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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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 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 4 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 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 6 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) (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 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? 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 | 21 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 Advanced Plasma and Laser 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 26 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 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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 28 Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) 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