強レーザー場中の原子

強レーザー場中の原子

Atom in an intense laser field

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]

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)

How intense is an intense laser field?

強レーザー場とは

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

Intensity 強度 10 ¹³ 〜 10 ¹⁵ W/cm ²

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

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

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)

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

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

１光子電離（光電効果）

I

基底状態

E = 0

€

! ω

€

E

= ! ω − I

1905

I

: Ionization potential

Condition for ionization

€

! ω > I

Ionization rate

R I

I : Light intensity

Single-photon ionization (photoelectric effect)

Kinetic energy of the ejected electron

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

１光子電離

I

ground state 基底状態 ionization

電離

€

! ω

€

E

= ! ω − I

€

d

dt C

(t )

= 2 π

! γ

= π

2 ! µ

E

€

µ

= ∫ ϕ

^z ϕ

^d

^{r = i z j}

Ionization rate (transition probability per  unit time) 単位時間当たりの遷移確率

€

ϕ

= 2e

× 1 4 π ϕ

= 2

1 − e

1

+ n ʹ′

kr 3 e

× F (i n ʹ′ + 2,4,2ikr) × 3

4 π ^cos θ

Photon energy 

(eV)

Ionization rate

Single-photon ionization 

8

１光子電離の強度依存性

10⁸ W/cm²

2 10⁸ W/cm² 2

Ionization

€

∝ ^Intensity

optical effect)

Intensity-dependence of single-photon ionization

9

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

多光子電離

I

€

! ω

€

! ω

€

! ω I

基底状態

E = 0

€

! ω

€

! ω < I

強度

イオン化に必要な光子数

放出された電子の運動エネルギー

R I

N = I

+ 1 E

= N I

What was believed till 1970‘s. 1970

非線形光学効果 MULTIPHOTON IONIZATION

Intensity

Ground state

Kinetic energy of the ejected electron Ionization rate

Number of photons necessary for ionization

Nonlinear optical phenomena

例：３光子電離

I

€

! ω

Hydrogen atom

€

I

= 13.6 eV

非線形光学応答(nonlinear optical  effect)

Ionization

€

∝ ^Intensity

3

n-photon ionization Ionization

€

∝ ^Intensity

n

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

Pulse duration  40fs

Example: 3-photon ionization

11 Peak intensity

Io n iza ti o n

Photoelectron  energy

€

E

= 3 ! ω − I

Power low confirmed for different target atoms

1965 〜 1975 I < 10 ¹³ W/cm ²

R

=

= I/

Experimental verification of the power low of ionization rate

Ionization rate

Xe Hg

I

zli-

k= ^~

7.44+ 0.77.: ^~ ke6.3+0.7

-I I i I I ) I I I I II I I I I I I I I

29.0 29.5 30.0 29.5 30.0 log Fo

FIG.

1.

Log-log plot of the ion-production rate vs. laser

peak flux. [Chin et al, Phys. Rev. 188, 7 (1969)] Protopapas et al., Rep. Prog. Phys. 60, 389 (1997)

超閾電離の発見

All the previous experiments only measured the total ionization yield

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。

Pierre Agostini

CEA-Saclay, France

Phys. Rev. Lett. 42, 1127 (1979)

超閾電離

波長

532nm = 2.33 eV I

(Xe) = 12.1298 eV N = 6

E

= N I

= 1.86 eV

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

Another photon absorbed after 6- photon ionization?

６光子電離の後で もう１光子吸収？

Discovery of above-threshold ionization (ATI)

ATI

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

自由電子は光子を吸えない

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

→

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

のみ、

free-free

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？

エネルギー保存 運動量保存

p

2 + n = p

2

= c | k | p

+ n k = p

A free electron cannot absorb photons

Energy conservation

Momentum conservation

より高強度の実験

Now certain that ATI is due to free-free transition

ATI

free-free

Kruit et al., Phys. Rev. A 28, 248 (1983) Group of FOM (Amsterdam)のグループ

MacIlrath et al., Phys. Rev. A 35, 4611 (1987) Group of AT&T Bell Lab.のグループ

E

= (N + S ) I

波長

1064 nm Xe gas

最小限必要な光子数 余分の光子数

Experiments with higher intensity

wavelength

Minimum

Extra photons

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

ATI の強度依存性

Comparable peak heights → non-perturbative

吸収光子数によらず、ピークの高さが同程度

→

低次の吸収ピークが消える（

peak suppression at low orders

Kruit et al., Phys. Rev. A 28, 248 (1983)

FOM (アムステルダム)のグループ

MacIlrath et al., Phys. Rev. A 35, 4611 (1987) AT&Tベル研のグループ

At high intensity

Intensity dependence

高次の摂動論

i t = (H

+ H

)

H

= e

n

i=1

r

· E(t)

H

= e m

n

i=1

p

· A(t) + ne

2m A

(t)

LENGTH FORM VELOCITY FORM

N

= 2 2e

0

c

N

f

M

断面積

cm ^2N s ^N-1

M

=

j ,j ,···,j

i | x | j j | x | j · · · j | x | f

(E

+ E

)(E

+ 2 E

) · · · (E

+ (N 1) E

)

High-order perturbation theory

cross section

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

水素原子の (N+S) 光子電離の断面積 (cm ^2(N+S) /W ^N+S /s)

最小限必要な光子数 N 最小限必要な光子数 N 最小限必要な光子数 N 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

Gontier and Trahin, J. Phys. B 13, 4383 (1980)

S=0と1が同じに

なる強度 (W/cm²) 4.89×10¹³ 1.51×10¹³ 5.80×10¹² 3.53×10¹²

非摂動論的になる強度の目安

長波長ほど低強度 ^{実験と整合}

(N+S)-photon ionization cross section of a hydrogen atom

Equal cross section for

S=0 and 2

Intensity at which the interaction becomes non-perturbative

longer wavelength → lower intensity

Consistent with experiments

非摂動論的？

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？

NUCLEAR COULOMBIC FORCE 原子核からのクーロン力

LASER ELECTRIC FORCE レーザー電界からの力

＝

a 0 e ² eE

4 ₀ a ² ₀

I = 3.51 10 ¹⁶ W/cm ²

？

Non-perturbative?

From another view point 別の観点から 見てみよう

PLASMA

プラズマ

電磁波中の荷電粒子

E(r, t) = 1

2 [E

(r, t)e

+ c.c.] = | E

| cos( t + ) B(r, t) = 1

2 [B

(r, t)e

+ c.c.] = | B

| cos( t + ) r(t) = R(t) + r(t)

Macroscopic drift motion

マクロなドリフト運動

Microscopic oscillation

ω

Slowly varying envelope

振動数

ω

r(t) = r

e

+ c.c.

| r

· E

| | E

|

Charged particle in an electromagnetic wave

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

r(t) = R(t) + r(t)

δr

E

, B

r(t) = r

e

+ c.c.

| r

· E

| | E

|

| r

· B

| | B

| v(t) = V(t) + v(t)

Non-relativistic electron velocity

電子の速度は非相対論的

V B

E

OSCILLATION AMPLITUDE

mass m, charge q

v(t) = v

e

+ c.c.

m ˙ v = qE(r, t)

v

= iqE

2m

v = ˙ r ^r

⁼

qE

2m

E(r, t) = B(r, t)

t ^B

⁼

E

i

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) B(R + r, t)]

q [E(R, t) + r · E(R, t) + V B(R, t) + v B(R, t)]

F q

2 ( r

· E

+ v

B

+ c.c.)

= q

4m

[E

· E

+ E

( E

) + c.c.] = q

4m

| E

|

F = U

(R, t) U

(R, t) = q

| E

(R, t) |

4m

PONDEROMOTIVE POTENTIAL (ENERGY)

ポンデロモーティブポテ ンシャル（エネルギー）

Force acting on the charged particle

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

ポンデロモーティブ力（動重力）

Potential force

Proportional to the laser intensity

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)

A charged particle in a laser field has an energy of U

by

default.

U

ルギーを持っている。

F = U

(R, t) U

(R, t) = q

| E

(R, t) |

4m

J.H. Eberly et a!., Above-Threshold Ionization 351

All told, in an inhomogeneous electromagnetic field the Lorentz force acting on a free electron has a quasi-static so-called ponderomotive component that points against the gradient of the cycle-averaged intensity [74—76].Although this is not always explicitly recognized, the magnetic component of the Lorentz force must be taken into account even in the non-relativistic derivation, or else the pondero- motive force may incorrectly appear to depend on the polarization of the field.

The ponderomotive force is clearly the negative of the gradient of a ponderomotive potential

V~= e2 ² I~I2, (6.6)

4mw

which is nothing but the cycle-averaged kinetic energy in the micromotion. This same quantity was called the jitter energyW~in section 4. When an electron adiabatically leaves a steady laser beam, all of its kinetic energy in the quiver motion is simply converted into translational kinetic energy by “sliding down” the potential hill, as shown in fig. 6. However, if the laser pulse duration is short enough, the potential V~collapses quickly and there will be no ponderomotive acceleration.

These effects were demonstrated in recent experiments [77].A pulsed laser beam was focused in the region between a source of the electrons and an electron detector, as shown in fig. 7. The arrivals of the electrons at the detector were consistent with the predicted effects of the ponderomotive potential of the focused laser pulse.

Ponderomotive effects on a bound electron are certainly quite complicated and have never been I (x, y)

380J~_

247 FocusJTh

~ ~ 180 ^{J “—}

113 ^{~•____ -..._ — - ~} ~t.

I .~ 47

/

__ ^/

I / ^{~‘ —87} —.---____-____

/ / ^,~.“

I ^_______ I ~—153.~__-.-~~‘\..._.______

I / ~-187

I / ‘~—253

~353 ________________

0.2 0.4 0.6 0.8

• eV

x

Fig. 6. Two hypothetical photoelectron trajectories under the in- Fig. 7. Direct observation of ponderomotive scattering of free dcc- fluence of ponderomotive acceleration. If the photoelectrons were trons by a light intensity gradient. The drawing shows a pulse of liberated with zero velocity, the distribution would be isotropic in the electrons approaching a pulsed-laser focus. The data show the effect x—yplane perpendicular to the laser beam axis. of changing the delay of the arrival of the laser pulse relative to the arrival of the electrons at the focus. Curves labeled 380 and 247 show no effect as the laser and electron pulses miss each other. Curves180 and 113 show that the electrons are beginning to overlap with the laser pulse and have a higher energy due to acceleration by its leading edge. Curves 47 and —20 correspond to closely timed arrival at the focus and show no electrons arriving at the detector due to strong ponderomotive scattering. The curves —87 to —187 show electron energy loss due to ponderomotive deceleration in the tail of the laser pulse, and curves —253 and —353 correspond to no electron-laser overlap again. [Private communication from PH. Bucksbaum; see also ref. [77].]

|E₀(R, t)|² I(R, t)

Ponderomotive force

PONDEROMOTIVE POTENTIAL (ENERGY)

ポンデロモーティブポテ ンシャル（エネルギー）

Ponderomotive energy from a microscopic view point

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

U

ミクロな視点からみた

Motion of a charge particle (mass m, charge q) in an oscillating electric field

m,

q

+ drift 並進運動

Energy of quiver motion (jitter motion)

Time average

時間平均

1

2 mv

= q

E

4m

= U

For an electron

U

(eV) = e

E

4m

= 9.337 10

I (W/cm

)

(µm) E (t) = E

sin t

m v ˙ = qE

sin t v = qE

m cos t

1

2 mv

= q

E

2m

cos

t

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

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

U

(eV) = e

E

4m

= 9.337 10

I (W/cm

)

(µm)

I

+U

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 U

Peak suppression due to ponderomotive shift

Effective ionization potential = I

+U

実効的なイオン化ポテンシャルが

I

+U

J.H. Eberly et a!., Above-Threshold Ionization 351

All told, in an inhomogeneous electromagnetic field the Lorentz force acting on a free electron has a quasi-static so-called ponderomotive component that points against the gradient of the cycle-averaged intensity [74—76].Although this is not always explicitly recognized, the magnetic component of the Lorentz force must be taken into account even in the non-relativistic derivation, or else the pondero- motive force may incorrectly appear to depend on the polarization of the field.

The ponderomotive force is clearly the negative of the gradient of a ponderomotive potential

V~= e2 ² I~I2, (6.6)

4mw

which is nothing but the cycle-averaged kinetic energy in the micromotion. This same quantity was called the jitter energyW~in section 4. When an electron adiabatically leaves a steady laser beam, all of its kinetic energy in the quiver motion is simply converted into translational kinetic energy by “sliding down” the potential hill, as shown in fig. 6. However, if the laser pulse duration is short enough, the potential V~collapses quickly and there will be no ponderomotive acceleration.

These effects were demonstrated in recent experiments [77].A pulsed laser beam was focused in the region between a source of the electrons and an electron detector, as shown in fig. 7. The arrivals of the electrons at the detector were consistent with the predicted effects of the ponderomotive potential of the focused laser pulse.

Ponderomotive effects on a bound electron are certainly quite complicated and have never been I (x, y)

380J~_

247 FocusJTh

~ ~ 180 ^{J “—}

113 ^{~•____ -..._ — - ~} ~t.

I .~ 47

/

__ ^/

I / ^{~‘ —87} —.---____-____

/ / ^,~.“

I ^_______ I ~—153.~__-.-~~‘\..._.______

I / ~-187

I / ‘~—253

~353 ________________

0.2 0.4 0.6 0.8

• eV

x

Fig. 6. Two hypothetical photoelectron trajectories under the in- Fig. 7. Direct observation of ponderomotive scattering of free dcc- fluence of ponderomotive acceleration. If the photoelectrons were trons by a light intensity gradient. The drawing shows a pulse of liberated with zero velocity, the distribution would be isotropic in the electrons approaching a pulsed-laser focus. The data show the effect x—yplane perpendicular to the laser beam axis. of changing the delay of the arrival of the laser pulse relative to the arrival of the electrons at the focus. Curves labeled 380 and 247 show no effect as the laser and electron pulses miss each other. Curves180 and 113 show that the electrons are beginning to overlap with the laser pulse and have a higher energy due to acceleration by its leading edge. Curves 47 and —20 correspond to closely timed arrival at the focus and show no electrons arriving at the detector due to strong ponderomotive scattering. The curves —87 to —187 show electron energy loss due to ponderomotive deceleration in the tail of the laser pulse, and curves —253 and —353 correspond to no electron-laser

Number of photons necessary for ionization

E

= [n (I

+ U

)] + U

= n I

Observed electron energy

n I

+ U

27

Effective ionization potential = I

+U

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

Bound electrons 束縛電子の場合

Negative for the ground state

→dipole trap

Positive for Rydberg atoms and free electrons

→

Lorentz oscillator model m x ¨ = eE

cos t m

x

x = x

cos t

Quantum mechanically, AC-stark effect

量子力学的には： AC シュタルクシフトに対応

E = e

E

4

2

| µ

|

2 2

ni

= 1

4 ( )E

2nd-order perturbation

theory

Electric dipole polarizability

E

e

E

4m

I

E

U

= e

E

4m

I U

| E

|

<< U

= e

m(

) E = e

E

4m(

)

リュードベリ原子は、強レーザーパルスから逃げる

online-only Methods). Recalling the key point of our investigation, that the electron remains bound after interaction with the laser pulse, we are able to observe the centre-of-mass motion. Furthermore, by solving the coupled Lorentz equations for the electron and the ion, including the Coulomb potential, we can directly reproduce the cap- ture process into bound Rydberg orbits and the force on the centre of mass.

According to our model we can rewrite equation (1) for the centre- of-mass positionRof the neutral atom:

MRR€ð Þt ~{ 1

4mev²+j jE0² ð2Þ Here,Mandme(51 atomic unit) are the masses of the atom and the electron, respectively, andRR€ð Þt is the second derivative of the centre- of-mass positionRwith respect to time. To calculate the pondero- motive force explicitly, we assume a linearly polarized laser beam with a Gaussian spatial intensity distribution, which reads, in cylin- drical coordinates:

Ið Þr ~jE0ð Þrj²~I0 1z z z0

! "2!{1

exp{2r²

r₀² ð3Þ where r0~w0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1zðz=z0Þ² q

, w₀ is the beam waist. Evaluating the gradient in equation (2) with the intensity distribution given by equation (3), we obtain, for the radial component of the centre-of- mass position perpendicular to the laser beam direction:

€

rr tð Þ~Ið ÞR Mv²

r tð Þ

r₀² f tð Þ ð4Þ wheref(t) is the laser pulse envelope, which we assume to be of the formf(t)5exp(–t²/t²), wheretis the pulse width. From equation (4)

we find that the maximum force along the radial direction scales as r₀^{1. Similarly, one can show that it scales asz^{1₀ along the laser beam direction. Because the Rayleigh lengthz0is typically a factor of 100 larger than the beam waistr₀, the gradient and thus the ponderomo- tive force in the laser beam direction is much smaller than in the radial direction and can be neglected. (However, the situation would be very different if we used a short-pulse standing-wave laser field.

We would then obtain a strong periodic intensity gradient on the scale of the laser wavelength, and might expect to see the Kapitza–

Dirac effect for neutral atoms in an intense standing-wave laser field instead of electrons¹⁹).

To solve equation (4), we assume that the neutral atom does not move significantly during the laser pulse. Hence, we setr(t)5ron the right-hand side of the equation, which allows us to solve equation (4) analytically for any initial position of an atom in the laser beam. We will concentrate our analysis on atoms located at the half beam size r₀/2, which experience the maximum force. Solving equation (4) for these conditions by integrating over the full laser pulse, we find the maximum velocityv_max(z):

vmaxð Þz ~ I0

2Mv²w0

ffiffiffip

p expð{0:5Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1z _z^z

0

$ %2

r 3 t ð5Þ

If we evaluate equation (5) at the focal plane for He atoms exposed to our focused laser beam at maximum intensity, we obtain a velocity of about 55 m s²¹from which accelerations of about 2310¹⁴gcan be deduced.

This exceeds the typical acceleration (deceleration) of neutral atoms²⁰or molecules in external fields^21,22. Compared to laser-cooling experiments in a continuous-wave laser field, for instance, which are

1,000 0

–60 –40 –20 0

v (m s^–1)

20 40 60

0.0 0.5

b a

c

Relative laser intensity 1.0

0 –6

–4 –2 0 2 4 6

14 7

0 r_D (mm)

z (mm)

–7

–14 5

He* yield (arbitrary units) 10 5

10

100 10 1 0.1

He* yield (arbitrary units)

Figure 1|Deflection of neutral He atoms after interaction with a focused laser beam. a, Distribution of excited He*atoms on the detector (colour scale, in number of atoms). The laser beam direction is indicated by the arrow.b, Cut through the atom distribution along the laser beam axis (zaxis) atr_D50 mm (black curve) and full projection onzaxis (dashed red curve) and intensity along thezaxis in units of the laser peak intensity

I056.9310¹⁵W cm^–2(blue curve).c, Cuts through the distribution at z50 mm (red curve) andz5 22.7 mm (black curve). The black curve shows the velocity distribution of excited neutral atoms at a position unaffected by the ponderomotive force, showing essentially the ‘natural’ velocity spread, while the red curve shows the velocity gain through the ponderomotive force.

LETTERS NATURE|Vol 461|29 October 2009

E

U

= e

E

4m

I

質量に反比例 原子核へのポンデロモー ティブ力は無視できる

原子全体は電子に働く力 に引っ張られる

LETTERS

Acceleration of neutral atoms in strong short-pulse laser fields

U. Eichmann^1,2, T. Nubbemeyer¹, H. Rottke¹& W. Sandner^1,2

A charged particle exposed to an oscillating electric field experi- ences a force proportional to the cycle-averaged intensity gradient.

This so-called ponderomotive force¹plays a major part in a variety of physical situations such as Paul traps^2,3for charged particles, electron diffraction in strong (standing) laser fields^4–6 (the Kapitza–Dirac effect) and laser-based particle acceleration^7–9. Comparably weak forces on neutral atoms in inhomogeneous light fields may arise from the dynamical polarization of an atom^10–12; these are physically similar to the cycle-averaged forces. Here we observe previously unconsidered extremely strong kinematic 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 mag- nitude as high as 10¹⁴times the Earth’s gravitational accelera- tion,g. To our knowledge, this is by far the highest observed acceleration on 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 concerning atomic ionization dynamics in strong laser fields. First, neutral atoms can survive a strong laser field in a (long-lived) excited state¹³, 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, the excitation process can be viewed as a fru- strated tunnel ionization¹⁴within the three-step model for strong- field ionization¹⁵.

In the first step, the electron tunnels in the close vicinity of the maximum electric field of a laser cycle. The liberated electron is then driven by the laser field with an amplitude that slowly decreases with decreasing pulse intensity; in this way an active damping of the elec- tronic motion takes place. After the laser pulse the electron is left with a drift energy too low to overcome the Coulomb potential of the ion and is recaptured into a Rydberg state. The quivering quasi-free electron experiences the ponderomotive force during the laser pulse owing to the intensity gradient in the focused laser beam. We will show here that the quiver motion of the electron is partially converted into centre-of-mass motion of the neutral atom, leading to a sub- stantial acceleration. This results in a measurable momentum trans- fer to the atom despite the short interaction time in the femtosecond range. Remarkably, the ponderomotive effect is typically estimated to be negligible for these conditions^16,17with, however, a few excep- tions¹⁸. We note that the investigation relies on the highly selective process of excitation of neutrals in a strong laser field, where kin- ematic effects are imparted only through the gradient of the laser field.

In the experiment we excite neutral He atoms in an effusive atomic beam using a perpendicularly intersecting focused laser beam. Using

the detection technique (see the Methods) we measure the distri- bution of excited He atoms on a detector as shown in Fig. 1. If, during the laser pulse, no momentum is transferred to the atoms, we would expect a slightly enlarged projected image of the (laser-intensity- dependent) distribution of excited atoms in the laser beam on the detector, that is, a distribution that extends along the laser beam direction (zaxis), typically within the Rayleigh length, but with a very narrow radial distribution (rDaxis) 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 focal plane (z50) that obviously stems from a deflecting radial force during the laser pulse. In Fig. 1b the cut along thezaxis (black curve) shows two maxima at roughly half the laser peak intensityI0/2, where the net production rate of excited helium atoms He*is apparently maxi- mum, whereas the He*signal atI0shows a pronounced minimum.

However, the loss of neutral excited atoms is largely due to their radial deflection. The full projection (red dashed curve) shows only a slight decrease in signal, indicating that even at the highest inten- sities He atoms are excited. The data are taken at a low beam target pressure of^5|10^{7mbar. The radial deflection 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 that the radial distribution is unaltered whether the linear polariza- tion of the laser beam is in the direction of the atomic beam or perpendicular to it. In this respect the intensity-dependent force very much resembles the ponderomotive force acting on charged part- icles. The question arises whether we can conclude that the ponder- omotive force is responsible for the observed centre-of-mass motion of the neutral particle.

To shed light on the underlying process we first recall that the ponderomotive forceFpon a charged particle is given by (all equa- tions are in atomic units):

Fp~{ q²

4mv²+j jE0² ð1Þ

Here,mandqare the mass and the charge of the particle, respectively, E(r,t)5E0(r,t)expivtis the electric field,vis the field frequency andE0(r,t) is the slowly varying field amplitude. Hence, in view of our frustrated tunnel ionization model, both the ionic core and the electron experience a mass-dependent ponderomotive force during the laser pulse. As a consequence of the mass dependency, however, the ionic core remains practically unaffected while the electron experiences a non-negligible ponderomotive force. This, in turn, means that to the first approximation a ponderomotive force acts directly on the centre-of-mass motion of the atom and leaves the recapturing process unaffected. This can be shown more rigorously:

we derive the centre-of-mass motion from the Lorentz force (see the

1Max-Born-Institute, Max-Born-Strasse 2a, 12489 Berlin, Germany.²Institut fu¨r Optik und Atomare Physik, Technische Universita¨t Berlin, 10632 Berlin, Germany.

Vol 461|29 October 2009|doi:10.1038/nature08481

1261 Macmillan Publishers Limited. All rights reserved

©2009

Nature 461, 1261-1264 (29 October 2009)

29

A Rydberg atom escapes from an intense laser beam

proportional to 1/m U

for the nucleus negligible Atom pulled by an electron

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

非摂動論的であることのめやす

Order of magnitude and trend consistent オーダーと波長依存性がよく合っている。

PEAK SUPPRESSION 低次のピークが消える

U _p E ₀ ² 4m ³

e ²

Up

530 nm 650 nm 910 nm 1082 nm

Gontier and Trahin 4.89×10¹³ 1.51×10¹³ 5.80×10¹² 3.53×10¹²

8.9×10¹³ 4.8×10¹³ 1.8×10¹³ 1.0×10¹³

A measure of non-perturbativeness

非摂動論的？

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

Why low-order peaks are suppressed?

NUCLEAR COULOMBIC FORCE 原子核からのクーロン力

LASER ELECTRIC FORCE レーザー電界からの力

＝

a 0 e ² eE

4 ₀ a ² ₀

I = 3.51 10 ¹⁶ W/cm ²

？

Non-perturbative?

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。

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.

wavelength 1064 nm

Xe gas

Above-threshold ionization (ATI)

roughly at 10 ¹³ ~10 ¹⁴ W/cm ² intensity in the near-infrared (NIR)

トンネル電離

電子は、光子ではなく、電界を 感じてる！

レーザー電界

原子核ポテン シャル 電子

トンネル効果

V (r, t) = e ² 4 ₀

1

r + ezE (t)

Tunneling ionization

The electron sees a field  rather than photons!

Laser electric field

Nuclear potential e

Laser electric field

Tunneling

At even higher intensity (>10 ¹⁴ W/cm ² ), another mechanism

of ionization takes place.

トンネル電離

レーザー電場

電子 トンネル

効果

原子核ポテン シャル

Tunneling ionization

Conditions of tunneling ionization

Tunneling rate W is high enough

W I

exp 4 2 3

I

E Field should be sufficiently strong

Field oscillation is slow enough

electron velocity v 2I

barrier thickness

e

Tunneling Laser electric field

Nuclear potential d I

/E

time scale of tunneling

d/v time scale of laser oscillation

1/2

tun

<

osc

= I

2U

< 1

Keldysh parameter

Xe (Ip=12.13 eV), wavelength1064nm, about 5.7 10¹³ W/cm²

= I _p 2U _p

Keldysh parameter

> 1 : Multi-photon regime

1 : Tunneling regime

= 1

Keldysh parameter

Conditions of tunneling ionization

Tunneling rate W is high enough

W I

exp 4 2 3

I

E Field should be sufficiently strong Field oscillation is slow enough

= I

2U

1 : Tunneling regime > 1 : Multi-photon regime

Don’t forget this!

Typical misunderstanding

terahertz radiation with 1 THz frequency and 2 MV/cm field strength U

= 44 eV tunneling ionization? NO!

only 5×10

W/cm

too weak for tunneling

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

I > 10

W/cm

I > 10

W/cm

I > 10

W/cm

Photon 光子

Change of ionization mechanism with laser intensity レーザー強度によるイオン化の変化

€

I > 10

W/cm

€

I > 10

W/cm

€

I > 10

W/cm

Photon 光子 Electric Field 電界

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

Xe (Ip=12.13 eV), 波長1064nmで、5.7 10¹³ W/cm²程度

= I _p

2U _p ^{ケルディッシュ (Keldysh) パラメーター}

> 1 ^{: 多光子領域} 1 ^{: トンネル領域}

= 1

トンネル領域

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)

k = mv( ) = e[A( ) A(t

)] = eA(t

)

トンネル電離後の電子の経路

レーザー電界

原子核ポテン シャル 電子

トンネル効果

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）

E (t)

v(t ) = 0 m v ˙ = eE(t)

A(t) = E (t)dt

Electric field F(t)

3 2

1 0

Vector potential A(t)

Electric field Vector potential

- k

同じエネルギーに 複数の経路が寄与

干渉

t

mv(t) = e

t t_r

E(t) = e[A(t) A(t

)]

電子経路の量子力学的干渉

個々の経路 i の持つ位相 ^{exp iS(t}

(i)r

)

作用 (action) ^{S (t) = dtL =}

^{dt (k +} _2m ^{eA(t ))}

^{+ I}

光電子の運動量分布 P (k)

i

exp iS (t

)

= 2U

1 + cos(2 t)

2 t 3

4 sin(2 t) + I

t

Electric field F(t)

3 2

1 0

Time (optical cycle)

Vector potential A(t)

Electric field Vector potential

- k

j = 1 j = 2 j = 3

Unit cell

t^(1,1)_r t^(2,1)_r

Intracycle interference

cos

S

2

exp iS(t

)

サイクル内干渉 サイクル間干渉

S = S(t

) S(t

)

V(z, t) = exp i(k+eA(t))z iS(t) Volkov波動関数

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

P (k)

i

exp iS(t

)

Electric field F(t)

3 2

1 0

Time (optical cycle)

Vector potential A(t)

Electric field Vector potential

- k

j = 1 j = 2 j = 3

Unit cell

t^(1,1)_r t^(2,1)_r

Intracycle interference

cos

S

2

exp iS (t

)

サイクル内干渉サイクル間干渉

S = S (t

) S (t

)

t

= t

+ 2

(j 1)

j

exp iS(t^(1j)r )

1 + exp(i / ) + exp(2i / ) +· · ·

ピーク（干渉が強め合う）の条件

= 2 n

整数

= S(t

+ 2 / ) S (t

) = 2 k

2m + U

+ I

k

2m + U

+ I

= n

E

= n I

U

光電子スペクトルの離散的な 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

E _kin = n I _p U _p

Electric field F(t)

3 2

1 0

Time (optical cycle)

Vector potential A(t)

Electric field Vector potential

- k

j = 1 j = 2 j = 3

Unit cell

t^(1,1)_r t^(2,1)_r

Intracycle interference

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

まとめ

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

強度 10 ¹³ 〜 10 ¹⁵ W/cm ² のレーザー場中のイオン化

まとめ

ポンデロモーティブエネルギーが重要なパラメーター U p 〜 I p 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

Electric field F(t)

3 2

1 0

Time (optical cycle)

Vector potential A(t)

Electric field Vector potential

- k

Phys. Rev. Lett. 71, 1994 (1993)