强激光场与原子分子相互作用 - capt称为volkov态 2008.4.24 capt...
TRANSCRIPT
-
2008.4.24CAPT
陈 京
北京应用物理与计算数学研究所
强激光场与原子分子相互作用
-
2008.4.24CAPT
提纲
背景介绍原子电离中的新问题分子的电离分子的高次谐波产生结论
-
2008.4.24CAPT
背景介绍
强激光和原子分子相互作用,主要是多光子过程。新的现象:阈上电离 Above-threshold ionization (ATI), 高次谐波产生 high harmonic generation(HHG), 非次序双(多)电离 Non-sequential double (multiple) ionization (NSDI)最近对于分子的研究成为热点:将分子与具有相似电离能的原子进行比较
-
2008.4.24CAPT
非微扰的特性
-
2008.4.24CAPT
电离过程的分类
p
pU
I2=γ
Keldysh参数1>γ
1
-
2008.4.24CAPT
隧穿出来后的运动方程
设电离时刻t=t0,vi=xi=0
-
2008.4.24CAPT
-
2008.4.24CAPT
某些分子在单电离过程中的”抑制“现象有电离抑制:O2:Xe、 S2:Xe、SO:Xe、 NO:Xe,和 D2:Ar无电离抑制:N2:Ar、 F2:Ar 和 CO:Kr
C. Guo, Phys. Rev. Lett 85, 2276 (2000)
-
2008.4.24CAPT
C. Guo et al., Phys. Rev. A 58, R4271 (1998)
双电离
-
2008.4.24CAPT
C. Cornaggia and Ph.HeringPhys. Rev. A 62, 023403 (2000)
-
2008.4.24CAPT
E. Eremina, X. Liu, H. Rottke, W. Sandner, M.G. Schatzel, A. Dreischuh, G.G. Paulus, Walther,R. Moshammer, and J. UllrichPhys. Rev. Lett 92, 173001 (2004)
动量分布
-
分子取向对双电离的影响
D. Zeidler et al. Phys.Rev.Lett. 95, 203003 (2005)
平行情况下一三象限多电子关联更加显著
为什么 ?动量提供重要的信息
-
2008.4.24CAPT
分子钟
H. Niikura et al., Nature 421, 826 (2003)
-
2008.4.24CAPT
gas jet
x-raysAmplifiedfemtosecond laser pulse
Coherent, ultrashort-pulse, low-divergence, x-ray beam generated by focusing a femtosecond laser in a gas jet
Harmonic orders > 300, photon energy > 500 eV, observed to date
Highest-order nonlinear-optical processes observed to date
-
2008.4.24CAPT
High harmonics in both domains
Possible E-field vs. time
Spectrum
A measured HHG spectrum:
And the field vs. time from a high-intensity, non-perturbative model:
S(ω) 2ωLaser
ω
ωLaser
E(t)
t
1/2ωLaser
-
2008.4.24CAPT
Typical HHG spectrum
Three regimes: tunnelling (perturbative) plateau
cutoff regime
-
2008.4.24CAPT
The cut-off wavelength depends on the medium.
oΔ
10
100
1000
10 30
Cut
-off
harm
onic
ord
er
Ionization potential (eV)
o experimental resultsΔ calculated results (ADK model)
XeKr
Ar
Ne
He
20
oΔ
oΔ
oΔ
oΔ
hυcutoff = Ip + 3.2Up
ionization potentialof atom
Up ∝ I λ2quiver energy of e-
-
2008.4.24CAPT
In He, it’s possible to generate x-rays in the water window.
Z. Chang et al, Phys. Rev. Lett. 79, 2967 (1997)C. Spielmann
et al, Science 278, 661 (1997)
Cutoff of Spectrometer
Inte
nsity
(arb
itrar
y un
its)
199163 211 221
water window
C edge
Harmonic order179
Coherent < 10fs x-ray generation in He to 2.7 nm
4 nm5 nm 3.5 nm
Application
-
2008.4.24CAPT
apt with 170as at Lund --simulationAttosecond pulse
-
2008.4.24CAPT
Single attosecond pulse
-
2008.4.24CAPT
非微挠处理有以下解析理论:Ammosov, Delone and Krainov (ADK) 理论隧穿过程
Keldysh, Faisal and Reiss (KFR) 理论量子S-矩阵理论
量子电动力学理论
数值的方法有:经典蒙特卡罗模拟含时薛定鄂方程的数值解Floquet 理论
-
2008.4.24CAPT
弱光和原子相互作用,主要是单光子过程。单光子激发、单光子电离(光电效应)理论上用微扰论处理
0
1 ( ) ( )2
1 ( )2
( )I
H V e tm
H Vm
H e t
= + + ⋅
= +
= ⋅
2
2
p r r E
p r
r E
其中外场可以表示为
( ) sin .t tω= 0E E
关于KFR理论
-
2008.4.24CAPT
利用含时微扰论可以计算跃迁率
22 ( ).I iw H kπ ψ ψ ρ′=h
f
对基态氢原子得到其微分电离率
620
23
2700
23
)1(cos64
akaEemk
ddw
+=
Ω hπθ
其中
pImk
−= ωhh2
22
-
2008.4.24CAPT
强激光和原子相互作用,主要是多光子过程。新的现象:阈上电离 Above-threshold ionization (ATI),
非次序双(多)电离 Non-sequential double (multiple) ionization (NSDI),高次谐波产生 high harmonic generation(HHG),相干阿秒脉冲
pk IsnE −+= ωh)(
-
2008.4.24CAPT
S-矩阵的两种形式
),( )(+∞→
= iBffiS ψψlimt
薛定鄂方程
0)( 0 =−−−∂ ψABt VVHi
),( )( BiffiS ψψ−
−∞→
= limt
0)( 0 =−−∂ AAt VHi ψ
0)( 0 =−−∂ BBt VHi ψ
假设我们已知以下两个方程的解
一.非微扰处理
-
2008.4.24CAPT
利用第一种形式
))()(),()(),((
),()1(
2)(
221)(
1121
1 1
ttVttGtVtdtdti
VdtiS
iABAB
tBABfi
f
if
++∫∫∫
−
−=−
ψψ
ψψ
此既传统的以VA为微扰项的微扰论展开。对强激光场中的原子电离过程,微扰论的处理给出
nnn Iσ=Γ
-
2008.4.24CAPT
更为方便的是利用第二种形式
))()(),(),((
),()1(
2221)(
1)(
21
1 1
ttVttGVtdtdti
VdtiS
i
if
BAABf
tBAAfi
ψψ
ψψ
∫∫∫
+−−
−=−
相互作用哈密顿量
me
mieVA 2
)( 2 2AA+
∇−⋅−=
有解
)),(2
exp(12
2/1 ττψ prp At
A Vdimpii
V ∫ ∞−−−⋅=称为Volkov态
-
2008.4.24CAPT
得到线极化的激光场中原子的电离率
)2
,()(
)()(
2/122
2/12)2()2(
0
2
2/153
zzJ
znznddw
ni
nn
m
−×
Ε−−−= Β∞
=∑
Ω
αψ
πω
p)
其中
θωα
ω
ω
cos)(2
,)(2
,4/
2/1
2
320
2
pm
Eznm
p
mEez
B
−=
−−=
=
-
2008.4.24CAPT
数值计算方法
-
2008.4.24CAPT
时间演化的计算
Splitting-operator
Alternating direction implicit Peaceman-Rachford) method
)()2/1()2/1)(2/1()2/1()( 1001 ttiHtiHtiHtiHtt II ψψ Δ−Δ+Δ−Δ+=Δ+
−−
-
2008.4.24CAPT
系统的初态可以用虚时演化的方法得到
∑=i
iii Ect )exp()( τϕψ
系统的偶极矩
高次谐波
-
2008.4.24CAPT
计算结果
-
2008.4.24CAPT
原子电离中的新问题
原子单电离过程中低能光电子的动量分布
R. Moshammer et al.,PRL 91, 113002 (2003)
理论预言 实验验证
J. Chen and C. H. Nam,PRA 66, 053415 (2002)
-
2008.4.24CAPT
单电离过程中的阈上电离(ATI)电子能谱结构
无再散射: 2UP考虑再散射:10UP
B. Sheehy et al., Phys. Rev. A58, 3942 (1998)
解运动方程
-
2008.4.24CAPT
更精密的实验结果
一个最鲜明的共同的特
征:各峰的位置基本不随强度的变化而移动。
p//
A Rudenko et al.,JPB 37, L407 (2004)
-
2008.4.24CAPT
( )
0
jfi fi
j
S S∞
=
= ∑
(0)0( ) p
iI tfi pS i dt t V eχ ϕ
∞
−∞= − ∫
'(1)0' ( ) ( , ') ( ') p
t iI tfi p f AS i dt dt t VG t t V t eχ ϕ
∞
−∞ −∞= − ∫ ∫
S矩阵理论的计算
直接电离项
散射项
-
2008.4.24CAPT
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.80.0
2.0x10-5
4.0x10-5
6.0x10-5
8.0x10-5
1.0x10-4
1.2x10-4
1.4x10-4
1.6x10-4
dw/d
p || (
arb.
units
)
P|| (a.u.)
I=0.60 PW/cm2
I=0.65 PW/cm2
I=0.70 PW/cm2
I=0.80 PW/cm2
I=0.84 PW/cm2
与实验比较
理论 实验
-
2008.4.24CAPT
Distributions in a constant-amplitude field without the term 2 21( ')p p λ− +v v
Fig.10
PRA 77, 033413 (2008)
-
2008.4.24CAPT
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.0
0.1
0.2
0.3
0.4
0.5
0.6
dW/d
p || (
arb.
units
)
p||
6d14 6.3d14 6.5d14 7d14 7.2d14
Rescaling the ionization rate
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
dW/d
p || (
arb.
uni
ts)
p||
I= 6d14
-
2008.4.24CAPT
分子的单电离
分子电离过程中的“抑制”效应J. Muth-Böhm et al. Phys.Rev.Lett. 85, 2280 (2000)
分子波函数LCAO-MO
分子的电离
-
2008.4.24CAPT
分子的单电离
J. Muth-Böhm et al. Phys.Rev.Lett. 85, 2280 (2000)
-
2008.4.24CAPT
-
2008.4.24CAPT
No suppression for F2!
-
2008.4.24CAPT
场强不同时,起主要贡献的轨道
不一样。
Xi Chu and Shih-I Chu, Phys. Rev. A 70, 061402 (2004)Daniel Dundas and Jan M. Rost,Phys.Rev. A 71, 013421 (2005)
-
2008.4.24CAPT
双中心ADK模型
在椭球坐标中写出分子的哈密顿量
其中坐标定义为
拉普拉斯算符
( )224
21
μλλ−
−Δ−=R
H
Rrr
Rrr baba −=+= μλ ,
( ) ( ) ( ) ( )( ) ⎥⎦⎤
⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−−−
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
−∂∂
−=Δ
φμλμλ
φμμ
μλλ
λμλ 2222
22222 11
114R
-
2008.4.24CAPT
考虑沿分子轴方向的外加电场
薛定鄂方程为
FzV =
( ) 0)24
21( 22 =−+−
+Δ ψλμμλ
λ FRER
-
2008.4.24CAPT
写出方向的一维薛定鄂方程为
( )( ) 0)
1411
12
111
21
21((
2
223
22
2
22222
2
2
=−−
−−
−−
−+
−+
−++
χλ
μλλμλ
λλ
λλλχ
FRm
RAERERdd
采用LCAO-MO近似。对氢分子有
)2
cosh(2)( 211 μκψ
λκκκ ReceecR
rri
ba−−− =+=
πκ 2
3
1 )1(21
sc
+=
-
2008.4.24CAPT
利用WKB方法可以计算出在λ很大的条件下
]3
)2(2exp[1)2
(cosh)(
)2( 2/302
22
2/50
212
μμμκχ κ
FEeR
pRFEc R −= −
其中2/1
0 )2( μλRFEp −=
最后得到分子在静电场中的电离率
μμμ
μκπ κ dFERe
FREcw Rs ]3
)2(2exp[1)2
(cosh)2(22/3
02
21
02
2/50
21 −= ∫−
-
2008.4.24CAPT
进一步得到氢分子在交变电场中的电离率
μμ
μμκπ κ dFERe
F
REcw Ra ]3)2(2exp[)
2(cosh)2()3(2
2/302
121
023
02
121
−= ∫−
不同的分子有不同的轨道波函数。对氮分子,其最外层的3σgπ轨道可以表示为
))2
sinh()2
cosh((Re 2 μκμκλμψλκ RRC
R
i +=−
πκ 2
5
1 )1(21
sc
+=
-
2008.4.24CAPT
对氧分子和氟分子,其最外层的1πg轨道可以表示为
φμλμκψλκ
cos))1)(1)((2
sinh(Re 2/1222 −−=− RC
R
i
得到氮分子在交变电场中的电离率
μμ
μκμκμμπ κ
dFE
RRReF
Ecw Ra
]3
)2(2exp[
))2
(sinh)2
(cosh(4
)2()3(2
2/30
22211
0
3
23
02
121
−×
+= ∫−
-
2008.4.24CAPT
这两种分子在交变电场中的电离率
μμ
μκμμπ κ
dFE
RReF
Ecw Ra
]3
)2(2exp[
))2
(sinh)1(4
)2()3(2
2/30
22211
0
3
23
02
121
−×
−= ∫−
-
2008.4.24CAPT
部分计算结果
PRA 76, 013418 (2007)PRA 76, 023401 (2007)
-
2008.4.24CAPT
分子的双电离
大家相信分子的非次序双电离过程中“再散射”机制仍然起主要作用。由于分子的复杂性,双电离率的计算是非常困难的。
Ionelectron
e1
e2
-
2008.4.24CAPT
核心思想
用具有一定初始分布的电子对代替量子力学中的波包,
用大量电子对的经典运动近似模拟波包的扩散和碰撞等动力学过程.
NSDI的半经典理论
-
2008.4.24CAPT
牛顿运动方程
-
2008.4.24CAPT
电子对初始分布(a)隧穿区
隧穿电子e1e1e2
{{
位置
动量
束缚电子e2 单电子微正则分布(SMD)
位置
+
动量{
-
2008.4.24CAPT
电子对初始分布(b)越垒区
双电子微正则分布(DMD)
-
2008.4.24CAPT
上述初始条件确定以后就可以利用牛顿方程准确“预言”电子对的“命运”并从中获取双电离事例. 每个事例具有不同的权重,由分子ADK公式给定:
对整个激光脉冲时间和作用空
间上具有不同取向的分子加权
求和,得到的离子信号强度能
够直接和实验进行比较.
-
2008.4.24CAPT
Excellent Agreement!
Phys.Rev.Lett. 99, 013003, 2007
-
2008.4.24CAPT
i ii iii
-
2008.4.24CAPT
Time Delay=奇数个半周期 正关联Time Delay=偶数个半周期 负关联椭圆形分布的长轴基本不变,短轴不断变长,最终将趋向于圆形分布,意味着动量关联随着时间延迟加长而逐渐消失。
-
2008.4.24CAPT
取向效应
-
2008.4.24CAPT
-
2008.4.24CAPT
-
2008.4.24CAPT
双电离比率:平行约为垂直的2倍,为什么?
-
2008.4.24CAPT
电子动量关联
-
2008.4.24CAPT
PRA 76, 023401 (2007)
-
2008.4.24CAPT
半经典理论可以对分子非次序双电离过程给出定性正确
的描述。
但半经典理论无法给出与实验一致的电子的动量关联。
主要原因很可能是在电离过程的第二步,即激发和场致
电离过程中没有考虑量子效应。
将第二个电子的隧穿过程加入半经典理论有可能给出更
正确的描述。
一些初步结论
-
2008.4.24CAPT
模型的改进
-2 -1 0 1 2-2
-1
0
1
2
p1
p 2
09.000E-40.0018000.0027000.0036000.0045000.0054000.0063000.0072000.0081000.0090000.0099000.010800.011700.012600.013500.014400.015300.016200.017100.01800
V-P
-2 -1 0 1 2-2
-1
0
1
2
p1
p 2
00.0011000.0022000.0033000.0044000.0055000.0066000.0077000.0088000.0099000.011000.012100.013200.014300.015400.016500.017600.018700.019800.020900.02200
P-V
-
2008.4.24CAPT
氘分子的电离与解离
H. Niikura et al., Nature 421, 826 (2003)
-
2008.4.24CAPT
Niikura等对于激光场中氘分子电离的解释
-
2008.4.24CAPT
碰撞时间差与解离时对应的的离子间距
12 ( )D
ER t+
=Δ
0it t tΔ = −
-
2008.4.24CAPT
半经典计算中的拟和处理
利用碰撞时间差与解离时对应的的离子间距计算碰撞后离子的动能时,处理电子有关信息时我们考虑其为Gaussian波包,波包中心位置的选取随穿电子的位置。
核的运动能够影响电离的过程,在氘的解离过程中我们考虑了五个平衡位置(分别为R=1.4,1.6,1.8,2.0,2.1a.u.)
-
2008.4.24CAPT
计算中的相关参数
不同波长下的激光场中返回电子获得的最大动能,
D2+的电离能,
800nmλ = 0.057 . .a uω =
3.17k pE U=2
24pEUω
=
1.04 . .kE a u=
1200nmλ = 2.35 . .kE a u=
14 21.5 10E Wcm−= ×
1850nmλ =0.038 . .a uω =0.024 . .a uω = 5.59 . .kE a u=
..1.1 uaI p =
-
2008.4.24CAPT
双电离的两种不同轨道
-
2008.4.24CAPT
碰撞时间和双电离发生时间的分布
-
2008.4.24CAPT
理论计算和实验的比较
12 ( )D
ER t+
=Δ
( ) 800a nmλ =( ) 1200b nmλ =( ) 1850c nmλ =
-
2008.4.24CAPT
1、再散射过程中,波长较长的激光场能够给予返回的电子(隧穿电子)足够的能量使之在与母核作用后,使另外电子(束缚电子)在碰撞后发生直接电离;对于波长较短的激光场,隧穿电子获得的能量小于束缚电子的电离能,束缚电子更易在被激发到激发态后再由场将其电离。
-
2008.4.24CAPT
2、对于波长较长的激光外场( ),隧穿电子在外场中运动的时间较长(激光脉冲的周期较长),不易在第一次返回核附近发生电离(原子核的横向距离较大),通过Coulomb聚焦才会与原子核发生碰撞,碰撞时刻多发生在第一个光学周期之后;由于电子在外场中获得的能量大于氘的电离能,碰撞后束缚电离能够直接电离,故碰撞电离占主导地位,双电离的时刻和碰撞时刻分布差别很小。但对于波长较短的激光外场( ),隧穿电子虽然一般在第一次返回到核附近就能与原子核发生碰撞,但由于其在外场中获得的动能不能使氘核发生电离仅能将束缚电子激发到高激发态,再由外场将其电离,所以双电离的时刻多发生在一个光学周期以后。
k pE I>
k pE I<
PRA 76, 013418 (2007)
-
2008.4.24CAPT
NSDI的量子理论
强场中的N-原子分子系统满足薛定谔方程
定义
Rn1 n2
e1e2r12
把原子的S-矩阵理论推广到了研究分子的NSDI问题上:
该系统的哈密顿量是
-
2008.4.24CAPT
即,如下方程的解是已知的
相应的格林函数满足方程:
我们知道外场中的分子在不考虑电子-电子间的相互作用时,以及孤立分子系统的薛定谔方程的解是已知的。
-
2008.4.24CAPT
S-matrix 的两种形式如下( )lim( ( ), ( ))fi Bf itS t tψ ψ+
→∞= ( )lim ( ( ), ( ))fi f BitS t tψ ψ
−
→−∞=
故
,
则强场中的N-原子分子系统的薛定谔方程的解可以用格林函数表示如下:
-
2008.4.24CAPT
用到线性叠加原子轨道波函数构造分子轨道波函数(LCAO-MO):
‘shake-off’ (SO) process ‘rescattering ’ process
NSDI 的两种竞争机制:
or
-
2008.4.24CAPT
在研究重分子的非次序双电离时,近似可以忽略整个过程中原子核的运动,假定末时刻关掉外场,则非次序双电离的总电离率为:
其中
Ip1 Ip2和 是电子1和电子2的电离能,
线偏光:
Up 是有质动能, 是电子在场的极化方向的振动半径
-
2008.4.24CAPT
应用以上的理论研究双原子分子N2和O2的非次序双电离。我们先假定N2和O2有相同的电离能和核间距(均采用N2的电离能与核间距),看分子结构的不同是否会对双电离率产生影响
激光波长-800nm,场的极化方向与分子的键轴方向平行。
10-17
10-15
10-13
10-11
10-9
1 1.5 2 2.5 3 3.5 410-14
10-12
10-10
10-8
10-6
(a)
N2 O2 Ar
N2 O2 Ar
(b)
Rat
e (a
b.un
it)
Intensity (1014w/cm2)
-
2008.4.24CAPT
考察N2,O2分子第一个电离电子的角分布 。
激光采用线偏光,场的极化方向与分子轴向
一致,考察电子出射方向上电离率的大小。
场强为1*1014[W/cm2]
0 20 40 60 80-0.00002
0.00000
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
0.00016
angle between p and E -- α
sing
le io
niza
tion
rate
N2 angle distribution
0 20 40 60 80
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
O2 angle distribution
angle between p and E -- α
sing
le io
niza
tion
rate
R(E)p
-
2008.4.24CAPT
C. Guo et al., Phys. Rev. A 58, R4271 (1998)
10-9
10-8
10-7
10-6
10-5
10-4
1 1.5 2 2.5 3 3.5 4
10-5
10-4
10-3
10-2
N2 O2 Ar
(a)
Rat
io
N2 O2 Ar
(b)
Intensity (1014w/cm2)
-
2008.4.24CAPT
S-矩阵的第二项(非次序双电离项)相比较第一项(Shake off项)占主导地位,我们主要考察非次序双电离项,即考虑电子-电子间的相互作用。我们发现当双原子分子N2和O2采用相同的电离能和核间距时,即只考虑分子的结构对非次序双电离率的影响,O2分子的双电离率明显被抑制了。考察分子非次序双电离的第一步电离过程发现,N2分子总是在场方向电离率最大,而O2分子则在偏离场方向一定的角度电离率最大,从而当场反向时,大部分O2分子中的第一个电离电子回不到核附近与其它电子相互作用,从而导致双电离率的下降。
Submitted to PRA
-
2008.4.24CAPT
结论分子的单电离也是一个复杂的过程,仍存在很多问题
半经典理论能对NSDI给出比较好的描述电子动量关联需要进一步考虑过程中的量子效应
量子S矩阵理论对分子双电离的研究亟需开展两种理论是互补的:
半经典:更清晰和明确的物理图像
量子: 干涉效应,多光子区域对电离加解离的研究:核运动的考虑是一个困难
-
2008.4.24CAPT
High Order Harmonic Generation (HOHG) Nonlinear response of atom/molecule to
strong laser field Coherent X-rays resource, attosecond
pulsesInternal structure of atom/molecule
-
2008.4.24CAPT
Nonlinear response of atom/molecule to strong laser field
Ion electronx-ray
-
2008.4.24CAPT
Modeling high harmonics electronlaser field
electron
laser field
electron
laser field
The potential due to the nucleus in the absence of the intense laser field:
electron
But the laser field is so intense that it highly distorts the potential!
U x( ) =−e2
4πεo x− eEx
-
2008.4.24CAPT
Three-step picture of HHG
RecombinationPropagation
-Wb
ωXUVEn
ergy
τ
x
τb
0
Laser electric field
Ionization
-
2008.4.24CAPT
Electron Wave-packet Model of HHG
Electron now represented by its wave-function, ψ
Part of the wave-function, ψg, remains in the ground state
Remaining part of wave-function, ψc, interferes with ψg on re-collision
This interference effect leads to an electron charge oscillation, and the emission of High Harmonic radiation
-
2008.4.24CAPT
利用高次谐波对分子轨道成像
J. Itatani et al., Nature 432, 867 (2003) 氮分子外层轨道波函数
-
2008.4.24CAPT
Methods:Nonperturbative approaches for HHG:Models:Quantum mechanical theory developed by Lewentein et al. and Becker et al., so called ‘Lewenstein model’Quantum electrodynamic theory
Ab initio methods:Classical simulationEvolution of time-dependent Schrödinger EquationFloquet theory
-
2008.4.24CAPT
Molecular HHG
Multi-center, symmetry of orbitals, alignment effect
Interference effect, extension of cutoff
-
2008.4.24CAPT
Difference between atomic and molecular HHGFor diatomic molecules, there are four kinds of trajectories
-
2008.4.24CAPT
Molecular Lewenstein modelStrong-field-approximation1) Only ground state is considered 2) The depletion of the ground state is ignored3) In the continuum, the electron can be treated as a free particle moving in the electric field without considering Coulomb potential
-
2008.4.24CAPT
The wavefunction is expanded as:
3( ) ( ( ) 0 ( , ) ),piI tt e a t d pb tψ = + ∫ p p (1)
The Schrodinger equation is2 2 ( )( , ) ( ( ) ) ( , ) ( ) 0 ,
2 2p x xp A tb t i I A t p b t iA t p= − + + − +p p p (2)
Where A(t) is the laser field
( ) ( )sin .A t f t t= (3)
-
2008.4.24CAPT
The solution of Eq. (2) can be written as
The x component of the dipole acceleration is calculated by
0( , ) ( ) 0 exp[ (( ( )) / 2 )],
t t
x ptb t i dt A t p i dt A t I
′′ ′ ′′ ′′= − − +∫ ∫p p p
(4)
( ) .x xv t x Eψ ψ ψ ψ= =
Neglecting the contribution from the continuum-continuum part
23
0( ) ( ) ( ) ( ) exp[ ( , , )] . .,
t
xv t i d p dt A t E t iS t t c cφ ′ ′ ′= − +∫ ∫p p
(5)
(6)
-
2008.4.24CAPT
where
).)]([21(),,( 2 p
t
tItAtdttS +−′′=′ ∫ ′ pp
Adopting LCAO-MO approximation for molecular wavefunction, we have
)7(
).)(()( 2/2/ RpRppp ⋅⋅− ±Φ= iii eeφ
.,.)],,(exp[)],,(exp[
)],,(exp[2)(()()()(
21
00
23
ccttiSttiS
ttiStEtAtdpditvt
ix
+′−±′−±
′−′′Φ= ∫∫pp
pp
)8(
-
2008.4.24CAPT
Only consider the contribution from the stationary pointsof the classical action
which gives
.),,(),,(,),,(),,(
),,,(),,(
2
1
0
RpppRppp
pp
⋅+′=′⋅−′=′
′=′
ttSttSttSttSttSttS
)9(
,0),,( =′∇ ttSi pp)10(
,)(1
),(1
2,1
0
R±′′′′′−
=
′′′′′−
=
∫
∫
′
′
tAtdtt
p
tAtdtt
p
t
txc
t
txc
)11(
With pyc=pzc=0. Assuming R is parallel to E.
-
2008.4.24CAPT
The HHG spectra of H2. I=1 × 1014W/cm2, λ=1064nm.
60 optical cycles20 opitcal cycles
-
2008.4.24CAPT
Three-peak structure
Result of ab initio calculation PRA 63, 023411(2002) Atomic HHG
-
2008.4.24CAPT
Time-frequency analysis:
Wigner distribution for analysis (J. H. Kim et al. PRA, 63, 053403(2001)).
.)()(1),( 2 tdettDttDtW ti ′′+′−= ′−∗∞
∞−∫ω
πω )10(
The frequency variation
.tI
It ∂∂
∂Φ∂
=∂Φ∂
=δω )11(
.))]([21( 2 tNItAtd p
t
tω−+−′′=Φ ∫ ′ p
Here Φ is the dynamic phase of the Nth harmonic
-
2008.4.24CAPT
MoleculeLong and short trajectories have different dependence on variation of intensity. At center, ∂I/ ∂ t =0, δω =0. Leaving center, two curves separate.
Wigner distribution Semiclassical calculation
-
2008.4.24CAPT
Two curves will meet at the time when the intensity decreases to the cut-off intensity of the Nth harmonic and compose a two-loop structure. Two curves interfere destructively or constructively.Atom
The motion of the deuteron
Semiclassical calculationWigner distribution
-
2008.4.24CAPT
Wavelet analysis
),(1)(α
βααβ
−=
tWtT
where W(t) is the Morlet’s wavelet
)12(
.)( 2/2
0 tti eetW −= ω )13(
-
2008.4.24CAPT
Δt Δω~1When α is small, Δω is too large to distinguish the subpeaks.When α=3, the time profiles clearly reach maximum at three different times.
-
2008.4.24CAPT
The appearance of three-peak structure requires:
The intensity should be lower than the saturation intensity. Otherwise, a pure blue shift will appear.
The pulse duration cannot be too shortIf ∂I/ ∂ t is too large, |δω |>1. Curves from different
order harmonics overlap too much and give rise to more complicate interference pattern.
-
2008.4.24CAPT
Harmonic spectrum of H2+ with large internuclear distance R=16 a.u. (Ip=0.5625 a.u.)
-
2008.4.24CAPT
Extension of cut-off due to large internuclear distance
One-center term spectrum coincides with total spectrum in the low energy part.
Two-center term spectrum coincides with total spectrum in the high energy part.
-
2008.4.24CAPT
The action
Applying saddle point method
For low order harmonic
-
2008.4.24CAPT
The kinetic energy vsrecombination time
Solid line and dotted line: trajectory of scenario C with v0=±(2Ip/me)1/2 and 0, respectively. Dash line and dash dot line: trajectory of scenario A with v0=±(2Ip/me)1/2 and 0, respectively.
-
2008.4.24CAPT
Exact value Ek=5.33a.u., giving cutoff order 31Without the tunnelling effect, Ek=4.53a.u., giving 28.
Former one is more consistent with the spectrum.
-
2008.4.24CAPT
Lewtenstein model is extended to molecular HHG.
Wigner distribution and wavelet analysis are used to analyse the fine structure of the spectra.The quite universal subpeak structure is due to interference between long and short trajectories.
Summary
JPB 39, 4747 (2006)
-
2008.4.24CAPT
When the internuclear distance is small, both spectra are very close for atom and molecule.When the internuclear distance is large, obvious extension of cut-off can be observed in the spectrum.
In the spectrum, one-center term gives main contribution to the low energy part.Two-center term mainly contributes to the high energy part.
-
2008.4.24CAPT
Beyond Lewenstein Model
Invalidation oftwo-level
approximation
-
2008.4.24CAPT
Reason for the development of HHG theory on molecule
(a) The continuum states that is ignored in the two-level model and become important for intense laser case.(b) With larger internuclear separation, the strong coupling of the ground state and the first excited state should be considered in Lewenstein model.
-
2008.4.24CAPT
TheoryGround assumption on our theory
(a) Except the ground state and the first excited state,the contribution from other bound states can be neglected;(b) The depletion of the ground state and the first excited state is small; (c) In the continuum, the electron can be treated as a free particle moving in the electric field without considering Coulomb potential
-
2008.4.24CAPT
Our analysis is divided into two cases:(a)near-resonance region of intermediate internuclear distance (b)the strong-coupling region of large internuclear distance. We will derive the time-dependent amplitudes of the ground, excited and continuum states, then calculate dipolar moments and their Fourier transformation, then obtain the analytic expressions of the amplitudes of HOHG for the whole range of harmonic order .
-
2008.4.24CAPT
The time-dependent wave functions can be expanded as
While the ionization is weak, by neglecting the depletion of the ground and the first excited state, the formulation of a(t)and b(t) can be obtained by a two-level approximation.
0iE tp(t) e [a(t) 0 b(t) 1 dpc (t) p ]ψ = + + ∫
r r
-
2008.4.24CAPT
The time-dependent dipole moment is
where
0 1D(t) (t) p (t) D (t) D (t)ψ ψ= = +r
t * iS(p,t,t )0 0
ˆD (t) i dp dt a (t) 0 p p A(t ) p a(t ) p 0 +b(t ) p 1 e ′−′ ′ ′ ′= ⋅ ⎡ ⎤⎣ ⎦∫ ∫rrr r r r r r
t * iS(p,t,t )1 0
ˆD (t) i dp dt b (t) 1 p p A(t ) p a(t ) p 0 +b(t ) p 1 e ′−′ ′ ′ ′= ⋅ ⎡ ⎤⎣ ⎦∫ ∫rrr r r r r r
t2
0tS(p, t,t ) [(p A ( t )) / 2 E ]dt
′′ ′′ ′′= − +∫
rr r
-
2008.4.24CAPT
denotes the transition back to the ground state, and
denotes the transition back to the first excited state.
0D (t)
1D (t)
-
2008.4.24CAPT
(a) Intermediate R with 1 0E E / 1ω− ≈
-
2008.4.24CAPT
Harmonic spectrum calculated from our model
-
2008.4.24CAPT
The splitting separation get from the model prediction and from 1D time-dependent exact calculation
(a) R=5.2
(b) R=6
-
2008.4.24CAPT
Influence of initial condition and internuclear distance
-
2008.4.24CAPT
(a) larger R with Cutoff law at large R
1 0E E / 0ω− ≈
-
2008.4.24CAPT
1)Our model is capable to produce harmonic structure for the whole range of harmonic order, including the molecular plateau due to CR transition and the atomic-like plateau for a long-wavelength excitation, and agrees well with the numerical results from directly solving the Schrodinger equation.2) Our theory identifies the role of the CR states in the fine structure of harmonic spectrum and shows that the harmonic generation in molecular system can be effectively controlled through CR states by adjusting the internuclear distance.
Summary
PRA 74, 063405 (2006)
-
2008.4.24CAPT
规范问题:两种规范长度规范:速度规范:对数值方法,原则上没有差别
对各种模型,由于近似,不同的规范可能得到不同的结果。
弱场与物质相互作用:两能级或三能级系统微扰计算
长度规范更方便
-
2008.4.24CAPT
强场与物质相互作用
多光子过程、非微扰处理
强场近似(Strong-Field-Approximation)结果依赖于规范的选取
对原子分子(小核间距)的电离过程,长度规范更为准确
对高次谐波过程:原子 长度规范更好
分子 速度规范才能给出正确的结果
-
2008.4.24CAPT
长度规范的困难和解决方案在长度规范下的哈密顿量
强场分子高次谐波假设:
a)束缚态中只考虑基态b)基态的衰减可以忽略c)电离后的电子不与核作用
-
2008.4.24CAPT
波函数可以写为
偶极距
正则动量
-
2008.4.24CAPT
分子的基态波函数采用LCAO-MO近似
则跃迁矩阵元可以写为
其中
-
2008.4.24CAPT
正比于R的项引起平移不变性的破坏对原子,可以将核放在原点上来克服这一困难
对分子,大核间距将带来严重的困难
假设:
即连续态与原子波函数正交
-
2008.4.24CAPT
得到跃迁矩阵元为
则偶极距可以写为
其中S0为作用量
-
2008.4.24CAPT
利用鞍点法对t和t’的积分进行近似计算
对第一步隧穿电离过程
对频率为Ω的谐波发射过程
当R很大时,隧穿过程将不存在,而谐波的截止频率将可以变得很大,这都与数值计算不符。
-
2008.4.24CAPT
对于速度规范,由于矩阵元中只有P没有P-A,所以以上困难不存在。
克服困难的途径。。。
当核间距R增大时,基态与第一激发态的能量差将接近光子能量当核间距R很大时,两个态近兼并都说明激发态不能忽略
-
2008.4.24CAPT
对SFA 进行修正:a)束缚态中只考虑基态和第一激发态b)基态和第一激发态的衰减可以忽略c)电离后的电子不与核作用则波函数可以写为
展开系数满足的方程为
-
2008.4.24CAPT
得到解
这里
基态与第一激发态波函数
-
2008.4.24CAPT
得到
当R很大时,如一维氢分子离子R=16 a.u.,能级差为10-6 a.u.可以假设能级差为零,得到
其中d为两能级间的偶极矩阵元
-
2008.4.24CAPT
得到
当R很大时,有d=R/2,得到
-
2008.4.24CAPT
最后得到长度规范下的偶极距
原有的非物理的能量移动被基态与第一激发态的强耦合引起的能量移动所消除而不再出现。
-
2008.4.24CAPT
计算结果
氢分子, nmcmWIuaR 800,/103.,.30 214 =×== λ
-
2008.4.24CAPT
如果在速度规范下同样考虑第一激发态
由于在兼并的条件下
因此第一激发态对谐波谱没有影响!
-
2008.4.24CAPT
基态能量
耦合后能量
在修正后的强场近似理论中,当核间距很大时,基态和第一激发态的强耦合使电子的能量随场移动,从而克服了原有的困难。
-
2008.4.24CAPT
另外一种考虑:
位移的原子势
方程
解是
对分子势
当a很大时
-
2008.4.24CAPT
在对分子高次谐波的处理中,长度规范是否仍然比速度规范优越?
对第一步电离的处理,长度规范比速度规范更准确。
从理论的自洽性来说,速度规范对束缚态之间耦合的描述没有长度规范准确。
问题:
对于小的核间距到底该如何处理?
Summary
PRA 75, 041402 (2007)PRA 76, 033403 (2007)
-
2008.4.24CAPT
Interference effect in MHHG
2D H2+ in a laser pulse with I=5x1014W/cm2. θ=40°
-
2008.4.24CAPT
The physical picture
-
2008.4.24CAPT
21 rkrk ⋅⋅ + ii ee
2/)12()( 21 π+=−⋅ mrrk
The interference term
The destructive interference
Constructive interference
-
2008.4.24CAPT
The energy
Considering the acceleration effect of the potential, the energy
-
2008.4.24CAPT
Experimental demonstration
T. Kanai et al. Nature 435, 470 (2005) C. Vozzi et al. PRL 95, 153902 (2005)
CO2
-
2008.4.24CAPT
The dipole acceleration
.),,(),,(,),,(),,(
),,,(),,(
2
1
0
RkkkRkkk
kk
⋅+′=′⋅−′=′
′=′
ttSttSttSttSttSttS
-
2008.4.24CAPT
After integrating over time
(a) I=1.5x1014W/cm2
(b)(c) I=2x1014W/cm2
(d) I=3x1014W/cm2
-
2008.4.24CAPT
AnalysisContribution from a wide range of momentum
-
2008.4.24CAPT
Shift of the suppression regime
(a)(b) I=2x1014W/cm2
(c)(d) I=3x1014W/cm2
-
2008.4.24CAPT
Under which condition the previous single electron momentum picture is available?
α=21 a.u. for I=1.5x1014W/cm2
-
2008.4.24CAPT
SummaryA group of interference effects with each produced by an independent electron de Broglie wave on the Yang’s slit (multiple atomic sites) and weighed by the momentum state probabilityBoth enhancement and suppression can occur to the molecular HHG due to the interference effectThe previous simple model is only an approximation of our theory at high laser intensity where the electron’s quiver motion amplitude is much bigger than the internuclear distance.
-
2008.4.24CAPT
结论强激光与物质相互作用是一个快速发展的领域
有待研究的问题
超强激光场 相对论效应
核过程
多电子效应 Single-active-electron 近似的适用性复杂分子的双电离 如O2
分子中的干涉效应 谐波、光电子谱中的干涉
理论方法的发展
模型 数值计算如含时Dirac方程的解
218 cm/W10I >
-
2008.4.24CAPT
Collaborators
刘杰研究员
傅立斌研究员
陈式刚院士
贺贤土院士
王兵兵副研究员(中科院物理所)
柳晓军研究员(武汉数物所)
李卫东教授 (山西大学)颜君研究员
Dr. 范靖云 (NIST, US)Dr. Wilhelm Becker (MBI, Germany)Dr. D. B. Milosevic (MBI, Germany)
陈彦军叶地发郭丽李燕贾欣燕王官奇
-
2008.4.24CAPT
Thank you for your attention!