Introduction tostrong-field physics
Hidetoshi Taya
Research area: Theoretical physics (particle/nuclear theory)

Today’s talk
Q: What happens if we make light (or “field” in general) stronger and stronger ?
LEDLaser welding

Cond.-mat(THz laser)

Guiness record???

?
HERCULES @ USA (2008)102

𝐼 ∼ 10−5 W/cm2

106 W/cm2

10W/cm1022 W/cm2or 𝑒𝐸 ∼ 10−1 eVΤcm

104 eVΤcm

106 eVΤcm

1012 eVΤcm∼ 10−3 eV 2

10−1 eV 2

1 eV 2

1 keV 2

Daily lifeIndustry

ScienceOur limit
at the present

Today’s talk
Q: What happens if we make light (or “field” in general) stronger and stronger ?
LEDLaser welding

Cond.-mat(THz laser)

Guiness record???

?
HERCULES @ USA (2008)102

𝐼 ∼ 10−5 W/cm2

106 W/cm2

10W/cm1022 W/cm2

or 𝑒𝐸 ∼ 10−1 eVΤcm104 eVΤcm

106 eVΤcm

1012 eVΤcm∼ 10−3 eV 2

10−1 eV 2

1 eV 2

1 keV 2

Daily life

Industry

Our limit
at the present

Science

A: Our vacuum shall be destroyed and gives us nontrivial responses(≈ “something” even from “nothing” )⇒ Purpose of this talk: Review such physics of strong lightContents
1. Overview of strong-field physics- becoming a hot topic in physics, thanks to the recent developments in exp.

- the Schwinger effect = a process to create “something” from “nothing”

2. Application of the Schwinger effect⇒ High-harmonic generation from the vacuum[HT, Hongo, Ikeda (2021)]- first prediction of HHG from the vacuum- a good example of “interdisciplinarity”
of strong-field physics⇒ your expertise should be useful in
my area and vice versa !

3. SummaryvacuumContents
1. Overview of strong-field physics- becoming a hot topic in physics, thanks to the recent developments in exp.

- the Schwinger effect = a process to create “something” from “nothing”

2. Application of the Schwinger effect⇒ High-harmonic generation from the vacuum[HT, Hongo, Ikeda (2021)]- first prediction of HHG from the vacuum- a good example of “interdisciplinarity”of strong-field physics⇒ your expertise should be useful in
my area and vice versa !

3. Summaryvacuum

What if field becomes strong ?
finalinitialtime evol.

No fieldWhat if field becomes strong ?
final

initialtime evol.

No field

Weak field

Strong fieldWhat if field becomes strong ?
final

initialtime evol.

𝑭No field

Weak field

Strong field

Only minor changes⇒ Perturbative⇒ Very well understoodin both exp.& theor.
g−2ex.) Electron (anomalous) magnetic moment 𝑎 ≔2
≈ Electron energy shift in a weak magnetic field𝑎 theor. = 1159652182.03 … × 10−12
𝑎 exp.
= 1159652180.73 … × 10−12[Aoyama, Kinoshita, Nio (2017)]

What if field becomes strong ?
final

initial𝑭
time evol.

𝑭𝑭𝑭No field

Weak field

𝑭…

Strong field

Only minor changes

Big change !

⇒ Perturbative⇒ Non-Perturbative⇒ Very well understoodin both exp.& theor.

⇒ Not understood wellg−2ex.) Electron (anomalous) magnetic moment 𝑎 ≔2
≈ Electron energy shift in a weak magnetic field𝑎 theor. = 1159652182.03 … × 10−12
𝑎 exp.
= 1159652180.73 … × 10−12[Aoyama, Kinoshita, Nio (2017)]

If field becomes strong, physics becomes totally different & nontrivial

When is field “strong” ?
Strong-field condition:
To significantly modify the original system with typical energy Δ,
the field must be more energetic than Δ⇒ Strong-field condition: 𝚫 < (𝐞𝐧𝐞𝐫𝐠𝐲 𝐬𝐜𝐚𝐥𝐞 𝐨𝐟 𝐭𝐡𝐞 𝐟𝐢𝐞𝐥𝐝)

When is field “strong” ?
Strong-field condition:
To significantly modify the original system with typical energy Δ,
the field must be more energetic than Δ⇒ Strong-field condition: 𝚫 < (𝐞𝐧𝐞𝐫𝐠𝐲 𝐬𝐜𝐚𝐥𝐞 𝐨𝐟 𝐭𝐡𝐞 𝐟𝐢𝐞𝐥𝐝)

Estimate of the minimum field strength:Standard Model

The (matter) particle having the minimum energy・ Charged ⇒ Couples toelectromagnetic (EM) field・ 𝒆𝑬 𝐌𝐞𝐕 𝟐 > 𝒎𝟐𝒆 = 𝟎. 𝟓𝟏𝟏 𝐌𝐞𝐕 𝟐≈ 𝑶(𝟏𝟎𝟐𝟖 𝐖/𝐜𝐦𝟐 )When is field “strong” ?
Strong-field condition:
To significantly modify the original system with typical energy Δ,
the field must be more energetic than Δ⇒ Strong-field condition: 𝚫 < (𝐞𝐧𝐞𝐫𝐠𝐲 𝐬𝐜𝐚𝐥𝐞 𝐨𝐟 𝐭𝐡𝐞 𝐟𝐢𝐞𝐥𝐝)

Estimate of the minimum field strength:Standard Model

The (matter) particle having the minimum energy・ Charged ⇒ Couples toelectromagnetic (EM) field・ 𝒆𝑬 𝐌𝐞𝐕 𝟐 > 𝒎𝟐𝒆 = 𝟎. 𝟓𝟏𝟏 𝐌𝐞𝐕 𝟐≈ 𝑶(𝟏𝟎𝟐𝟖 𝐖/𝐜𝐦𝟐 )∴ EXTREMELY strong⇒ Impossible to realize exp.
in 20th century

Recent availability of strong EM fields
The situation changing: becoming able to create/observe strong light∴ NOW is the BEST time to study strong-field physics

Recent availability of strong EM fields
The situation changing: becoming able to create/observe strong light

Laser intensity 𝑰 (𝐖/𝐜𝐦𝟐 )High-power laser
Strong regime (𝒆𝑬Τ𝒎𝟐𝒆 > 𝟏)

・ Heavy-ion collisions
RIC (2000~), LHC (2012~),FAIR/NICA/HIAF/J-Parc-HI/… (20XX~)
𝐼 ∼ 1035 W/cm2(𝑒𝐸, 𝑒𝐵 ∼ 𝑚𝜋2 ∼ 140 MeV 2 )●●Latest: ELI・ Electron collider + Laser●Guiness record (2008年)
HERCULES @ USA●Extreme physical systems

Start soon: LUXE @ DESY, FACET-II @ SLAC𝐼 ∼ 1029 W/cm2(𝑒𝐸, 𝑒𝐵 > 𝑚𝑒2 ∼ 1 MeV 2 )・ Magnetar

Invention of laser(Nobel prize in 1964)

Invention of CPA(Nobel prize in 2018)

Suzaku (2005~2015), NICER (2017~)
XL-Calibur (2018~), IXPE (2021~), …

𝐼 ∼ 1029 W/cm2(𝑒𝐸, 𝑒𝐵 > 𝑚𝑒2 ∼ 1 MeV 2 )∴ NOW is the BEST time to study strong-field physics

A bit more on heavy-ion collisions
Low-energy ( 𝒔𝐍𝐍 = 𝟐 − 𝟏𝟎) HIC is interesting among other strong-field systems
・ Sales point: The only system that has supercritical 𝐹: = 𝑬𝟐 − 𝑩𝟐 > 0, 𝐺 ∶= 𝑬 ⋅ 𝑩 ≠ 0

Field profile
strength

high-power laser

magnetar

High-energy HIC

Low-energy HIC(usually) 𝐹 = 𝐺 = 0

𝐹 < 0, 𝐺 = 0

𝐹 < 0, 𝐺 = 0

𝐹 > 0, 𝐺 ≠ 0

subcriticalsuper-longsupercriticalsuper-long

far-supercriticalsuper-shortsupercriticallonglifetime

E・B<0yimpact para. bx Bb = 0 fm

z

E・B>03 fm

6 fm

𝐺 = 𝑬⋅𝑩E@ 𝑠NN = 5.2 GeV

𝐹 = 𝑬𝟐 − 𝑩𝟐・ Numerical estimation of the EM profile in low-energy HIC[HT, Nishimura, Ohnishi (2024)] [HT (in prep)]

・ So, I’m interested in this and wanna study this further:⇒ Chiral XXX ? Axion electrodynamics? Novel QCD phase? Let’s discuss if interested

Recent availability of strong EM fields
The situation changing: becoming able to create/observe strong light

Laser intensity 𝑰 (𝐖/𝐜𝐦𝟐 )High-power laser
Strong regime (𝒆𝑬Τ𝒎𝟐𝒆 > 𝟏)

・ Heavy-ion collisions
RIC (2000~), LHC (2012~),FAIR/NICA/HIAF/J-Parc-HI/… (20XX~)
𝐼 ∼ 1035 W/cm2(𝑒𝐸, 𝑒𝐵 ∼ 𝑚𝜋2 ∼ 140 MeV 2 )●●Latest: ELI・ Electron collider + Laser●Guiness record (2008年)
HERCULES @ USA●Extreme physical systems

Start soon: LUXE @ DESY, FACET-II @ SLAC𝐼 ∼ 1029 W/cm2(𝑒𝐸, 𝑒𝐵 > 𝑚𝑒2 ∼ 1 MeV 2 )・ Magnetar

Invention of laser(Nobel prize in 1964)

Invention of CPA(Nobel prize in 2018)

Suzaku (2005~2015), NICER (2017~)
XL-Calibur (2018~), IXPE (2021~), …

𝐼 ∼ 1029 W/cm2(𝑒𝐸, 𝑒𝐵 > 𝑚𝑒2 ∼ 1 MeV 2 )∴ NOW is the BEST time to study strong-field physics

What can happen with strong EM field ?
When 𝒆𝑬 > 𝒎𝟐𝒆 , many non-trivial phenomena have been predicted to occur
Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, Phys. Rept. (2023)]

Birefringence ofphoton in vacuum

Photon splitting

Hadron mass change(for 𝑒𝐵 > Λ2QCD )QCD phase diagram(for 𝑒𝐵 > Λ2QCD )[D’Elia et al. (2022)]～[HT (2015)]

Among others, the (Sauter-)Schwinger effect is the most intriguing

Sauter (1931), Schwinger (1951)

Particles are spontaneously produced from the vacuum (≈ “sthg” from “nothing”)

What can happen with strong EM field ?
When 𝒆𝑬 > 𝒎𝟐𝒆 , many non-trivial phenomena have been predicted to occur
Review: [Fedotov, Ilderton, Karbstein, King, Seipt, HT, Torgrimsson, Phys. Rept. (2023)]

Birefringence ofphoton in vacuum

Photon splitting

Hadron mass change(for 𝑒𝐵 > Λ2QCD )QCD phase diagram(for 𝑒𝐵 > Λ2QCD )[D’Elia et al. (2022)]～[HT (2015)]

Among others, the (Sauter-)Schwinger effect is the most intriguing

Sauter (1931), Schwinger (1951)

Particles are spontaneously produced from the vacuum (≈ “sthg” from “nothing”)
Our vacuum is not an empty space,but has a structure similar to semi-conductor(QED)

Vacuum

CAN observe

Particle production !

(≈ conduction band)

energyDirac (1930)

Then, real particles will be produced out ofthe vacuum by strong enough field

momentum

Dirac sea

gap = 2m

EM field
CANNOT observe(≈ valence band)

Sauter (1931) Schwinger (1951)

Contents
1. Overview of strong-field physics- becoming a hot topic in physics, thanks to the recent developments in exp.

- the Schwinger effect = a process to create “something” from “nothing”

2. Application of the Schwinger effect⇒ High-harmonic generation from the vacuum[HT, Hongo, Ikeda (2021)]- first prediction of HHG from the vacuum- a good example of “interdisciplinarity”
of strong-field physics⇒ your expertise should be useful in
my area and vice versa !

3. SummaryvacuumAnalogue of the Schwinger effectAnalogous particle production occurs in various physical systems
・ ex: Hawking radiation (strong gravity), dynamical Casimir effect (strong oscillation),(p)reheating in the early Universe (strong inflaton field), ...

・ Landau-Zener effect in materials would be the most prominent

Landau (1932), Zener (1932)

Analogue of the Schwinger effectAnalogous particle production occurs in various physical systems
・ ex: Hawking radiation (strong gravity), dynamical Casimir effect (strong oscillation),(p)reheating in the early Universe (strong inflaton field), ...

・ Landau-Zener effect in materials would be the most prominent(QED)

Landau (1932), Zener (1932)

Ground state of materials(e.g., semi-conductor)

VacuumPhysical states

gap = 2m= O(1 MeV)

Dirac seaConduction
bandv.s.

gap = Δ= O(1 eV)
Valenceband

The responses of the vacuum and ground state by an external EM field should be similarAnalogue of the Schwinger effectAnalogous particle production occurs in various physical systems
・ ex: Hawking radiation (strong gravity), dynamical Casimir effect (strong oscillation),(p)reheating in the early Universe (strong inflaton field), ...

・ Landau-Zener effect in materials would be the most prominent(QED)

Landau (1932), Zener (1932)

Ground state of materials(e.g., semi-conductor)

VacuumPhysical states

gap = 2m
= O(1 MeV)

Conduction
band

gap = Δ= O(1 eV)v.s.

Valenceband

Dirac sea

The responses of the vacuum and ground state by an external EM field should be similar
Indeed, the particle prediction rate and the excitation rate obey the same exponential formula2

ΓSchwinger = # exp[ # 𝑚 ൗ𝑒𝐸 ]2

ΓLandau−Zener = # exp[ # Δ ൗ𝑒𝐸 ]Schwinger effect has not been verified yet
BUT, Landau-Zener transition has been observed and utilized, e.g., to design devices⇒ Importing ideas from other areas of physics is quiteto better understand the Schwinger effect (or strong-field physics in general)As such idea: High-harmonic generation from the vacuum

・ HHG developed in optics & solid-state physics- Observed with various materials: gas (1988), liquid (2014), solid (2011), ...
- Many applications: laser pointer, high-speed switching, ...
- Nobel prize in 2023

As such idea: High-harmonic generation from the vacuum

EM-field

Ground state

Acceleration

Landau-Zener

Photon emission

・ HHG developed in optics & solid-state physics[Ghimire et al., (2011)]- Observed with various materials: gas (1988), liquid (2014), solid (2011), ...
- Many applications: laser pointer, high-speed switching, ...
- Nobel prize in 2023

・ Microscopic mechanism: Landau-Zener is the key- Theory for band materials developed recently [Vampa et al. (2014)](cf. 3 step model for gas is insufficient for band materials) [Corkum (1993)] [Lewenstein et al. (1994)]- But still incomplete and remains a hot topic, e.g., lack of analytical formulation

As such idea: High-harmonic generation from the vacuum

EM-field

Ground stateVacuum

Acceleration

Landau-Zener

Photon emission

Schwinger・ HHG developed in optics & solid-state physics[Ghimire et al., (2011)]- Observed with various materials: gas (1988), liquid (2014), solid (2011), ...
- Many applications: laser pointer, high-speed switching, ...
- Nobel prize in 2023

・ Microscopic mechanism: Landau-Zener is the key- Theory for band materials developed recently [Vampa et al. (2014)](cf. 3 step model for gas is insufficient for band materials) [Corkum (1993)] [Lewenstein et al. (1994)]- But still incomplete and remains a hot topic, e.g., lack of analytical formulationAnalogy b/w Schwinger & Landau-Zener ⇒ natural to expect HHG from the vacuum

How to formulate vacuum HHG ?
Observable: Photon spectrum ⇒ solving of a differential equation

How to formulate vacuum HHG ?
Observable: Photon spectrum ⇒ solving of a differential equation
Maxwell eq. ⇒ Photon spectrum is determined by current

𝐽𝜈 = 𝜕𝜇 𝐹𝜇𝜈 ∼ 𝜕 2 𝐴𝜈

d𝑁photon⇒ 𝜔= 𝜔 × F. T. 𝐽 2d𝜔How to formulate vacuum HHG ?
Observable: Photon spectrum ⇒ solving of a differential equation
Maxwell eq. ⇒ Photon spectrum is determined by current

𝐽𝜈 = 𝜕𝜇 𝐹𝜇𝜈 ∼ 𝜕 2 𝐴𝜈

d𝑁photon⇒ 𝜔= 𝜔 × F. T. 𝐽 2d𝜔

current produced by the Schwinger effect:
ത෠ 𝜇 𝜓෠𝐽 = 𝐽መ = 𝜓𝛾How to formulate vacuum HHG ?
Observable: Photon spectrum ⇒ solving of a differential equation
Maxwell eq. ⇒ Photon spectrum is determined by current

𝐽𝜈 = 𝜕𝜇 𝐹𝜇𝜈 ∼ 𝜕 2 𝐴𝜈

d𝑁photon⇒ 𝜔= 𝜔 × F. T. 𝐽 2d𝜔

current produced by the Schwinger effect:
ത෠ 𝜇 𝜓෠𝐽 = 𝐽መ = 𝜓𝛾determined by the Schrodinger eq. (the Dirac eq.) under the strong field𝑖ℏ𝜕𝑡 𝜓෠ = 𝐻 𝜓෠ = 𝐻free + 𝑉strong field 𝜓෠( )How to formulate vacuum HHG ?
Observable: Photon spectrum ⇒ solving of a differential equation
Maxwell eq. ⇒ Photon spectrum is determined by current

𝐽𝜈 = 𝜕𝜇 𝐹𝜇𝜈 ∼ 𝜕 2 𝐴𝜈

d𝑁photon⇒ 𝜔= 𝜔 × F. T. 𝐽 2d𝜔

current produced by the Schwinger effect:
ത෠ 𝜇 𝜓෠𝐽 = 𝐽መ = 𝜓𝛾determined by the Schrodinger eq. (the Dirac eq.) under the strong field𝑖ℏ𝜕𝑡 𝜓෠ = 𝐻 𝜓෠ = 𝐻free + 𝑉strong field 𝜓෠( )∴ We can take two strategies to discuss vacuum HHG:
(1) Numerically solve Eq. () (sometimes called TDSE method)(2) Analytically solve Eq. () by using some “wise” mathematical method⇒ In this work: Exact WKB method

Exact WKB method
Exact WKB = a wise method to solve ODE with a small parameter= ”usual” WKB + Borel resum.
[Voros (1983)][Pham, Dillinger, Delabaere,Aoki, Koike, Takei, …][Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)][HT, Fujimori, Misumi, Nitta, Sakai (2021)]

Exact WKB method
Exact WKB = a wise method to solve ODE with a small parameter= ”usual” WKB + Borel resum.
[Voros (1983)][Pham, Dillinger, Delabaere,Aoki, Koike, Takei, …][Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

A perturbation theory w.r.t. ℏ (or adiabatic approx.)𝑡 ≡ ℏ𝜏

・ Consider 0 = ℏ2 𝜕𝑡2 + 𝑄 𝑡 𝜙 𝑡i𝜕𝜏2 + 𝑄 ℏ𝜏 𝜙 𝜏⇔𝑡⇒ 𝜙± 𝑡; ℏ ≔ exp ∓ ‫ 𝑡׬‬d𝑡 ′ 𝑄 𝑡 ′ℏ𝑛× σ∞
𝑛=0 𝜓±,𝑛 (𝑡)ℏ

0th order = plane wave
i~ exp[∓ 𝑄𝑡]

Perturbation w.r.t. ℏ0ℏ

・ WKB expansion makes sense if the perturbative part is convergent
・ However, 𝜓±,𝑛 ~𝑛! in general (e.g., Airy function 𝑄 𝑡 ∝ 𝑡)
⇒ WKB expansion has zero radius of convergence ⇒ ill-defined !!!
[HT, Fujimori, Misumi, Nitta, Sakai (2021)]

Exact WKB method
Exact WKB = a wise method to solve ODE with a small parameter= ”usual” WKB + Borel resum.
[Voros (1983)][Pham, Dillinger, Delabaere,Aoki, Koike, Takei, …][Jeffery (1924)] [Wentzel (1926)][Kramers (1926)] [Brillouin (1926)][Ecalle (1981)]

A resummation scheme for factorially divergent ~n! series𝑛・ Consider the div. part of WKB expansion 𝜓± 𝑡; ℏ ≔ σ∞
𝑛=0 𝜓±,𝑛 (𝑡)ℏ∞

① Construct “Borel transformation”: 𝐵 𝜓± 𝑡; 𝜂 ≔ ෍∞𝑛

𝜓±,𝑛 (𝑡) 𝑛𝜂𝑛!

d𝜂 −𝜂/ℏe𝐵[𝜓± ](𝑡; 𝜂)② Laplace trans. gives “Borel sum”: Ψ± 𝑡; ℏ ≔ නℏ0・ Ψ± is well-defined and is a natural analytic continuation of 𝜓±⇒ 𝚿± gives a well-defined version of the WKB solution !
! on the real axis, Ψ± is ill-defined・ Note 1: If 𝐵 𝜓± has singularities⇒ but, the “ill-defined-ness” can be used to quantify
the Stokes phenomenon of the WKB solution
Note 2: in practice, some approximations need to be used in Borel resum⇒ in this work: only leading-order n! div. is considered (Airy approx.)[HT, Fujimori, Misumi, Nitta, Sakai (2021)]

Result (1/2)
Setup: A monochromatic E-field 𝐸(𝑡) = 𝐸0 cos(Ω𝑡) applied onto QED vacuum
What I did: Compute harmonic spectrum numerically and analytically with WKBResult (1/2): Numerical
Setup: A monochromatic E-field 𝐸(𝑡) = 𝐸0 cos(Ω𝑡) applied onto QED vacuum
What I did: Compute harmonic spectrum numerically and analytically with WKBHarmonic spectrum

Weak field:

𝒎𝛀
= 𝟏𝟎𝟎
𝒆𝑬𝟎

Result (1/2): Numerical
Setup: A monochromatic E-field 𝐸(𝑡) = 𝐸0 cos(Ω𝑡) applied onto QED vacuum
What I did: Compute harmonic spectrum numerically and analytically with WKB𝒎𝛀
= 𝟏𝟎𝟎𝒆𝑬𝟎

Strong field:

Harmonic spectrum

Weak field:

・ High harmonics appears from the vacuum when field becomes strong !
- structure similar to solid HHG (e.g., plateau, cutoff)𝒎𝛀= 𝟏/𝟒𝒆𝑬𝟎

Result (2/2): Analytical
Setup: A monochromatic E-field 𝐸(𝑡) = 𝐸0 cos(Ω𝑡) applied onto QED vacuum
What I did: Compute harmonic spectrum numerically and analytically with WKB
𝒎𝛀
= 𝟏/𝟒
𝒆𝑬𝟎

Harmonic spectrum

Strong field:

・ High harmonics appears from the vacuum when field becomes strong !
- structure similar to solid HHG (e.g., plateau, cutoff)・ Analytical WKB calculation is good at the plateau but bad after the cutoff⇒ useful to get a deeper insight of HHG in the agreement region
e.g., determination of the cutoff location (→ next slide)Physics picture: Interband interference
Acceleration

EM-field

VacuumSchwinger

Photon emission

・ Total wavefunction = superposition of the two energy states:𝜓 = 𝜓+ + 𝜓− = # e−i‫ 𝜖 ׬‬d𝑡 + # e+i‫ 𝜖 ׬‬d𝑡w/ 𝜖 ∼𝑚2 + 𝒑𝟐 ∼𝑒𝐸0cos(Ω𝑡)Ω𝑚2 + (𝑒𝐴(𝑡))2Physics picture: Interband interference
Acceleration

EM-field

VacuumSchwinger

Photon emission

・ Total wavefunction = superposition of the two energy states:𝜓 = 𝜓+ + 𝜓− = # e−i‫ 𝜖 ׬‬d𝑡 + # e+i‫ 𝜖 ׬‬d𝑡w/ 𝜖 ∼𝑚2 + 𝒑𝟐 ∼𝑒𝐸0cos(Ω𝑡)Ω

𝑚2 + (𝑒𝐴(𝑡))2⇒ current can be decomposed into intra- and inter-band contributions
ത𝐽 ∼ 𝜓𝛾𝜓∼ # 𝜓ത± 𝜓± + # 𝜓ത± 𝜓∓ ∼ # e𝑖‫𝜖 ׬‬−𝜖 d𝑡 + # 𝑒 𝑖‫𝜖 ׬‬+𝜖 d𝑡 ∼ # + # e2i‫ 𝜖 ׬‬d𝑡
⇒ the intra-band interference has higher frequency ⇒ source of HHG- is different from the naïve 3 step model in gas HHG, but is consistent w/ solid HHG[Vampa et al. (2014)]

Harmonic spectrum

Physics picture: Interband interference

Harmonic order・ Total wavefunction = superposition of the two energy states:𝜓 = 𝜓+ + 𝜓− = # e−i‫ 𝜖 ׬‬d𝑡 + # e+i‫ 𝜖 ׬‬d𝑡w/ 𝜖 ∼𝑚2 + 𝒑𝟐 ∼𝑒𝐸0cos(Ω𝑡)Ω𝑚2 + (𝑒𝐴(𝑡))2⇒ current can be decomposed into intra- and inter-band contributionsത𝐽 ∼ 𝜓𝛾𝜓∼ # 𝜓ത± 𝜓± + # 𝜓ത± 𝜓∓ ∼ # e𝑖‫𝜖 ׬‬−𝜖 d𝑡 + # 𝑒 𝑖‫𝜖 ׬‬+𝜖 d𝑡 ∼ # + # e2i‫ 𝜖 ׬‬d𝑡
⇒ the intra-band interference has higher frequency ⇒ source of HHG- is different from the naïve 3 step model in gas HHG, but is consistent w/ solid HHG[Vampa et al. (2014)]

Harmonic spectrum

Physics picture: Interband interference𝝎 = 𝟐𝒆𝑬𝟎 /𝛀
Harmonic order・ Total wavefunction = superposition of the two energy states:𝜓 = 𝜓+ + 𝜓− = # e−i‫ 𝜖 ׬‬d𝑡 + # e+i‫ 𝜖 ׬‬d𝑡w/ 𝜖 ∼𝑚2 + 𝒑𝟐 ∼𝑒𝐸0cos(Ω𝑡)Ω𝑚2 + (𝑒𝐴(𝑡))2⇒ current can be decomposed into intra- and inter-band contributions
ത𝐽 ∼ 𝜓𝛾𝜓∼ # 𝜓ത± 𝜓± + # 𝜓ത± 𝜓∓ ∼ # e𝑖‫𝜖 ׬‬−𝜖 d𝑡 + # 𝑒 𝑖‫𝜖 ׬‬+𝜖 d𝑡 ∼ # + # e2i‫ 𝜖 ׬‬d𝑡
⇒ the intra-band interference has higher frequency ⇒ source of HHG- is different from the naïve 3 step model in gas HHG, but is consistent w/ solid HHG[Vampa et al. (2014)]

・ Cutoff law: F. T. 𝐽inter ∼ ‫ ׬‬d𝑡 exp −i ‫ ׬‬d𝑡 𝜔 − 2𝜖⇒ 𝜔cutoff ∼ max 2𝜖 𝑡 ∼ 2𝑒𝐸0 /Ω𝑡- 𝜔cutoff ∝ 𝐸0 is consistent w/ solid HHG, while 𝜔cutoff ∝ Ω−1 is our prediction[Ghimire et al., (2011)]

Contents
1. Overview of strong-field physics- becoming a hot topic in physics, thanks to the recent developments in exp.

- the Schwinger effect = a process to create “something” from “nothing”

2. Application of the Schwinger effect⇒ High-harmonic generation from the vacuum[HT, Hongo, Ikeda (2021)]- first prediction of HHG from the vacuum- a good example of “interdisciplinarity”
of strong-field physics⇒ your expertise should be useful in
my area and vice versa !

3. Summaryvacuum

SummaryQ: What happens if we make light (or “field” in general) stronger and stronger ?
LEDMagnifying glass

Laser welding

THz Laserin solid-state phys.

???

?
𝐼 ∼ 10−5 W/cm2

102 W/cm2

106 W/cm2

1010 W/cm2> 1028 W/cm2

A: Our vacuum shall be destroyed and gives us nontrivial responses
・ Why interesting: Unexplored nonpert. regime ⇒ many nontrivial phenomena we’ve never seen
・ Why timely: Recent availability of strong fields
e.g., high-power laser, magnetar, heavy-ion colls

Example phenomenon: HHG from the vacuum
・ First prediction of HHG from the vauum
・ An example of “interdisciplinarity” strong-field physics⇐ borrowed ideas fromoptics, solid-state physics, & math[HT, Hongo, Ikeda (2021)]

Exact-WKB recipe for Stokes phenomenon
Step 1: Draw a Stokes graph・ Laplace trans. of 𝐵 𝜓± (𝑡; 𝜂) hits singularities(non-Borel summable) when locatedon Stokes lines in the t-plane
一 : Stokes lines𝑡𝑡 ∈ ℂ 0 = Im i න d𝑧

● : turning points

𝑄 𝑧 }

𝑡tp

𝑄 𝑡tp = 0Step 2: Compute Borel sum 𝚿± at each Stokes region
・ Borel sum is well-defined and computable in each region separated by Stokes lines∞d𝜂 −𝜂/ℏ
i 𝑡 ′Ψ± = නe𝐵[𝜓± ](𝑡; 𝜂) ∼ exp ∓ න d𝑡 𝑄 𝑡 ′ℏ 𝑡00 ℏ× (1 + 𝑂(ℏ)) at each Stokes region

Step 3: Compute the Stokes constants 𝜶 and 𝜷・ Whenever crosses Stokes lines, Ψ± jumps discontinuously (Stokes phenomenon)⇒ The discontinuity is given by the integral of singularities of 𝐵 𝜓±Ψ+ region 𝐴 = 𝛼 Ψ+ region 𝐵 + 𝛽 Ψ− region 𝐵∼රd𝜂sing.onℏ
a Stokes linee−𝜂/ℏ 𝐵[𝜓± ](𝑡; 𝜂)Application to high-harmonic generation (2/2)
・ WKB works more in the deep non-perturbative regime 𝑬𝟎 → 𝐥𝐚𝐫𝐠𝐞, 𝛀 → 𝐬𝐦𝐚𝐥𝐥
・ Demonstration: magnitude of the harmonic peak at 𝝎/𝛀 = 𝟗
Dependence on strength 𝐸0

Dependence on frequency Ωanalytical with WKBnumerical

・ Lessons:(1) WKB makes it easier to analyze the non-perturbative regime(2) Saturation & oscillation of the harmonic intensity⇒ consistent with recent semi-conductor exp.

[Xia et al., (2020)](but only 𝐸0 -dep. is measured and Ω-dep. is our prediction)

Peripheral case
[Nishimura (Osaka), Ohnishi (Kyoto), HT, in progress]

𝑒 2 𝐹 = |𝑒𝑬|2 − 𝑒𝑩 2Non-central collision ⇒ B & E・B are produced

E・B<0yimpact para. bx B
E・B>050 MeV 450 MeV 40 MeV 40 MeV 40 MeV 4− 50 MeV 4− 50 MeV 4− 50 MeV 45 fm

b = 0 fmz
𝑒 2 𝐺 = 𝑒𝑬 ⋅ 𝑒𝑩E50 MeV 410 fm

50 MeV 450 MeV 450 MeV 40 MeV 40 MeV 40 MeV 4− 50 MeV 4− 50 MeV 4− 50 MeV 4Preliminaryresults@ ECM = 7.7 GeV・ 𝑒𝑬 ⋅ 𝑒𝑩 = 𝑂 50 MeV 4 ⇒ Non-negligible for QCD (Enough strong for electron and current quarks)
・ Perhaps, a nice place to study chiral-anomaly-related stuffs
𝑒𝑬 ⋅ 𝑒𝑩𝑚2ex) chirality production via the Schwinger mechanism 𝑁5 = 𝑉𝑇exp −𝜋2𝜋 2𝑒𝐸