 0
e−2𝑧𝑧 /3e+2𝑧𝑧 /3
⇒ Ai z ~ 𝑐𝑐−
+ 𝑐𝑐+
𝑧𝑧1/4
𝑧𝑧1/4
・ In Phys: The mixing means production of particles from the vacuum†at 𝑡𝑡 ∼ −∞𝜓𝜓� ∼ e−i𝜔𝜔𝒑𝒑 𝑡𝑡 𝑎𝑎�in + e+i𝜔𝜔𝒑𝒑 𝑡𝑡 𝑏𝑏�in†⇒ 𝜓𝜓� ∼ (𝛼𝛼e−i𝜔𝜔𝒑𝒑𝑡𝑡 +𝛽𝛽e+i𝜔𝜔𝒑𝒑 𝑡𝑡 )𝑎𝑎�in + (𝛼𝛼 ∗ e+i𝜔𝜔𝒑𝒑 𝑡𝑡 + 𝛽𝛽 ∗ e−i𝜔𝜔𝒑𝒑𝑡𝑡 )𝑏𝑏�in††= e−i𝜔𝜔𝒑𝒑 𝑡𝑡 (𝛼𝛼 𝑎𝑎�in + 𝛽𝛽 ∗ 𝑏𝑏�in) + e+i𝜔𝜔𝒑𝒑𝑡𝑡 (𝛽𝛽 𝑎𝑎�in + 𝛼𝛼 ∗ 𝑏𝑏�in)Re[𝑧𝑧]

whenever Stokes pheno.
occur at 𝑡𝑡 ≠ −∞𝑏𝑏�𝑡𝑡†††�2⇒ Nice0; in 𝑎𝑎�way
�𝑡𝑡 0;toin include= 0; in 𝑏𝑏�①
in ???
= 𝛽𝛽 ⇒≠ 0Exact WKB !!!
𝑡𝑡 𝑎𝑎
𝑡𝑡 𝑏𝑏𝑡𝑡 &0;②
𝑎𝑎�𝑡𝑡

𝑐𝑐− ≠ 0, c+ = 0

What are needed for the formulation ?

① Need to know wavefunction 𝜓𝜓±∵ Photon spectra are given i.t.o. current, which is expressed i.t.o. wavefunctions 𝜓𝜓±
d3 𝒑𝒑
d𝑁𝑁
2
�
𝜇𝜇
𝐽𝐽 𝑡𝑡 ≔ 0; in 𝜓𝜓𝛾𝛾 𝜓𝜓� 0; in = �𝜓𝜓�−,𝒑𝒑 𝛾𝛾 𝜇𝜇 𝜓𝜓−,𝒑𝒑where𝜔𝜔∼ 𝜔𝜔𝐽𝐽̃ 𝜔𝜔
3
2𝜋𝜋
d𝜔𝜔
� 𝒙𝒙) = � d3 𝒑𝒑
𝜓𝜓(𝑡𝑡,

ei𝒑𝒑⋅𝒙𝒙(in)(in)†
𝜓𝜓 (𝑡𝑡)𝑎𝑎�𝒑𝒑 + 𝜓𝜓−,𝒑𝒑 (𝑡𝑡)𝑏𝑏�−𝒑𝒑
2𝜋𝜋 3/2 +,𝒑𝒑with B.C.

lim 𝜓𝜓±,𝒑𝒑 ∝ e∓i𝜔𝜔𝒑𝒑 𝑡𝑡 = plane waves

𝑡𝑡→−∞② Need to include Stokes phenomenon of 𝜓𝜓±∵ Particle prod. is the origin of HHG and is interpreted as a Stokes phenomenon of ODEs・ In Math: Stokes phenomenon ~ Sudden mixing up of positive & negative frequency solutions
Ex.) Airy function: 0 = 𝜕𝜕𝑧𝑧2 − 𝑧𝑧 Ai 𝑧𝑧

3⁄2

Im[𝑧𝑧]

3⁄2

𝑐𝑐− ≠ 0, c+ ≠ 0
e−2𝑧𝑧 /3e+2𝑧𝑧 /3
⇒ Ai z ~ 𝑐𝑐−
+ 𝑐𝑐+
𝑧𝑧1/4
𝑧𝑧1/4
・ In Phys: The mixing means production of particles from the vacuum†𝜓𝜓� ∼ e−i𝜔𝜔𝒑𝒑 𝑡𝑡 𝑎𝑎�in + e+i𝜔𝜔𝒑𝒑 𝑡𝑡 𝑏𝑏�inat 𝑡𝑡 ∼ −∞†⇒ 𝜓𝜓� ∼ (𝛼𝛼e−i𝜔𝜔𝒑𝒑𝑡𝑡 +𝛽𝛽e+i𝜔𝜔𝒑𝒑 𝑡𝑡 )𝑎𝑎�in + (𝛼𝛼 ∗ e+i𝜔𝜔𝒑𝒑 𝑡𝑡 + 𝛽𝛽 ∗ e−i𝜔𝜔𝒑𝒑𝑡𝑡 )𝑏𝑏�in††= e−i𝜔𝜔𝒑𝒑 𝑡𝑡 (𝛼𝛼 𝑎𝑎�in + 𝛽𝛽 ∗ 𝑏𝑏�in) + e+i𝜔𝜔𝒑𝒑𝑡𝑡 (𝛽𝛽 𝑎𝑎�in + 𝛼𝛼 ∗ 𝑏𝑏�in)Re[𝑧𝑧]

whenever Stokes pheno.
occur at 𝑡𝑡 ≠ −∞𝑏𝑏�𝑡𝑡†††�2⇒ Nice0; in 𝑎𝑎�way
�𝑡𝑡 0;toin include= 0; in 𝑏𝑏�①
in ???
= 𝛽𝛽 ⇒≠ 0Exact WKB !!!
𝑡𝑡 𝑎𝑎
𝑡𝑡 𝑏𝑏𝑡𝑡 &0;②
𝑎𝑎�𝑡𝑡

𝑐𝑐− ≠ 0, c+ = 0

Any nice analytical method for ① & ② ? ⇒ Exact WKB !

[Voros (1983)][Pham, Dillinger, Delabaere,
Aoki, Koike, Takei, …]Exact WKB method[Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

Exact WKB = ”usual” WKB + Borel resum.

[Voros (1983)][Pham, Dillinger, Delabaere,
Aoki, Koike, Takei, …]Exact WKB method[Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

Exact WKB = ”usual” WKB + Borel resum.
A perturbation theory w.r.t. ℏ (or adiabatic approx.)

・ Consider 0 = ℏ2 𝜕𝜕𝑡𝑡2 + 𝑄𝑄 𝑡𝑡 𝜙𝜙 𝑡𝑡
i𝑡𝑡 ≡ ℏ𝜏𝜏𝑡𝑡𝜕𝜕𝜏𝜏2 + 𝑄𝑄 ℏ𝜏𝜏 𝜙𝜙 𝜏𝜏

⇔⇒ 𝜙𝜙± 𝑡𝑡; ℏ ≔ exp ∓ ∫𝑡𝑡 d𝑡𝑡 ′ 𝑄𝑄 𝑡𝑡 ′ℏ0

0th order = plane wavei
~ exp[∓ 𝑄𝑄𝑡𝑡]ℏ∞× ∑𝑛𝑛=0𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ𝑛𝑛
Perturbation w.r.t. ℏ[Voros (1983)][Pham, Dillinger, Delabaere,
Aoki, Koike, Takei, …]Exact WKB method[Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

Exact WKB = ”usual” WKB + Borel resum.
A perturbation theory w.r.t. ℏ (or adiabatic approx.)

・ Consider 0 = ℏ2 𝜕𝜕𝑡𝑡2 + 𝑄𝑄 𝑡𝑡 𝜙𝜙 𝑡𝑡
i𝑡𝑡 ≡ ℏ𝜏𝜏𝑡𝑡𝜕𝜕𝜏𝜏2 + 𝑄𝑄 ℏ𝜏𝜏 𝜙𝜙 𝜏𝜏

⇔⇒ 𝜙𝜙± 𝑡𝑡; ℏ ≔ exp ∓ ∫𝑡𝑡 d𝑡𝑡 ′ 𝑄𝑄 𝑡𝑡 ′ℏ0

0th order = plane wavei
~ exp[∓ 𝑄𝑄𝑡𝑡]ℏ∞× ∑𝑛𝑛=0𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ𝑛𝑛
Perturbation w.r.t. ℏ・ WKB expansion makes sense if the perturbative part is convergent
・ However, 𝜓𝜓±,𝑛𝑛 ~𝑛𝑛! in general (e.g., Airy function 𝑄𝑄 𝑡𝑡 ∝ 𝑡𝑡)⇒ WKB expansion has zero radius of convergence ⇒ ill-defined !!!

[Voros (1983)][Pham, Dillinger, Delabaere,
Aoki, Koike, Takei, …]Exact WKB method[Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

Exact WKB = ”usual” WKB + Borel resum.
A resummation scheme for factorially divergent ~n! series𝑛𝑛・ Consider the div. part of WKB expansion 𝜓𝜓± 𝑡𝑡; ℏ ≔ ∑∞𝑛𝑛=0 𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ∞① Construct “Borel transformation”: 𝐵𝐵 𝜓𝜓± 𝑡𝑡; 𝜂𝜂 ≔ �② Laplace trans. gives “Borel sum”:∞𝑛𝑛𝜓𝜓±,𝑛𝑛 (𝑡𝑡) 𝑛𝑛𝜂𝜂𝑛𝑛!

d𝜂𝜂 −𝜂𝜂/ℏΨ± 𝑡𝑡; ℏ ≔ �e𝐵𝐵[𝜓𝜓± ](𝑡𝑡; 𝜂𝜂)ℏ0[Voros (1983)][Pham, Dillinger, Delabaere,
Aoki, Koike, Takei, …]Exact WKB method[Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

Exact WKB = ”usual” WKB + Borel resum.
A resummation scheme for factorially divergent ~n! series𝑛𝑛・ Consider the div. part of WKB expansion 𝜓𝜓± 𝑡𝑡; ℏ ≔ ∑∞𝑛𝑛=0 𝜓𝜓±,𝑛𝑛 (𝑡𝑡)ℏ∞① Construct “Borel transformation”: 𝐵𝐵 𝜓𝜓± 𝑡𝑡; 𝜂𝜂 ≔ �② Laplace trans. gives “Borel sum”:∞𝑛𝑛𝜓𝜓±,𝑛𝑛 (𝑡𝑡) 𝑛𝑛𝜂𝜂𝑛𝑛!

d𝜂𝜂 −𝜂𝜂/ℏΨ± 𝑡𝑡; ℏ ≔ �e𝐵𝐵[𝜓𝜓± ](𝑡𝑡; 𝜂𝜂)ℏ0・ Ψ± is well-defined, unless 𝐵𝐵 𝜓𝜓± has singularities on real axis
・ Asymptotic expansion of Ψ± = 𝜓𝜓±⇒ Ψ± is a natural analytic continuation of 𝜓𝜓±⇒ 𝚿𝚿± gives a well-defined version of the WKB solution !

・ Singularities of 𝑩𝑩 𝝍𝝍± can be used to describe Stokes phenomenon

Exact-WKB recipe for Stokes phenomenon
Step 1: Draw a Stokes graph・ Laplace trans. of 𝐵𝐵 𝜓𝜓± (𝑡𝑡; 𝜂𝜂) hits singularities(non-Borel summable) when locatedon Stokes lines in the t-plane
一 : Stokes lines𝑡𝑡

𝑡𝑡 ∈ ℂ 0 = Im i � d𝑧𝑧● : turning points

𝑄𝑄 𝑡𝑡tp = 0

𝑡𝑡tp

𝑄𝑄 𝑧𝑧 }Exact-WKB recipe for Stokes phenomenon
Step 1: Draw a Stokes graph・ Laplace trans. of 𝐵𝐵 𝜓𝜓± (𝑡𝑡; 𝜂𝜂) hits singularities(non-Borel summable) when locatedon Stokes lines in the t-plane
一 : Stokes lines𝑡𝑡

𝑡𝑡 ∈ ℂ 0 = Im i � d𝑧𝑧● : turning points

𝑡𝑡tp

𝑄𝑄 𝑧𝑧 }

𝑄𝑄 𝑡𝑡tp = 0

Step 2: Compute Borel sum 𝚿𝚿± at each Stokes region

・ Borel sum is well-defined and computable in each region separated by Stokes lines∞d𝜂𝜂 −𝜂𝜂/ℏ
i 𝑡𝑡 ′
Ψ± = �
𝐵𝐵[𝜓𝜓± ](𝑡𝑡; 𝜂𝜂) ∼ exp ∓ � d𝑡𝑡 𝑄𝑄 𝑡𝑡 ′
e
ℏ
ℏ 𝑡𝑡0
0

× (1 + 𝑂𝑂(ℏ)) at each Stokes region

Exact-WKB recipe for Stokes phenomenon
Step 1: Draw a Stokes graph・ Laplace trans. of 𝐵𝐵 𝜓𝜓± (𝑡𝑡; 𝜂𝜂) hits singularities(non-Borel summable) when locatedon Stokes lines in the t-plane
一 : Stokes lines𝑡𝑡

𝑡𝑡 ∈ ℂ 0 = Im i � d𝑧𝑧● : turning points

𝑡𝑡tp

𝑄𝑄 𝑧𝑧 }

𝑄𝑄 𝑡𝑡tp = 0

Step 2: Compute Borel sum 𝚿𝚿± at each Stokes region

・ Borel sum is well-defined and computable in each region separated by Stokes lines∞d𝜂𝜂 −𝜂𝜂/ℏ
i 𝑡𝑡 ′
Ψ± = �
𝐵𝐵[𝜓𝜓± ](𝑡𝑡; 𝜂𝜂) ∼ exp ∓ � d𝑡𝑡 𝑄𝑄 𝑡𝑡 ′
e
ℏ
ℏ 𝑡𝑡0
0

× (1 + 𝑂𝑂(ℏ)) at each Stokes region

Step 3: Compute the Stokes constants 𝜶𝜶 and 𝜷𝜷

・ Whenever crosses Stokes lines, Ψ± jumps discontinuously (Stokes phenomenon)⇒ The discontinuity is given by the integral of singularities of 𝐵𝐵 𝜓𝜓±Ψ+ region 𝐴𝐴 = 𝛼𝛼 Ψ+ region 𝐵𝐵 + 𝛽𝛽 Ψ− region 𝐵𝐵

∼�

d𝜂𝜂

sing.onℏa Stokes linee−𝜂𝜂/ℏ 𝐵𝐵[𝜓𝜓± ](𝑡𝑡; 𝜂𝜂)Contents

I. Introduction to HHG & Exact WKB
II. Application of Exact WKB to HHG
III. Summary

Strong-field QED wavefunction within EWKB
Apply Exact WKB to Dirac eq.: 0 = iℏ𝛾𝛾 0 𝜕𝜕𝑡𝑡 − 𝜸𝜸 ⋅ 𝒑𝒑 − 𝑒𝑒𝑨𝑨 − 𝑚𝑚 𝜓𝜓 𝑡𝑡
Step 1: Draw a Stokes graph
Step 2: Compute Borel sum Ψ± at each Stokes region
Step 3: Compute the Stokes constants 𝛼𝛼 and 𝛽𝛽
At the leading order in ℏStrong-field QED wavefunction within EWKB
Apply Exact WKB to Dirac eq.: 0 = iℏ𝛾𝛾 0 𝜕𝜕𝑡𝑡 − 𝜸𝜸 ⋅ 𝒑𝒑 − 𝑒𝑒𝑨𝑨 − 𝑚𝑚 𝜓𝜓 𝑡𝑡
Step 1: Draw a Stokes graph
Step 2: Compute Borel sum Ψ± at each Stokes region
Step 3: Compute the Stokes constants 𝛼𝛼 and 𝛽𝛽Ψ−,𝒑𝒑 𝑡𝑡 = 𝑣𝑣𝒑𝒑−𝑒𝑒𝑨𝑨 e
where

𝑡𝑡𝑛𝑛cr =𝑡𝑡+i ∫ 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨

At the leading order in ℏ− � Θ 𝑡𝑡 − 𝑡𝑡𝑛𝑛cr
𝑛𝑛
𝑡𝑡

𝜎𝜎𝑛𝑛
𝑡𝑡
𝑛𝑛 − ℏ−1 e𝑢𝑢𝒑𝒑−𝑒𝑒𝑨𝑨 e−i ∫ 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨

1 + 𝑂𝑂 ℏ

2− Re 𝜎𝜎+ 𝑂𝑂(e ℏ)𝑡𝑡 ∈ ℝ 0 = Im i � 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨 } is the crossing b/w 𝑛𝑛-th Stokes line and the real axis
tp
tp

𝑡𝑡𝑛𝑛

𝜎𝜎𝑛𝑛 = −2i �

0

𝑡𝑡𝑛𝑛

d𝑡𝑡 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨 is the “instanton” action associated to 𝑛𝑛-th Stokes line

Strong-field QED wavefunction within EWKB
Apply Exact WKB to Dirac eq.: 0 = iℏ𝛾𝛾 0 𝜕𝜕𝑡𝑡 − 𝜸𝜸 ⋅ 𝒑𝒑 − 𝑒𝑒𝑨𝑨 − 𝑚𝑚 𝜓𝜓 𝑡𝑡
Step 1: Draw a Stokes graph
Step 2: Compute Borel sum Ψ± at each Stokes region
Step 3: Compute the Stokes constants 𝛼𝛼 and 𝛽𝛽Ψ−,𝒑𝒑 𝑡𝑡 = 𝑣𝑣𝒑𝒑−𝑒𝑒𝑨𝑨 e
where

𝑡𝑡𝑛𝑛cr =𝑡𝑡+i ∫ 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨

At the leading order in ℏ− � Θ 𝑡𝑡 − 𝑡𝑡𝑛𝑛cr
𝑛𝑛
𝑡𝑡

𝜎𝜎𝑛𝑛
𝑡𝑡
𝑛𝑛 − ℏ−1 e𝑢𝑢𝒑𝒑−𝑒𝑒𝑨𝑨 e−i ∫ 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨

1 + 𝑂𝑂 ℏ

2− Re 𝜎𝜎+ 𝑂𝑂(e ℏ)𝑡𝑡 ∈ ℝ 0 = Im i � 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨 } is the crossing b/w 𝑛𝑛-th Stokes line and the real axis
tp
tp

𝑡𝑡𝑛𝑛

𝜎𝜎𝑛𝑛 = −2i �

0

𝑡𝑡𝑛𝑛

d𝑡𝑡 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨 is the “instanton” action associated to 𝑛𝑛-th Stokes line

Negative plane wave
(0th WKB sol.)

Strong-field QED wavefunction within EWKB
Apply Exact WKB to Dirac eq.: 0 = iℏ𝛾𝛾 0 𝜕𝜕𝑡𝑡 − 𝜸𝜸 ⋅ 𝒑𝒑 − 𝑒𝑒𝑨𝑨 − 𝑚𝑚 𝜓𝜓 𝑡𝑡
Step 1: Draw a Stokes graph
Step 2: Compute Borel sum Ψ± at each Stokes region
Step 3: Compute the Stokes constants 𝛼𝛼 and 𝛽𝛽Ψ−,𝒑𝒑 𝑡𝑡 = 𝑣𝑣𝒑𝒑−𝑒𝑒𝑨𝑨 e
where

𝑡𝑡𝑛𝑛cr =𝑡𝑡+i ∫ 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨

At the leading order in ℏ− � Θ 𝑡𝑡 − 𝑡𝑡𝑛𝑛cr
𝑛𝑛
𝑡𝑡

𝜎𝜎𝑛𝑛
𝑡𝑡
𝑛𝑛 − ℏ−1 e𝑢𝑢𝒑𝒑−𝑒𝑒𝑨𝑨 e−i ∫ 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨

1 + 𝑂𝑂 ℏ

2− Re 𝜎𝜎+ 𝑂𝑂(e ℏ)𝑡𝑡 ∈ ℝ 0 = Im i � 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨 } is the crossing b/w 𝑛𝑛-th Stokes line and the real axis
tp
tp

𝑡𝑡𝑛𝑛

𝜎𝜎𝑛𝑛 = −2i �

0

𝑡𝑡𝑛𝑛

d𝑡𝑡 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨 is the “instanton” action associated to 𝑛𝑛-th Stokes line

Negative plane wave
(0th WKB sol.)# of Stokes
lines crossed

Positive plane wave
(0th WKB sol.)

Stokes const.
⇒ Reproduces the known semi-classical formulas for the particle prod. # at out-statee.g., Worldline instanton method [Dunne, Schubert, …];
Steepest descent analyses [Izykson-Zuber, Dykene-Davis-Pechuaks]
Imaginary-time method [Popov]; Divergent asymptotic series method [Berry-Dingle]

Strong-field QED wavefunction within EWKB
Apply Exact WKB to Dirac eq.: 0 = iℏ𝛾𝛾 0 𝜕𝜕𝑡𝑡 − 𝜸𝜸 ⋅ 𝒑𝒑 − 𝑒𝑒𝑨𝑨 − 𝑚𝑚 𝜓𝜓 𝑡𝑡
Step 1: Draw a Stokes graph
Step 2: Compute Borel sum Ψ± at each Stokes region
Step 3: Compute the Stokes constants 𝛼𝛼 and 𝛽𝛽Ψ−,𝒑𝒑 𝑡𝑡 = 𝑣𝑣𝒑𝒑−𝑒𝑒𝑨𝑨 e
where

𝑡𝑡𝑛𝑛cr =𝑡𝑡+i ∫ 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨

At the leading order in ℏ− � Θ 𝑡𝑡 − 𝑡𝑡𝑛𝑛cr
𝑛𝑛
𝑡𝑡

𝜎𝜎𝑛𝑛
𝑡𝑡
𝑛𝑛 − ℏ−1 e𝑢𝑢𝒑𝒑−𝑒𝑒𝑨𝑨 e−i ∫ 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨

1 + 𝑂𝑂 ℏ

2− Re 𝜎𝜎+ 𝑂𝑂(e ℏ)𝑡𝑡 ∈ ℝ 0 = Im i � 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨 } is the crossing b/w 𝑛𝑛-th Stokes line and the real axis
tp
tp

𝑡𝑡𝑛𝑛

𝜎𝜎𝑛𝑛 = −2i �

0

𝑡𝑡𝑛𝑛

d𝑡𝑡 𝜔𝜔𝒑𝒑−𝑒𝑒𝑨𝑨 is the “instanton” action associated to 𝑛𝑛-th Stokes line

Negative plane wave
(0th WKB sol.)# of Stokes
lines crossed

Positive plane wave
(0th WKB sol.)

Stokes const.
⇒ Reproduces the known semi-classical formulas for the particle prod. # at out-statee.g., Worldline instanton method [Dunne, Schubert, …];
Steepest descent analyses [Izykson-Zuber, Dykene-Davis-Pechuaks]
Imaginary-time method [Popov]; Divergent asymptotic series method [Berry-Dingle]𝚿𝚿−,𝒑𝒑 enables us to discuss various processes in the presence of vacuum particle prod.

HHG: Harmonic spectrum
2d𝑁𝑁
Setup: A monochromatic E-field 𝐸𝐸(𝑡𝑡) = 𝐸𝐸0 cos(Ω𝑡𝑡)
Note that 𝜔𝜔 d𝜔𝜔 ∼ 𝜔𝜔𝐽𝐽̃ 𝜔𝜔
What I computed:
� 𝜇𝜇 𝜓𝜓� 0; in
VEV of current 𝐽𝐽̃ 𝜔𝜔 = F. T. 𝐽𝐽(𝑡𝑡) = F. T. 0; in 𝜓𝜓𝛾𝛾

using numerics and analytically with the EWKB wavefunc. Ψ−,𝒑𝒑 at LO in ℏ

HHG: Harmonic spectrum
2d𝑁𝑁
Setup: A monochromatic E-field 𝐸𝐸(𝑡𝑡) = 𝐸𝐸0 cos(Ω𝑡𝑡)
Note that 𝜔𝜔 d𝜔𝜔 ∼ 𝜔𝜔𝐽𝐽̃ 𝜔𝜔
What I computed:
� 𝜇𝜇 𝜓𝜓� 0; in
VEV of current 𝐽𝐽̃ 𝜔𝜔 = F. T. 𝐽𝐽(𝑡𝑡) = F. T. 0; in 𝜓𝜓𝛾𝛾

using numerics and analytically with the EWKB wavefunc. Ψ−,𝒑𝒑 at LO in ℏ

weak & fast 𝛾𝛾Keldysh ≔

𝑚𝑚Ω
= 100
𝑒𝑒𝐸𝐸0

strong & slow 𝛾𝛾Keldysh = 1/4

2𝜋𝜋𝛾𝛾Keldysh

・ Odd high-harmonic peaks appear as decreasing 𝜸𝜸𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊

2𝜋𝜋𝛾𝛾Keldysh⇒ Plateau (cutoff) appears when the physics becomes nonperturbative (perturbative)

HHG: Harmonic spectrum
2d𝑁𝑁
Setup: A monochromatic E-field 𝐸𝐸(𝑡𝑡) = 𝐸𝐸0 cos(Ω𝑡𝑡)
Note that 𝜔𝜔 d𝜔𝜔 ∼ 𝜔𝜔𝐽𝐽̃ 𝜔𝜔
What I computed:
� 𝜇𝜇 𝜓𝜓� 0; in
VEV of current 𝐽𝐽̃ 𝜔𝜔 = F. T. 𝐽𝐽(𝑡𝑡) = F. T. 0; in 𝜓𝜓𝛾𝛾

using numerics and analytically with the EWKB wavefunc. Ψ−,𝒑𝒑 at LO in ℏ

weak & fast 𝛾𝛾Keldysh ≔

𝑚𝑚Ω
= 100
𝑒𝑒𝐸𝐸0

strong & slow 𝛾𝛾Keldysh = 1/4

2𝜋𝜋𝛾𝛾Keldysh

2𝜋𝜋𝛾𝛾Keldysh

・ Odd high-harmonic peaks appear as decreasing 𝜸𝜸𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊⇒ Plateau (cutoff) appears when the physics becomes nonperturbative (perturbative)

・ LO Exact WKB is very GOOD for the plateau, but is BAD after the cutoff⇒ LO approx. is GOOD for nonpert. processes but BAD for pert. processes (e.g., multi-photon)

HHG: Harmonic intensity✔ Magnitude of the harmonic peak at 𝝎𝝎/𝛀𝛀 = 𝟗𝟗
Dependence on strength 𝐸𝐸02𝜋𝜋𝛾𝛾Keldysh

Dependence on frequency Ωanalytical with the EWKB wavefunc. Ψ−,𝒑𝒑 at LO in ℏ

numerics

2𝜋𝜋𝛾𝛾Keldysh

・ In the nonpert. regime (strong 𝑬𝑬𝟎𝟎 & small 𝛀𝛀), the harmonic intensity:- saturates ⇒ resulting in the plateau- oscillates ⇒ consistent with recent semi-conductor exp.

[Xia et al., (2020)](but only 𝐸𝐸0 -dep. is measured and Ω-dep. is our prediction)

・ The saturation & oscillation are nicely captured by LO Exact WKBExact WKB is a nice analytical approach for nonpertubative processes
in the presence of vacuum particle production

Contents

I. Introduction to HHG & Exact WKB
II. Application of Exact WKB to HHG
III.Summary

Summary
Problem: High-harmonic generation (HHG) from the vacuum by strong
time-dependent & spatially-homogeneous electric field
Approach: Exact WKB method✔ Exact WKB is a nice analytical method to construct wavefunc. under strong
fields, including Stokes phenomenon (i.e., dynamics of vacuum particle prod.)✔ Particle production from the vacuum leads to high-harmonic generation:
・ Only odd harmonics are generated𝑚𝑚Ω

・ Plateau structure, whose cutoff is set by the Keldysh parameter 𝛾𝛾Keldysh = 𝑒𝑒𝐸𝐸・ Saturation and oscillation in the nonperturbative regime 𝛾𝛾Keldysh < 1~✔ The nonperturbative features of HHG are nicely reproduced by Exact WKBeven with LO treatment in ℏ0

⇒ Exact WKB (or WKB) is a powerful tool to describe strong-field QED phenomena !
cf. Antonino’s talk (w/o Stokes pheno. but w/ spatial inhomo.)

For more details: [1] HT, Fujimori, Misumi, Nitta, Sakai, JHEP 03, 082 (2021) [2010.16080][2] HT, Hongo, Ikeda, 2105.12446(Note that [2] is written for cond-mat.,but the basic idea/calculation is the same for strong-field QED)Intuitive picture

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 =than tunneling ; 𝑁𝑁 is the largest at Ω ∼ 2𝑚𝑚
𝒎𝒎𝛀𝛀
𝒆𝒆𝑬𝑬
, 𝝂𝝂 ≡ 𝛀𝛀𝟐𝟐𝟎𝟎 control the interplay
𝒆𝒆𝑬𝑬𝟎𝟎・ 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)

Intra- and inter-band contributions

Analytical expression for the current