Enhancementof chirality production from the vacuum
by time-dependent electromagnetic fields
Hidetoshi Taya(RIKEN iTHEMS)
Ref: HT, Phys.Rev.Res. 2 (2020) 2, 023257 [2003.08948]
Fukushima, Hidaka, Shimazaki, HT, in preparation

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Summary

Background

Chirality production from the vacuum for massive fermions⇒ well understood for slow EM fields
・ Driven by the Schwinger mechanism
・ Exponentially suppressed ⇒ Difficult to be observed in experiments
Problem
Idea
MethodEnhancement mechanism for chirality production ?
Use time-dependent fast EM fields(1) Perturbation theory in the Furry picture(2) Floquet approach (high-frequency expansion)

Result & Message
Chirality production is significantly enhanced if EM fields are fast
⇒ worthwhile to investigate chirality-related phenomena in strong-field QED

1. Introduction
2. Enhancement of chirality production
by dynamically assisted Schwinger mech.
3. Enhancement of chirality production
by perturbative mechanism with fast E4. Summary

Chirality production✔ Chirality is produced through anomaly when 𝑬𝑬 ⋅ 𝑩𝑩 ≠ 0・ Microscopically, the interplay b/w Schwinger mech. by E-field & Landau quant. by B-field
momentumspin-E-field
B-field⇒ chirality ~ helicity = +𝟐𝟐 × 𝑵𝑵𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐢𝐢𝐢𝐢 𝐋𝐋𝐋𝐋𝐋𝐋+momentum
spin[Fukushima, Warringa,Kharzeev, Copinger,Shi, Ambjorn, Greensite,Peterson, HT, …]

Chirality production✔ Chirality is produced through anomaly when 𝑬𝑬 ⋅ 𝑩𝑩 ≠ 0・ Microscopically, the interplay b/w Schwinger mech. by E-field & Landau quant. by B-fieldmomentum
spin-E-field+B-fieldmomentum⇒ chirality ~ helicity = +𝟐𝟐 × 𝑵𝑵𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐢𝐢𝐢𝐢 𝐋𝐋𝐋𝐋𝐋𝐋

spin[Fukushima, Warringa,Kharzeev, Copinger,Shi, Ambjorn, Greensite,Peterson, HT, …]・ 𝑬𝑬 ⋅ 𝑩𝑩 ≠ 0 is realized in various physical situations and leads to interesting phenomenaex.1) In particle & cond-mat physics: Anomalous transport phenomena (e.g., Chiral Magnetic Effect,ex.2) In laser physics, 𝑬𝑬 ⋅ 𝑩𝑩 ≠ 0 can be realized
by a collision of two counter propagating standing waves
0𝑨𝑨1 ∝ 1 sin 𝜔𝜔 +𝑥𝑥 − 𝑡𝑡00, 𝑨𝑨2 ∝ sin 𝜙𝜙
cos 𝜙𝜙sin 𝜔𝜔 −𝑥𝑥 − 𝑡𝑡・ is important for get a deeper understanding of chiral anomaly

Chiral Separation Effect,Chiral Vortical Effect, …)⇒ 𝑬𝑬 ⋅ 𝑩𝑩 =cos 𝜙𝜙
1 + sin 𝜙𝜙Chirality production✔ Chirality is produced through anomaly when 𝑬𝑬 ⋅ 𝑩𝑩 ≠ 0・ Microscopically, the interplay b/w Schwinger mech. by E-field & Landau quant. by B-field
momentumspin

E-field-+B-field

momentum⇒ chirality ~ helicity = +𝟐𝟐 × 𝑵𝑵𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐢𝐢𝐢𝐢 𝐋𝐋𝐋𝐋𝐋𝐋

spin[Fukushima, Warringa,Kharzeev, Copinger,Shi, Ambjorn, Greensite,Peterson, HT, …]・ 𝑬𝑬 ⋅ 𝑩𝑩 ≠ 0 is realized in various physical situations and leads to interesting phenomenaex.1) In particle & cond-mat physics: Anomalous transport phenomena (e.g., Chiral Magnetic Effect,

Chiral Separation Effect,
Chiral Vortical Effect, …)

ex.2) In laser physics, 𝑬𝑬 ⋅ 𝑩𝑩 ≠ 0 can be realized
by a collision of two counter propagating standing waves
0𝑨𝑨1 ∝ 1 sin 𝜔𝜔 +𝑥𝑥 − 𝑡𝑡
00, 𝑨𝑨2 ∝ sin 𝜙𝜙cos 𝜙𝜙sin 𝜔𝜔 −𝑥𝑥 − 𝑡𝑡

・ is important for get a deeper understanding of chiral anomaly⇒ 𝑬𝑬 ⋅ 𝑩𝑩 =cos 𝜙𝜙
1 + sin 𝜙𝜙✔ Chirality production is suppressed strongly by mass2

∵ 𝑁𝑁pair in LLL ∝ e−# 𝑚𝑚 /𝑒𝑒𝑒𝑒 ⇒ Tiny effects… Difficult to observe…

Q: Any way to avoid the mass suppression ?

How to enhance chirality production ?
Idea: Increase the number of pairs 𝑵𝑵𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐢𝐢𝐢𝐢 𝐋𝐋𝐋𝐋𝐋𝐋 ⇒ Use fast E-fieldHow to enhance chirality production ?
Idea: Increase the number of pairs 𝑵𝑵𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐢𝐢𝐢𝐢 𝐋𝐋𝐋𝐋𝐋𝐋 ⇒ Use fast E-field✔ Physics of particle prod. changes depending on frequency 𝛀𝛀 of E-field(1) Ω small ⇒ Non-perturbative tunneling (Schwinger mechanism)
𝐸𝐸

(2)

𝑥𝑥

2𝑚𝑚2𝑚𝑚/𝑒𝑒𝑒𝑒gap
gap
𝑁𝑁 ∼ exp −# × height × lengh⇒ Strong exponential suppression
𝑬𝑬

(3)

𝑚𝑚2
∼ exp −𝜋𝜋𝑒𝑒𝑒𝑒

How to enhance chirality production ?
Idea: Increase the number of pairs 𝑵𝑵𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐢𝐢𝐢𝐢 𝐋𝐋𝐋𝐋𝐋𝐋 ⇒ Use fast E-field✔ Physics of particle prod. changes depending on frequency 𝛀𝛀 of E-field(1) Ω small ⇒ Non-perturbative tunneling (Schwinger mechanism)
𝐸𝐸

2𝑚𝑚𝑥𝑥

gap
gap
𝑁𝑁 ∼ exp −# × height × lengh

2𝑚𝑚/𝑒𝑒𝑒𝑒𝑚𝑚2∼ exp −𝜋𝜋𝑒𝑒𝑒𝑒⇒ Strong exponential suppression(2) Ω large ⇒ Perturbative one- (or multi-) photon pair production𝑬𝑬Ω(3)

𝑁𝑁 ∼𝑬𝑬2𝑒𝑒𝑒𝑒∝
𝑚𝑚22⇒ Weak power suppressionex) Sauter field 𝑒𝑒𝑒𝑒 𝑡𝑡 = 𝑒𝑒𝐸𝐸0 /cosh2 (Ω𝑡𝑡)
𝑁𝑁𝛾𝛾 ≪ 1𝜈𝜈 ≫ 1𝛾𝛾 ≫ 1𝜈𝜈 ≪ 1

Ω/𝑚𝑚

How to enhance chirality production ?
Idea: Increase the number of pairs 𝑵𝑵𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝐢𝐢𝐢𝐢 𝐋𝐋𝐋𝐋𝐋𝐋 ⇒ Use fast E-field✔ Physics of particle prod. changes depending on frequency 𝛀𝛀 of E-field(1) Ω small ⇒ Non-perturbative tunneling (Schwinger mechanism)
𝐸𝐸

2𝑚𝑚

𝑥𝑥gap
gap
𝑁𝑁 ∼ exp −# × height × lengh

2𝑚𝑚/𝑒𝑒𝑒𝑒𝑚𝑚2∼ exp −𝜋𝜋𝑒𝑒𝑒𝑒⇒ Strong exponential suppression(2) Ω large ⇒ Perturbative one- (or multi-) photon pair production
𝑬𝑬Ω𝑁𝑁 ∼𝑬𝑬2𝑒𝑒𝑒𝑒∝
𝑚𝑚22⇒ Weak power suppressionex) Sauter field 𝑒𝑒𝑒𝑒 𝑡𝑡 = 𝑒𝑒𝐸𝐸0 /cosh2 (Ω𝑡𝑡)
𝑁𝑁𝛾𝛾 ≪ 1𝜈𝜈 ≫ 1𝛾𝛾 ≫ 1𝜈𝜈 ≪ 1

Ω/𝑚𝑚(3) Superposition of large & small Ω’s ⇒ dynamically assisted Schwinger mech.
gapgap𝑁𝑁 ∼ exp −# × height × lengh
reduced by pert.

⇒ enhancementex) 𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 + 0.01𝐸𝐸0 cos Ω𝑡𝑡
Semi-classicalresult

Schwinger resultfor const. E-field

Exactnumerics

1. Introduction
2. Enhancement of chirality production
by dynamically assisted Schwinger mech.
3. Enhancement of chirality production
by perturbative mechanism with fast E
4. Summary

Method: Perturbation theory in the Furry picture✔ Reason:
The most successful approach to
the dynamically assisted Schwinger mech.
[Torgrimsson, Schneider, Oertel, Schutzhold (2017)][Torgrimsson, Schneider, Schutzhold (2018)][HT, (2019)] [Huang, HT (2019)]ex) 𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 + 0.01𝐸𝐸0 cos Ω𝑡𝑡
Semi-classicalresultExact
numerics

Schwinger resultfor const. E-field

Furry picture

Method: Perturbation theory in the Furry picture✔ Reason:
The most successful approach to
the dynamically assisted Schwinger mech.
[Torgrimsson, Schneider, Oertel, Schutzhold (2017)][Torgrimsson, Schneider, Schutzhold (2018)][HT, (2019)] [Huang, HT (2019)]ex) 𝐸𝐸 𝑡𝑡 = 𝐸𝐸0 + 0.01𝐸𝐸0 cos Ω𝑡𝑡
Semi-classicalresultExact
numerics

Furry picture

Schwinger resultfor const. E-field✔ Extend the E-field case to 𝑬𝑬||𝑩𝑩 case & compute chiral density� , 𝑩𝑩� with perturbative weak fast ℰ
SETUP: Parallel strong slow 𝑬𝑬
� + ℰ 𝑡𝑡
𝑬𝑬 𝑡𝑡 = 𝑬𝑬�𝑩𝑩 𝑡𝑡 = 𝑩𝑩� �STEP 1: Solve Dirac eq. under 𝑬𝑬, 𝑩𝑩 non-perturbatively, and include
effects of ℰ perturbatively
�/ − 𝑚𝑚 𝜓𝜓� = 𝒆𝒆𝓐𝓐
/ 𝜓𝜓�𝑖𝑖𝑖𝑖/ − 𝒆𝒆𝑨𝑨∞(0)𝑥𝑥 + ∫−∞ d𝑦𝑦 4 𝑆𝑆R 𝑥𝑥, 𝑦𝑦 𝒆𝒆𝓐𝓐(𝑦𝑦)𝜓𝜓� 0 𝑦𝑦 + 𝑂𝑂(|𝒆𝒆𝓐𝓐|2 )⇒ 𝜓𝜓� 𝑥𝑥 = 𝜓𝜓�STEP 2: Compute VEV of chiral density operator� 5 𝜓𝜓� vac; in = 𝑂𝑂 1 + 𝑂𝑂 𝒆𝒆𝒆𝒆 1 + 𝑂𝑂 𝒆𝒆𝒆𝒆 2 + ⋯𝐽𝐽50 ≡ lim ∫ d𝒙𝒙3 vac; in 𝜓𝜓𝛾𝛾𝑡𝑡→∞

Result: Enhancement in chirality production[HT (2020)]

w/ dynamical assistancew/o dynamical assistance

𝐽𝐽50# factor✔ Analytical formula:𝑒𝑒𝑒𝑒 𝑡𝑡𝑒𝑒𝐸𝐸�=+ 0.001 cos Ω𝑡𝑡
𝑚𝑚2𝑚𝑚2𝑒𝑒𝑒𝑒 𝑡𝑡𝑒𝑒 𝐵𝐵�= 2𝑚𝑚2𝑚𝑚

22 ∞̃� 𝐵𝐵� −𝜋𝜋𝑚𝑚2𝐽𝐽50𝑒𝑒𝐸𝐸𝑒𝑒
2𝜋𝜋 𝑚𝑚2i 𝑚𝑚2i 𝜔𝜔2ℰ(𝜔𝜔)=; 2;) ]e 𝑒𝑒𝐸𝐸� × [1 +� d 𝜔𝜔.1 𝐹𝐹�1 (1 −2���2𝜋𝜋𝑇𝑇 𝑒𝑒𝐸𝐸�22𝑉𝑉𝑉𝑉𝑒𝑒
𝐸𝐸𝑒𝑒
𝐸𝐸𝐸𝐸0⇒ Huge enhancement by the dynamical assistance !!

・ chirality production becomes free from the exponential suppression
・ enhancement becomes more significant for more massive case

1. Introduction
2. Enhancement of chirality production
by dynamically assisted Schwinger mech.
3. Enhancement of chirality production
by perturbative mechanism with fast E
4. Summary

Method: Floquet approach (High-frequency expansion)✔ Reason:Quite successful in cond-mat community to describe high-frequency periodically driven systemsex.) Floquet engineering of quantum materials

Review: [Oka, Kiramura (2019)]

Method: Floquet approach (High-frequency expansion)✔ Reason:Quite successful in cond-mat community to describe high-frequency periodically driven systemsex.) Floquet engineering of quantum materials

Review: [Oka, Kiramura (2019)]✔ Floquet approach Review: [Bukov, D’Alessio, Polkovnikov (2015)]
Assumption: Hamiltonian is periodic in time 𝐻𝐻(𝑡𝑡, 𝑥𝑥) = 𝐻𝐻(𝑡𝑡 + 𝑇𝑇, 𝑥𝑥)

STEP 1: Use Floquet theorem to the time-translation unitary operator′′𝑈𝑈 𝑡𝑡, 𝑡𝑡 ′ ; 𝑥𝑥 = e−i𝐾𝐾(𝑡𝑡 ,𝑥𝑥) e+i𝐻𝐻F (𝑥𝑥)(𝑡𝑡−𝑡𝑡 ) e+i𝐾𝐾(𝑡𝑡,𝑥𝑥)

where 𝐾𝐾 𝑡𝑡, 𝑥𝑥 = 𝐾𝐾 𝑡𝑡 + 𝑇𝑇, 𝑥𝑥 and 𝐻𝐻F (𝑥𝑥) is time independent

STEP 2: Use high-frequency expansion to perturbatively determine 𝐻𝐻F & 𝐾𝐾
From −i𝜕𝜕𝑡𝑡 𝑈𝑈 𝑡𝑡, 𝑡𝑡 ′ ; 𝑥𝑥 = 𝐻𝐻 𝑡𝑡; 𝑥𝑥 𝑈𝑈(𝑡𝑡, 𝑡𝑡 ′ ; 𝑥𝑥) ,𝜕𝜕e+i𝐾𝐾(𝑡𝑡,𝑥𝑥) −i𝐾𝐾(𝑡𝑡,𝑥𝑥)
𝐻𝐻F 𝑥𝑥 = e
𝐻𝐻 𝑡𝑡, 𝑥𝑥 e+ i𝜖𝜖e𝜕𝜕𝜕𝜕Formally assuming that the time derivative is large, i.e., 𝜕𝜕𝑡𝑡 → 𝜖𝜖 −1 𝜕𝜕𝑡𝑡 (𝜖𝜖 ≪ 1)
and then determine 𝐻𝐻F , 𝐾𝐾 perturbatively in 𝜖𝜖+i𝐾𝐾 𝑡𝑡,𝑥𝑥−i𝐾𝐾 𝑡𝑡,𝑥𝑥−1

STEP 3: Use obtained 𝑈𝑈 to compute observables

Method: Floquet approach (High-frequency expansion)✔ Reason:Quite successful in cond-mat community to describe high-frequency periodically driven systemsex.) Floquet engineering of quantum materials

Review: [Oka, Kiramura (2019)]✔ Floquet approach Review: [Bukov, D’Alessio, Polkovnikov (2015)]
Assumption: Hamiltonian is periodic in time 𝐻𝐻(𝑡𝑡, 𝑥𝑥) = 𝐻𝐻(𝑡𝑡 + 𝑇𝑇, 𝑥𝑥)STEP 1: Use Floquet theorem to the time-translation unitary operator′′

𝑈𝑈 𝑡𝑡, 𝑡𝑡 ′ ; 𝑥𝑥 = e−i𝐾𝐾(𝑡𝑡 ,𝑥𝑥) e+i𝐻𝐻F (𝑥𝑥)(𝑡𝑡−𝑡𝑡 ) e+i𝐾𝐾(𝑡𝑡,𝑥𝑥)

where 𝐾𝐾 𝑡𝑡, 𝑥𝑥 = 𝐾𝐾 𝑡𝑡 + 𝑇𝑇, 𝑥𝑥 and 𝐻𝐻F (𝑥𝑥) is time independent

STEP 2: Use high-frequency expansion to perturbatively determine 𝐻𝐻F & 𝐾𝐾
From −i𝜕𝜕𝑡𝑡 𝑈𝑈 𝑡𝑡, 𝑡𝑡 ′ ; 𝑥𝑥 = 𝐻𝐻 𝑡𝑡; 𝑥𝑥 𝑈𝑈(𝑡𝑡, 𝑡𝑡 ′ ; 𝑥𝑥) ,𝜕𝜕e+i𝐾𝐾(𝑡𝑡,𝑥𝑥) −i𝐾𝐾(𝑡𝑡,𝑥𝑥)
𝐻𝐻F 𝑥𝑥 = e
𝐻𝐻 𝑡𝑡, 𝑥𝑥 e+ i𝜖𝜖e𝜕𝜕𝜕𝜕Formally assuming that the time derivative is large, i.e., 𝜕𝜕𝑡𝑡 → 𝜖𝜖 −1 𝜕𝜕𝑡𝑡 (𝜖𝜖 ≪ 1)
and then determine 𝐻𝐻F , 𝐾𝐾 perturbatively in 𝜖𝜖+i𝐾𝐾 𝑡𝑡,𝑥𝑥−i𝐾𝐾 𝑡𝑡,𝑥𝑥−1

STEP 3: Use obtained 𝑈𝑈 to compute observables✔ In this work:−i𝜕𝜕𝑥𝑥 + 𝑒𝑒𝐴𝐴0 − 𝑒𝑒𝐴𝐴1
Apply Floquet approach to chirality prod. in QED1+1: 𝐻𝐻 𝑡𝑡, 𝑥𝑥 ≔𝑚𝑚𝑚𝑚+i𝜕𝜕𝑥𝑥 + 𝑒𝑒𝐴𝐴0 + 𝑒𝑒𝐴𝐴1

Result: Enhancement in chirality production′′′𝑡𝑡
𝑡𝑡
𝑡𝑡
𝑡𝑡
10′
3 22′′′𝐽𝐽5 𝑡𝑡, 𝑥𝑥 = 𝜖𝜖 � d𝑡𝑡 𝑒𝑒𝑒𝑒 + 𝜖𝜖 𝜕𝜕𝑥𝑥 − 2𝑚𝑚 � d𝑡𝑡 � d𝑡𝑡 � d𝑡𝑡 ′′′ 𝑒𝑒𝑒𝑒 + 𝑂𝑂(𝜖𝜖 5 )𝜋𝜋✔ Completely different parameter dependence compared to the slow E-field caseslow case:

𝐽𝐽50 𝑡𝑡, 𝑥𝑥

2𝑚𝑚
1 𝑡𝑡 ′−𝜋𝜋′|𝑒𝑒𝑒𝑒(𝑡𝑡,𝑥𝑥)|= � d𝑡𝑡 𝑒𝑒𝑒𝑒 𝑡𝑡 , 𝑥𝑥 e𝜋𝜋⇒ No exponential suppression in the fast case
⇒ Enhancement by the fast E-field !!

(Even if E is weak, it produces huge chirality as long as it is fast)1. Introduction
2. Enhancement of chirality production
by dynamically assisted Schwinger mech.
3. Enhancement of chirality production
by perturbative mechanism with fast E
4. Summary

Summary

Background

HT, Phys.Rev.Res. 2 (2020) 2, 023257 [2003.08948]
Fukushima, Hidaka, Shimazaki, HT, in preparation

Chirality production from the vacuum for massive fermions⇒ well understood for slow EM fields
・ Driven by the Schwinger mechanism
・ Exponentially suppressed ⇒ Difficult to be observed in experiments
Problem
Idea
MethodEnhancement mechanism for chirality production ?
Use time-dependent fast EM fields(1) Perturbation theory in the Furry picture(2) Floquet approach (high-frequency expansion)

Result & Message
Chirality production is significantly enhanced if EM fields are fast
⇒ worthwhile to investigate chirality-related phenomena in strong-field QED