強レーザー場中の原子 - 石川顕一

Advanced Plasma and Laser Science
プラズマ・レーザー特論E
Kenichi Ishikawa (石川顕一)
http://ishiken.free.fr/english/lecture.html
[email protected]
強レーザー場中の原子
Atom in an intense laser field
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
References 参考文献
Laser fundmentals, Rabi oscillation レーザーの基礎・原理、
ラビ振動
William T. Silfvast, “Laser Fundamentals”
(Cambridge University Press)
霜田光一「レーザー物理入門」（岩波書店）
Atom in an intense laser field
M. Protopapas, C.H. Keitel and P.L. Knight, “Atomic
physics with super-high intensity lasers”, Rep. Prog. Phys.
60, 389–486 (1997)
2
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How intense is an intense laser field?
強レーザー場とは
Intensity 強度 1013∼1015 W/cm2
Intensity at which the interaction with an atom becomes
non-perturbative 原子との相互作用が非摂動論的になり始
める強度。
Effect of laser on the electron ∼ Effect of the nucleus on
the electron
レーザー場が電子におよぼす影響 ∼ 原子核が電子におよ
ぼす影響
3
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High-field phenomena
高強度場現象
Above-threshold ionization (ATI) 超閾電離
Ionization upon which an atom absorbs more photons
than minimum necessary. 必要以上の光子を吸収してイオ
ン化する過程
Tunneling ionization トンネル電離
Ionization by the tunneling effect rather than absorption of
photons トンネル効果によるイオン化
High-harmonic generation (HHG) 高次高調波発生
Generation of harmonics of very high orders 波長変換によ
って高次の倍波が発生する現象
4
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Key concepts
キーとなる概念
Ponderomotive energy ポンデロモー
ティブエネルギー (this week)
Quantum paths (trajectories) 量子経
路 (next week)
5
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Why is high-field phenomena
fascinating? 高強度場現象の魅力
We can look at a common phenomenon
from various view points. 同じ現象を、様々
な観点からとらえることができる。
Atomic physics meets plasma physics. 原子
物理とプラズマ物理の出会うところ
6
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Single-photon ionization (photoelectric effect)
１光子電離（光電効果）
1905年 Einstein アインシュタイン
E=0
IP
!ω
基底状態
€
Ip : Ionization potential
イオン化ポテンシャル
Kinetic energy of the ejected electron
放出された電子の運動エネルギー
Eel = !ω − I p
Condition for ionization イオン化の条件
!ω > I p
€
Ionization rate イオン化レート
R
I
€
I : Light intensity 光の強度
7
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Single-photon ionization
１光子電離
ionization Ionization rate (transition probability per
電離
unit time) 単位時間当たりの遷移確率
Ip
d
2π 2 π
2
2
C2 (t) =
γ =
µ12 E02
dt
!
2!
!ω
µ ij =
ground state 基底状態
€
Eel = !ω − I p
ϕ1s = 2e− r ×
€
1
4π
2
2
2 kr −ikr
ʹ′
1
+
n
e
−2 πn ʹ′
3
1− e
3
×F(inʹ′ + 2,4,2ikr) ×
cosθ
4π
ϕεp =
3
j
iz j
€
Ionization rate
€
∗
i
∫ ϕ zϕ d r =
€
Photon energy (eV)
8
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Intensity-dependence of single-photon ionization
１光子電離の強度依存性
2 108 W/cm2
2
108 W/cm2
Ionization
∝ Intensity
線形光学効果(linear
optical effect)
€
9
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MULTIPHOTON IONIZATION
What was believed till
1970‘s. 1970年代末まで信
多光子電離
じられていたこと
E=0
!ω
IP
IP
!ω
Intensity
€
強度
Ground state
基底状態
!ω < I p
€
!ω
€
€
LOW 弱
€ necessary for ionization
Number of photons
N=
イオン化に必要な光子数
Kinetic energy of the ejected electron
放出された電子の運動エネルギー
Ionization rate イオン化レート
!ω
Ekin = N
R
IN
HIGH 強
Ip
+1
Ip
Nonlinear optical phenomena
非線形光学効果
10
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Example: 3-photon ionization 例：３光子電離
Ip
€
Ionization
Photoelectron
energy
!ω
Pulse duration
40fs
Eel = 3!ω − I p
Hydrogen atom I p = 13.6 eV
€
3
Ionization
∝ Intensity
€
n-photon ionization
€
Ionization
€
n
∝ Intensity
Peak intensity
非線形光学応答(nonlinear optical
effect)
requires a bright source →
realized only with lasers
強い光源が必要 → レーザーの出現
によって初めて実現
11
Experimental verification of the
power low of ionization rate
1965∼1975
Ionization rate
I < 1013 W/cm2
RN =
N
N
= I/
Power low confirmed for different target atoms
Xe
Hg
I
zlik=
~
7.44+ 0.77.:
-I
k
~
e
6.3+0.7
I
I
29.0
i
I
I
)
I
29.5
I
I
I
I
30.0
I
I
I
I
I
I
I
29.5
I
30.0
log Fo
FIG. 1.
Log-log
of
ion-production
Log-log
plot
of the ion-production
raterate
vs. laser
peak flux. [Chin et al, Phys. Rev. 188, 7 (1969)]
Protopapas et al., Rep. Prog. Phys. 60, 389 (1997)
12
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Discovery of above-threshold ionization (ATI)
超閾電離の発見
Pierre Agostiniら（CEA-Saclay, France フランス原子力庁サクレー研究所）
All the previous experiments only measured the total ionization yield
それまでの実験はいずれも、トータルのイオン化収量を測定していた。
Agostini et al. measured the photoelecton energy spectrum for the first time.
初めて光電子のエネルギースペクトルを測定した。
= 2.33 eV Ip (Xe) = 12.1298 eV N = 6
波長532nm
A peak of energy higher than expected for 6-photon ionization
６光子電離で予想されるより高エネルギーの位置にもピークを発見
Ekin = N
Ip = 1.86 eV
ATI
超閾電離
Another photon
absorbed after 6photon ionization?
６光子電離の後で
もう１光子吸収？
Phys. Rev. Lett. 42, 1127 (1979)
13
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A free electron cannot absorb photons
自由電子は光子を吸えない
Energy conservation
エネルギー保存
p2i
+n
2
p2f
=
2
Momentum conservation
運動量保存
pi + n k = pf
= c|k|
Solutions exist only for n = 0 → A free electron can neither absorb
nor emit photons, because the momentum cannot be conserved
解があるのは、n=0の場合だけ→運動量保存が満たされないため、
自由電子は光子を吸収も放出もできない。
Free-free transition possible only near the ion which absorbs the
momentum difference 運動量の差を吸収してくれるイオンの近傍で
のみ、free-free遷移が可能
Does a rapidly-escaping electron have time to absorb a photon?
イオンから逃げていく電子が、光子を吸う暇があるのか？
14
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Experiments with higher intensity
より高強度の実験
wavelength
波長 1064 nm
Xe gas
Kruit et al., Phys. Rev. A 28, 248 (1983)
Group of FOM (Amsterdam)のグループ
Ekin =
Minimum
最小限必要な光子数
(N + S)
Ip
MacIlrath et al., Phys. Rev. A 35, 4611 (1987)
Group of AT&T Bell Lab.のグループ
余分の光子数 Extra photons
Now certain that ATI is due to free-free transition
ATIは、free-free遷移による光子吸収であることが確実に
15
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Intensity dependence ATIの強度依存性
MacIlrath et al., Phys. Rev. A 35, 4611 (1987)
AT&Tベル研のグループ
Kruit et al., Phys. Rev. A 28, 248 (1983)
FOM (アムステルダム)のグループ
At high intensity 高強度では
Comparable peak heights → non-perturbative
吸収光子数によらず、ピークの高さが同程度→非摂動論的
低次の吸収ピークが消える（peak suppression at low orders）
16
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High-order perturbation theory 高次の摂動論
i
t
= (H0 + HI )
n
HI =
ri
e
· E(t)
i=1
or または
HI =
LENGTH FORM
(N )
f
Mi
=
j ,j ,··· ,j
N
(Ei +
pi
i=1
ne2 2
· A(t) +
A (t)
2m
VELOCITY FORM
cross section
断面積
n
e
m
=
2
2e2
0c
N
(N )
Mi f
2
unit
単位
cm2NsN-1
f
i|x|j j |x|j · · · j |x|f
Ej )(Ei + 2
Ej ) · · · (Ei + (N
1)
Ej )
17
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(N+S)-photon ionization cross section of a hydrogen atom
水素原子の(N+S)光子電離の断面積 (cm2(N+S)/WN+S/s)
Gontier and Trahin, J. Phys. B 13, 4383 (1980)
最小限必要な光子数 N
余分の光子数 S
6 (530 nm)
8 (650 nm)
10 (910 nm)
12 (1082 nm)
0
1.39×10-69
1.49×10-97
4.51×10-123
3.46×10-149
1
2.84×10-83
9.85×10-111
7.78×10-136
9.81×10-162
2
2.92×10-97
2.53×10-124
5.35×10-149
1.10×10-174
3
2.80×10-111
5.84×10-138
2.61×10-162
1.08×10-187
4
2.66×10-125
1.35×10-151
1.89×10-175
9.87×10-201
5
2.32×10-139
2.75×10-165
1.04×10-188
8.91×10-214
4.89×1013
1.51×1013
5.80×1012
3.53×1012
S=0と1が同じに
なる強度 (W/cm2)
Equal cross section for
S=0 and 2
Intensity at which the interaction becomes non-perturbative
非摂動論的になる強度の目安
longer wavelength → lower intensity
長波長ほど低強度
実験と整合
Consistent with experiments
18
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Non-perturbative? 非摂動論的？
NUCLEAR COULOMBIC FORCE
原子核からのクーロン力
a0
＝
LASER ELECTRIC FORCE
？
レーザー電界からの力
e2
4
eE
2
a
0 0
I = 3.51
1016 W/cm
2
Why non-perturbative at much lower intensity
なぜ、これよりずっと低い強度で非摂動論的になるのか？
Why non-perturbative at lower intensity for longer
wavelength なぜ、長波長ほど、低強度で非摂動論的になる
のか？
Why low-order peaks are suppressed? なぜ、低次の光電子
ピークが消えるのか？
19
From another view
point 別の観点から
見てみよう
PLASMA
プラズマ
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Charged particle in an electromagnetic wave
電磁波中の荷電粒子
1
[E0 (r, t)e
2
1
B(r, t) = [B0 (r, t)e
2
E(r, t) =
r(t) = R(t) + r(t)
Macroscopic drift
motion
マクロなドリフト運動
i t
+ c.c.] = |E0 | cos( t + )
i t
+ c.c.] = |B0 | cos( t + )
Slowly varying envelope
振動数ωにくらべてゆっくり変化（エンベロープ）
Microscopic oscillation
ミクロな振動運動（振動数ω）
r(t) = r0 e
| r0 · E0 |
i t
+ c.c.
|E0 |
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
r(t) = R(t) + r(t)
r(t) = r0 e
i t
+ c.c.
δr0のスケールでは、E0, B0はほとんど変わらない。
v(t) = V(t) + v(t)
v(t) = v0 e
i t
m ˙v = qE(r, t)
v = ˙r
E(r, t) =
B(r, t)
t
| r0 ·
E0 |
B0 |
|E0 |
|B0 |
+ c.c.
Non-relativistic electron velocity
電子の速度は非相対論的
OSCILLATION AMPLITUDE
振動運動の振幅
mass m, charge q
| r0 ·
V
v0 =
B0 =
B0
iqE0
2m
E0
r0 =
qE0
2m 2
E0
i
22
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Force acting on the charged particle
荷電粒子に作用する力
F = q[E(r(t), t) + v(t)
B(r(t), t)]
= q[E(R + r, t) + (V + v)
q[E(R, t) + r ·
B(R + r, t)]
E(R, t) + V
B(R, t) + v
q
( r0 · E0 + v0 B0 + c.c.)
F
2
q2
E0 ) + c.c.] =
=
[E0 · E0 + E0 (
2
4m
F=
Up (R, t)
B(R, t)]
q2
4m
2
|E0 |2
q 2 |E0 (R, t)|2
Up (R, t) =
4m 2
PONDEROMOTIVE POTENTIAL
(ENERGY)
ポンデロモーティブポテ
ンシャル（エネルギー）
23
All told, in an inhomogeneous electromagnetic field the Lorentz force
the
intensity [74—76].
Although this is not always explicitly recognized, th
Lorentz force must be taken into account even in the non-relativistic d
motive force may incorrectly appear to depend on the polarization of t
The ponderomotive force is clearly the negative of the gradient of a
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Science so-called
(Kenichi ISHIKAWA)
for internal
use onlythat
(Univ.points
of Tokyo)
quasi-static
ponderomotive
component
against
Ponderomotive force
ポンデロモーティブ力（動重力）
V~=4mw
e2 2
F=
Up (R, t)
q 2 |E0 (R, t)|2
Up (R, t) =
4m 2
I~I2,
which is nothing but the cycle-averaged kinetic energy in the microm
called the jitter energy W~in section 4. When an electron adiabatically le
its kinetic energy in the quiver motion is simply converted into translat
down” the potential hill, as shown in fig. 6. However, if the laser puls
potential V~collapses quickly and there will be no ponderomotive acc
These effects were demonstrated in recent experiments [77].A pulsed
region between a source of the electrons and an electron detector, as sho
electrons at the detector were consistent with the predicted effects of t
the focused laser pulse.
Ponderomotive effects on a bound electron are certainly quite com
PONDEROMOTIVE POTENTIAL
(ENERGY)
ポンデロモーティブポテ
ンシャル（エネルギー）
I (x, y)
Potential force ポテンシャル力
|E0 (R, t)|2
Proportional to the laser intensity 電磁波の強度に比例
I(R, t)
~
Independent of the sign of charge (from the beam axis to
outside)
電荷の正負によらず向きが同じ（ビームの中心から外へ）
Higher energy for lighter particles (larger effect for electrons
than for nuclei and ions) 軽い粒子ほど大きなエネルギー
A charged particle in a laser field has an energy of Up by
default. 荷電粒子は、レーザー場中にただいるだけでUpのエネ
ルギーを持っている。
I
/
I
~
__
/
I
I
I
/
/
/
_______
.~
I
/
/
~‘
380J~_
247
180
113 ~•____
J
“—
-...
47
—87
—.---____-____
,~.“
•
x
Fig. 6. Two hypothetical photoelectron trajectories under the influence of ponderomotive acceleration. If the photoelectrons were
liberated with zero velocity, the distribution would be isotropic in the
x—y plane perpendicular to the laser beam axis.
~—153 .~__-.-~~‘\......_._____
~-187
‘~—253
~353 ________________
0.2 0.4 eV
0.6 0
Fig. 7. Direct observatio
trons by a light intensit
electrons approaching a p
of changing the delay of
arrival of the electrons at
no effect as the laser and
and 113 show that the e
laser pulse and have a hig
edge. Curves 47 and —2
focus and show no elect
ponderomotive scattering
energy loss due to ponde
pulse, and curves —253
overlap again. [Private
also ref. [77].]
24
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
ミクロな視点からみた
Ponderomotive energy from a
microscopic view point
Motion of a charge particle (mass m, charge q) in an oscillating electric field
振動電界中の質量m, 電荷qの荷電粒子の運動
E(t) = E0 sin t
mv̇ = qE0 sin t
qE0
cos t + drift 並進運動
m
Energy of quiver motion (jitter motion)のエネルギー
v=
q 2 E02
1
1
q 2 E02
2
Time
average
2
2
mv =
= Up
mv =
cos t
2
2
2
4m
2
2m
時間平均
For an electron 電子の場合
e2 E02
2 2
14
Up (eV) =
=
9.337
10
I(W/cm
) (µm)
4m 2
A charged particle in a laser field has an energy of Up by default.
電子（荷電粒子）は、レーザー場中にただいるだけでUpのエネルギーを持
っている。
25
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Peak suppression due to ponderomotive shift
低次のピークがなくなるのはポンデロ
モーティブシフトの効果
Effective ionization potential = Ip+Up
実効的なイオン化ポテンシャルがIp+Upになる。
e2 E02
2 2
14
Up (eV) =
=
9.337
10
I(W/cm
) (µm)
4m 2
長波長の方が起こりやすいことも説明できる。
Lower I for longer wavelength at fixed Up
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Effective ionization potential = Ip+Up
実効的なイオン化ポテンシャルがIp+Upになる。
J.H. Eberly et a!., Above-Threshold Ionization
All told, in an inhomogeneous electromagnetic field the Lorentz force a
quasi-static so-called ponderomotive component that points against the g
intensity [74—76].
Although this is not always explicitly recognized, the
Lorentz force must be taken into account even in the non-relativistic der
motive force may incorrectly appear to depend on the polarization of th
The ponderomotive force is clearly the negative of the gradient of a p
V~=4mw
e2 2
Number of photons necessary for ionization
イオン化に必要な光子数
n
I~I2,
which is nothing but the cycle-averaged kinetic energy in the micromot
called the jitter energy W~in section 4. When an electron adiabatically lea
its kinetic energy in the quiver motion is simply converted into translatio
down” the potential hill, as shown in fig. 6. However, if the laser pulse
potential V~collapses quickly and there will be no ponderomotive accel
These effects were demonstrated in recent experiments [77].A pulsed
region between a source of the electrons and an electron detector, as show
electrons at the detector were consistent with the predicted effects of th
focused
Ip the
+Ponderomotive
Up laser pulse.
effects on a bound electron are certainly quite comp
I (x, y)
Observed electron energy 観測される電子のエネルギー
~
Ekin = [n
(Ip + Up )] + Up = n
Ip
I
/
I
~
__
/
I
I
I
/
/
/
_______
.~
I
/
/
~‘
380J~_
247
180
113 ~•____
J
“—
-......_
47
—87
—.---____-____
,~.“
•
x
Fig. 6. Two hypothetical photoelectron trajectories under the influence of ponderomotive acceleration. If the photoelectrons were
liberated with zero velocity, the distribution would be isotropic in the
x—y plane perpendicular to the laser beam axis.
~—153 .~__-.-~~‘\......_.______
~-187
‘~—253
~353 ________________
0.2 0.4 eV
0.6 0.8
Fig. 7. Direct observation
trons by a light intensity
electrons approaching a pu
27delay of th
of changing the
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Bound electrons 束縛電子の場合
Quantum mechanically, AC-stark effect
量子力学的には：ACシュタルクシフトに対応
2nd-order perturbation
theory 摂動論から
Lorentz oscillator model
e2 E02
E=
4
mẍ =
x = x0 cos t
2
n
eE0 cos t
e2
=
m( 2
e2 E02
E=
4m( 2
Negative for the ground state
基底状態では負→dipole trap
2
ni |µin |
2
2
ni
Eg
1
( )E02
4
Electric dipole polarizability
m 02 x 電気双極子分極率
=
2
0)
2
0)
e2 E02
4m 02
I
Positive for Rydberg atoms and free electrons リュー
0
<< Up
ドベリ原子・自由電子では正→ビーム中心から逃げる
e2 E02
Up
| Eg |
ER Up =
I
2
4m
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リュードベリ原子は、強レーザーパルスから逃げる
A Rydberg atom escapes from an intense laser beam
proportional to 1/m
質量に反比例
LETTERS
I
Up for the nucleus negligible
原子核へのポンデロモー
ティブ力は無視できる
Vol 461 | 29 October 2009 | doi:10.1038/nature08481
Atom pulled by an electron
原子全体は電子に働く力
LETTERS
に引っ張られる
NATURE
| Vol 461 | 29 October 2009
Acceleration of neutral atoms in strong short-pulse
laser fields
Nature 461, 1261-1264 (29 October 2009)
c
He* yield (arbitrary units)
–60
10
–40
–20
v (m s–1)
0
20
40
60
U. Eichmann1,2, T. Nubbemeyer1, H. Rottke1 & W. Sandner1,2
5
A charged particle exposed to an oscillating electric field experiences a force proportional
to the cycle-averaged intensity gradient.
0
This so-called ponderomotive force1 plays a major part in a variety
of physical situations such as Paul traps2,3 for charged particles,
electron diffraction ina strong (standing) laser fields4–6 (the
Kapitza–Dirac effect) and laser-based particle acceleration7–9.
Comparably weak forces on neutral atoms in inhomogeneous light
fields may1,000
arise from the dynamical polarization of an atom10–12;
these are physically similar to the cycle-averaged forces. Here we
observe previously
unconsidered extremely strong kinematic
100
forces on neutral atoms in short-pulse laser fields. We identify
the ponderomotive force on electrons as the driving mechanism,
leading to ultrastrong
acceleration of neutral atoms with a mag10
nitude as high as 1014 times the Earth’s gravitational acceleration, g. To our knowledge, this is by far the highest observed
acceleration on
1 neutral atoms in external fields and may lead to
new applications in both fundamental and applied physics.
The investigation has become possible through two recent findings
0.1
concerning atomic
ionization dynamics in strong laser fields. First,
neutral atoms can survive a strong laser field in a (long-lived) excited
13
state , in which they can be detected directly in an atomic beam by
means of a standard electron or ion detector14. Thus, any momentum
transferred to the neutral atom can easily be detected. Second,
according to the physical picture behind the excitation process, the
excited electron behaves as a quasi-free electron during the laser
pulse. More precisely,–14
the excitation process
can be viewed
as a fru-7
0
–7
strated tunnel ionization14 within the three-step rmodel
for strongD (mm)
field ionization15.
first step,
the electron
in the with
close vicinity
of the
Figure 1 | Deflection In
of the
neutral
He atoms
aftertunnels
interaction
a focused
maximum electric
field of
a laser
cycle.on
The
liberated
electron
is then
of excited
He*
atoms
the
detector
(colour
laser beam. a, Distribution
by the
laser
fieldbeam
with an
amplitudeisthat
slowly decreases
scale, in number ofdriven
atoms).
The
laser
direction
indicated
by the with
decreasing pulse intensity; in this way an active damping of the electhe
atom
distribution
along
the
laser
beam
axis
(z axis)
arrow. b, Cut through
tronic motion takes place. After the laser pulse the electron is left with
the detection technique (see the Methods) we measure the distriintensity
bution of excited HeRelative
atoms onlaser
a detector
as shown in Fig. 1. If, during
the laser pulse,0.0
no momentum0.5
is transferred to1.0
the atoms, we would
6
expect a slightly enlarged projected image of the (laser-intensityb
dependent) distribution of excited atoms in the laser beam on the
detector, that is, a distribution that extends along the laser beam
direction (z axis), typically within the Rayleigh 4
length, but with a
very narrow radial distribution (rD axis) of the order of the size of
the laser beam waist.
In Fig. 1a, however, we see a strikingly large radial distribution of
excited atoms with a strong maximum in the laser 2focal plane (z 5 0)
that obviously stems from a deflecting radial force during the laser
pulse. In Fig. 1b the cut along the z axis (black curve) shows two
maxima at roughly half the laser peak intensity I00/2, where the net
production rate of excited helium atoms He* is apparently maximum, whereas the He* signal at I0 shows a pronounced minimum.
However, the loss of neutral excited atoms is largely due to their
radial deflection. The full projection (red dashed –2
curve) shows only
a slight decrease in signal, indicating that even at the highest intensities He atoms are excited. The data are taken at a low beam target
pressure of ^5|10{7 mbar. The radial deflection
–4 is unchanged
when we increase the target pressure by more than a factor of 30.
This excludes many-particle effects based on atom density or space
charge as an origin of our observations. Furthermore, we emphasize
–6 linear polarizathat the radial distribution is unaltered whether the
5 direction of 10
0 beam is in the
tion of the
the atomic beam or
14laser
yield
(arbitrary
units)
perpendicular to it.He*
In this
respect
the intensity-dependent
force very
much resembles the ponderomotive force acting on charged part15
–2
The3question
we can conclude
that the ponderIicles.
10 Warises
cm whether
(blue curve).
c, Cuts through
the distribution at
0 5 6.9
responsible
observed
centre-of-mass
zomotive
5 0 mmforce
(rediscurve)
and for
z 5the
22.7
mm (black
curve).motion
The black curve shows
of the
neutraldistribution
particle.
the
velocity
of excited neutral atoms at a position unaffected by
shed light on the underlying process we first recall that the
theTo
ponderomotive
force, showing essentially the ‘natural’ velocity spread,
ponderomotive force Fp on a charged particle is given by (all equaz (mm)
ER
e2 E02
Up =
4m 2
29
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
A measure of non-perturbativeness
非摂動論的であることのめやす
PEAK SUPPRESSION 低次のピークが消える
E02
Up
Gontier and Trahin
Up
4m
e2
3
530 nm
650 nm
910 nm
1082 nm
4.89×1013
1.51×1013
5.80×1012
3.53×1012
8.9×1013
4.8×1013
1.8×1013
1.0×1013
Order of magnitude and trend consistent
オーダーと波長依存性がよく合っている。
30
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Non-perturbative? 非摂動論的？
NUCLEAR COULOMBIC FORCE
原子核からのクーロン力
a0
＝
LASER ELECTRIC FORCE
？
レーザー電界からの力
e2
4
eE
2
a
0 0
I = 3.51
1016 W/cm
2
Why non-perturbative at much lower intensity
なぜ、これよりずっと低い強度で非摂動論的になるのか？
Why non-perturbative at lower intensity for longer wavelength なぜ、長波長
ほど、低強度で非摂動論的になるのか？
Why low-order peaks are suppressed? なぜ、低次の光電子ピークが消える
のか？
Explained by the ponderomotive energy ポンデロモーテ
ィブエネルギーでよく説明できる。
31
Above-threshold ionization (ATI)
roughly at 1013~1014 W/cm2 intensity in the near-infrared (NIR)
wavelength
1064 nm
Xe gas
Kruit et al., Phys. Rev. A 28, 248 (1983)
Group of FOM (Amsterdam)
MacIlrath et al., Phys. Rev. A 35, 4611 (1987)
Group of AT&T Bell Lab.
32
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Tunneling ionization
トンネル電離
At even higher intensity (>1014 W/cm2), another mechanism
of ionization takes place.
Laser electric field
レーザー電界
Nuclear potential
V (r, t) =
原子核ポテン
シャル
e- 電子
e2 1
+ ezE(t)
4 0r
トンネル効果
Tunneling
The electron sees a field
rather than photons!
電子は、光子ではなく、電界を
感じてる！
33
Tunneling ionization
トンネル電離
レーザー電場
原子核ポテン
シャル
電子
トンネル
効果
Conditions of tunneling ionization
Tunneling rate W is high enough
3/2
Ip exp
W
4 2 Ip
3 E
Field should be sufficiently strong
Field oscillation is slow enough
Laser electric field
electron velocity
v
2Ip
barrier thickness
d
Ip /E
d/v
time scale of tunneling
time scale of laser oscillation
tun
<
osc
tun
osc
Nuclear
potential
e-
Tunneling
1/2
tun
osc
=
Ip
<1
2Up
Keldysh parameter
35
Keldysh parameter
Keldysh parameter
=
>1
1
=1
Ip
2Up
: Multi-photon regime
: Tunneling regime
Xe (Ip=12.13 eV), wavelength1064nm, about 5.7 1013 W/cm2
36
Conditions of tunneling ionization
Tunneling rate W is high enough
3/2
W
Ip exp
4 2 Ip
3 E
Don’t forget this!
Field should be sufficiently strong
Field oscillation is slow enough
=
Ip
2Up
1
: Tunneling regime
>1
: Multi-photon regime
Typical misunderstanding
terahertz radiation with 1 THz frequency and 2 MV/cm field strength
Up = 44 eV
tunneling ionization?
NO!
only 5×109 W/cm2
too weak for tunneling
37
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Change of ionization mechanism with laser intensity
レーザー強度によるイオン化の変化
Photon 光子
I >1012 W/cm2
€
€
I >1013 W/cm2
I >1014 W/cm2
€
38
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Change of ionization mechanism with laser intensity
レーザー強度によるイオン化の変化
Photon 光子
Electric Field 電界
I >1012 W/cm2
€
€
I >1013 W/cm2
I >1014 W/cm2
€
39
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
トンネル電離でも光電子ス
ペクトルは離散的
=
Ip
ケルディッシュ(Keldysh)パラメーター
2Up
> 1 : 多光子領域
1 : トンネル領域
トンネル領域
=1
Xe (Ip=12.13 eV), 波長1064nmで、5.7 1013 W/cm2程度
40
なぜ、トンネル電
離でも光電子スペ
クトルは離散的な
のか？
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
トンネル電離後の電子の経路
イオン化後は原子核（イオン）からのクーロンポテンシャルを無視（高強度場近似）
レーザー電界 E(t)
mv̇ =
mv(t) =
原子核ポテン
シャル
電子
v(t ) = 0
時刻 tr でイオン化。初速ゼロ
eE(t)
t
e
E(t) = e[A(t)
A(tr )]
tr
トンネル効果
A(t) =
E(t)dt
ベクトルポテンシャル
最終的な（観測される）電子の速度（運動量） k = mv( ) = e[A( )
1
2
Time (optical cycle)
Vector potential A(t)
-k
0
eA(tr )
Vector potential
Electric field F(t)
Electric field
A(tr )] =
同じエネルギーに
複数の経路が寄与
干渉
3
42
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
電子経路の量子力学的干渉
(i)
個々の経路 i の持つ位相
作用(action)
exp
S(t) =
V
光電子の運動量分布
dt
1+
(z, t) = exp i(k + eA(t))z
cos(2 t)
2
P (k)
3
sin(2 t) + Ip t
4
t
exp
(i)
iS(tr )
2
i
Unit cell
j=2
Vector potential
j=3
-k
0 t(1,1) t(2,1)1
2
r
r
Intracycle
Time (optical cycle)
3
Vector potential A(t)
Electric field F(t)
j=1
Electric field
iS(t)
(k + eA(t ))2
+ Ip
2m
t
dtL =
= 2Up
Volkov波動関数
iS(tr )
cos2
S
2
サイクル内干渉
exp
(1j)
iS(tr )
2
j
サイクル間干渉
S = S(t(2,1)
)
r
S(t(1,1)
)
r
interference
43
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
サイクル内干渉とサイクル間干渉
(i)
P (k)
iS(tr )
exp
2
cos2
i
Unit cell
Electric field
j=2
interference
3
Vector potential A(t)
-k
0 t(1,1) t(2,1)1
2
r
r
Intracycle
Time (optical cycle)
exp
j
Vector potential
j=3
Electric field F(t)
j=1
S
2
2
(1j)
iS(tr )
サイクル内干渉 サイクル間干渉
S = S(t(2,1)
)
r
S(t(1,1)
)
r
(1,j)
= t(1,1)
+
サイクル間干渉 tr
r
2
(j
1)
レーザー電界の周期ごと
k2
2
= S(tr + 2 / ) S(tr ) =
+ Up + Ip
2m
(1j)
exp
j
iS(tr
)
1 + exp(i / ) + exp(2i / ) + · · ·
=2 n
ピーク（干渉が強め合う）の条件
k2
+ Up + Ip = n
2m
光電子の運動エネルギー
Ekin = n
Ip
整数
Up
44
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Ekin = n
Unit cell
j=2
Vector potential
0 t(1,1) t(2,1)1
2
r
r
Time (optical cycle)
3
Vector potential A(t)
-k
Intracycle
interference
Up
光電子スペクトルの離散的な
j=3
Electric field F(t)
j=1
Electric field
Ip
ピーク
電子経路のサイクル間干渉に
よる
トンネル電離が、レーザー
場の周期で起こるため
ポンデロモーティブシフトが
自然に出てくる
45
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
まとめ
強度 1013∼1015 W/cm2のレーザー場中のイオン化
超閾電離(Above-threshold ionization, ATI)
必要以上の光子を吸収してイオン化する過程（光子
の観点）
トンネル電離
トンネル効果によるイオン化（電磁波の観点）
光電子スペクトルは離散的なピークからなる
free-free遷移による光子の吸収（原子物理の観点）
トンネル電離で周期的に出てくる電子の干渉
46
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
まとめ
Vector potential
-k
0
1
2
Time (optical cycle)
Vector potential A(t)
Electric field F(t)
Electric field
3
ポンデロモーティブエネルギーが重要なパラメーター
Up∼Ipが「高強度レーザー場」のめやす
プラズマ物理の観点
レーザー場中での電子の運動を考えるのが有用
古典的な運動経路＋量子力学的な位相
Phys. Rev. Lett. 71, 1994 (1993)
47