arXiv:2303.13169

Fourier-limited attosecond pulse
generation from lightwave-driven
electrons in solids
光電場駆動された固体電子によるFourier限界アト秒パルス発生

Shohei Imai, Department of Physics, the University of Tokyo
今井渉平 (東大理)共同研究者：小野淳 (東北大理)

TQFT, August 28, 2023

Attosecond science and noneq. dynamics
Intense laser pulse

High harmonic generation (HHG)
Eg) Ne gas: 20th to 100th harmonics
J. Macklin et al., PRL 70, 766 (1993).

~100 as M. Hentschel et al.,Nature 414, 509 (2001).

Gas or solid

Strong-field induced
nonequilibrium dynamics
Presenter: S. Imai (UTokyo)

Attosecond pulse generation

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

1/14

Attosecond world
Time scale

Atto sec.

Electron dynamics in atomic scale

Femto sec.
Coherent dynamics in solids

Hydrogen atom

Orbital period ~ 160 as
Spin echo, Wikipedia

Neutral pion decaying

Relaxation of typical excited states ~ 10 fs

Lifetime ~ 85 as

Presenter: S. Imai (UTokyo)

Observation of many-bodydynamics before dephasing!

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

2/14

Attosecond science in solid materials
Time-profile of lightIntense THz laser

GaSeIncident light~ 7 fsEmission

M. Hohenleutner et al., Nature 523, 572 (2015).

Uncertainty relation~ 1 eV in typical solids

Presenter: S. Imai (UTokyo)

The only example: SiO2 472 as

M. Garg et al., Nature 538, 359 (2016).

Need for Fourier-limited pulse generation☐ Controlling the wave packet dynamics of electrons in solids
arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

3/14

Emission from wave packet dynamics
3-step model P. B. Corkum, PRL 71, 1994 (1993).
Excitation

Acceleration Recombination

Incident lightReversed wave (echo) emission
Incident light

Generated light

Generated light

Strong-field excitation (tunneling)→ Weak-field excitation

Echo generation from lightwavedriven electron in solids
SI, A. Ono, and S. Ishihara,
Phys. Rev. Res. 4, 043155 (2022).

Excitation

Driving

Recombination

Lightwave-driven dynamics of tunnel electron wave packets
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

4/14

Quantum tunneling by strong electric field
Landau–Zener tunneling (dielectric breakdown)
Schwinger effectTunneling probability

Presenter: S. Imai (UTokyo)

Need to clarify the transient dynamics
of tunnel probability amplitudes!
arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

5/14

Purpose
Attosecond pulse generation in solid materials☐ Controlling the wave packet dynamics of tunnel electrons
Excitation + Driving☐ What is the wave packet after tunneling?
Massive Dirac model in Sauter potential
Conduction band

Valence band

☐ How is the wave packet driven?
Sign change of effective mass

DiracBand insulatorSystematic optimization of driving pulse waveforms achieves Fourier-limited pulse.
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

6/14

Ansatz of tunneling electron wave packets
Tunneling occurs near
the gap minimum.

Conduction band

Electric field

DiracBand ins.

Electron–hole wavefunction

Vector potential

Valence band
N. B. Narozhnyi and A. I. Nikishov,Yadem. Fiz. 11: 1072-7 (1970).

Tunneling process and dispersion relation of Bloch electrons are considered.
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

7/14

Optimization of driving-pulse waveform
1. Assume an initial waveform of an external field

2. Fit the Sauter potential to 𝐸(𝜏)’s peak

Sauter potential

3. Calculate the wavefunction of tunneling electron–hole pairs

4. Modify the waveform to make the phase of the wavefunction constant in 𝑘by Gauss–Newton method
The analytical expression of the wavefunction facilitates the systematic optimization.
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

8/14

Model & Method
One-dimensional two-orbital tight-binding model

Energy band structure𝑒−Transfer integral

Dirac parameters

Onsite potentialℎ+Time-dependent external field
Peierls substitution

Vector potential

Excitation

Electric current operator

Driving

Numerical methods

Diagonalization (Wave number points 𝑁 = 1000)
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

9/14

Time profile of the optimized and emitted pulses
Optimized electric field

Electric current

Energy gap
Band widthAnsatz
Simulation
645 asTime
Our ansatz describes the transient state of tunneling electrons under optical driving.
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

10/14

Sensitivity of pulse width
FWHM of electric currents

Deviation

Fourier limit

Duration approaches Fourier limit.
Presenter: S. Imai (UTokyo)

Allowable error can be estimated.

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

11/14

Mechanism of pulse compression
Wave packet dynamics of particles with finite mass

Optical driving ofBloch electrons

Wave packet, WikipediaDynamic signchange of mass

Dynamic mass-sign change allows for the refocus of wave packets.
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

12/14

Real-space wave packet dynamics

Relative displacement

Central position

Refocusing of the electron wave packet can be achieved by optimal driving.
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

13/14

Summary☑ What is the wave packet of tunneling electrons?
Our ansatz reveals the waveform (wavefunction)of tunneling electrons.

Dynamic sign change of the mass shortens
the emitted pulse duration.

Electric current☑ How is the wave packet driven?

Optimized electric field
AnsatzSimulation

645 asTime☑ What is the optimal pulse waveform for attosecond pulses?
Systematic optimization of driving pulse waveforms achieves Fourier-limited pulse.
arXiv:2303.13169
Presenter: S. Imai (UTokyo)

arXiv:2303.13169. E-mail: shohei.imai@phys.s.u-tokyo.ac.jp

14/14