Vacuum HHG
in strong-field QED
Hidetoshi Taya (Keio U.)

This talkDiscuss vacuum high-harmonic generation (HHG) in strong-field QED
Naively …

vacuumVacuum HHG

vacuumTake-home messages:
・ Vacuum HHG surely occurs and semi-classical analysis can explain the basic features(e.g., cutoff law)・ Physics: Interband current induced by the Schwinger effect andquantum interference of the Schwinger effect are the essence

NB: Very preliminary, so comments/criticisms are welcome !
NB: This work is an extension of my previous work for 2d Dirac material [HT, Hongo, Ikeda (2021)]

Contents
1. Introduction

2. Theory
・ Numerical simulation ⇒ observe basic features of HHG

・ Semi-classical analysis ⇒ reveal the origin of HHG

3. Summary

Contents
1. Introduction

2. Theory
・ Numerical simulation ⇒ observe basic features of HHG

・ Semi-classical analysis ⇒ reveal the origin of HHG

3. Summary

Introduction to HHG
Actual HHG spectrum in a solidmaterialgasliquids

solidsplateau

cutoff𝜔out /𝜔inOriginally developed in optics and solid-state physics
・ First observation with gas (= one-band material) in 1988
⇒ Many applications: laser pointer, attosecond light source, …
Nobel prize in 2023

・ Recent development: observation with solids (= band materials) in 2011
⇒ Key features: plateau + cutoff + discretized peaks[Ghimire et al., (2011)]

QED analog ?
vacuumVery natural, if we admit the analogy b/w band-materials & QED vacuum
・ Many successful analogies: Schwinger effectLandau-Zener transition
Dynamically assisted Schwinger
Franz-Keldysh effect
Quantum interference of Schwinger
Stuckelberg-phase effect …

Not only interesting but also useful[T. Gaumnitz et al., (2017)]

・ The fastest light source of the zs (zepto second ~ 1/𝑚𝑒 ) order (cf. current world record: 43 as ~ 100 eV −1 )・ Application to physics near/beyond the pair creation threshold (e.g., dynamically assisted Schwinger)

Previous studies
・ This is not the first study: [Piazza, Hatsagortsyan, Keitel (2005)], [Fedotov, Narozhny (2006)], [Bohl, King, Ruhl (2015)] …
・ But, previous studies are based on constant-field (low freq.) approx. (e.g., EH Lagrangian)⇒ Problematic, since time dependence can change physics drastically !

Sales point: Proper inclusion of the time dependence of the input driving fieldContents
1. Introduction

2. Theory・ Numerical simulation ⇒ observe basic features of HHG

・ Semi-classical analysis ⇒ reveal the origin of HHG

3. Summary

Formulation
തSetup: QED in a spatially-homogeneous AC field 𝐸(𝑡)= 𝐸ത0 cos(Ω𝑡)ҧ 𝐴ҧ
classical driving field 𝐴=1
ҧ መ where 𝐴መ = 𝐴ҧ + 𝒜መ sourced by 𝐽
ℒ = 𝜓ത෠ i𝜕/ − 𝑒 𝐴/መ − 𝑚 𝜓෠ − 𝐹෠ 𝜇𝜈 𝐹෠𝜇𝜈 − 𝐽𝐴response
4Observable: Harmonic spectrum of the response EM fieldFormulation
തSetup: QED in a spatially-homogeneous AC field 𝐸(𝑡)= 𝐸ത0 cos(Ω𝑡)ҧ 𝐴ҧ
classical driving field 𝐴=1ҧ መ where 𝐴መ = 𝐴ҧ + 𝒜መ sourced by 𝐽ℒ = 𝜓ത෠ i𝜕/ − 𝑒 𝐴/መ − 𝑚 𝜓෠ − 𝐹෠ 𝜇𝜈 𝐹෠𝜇𝜈 − 𝐽𝐴response
4

Observable: Harmonic spectrum of the response EM field
Step 1: Write down EoMs
Electron: i𝜕/ − 𝑒 𝐴/ҧ − 𝑚 𝜓෠ = 𝑒 𝒜/መ 𝜓෠Driving field: 𝜕 2 𝐴ҧ 𝜇 = 𝐽𝜇ҧത෠ 𝜇 𝜓෠
Response: 𝜕 2 𝒜መ 𝜇 = 𝑒 𝜓𝛾ത෠ 𝜇 𝜓෠ ≠ 0
NB: The response can be macroscopically large 𝒜መ 𝜇 ≠ 0 since 𝑒𝜓𝛾⇒ focus on 𝒜መ 𝜇 = 𝑂(ℏ0 ) and neglect 𝑎ො ≔ 𝒜መ − 𝒜መ = 𝑂(ℏ)

Formulation
തSetup: QED in a spatially-homogeneous AC field 𝐸(𝑡)= 𝐸ത0 cos(Ω𝑡)ҧ 𝐴ҧ
classical driving field 𝐴=1
ҧ መ where 𝐴መ = 𝐴ҧ + 𝒜መ sourced by 𝐽
ℒ = 𝜓ത෠ i𝜕/ − 𝑒 𝐴/መ − 𝑚 𝜓෠ − 𝐹෠ 𝜇𝜈 𝐹෠𝜇𝜈 − 𝐽𝐴response
4

Observable: Harmonic spectrum of the response EM field
Step 1: Write down EoMs
Electron: i𝜕/ − 𝑒 𝐴/ҧ − 𝑚 𝜓෠ = 𝑒 𝒜/መ 𝜓෠ത෠ 𝜇 𝜓෠
Response: 𝜕 2 𝒜መ 𝜇 = 𝑒 𝜓𝛾

Driving field: 𝜕 2 𝐴ҧ 𝜇 = 𝐽𝜇ҧത෠ 𝜇 𝜓෠ ≠ 0NB: The response can be macroscopically large 𝒜መ 𝜇 ≠ 0 since 𝑒𝜓𝛾⇒ focus on 𝒜መ 𝜇 = 𝑂(ℏ0 ) and neglect 𝑎ො ≔ 𝒜መ − 𝒜መ = 𝑂(ℏ)

Step 2: Solve the EoMs with the Green function method
・Electron EoM ⇒ 𝜓෠ 𝑥 = 𝜓෠ 0 𝑥 + ∫ d4 𝑥 ′ 𝑆 𝑥, 𝑥 ′ 𝑒𝒜 𝑥 ′ 𝜓෠ 𝑥 ′ = 𝜓෠ 0 + 𝑂(𝑒)where i𝜕/ − 𝑒𝐴/ҧ − 𝑚 𝑆(𝑥, 𝑥′) = 𝛿 4 (𝑥 − 𝑥′) and i𝜕/ − 𝑒𝐴
/ ҧ − 𝑚 𝜓෠ 0 = 0𝜇 ෠ത෠・Response EoM ⇒ 𝒜መ 𝑥 = 𝒜መ 0 𝑥 + ∫ d4 𝑥 ′ 𝐺 𝑥, 𝑥 ′ 𝑒𝜓(𝑥′)𝛾𝜓(𝑥′)where 𝜕 2 𝐺(𝑥, 𝑥′) = 𝛿 4 (𝑥 − 𝑥′) and 𝜕 2 𝒜መ 0 = 0⇒ 𝒜መ = ∫ d4 𝑥 ′ 𝐺 𝑥, 𝑥 ′ 𝑒𝜓ത෠ 0 𝑥 ′ 𝛾 𝜇 𝜓෠ 0 𝑥 ′NB: The quantum part 𝑎ො = 𝒜መ − 𝒜መ corresponds to≈～～～～～ , which is kinematically disfavoredFormulation
തSetup: QED in a spatially-homogeneous AC field 𝐸(𝑡)= 𝐸ത0 cos(Ω𝑡)ҧ 𝐴ҧ
classical driving field 𝐴=1ҧ መ where 𝐴መ = 𝐴ҧ + 𝒜መ sourced by 𝐽
ℒ = 𝜓ത෠ i𝜕/ − 𝑒 𝐴/መ − 𝑚 𝜓෠ − 𝐹෠ 𝜇𝜈 𝐹෠𝜇𝜈 − 𝐽𝐴
response
4Observable: Harmonic spectrum of the response EM field

Step 3: Take the Fourier transformመ𝒜(𝑥)= ∫ d4 𝑥 ′ 𝐺 𝑥, 𝑥 ′ 𝑒 𝜓ത෠ 0 𝑥 ′ 𝛾 𝜇 𝜓෠ 0 𝑥 ′
F.T.

−i෫
መ
Harmonic spectrum: ℰሚ 𝜔 ≔ −𝑖𝜔 𝒜(𝜔)= F. T. 𝑒𝜓ത෠ 0 𝛾𝜓෠ 0𝜔

∴ Solve the Dirac eq. & do the Fourier integral ⇒ Get harmonic spectrum

Numerical observation
Use numerics to get an exact harmonic spectrum of vacuum HHG

Numerical observation
Use numerics to get an exact harmonic spectrum of vacuum HHG
Harmonic spectrum for 𝐸ത = 𝐸0 cos Ω𝑡Harmonic intensity ℰሚ 𝜔 /ΩWeak field (fixed ΩΤ𝑚 = 0.2)
𝑒𝐸0 /𝑚2 = 0.011𝑚Ω
(𝛾 ≔ 𝑎 = 𝑒𝐸 = 10)0Strong field𝑒𝐸0 /𝑚2 = 0.05
(𝛾 = 2)𝑒𝐸0 /𝑚2 = 0.15
(𝛾 = 0.666 …)

0Harmonic order 𝜔/Ω

Some features of vacuum HHG(1) Plateau + cutoff + discretized peaks, similarly to solid-state HHG(2) Cutoff increases with 𝐸0(3) Hamonic intensities do not necessary increase with 𝐸0(⇒ actually, it oscillates with 𝐸0 )𝑒𝐸0 /𝑚2 = 0.30
(𝛾 = 0.333 …)

Numerical observation
Use numerics to get an exact harmonic spectrum of vacuum HHG
Harmonic spectrum for 𝐸ത = 𝐸0 cos Ω𝑡Harmonic intensity ℰሚ 𝜔 /ΩWeak field (fixed ΩΤ𝑚 = 0.2)
𝑒𝐸0 /𝑚2 = 0.011𝑚Ω
(𝛾 ≔ 𝑎 = 𝑒𝐸 = 10)0Strong field𝑒𝐸0 /𝑚2 = 0.05
(𝛾 = 2)𝑒𝐸0 /𝑚2 = 0.15
(𝛾 = 0.666 …)

𝑒𝐸0 /𝑚2 = 0.30
(𝛾 = 0.333 …)

0Harmonic order 𝜔/Ω

Some features of vacuum HHG(1) Plateau + cutoff + discretized peaks, similarly to solid-state HHG(2) Cutoff increases with 𝐸0(3) Hamonic intensities do not necessary increase with 𝐸0(⇒ actually, it oscillates with 𝐸0 )Origins of those features ? ⇒ Give you an answer with semi-classical analysis

Semi-classical analysis (1/3)෡ 𝜸𝝍ഥ෡𝟎 ∝ 𝓔෩
Two contributions to the current 𝑱 = 𝒆𝝍𝟎・ Strong E field ⇒ Pair production⇒ Total mode function 𝜓0 𝑡 = in = 𝜓− → # × 𝜓− + # × 𝜓+・ VEV has two contr.: 𝐽 = 𝑒 𝜓ത− 𝛾𝜓− → # × 𝑒𝜓ത− 𝛾𝜓−+ # × 𝑒 𝜓ത+ 𝛾𝜓+→ Vacuum current (usually =0)→ Intraband current+ # × Re[𝑒𝜓ത+ 𝛾𝜓− ] → Interband current
・ Different time dynamical phases: 𝜓± ∝ e∓i𝑚𝑡⇒ Interband current has the fastest time dependence ∝ 𝑒 +2i𝑚𝑡⇒ the source of high harmonics !
NB: same idea in solid-state HHG[Vampa et al., (2014)]

Semi-classical analysis (2/3)
More quantitative argument: Intermediate particle picture + Semi-classical analysis

Semi-classical analysis (2/3)
More quantitative argument: Intermediate particle picture + Semi-classical analysis

Step 1: Introduce an intermediate particle picture
・Identify 𝜓± as the instantaneous solutions to the Dirac eq.
𝑒𝐸ത𝜓0 (𝑡) = 𝛼(𝑡) × 𝜓− (𝑡) + 𝛽 𝑡 × 𝜓+ (𝑡) where 𝐻Dirac 𝜓± = ± 𝑚2 + 𝒑 + Ω0 sin Ω𝑡2

𝜓± =: 𝐸± 𝜓± (⇒ 𝜓± ∝ 𝑒 −𝑖∫ d𝑡𝐸± )Semi-classical analysis (2/3)
More quantitative argument: Intermediate particle picture + Semi-classical analysis

Step 1: Introduce an intermediate particle picture
・Identify 𝜓± as the instantaneous solutions to the Dirac eq.
𝑒𝐸ത𝜓0 (𝑡) = 𝛼(𝑡) × 𝜓− (𝑡) + 𝛽 𝑡 × 𝜓+ (𝑡) where 𝐻Dirac 𝜓± = ± 𝑚2 + 𝒑 + Ω0 sin Ω𝑡2

𝜓± =: 𝐸± 𝜓± (⇒ 𝜓± ∝ 𝑒 −𝑖∫ d𝑡𝐸± )Step 2: Solve the evolution eqs. for the Bogoliubov coefficients 𝛼 & 𝛽・The evol. eqs. for 𝜓0 and 𝜓± are known ⇒ can write down evol. eq. for 𝛼 & 𝛽・Complicated ⇒ Not exactly solvable but analytically with some approx.
⇒ semi-classical method [Dumlu, Dunne (2010)] [HT et al. (2020)]
Figure stolen from [Shevchenko, Ashhab, Nori (2010)]

𝜓+ ∝ 𝑒 −𝑖∫ d𝑡𝐸+𝛽2⇐

𝜓− ∝ 𝑒 +𝑖∫ d𝑡𝐸−𝑡(1) Periodic jumps w/ freq. Ω ⇐ Pair production occurs periodically at around the minimum gap(2) 𝛽 does not increase monotonically but oscillates⇐ Quantum interference among pair productions (Stuckelberg-phase effect)⇒ max 𝛽 when most constructive ⇔ 2𝜋ℤ ∋ 𝐸+ − 𝐸− ∝ 𝑒𝐸ത0 /Ω ⇒ determine harmonic intensity

Semi-classical analysis (3/3)
More quantitative argument: Intermediate particle picture + Semi-classical analysis

Step 3: Calculate intra- and inter-band currentsℰ(𝑡) ∝ 𝐽(𝑡) = 𝐽intra (𝑡) + 𝐽inter (𝑡)2∼ 𝛽 𝑡2𝑒𝜓ത+ 𝑡 𝛾𝜓+ 𝑡 ∼ 𝛽 𝑡2

⇒ ℰሚ 𝜔 ∝ 𝐽ሚ 𝜔 = 𝐽ሚintra 𝜔 + 𝐽inter 𝜔~ ∫ d𝑡 𝑒 −i𝜔𝑡 𝛽 𝑡2ഥ0𝑒𝐸+2i∫ d𝑡 𝑚 + 𝒑+sin Ω𝑡Ω∼ 𝛼(𝑡)𝛽(𝑡)𝑒𝜓ത + (𝑡)𝛾𝜓− (𝑡) ∼ 𝛽 𝑡 𝑒2+ ∫ d𝑡 𝑒 −i𝜔𝑡 𝛽 𝑡 𝑒~ 𝛽 2 𝛿 𝜔 − Ω + 𝛽∫ d𝑡 𝑒 −i𝜔𝑡 𝑒+2i∫ d𝑡 𝑚2 + 𝒑+𝑒𝐸ത0sin Ω𝑡Ω𝑒𝐸ത
+2i∫ d𝑡 𝑚2 + 𝒑+ 0 sin Ω𝑡Ω

can be non-negligible only if 2 min 𝑚2 + 𝒑 +𝑒 𝐸ത0Ω

sin Ω𝑡222< 𝜔 < 2 max 𝑚2 + 𝒑 +⇒ 2𝑚 < 𝜔 < 2𝑒𝐸0 /Ω ⇒ determine the plateau and the cutoff𝑒𝐸ത0Ωsin Ω𝑡2

Semi-classical lessons v.s. numerics
Lessons from semi-classical analysis:
・ Harmonic intensity oscillates with 𝐸ത0 due to quantum interference of the pair production・ High harmonics originate from the interband current, as it is very fast 𝐽inter ∝ 𝑒ഥ0𝑒𝐸sin Ω𝑡Ω+2i∫ d𝑡 𝑚2 + 𝒑+・ Cutoff law min 𝐸+ − 𝐸− < 𝜔 < max 𝐸+ − 𝐸− ⇒ 2𝑚 < 𝜔 < 2𝑒𝐸0 /Ω
NB： Time dependence of the driving field is crucial; otherwise, no interference and no plateau

2Semi-classical lessons v.s. numerics
Lessons from semi-classical analysis:
・ Harmonic intensity oscillates with 𝐸ത0 due to quantum interference of the pair production・ High harmonics originate from the interband current, as it is very fast 𝐽inter ∝ 𝑒

ഥ0𝑒𝐸sin Ω𝑡Ω+2i∫ d𝑡 𝑚2 + 𝒑+2・ Cutoff law min 𝐸+ − 𝐸− < 𝜔 < max 𝐸+ − 𝐸− ⇒ 2𝑚 < 𝜔 < 2𝑒𝐸0 /Ω
NB： Time dependence of the driving field is crucial; otherwise, no interference and no plateau

All of those lessons are consistent with numerics !

𝜔 = 2𝑚𝑒𝐸0Ω
=0.3,= 0.1𝑚2𝑚

𝜔 = 2𝑒𝐸0 /Ω

totalinterintra

Harmonic order 𝜔/Ω

Harmonic intensity at 𝜔/Ω = 45 (with 𝑚Ω = 0.1)
Harmonic intensity ℰሚ 𝜔 /ΩHarmonic intensity ℰሚ 𝜔 /ΩHarmonic spectrum for

Field strength 𝑒𝐸0 /𝑚2

Contents
1. Introduction

2. Theory・ Numerical simulation ⇒ observe basic features of HHG・ Semi-classical analysis ⇒ reveal the origin of HHG

3. Summary

Summary
Discuss vacuum high-harmonic generation (HHG) in strong-field QED
Harmonic spectrum forvacuum

Harmonic intensity ℰሚ 𝜔 /ΩVacuum HHG

Take-home messages:

𝜔 = 2𝑚𝑒𝐸0Ω= 0.3, = 0.12𝑚𝑚

𝜔 = 2𝑒𝐸0 /Ω

total
interintra

Harmonic order 𝜔/Ω

・ Vacuum HHG surely occurs and semi-classical analysis can explain the basic features(e.g., cutoff law)・ Physics: Interband current induced by the Schwinger effect andquantum interference of the Schwinger effect are the essence

NB: Very preliminary, so comments/criticisms are welcome !
NB: This work is an extension of my previous work for 2d Dirac material [HT, Hongo, Ikeda (2021)]