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Be2+离子与激光电场的作用过程如图 1a所示。其中,T为一束红外飞秒激光脉冲的持续时间,τ为两束红外飞秒脉冲的时间间隔。图 1b为Be2+离子的二能级系统。|1〉代表Be2+离子的基态1s2 1S0,|2〉代表Be2+离子的激发态1s2p 1P1,其能级差Er=123.7eV,跃迁衰变宽度Γ21=1.2×1011s-1。首先,一束超短的X射线激光脉冲与离子系统相作用,电子由基态|1〉 (1s2 1S0)通过单光子吸收被激发到激发态|2〉(1s2p1P1),形成跃迁偶极子d(t),其中t代表时间。Be2+离子在激光场中可以由系统的偶极子响应描述。d(t)由偶极矩的期望值$\langle \mathit{\boldsymbol{\psi }}{(t)}\left| {\mathit{\boldsymbol{\hat d}}} \right|\mathit{\boldsymbol{\psi }}{(t)}\rangle $给出,其中$ {\mathit{\boldsymbol{\hat d}}} $为系统的跃迁偶极矩算符,$\left| {\mathit{\boldsymbol{\psi }}{\rm{(t)}}\rangle } \right. $为系统的波函数。对于一个二能级系统,其量子态的时间演化可以表示为:
$ \begin{array}{l} \left| {\mathit{\boldsymbol{\psi }}{\rm{(t)}}\rangle } \right. = {c_1}(t){\rm{exp( - i}}\frac{{{E_1}}}{\hbar }t{\rm{)}}\left| {1\rangle } \right. + {\rm{ }}\\ {c_2}\left( t \right){\rm{exp( - i}}\frac{{{E_2}}}{\hbar }t{\rm{)}}\left| {2\rangle } \right. \end{array} $
(1) Figure 1. a—schematic diagram of phase control implementation in Be2+ system b—energy level diagram of two-level system in Be2+ system
式中,$ \hbar $为约化普朗克常数,$ {\left| {{c_j}\left( t \right)} \right|^2}$为态|j〉的布居数,Ej表示相应态的能量本征值,由薛定谔方程给出,这里取j=1, 2。
跃迁偶极矩$\left\langle {j\left| {\mathit{\boldsymbol{\hat d}}} \right|j} \right\rangle $为零(j=1, 2),假设跃迁偶极矩矩阵元$\left\langle {2\left| {\mathit{\boldsymbol{\hat d}}} \right|1} \right\rangle $不存在时间依赖性。系统的偶极子振荡存在衰变宽度Γ,则偶极子振荡为:
$ \begin{array}{l} d\left( t \right) = {c_2}^*\left( t \right){c_1}\left( t \right){\rm{exp(i}}\frac{{{E_2}-{E_1}}}{\hbar }t{\rm{)}}\left\langle {2\left| {\mathit{\boldsymbol{\hat d}}} \right|1} \right\rangle {\rm{exp}}(-\frac{\mathit{\Gamma }}{2}t) + {\rm{c}}.{\rm{c}}. = \\ {c_2}^*\left( t \right){c_1}\left( t \right){\rm{exp(i}}\frac{{{E_{\rm{r}}}}}{\hbar }t{\rm{)}}\left\langle {2\left| {\mathit{\boldsymbol{\hat d}}} \right|1} \right\rangle {\rm{exp}}(-\frac{\mathit{\Gamma }}{2}t) + {\rm{c}}.{\rm{c}}. \end{array} $
(2) 式中,Er=E2-E1,c.c.为前一项的复共轭。假设未知的跃迁偶极矩阵元$ \left\langle {2\left| {\mathit{\boldsymbol{\hat d}}} \right|1} \right\rangle $为常数,系数c1(t)和c2*(t)由时间演化的微扰理论计算得到,上标*表示复共轭。激光脉冲在偶极子衰变过程引入一个微扰算符项$ \mathit{\boldsymbol{\hat W}}\left( \mathit{\boldsymbol{t}} \right)$,相应时间演化的薛定谔方程为:
$ {\rm{i}}\hbar \frac{\partial }{{\partial t}}\left| {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}{\rm{(t)}}\rangle } \right. = \left[ {{{\mathit{\boldsymbol{\hat H}}}_0} + \mathit{\boldsymbol{\hat W}}\left( \mathit{\boldsymbol{t}} \right)} \right]\left| {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}{\rm{(t)}}\rangle } \right. $
(3) 式中,${{{\mathit{\boldsymbol{\hat H}}}_0}} $为无激光脉冲时系统的哈密顿算符,(3)式的解可设为(1)式。
将(1)式代入(3)式,并投影到态|1〉和|2〉,为了方便,定义${W_{jk}} \equiv \left\langle {j\left| {\mathit{\boldsymbol{\hat W}}\left( \mathit{\boldsymbol{t}} \right)} \right|k} \right\rangle (j{\rm{, k}} = 1, 2) $。系数c1(t)和c2(t)可以由下面方程得出:
$ {\rm{i}}\hbar \frac{{{\rm{d}}{c_1}\left( t \right)}}{{{\rm{d}}t}} = {c_1}\left( t \right){W_{11}} + {c_2}\left( t \right){\rm{exp(}} - {\rm{i}}\frac{{{E_{\rm{r}}}}}{\hbar }t){W_{12}} $
(4) $ {\rm{i}}\hbar \frac{{{\rm{d}}{c_2}\left( t \right)}}{{{\rm{d}}t}} = {c_1}\left( t \right){\rm{exp(i}}\frac{{{E_{\rm{r}}}}}{\hbar }t){W_{12}} + {c_2}\left( t \right){W_{22}} $
(5) 假设Wjj(j=1, 2)为零,且微扰$ W\left( t \right) = \langle \mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( t \right)\left| {\mathit{\boldsymbol{\hat W}}\left( t \right)} \right|\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( t \right)\rangle $很小,可用迭代近似法求解上式。若无微扰存在,系统处于|1〉态:
$ \left\{ {\begin{array}{*{20}{c}} {{c_1}^{(0)}\left( t \right) = 1}\\ {{c_2}^{(0)}\left( t \right) = 0} \end{array}} \right. $
(6) 式中,上标(0)代表近似的级。为了计算1级近似,将(6)式代入(4)式及(5)式的右边得:
$ \left\{ \begin{array}{*{20}{l}} {{c}_{1}}^{(1)}({t})=1 \\ {{c}_{1}}^{(1)}({t})=-\frac{\text{i}}{\hbar }\int_{0}^{t}{\text{d}t\mathsf{'}\exp (\mathsf{i}\frac{{{E}_{\text{r}}}}{\hbar }t\mathsf{'})\left\langle 2|\overset{\wedge }{\mathop{\mathrm{W}}}\,\text{(}t\mathsf{'}\text{)}|1 \right\rangle } \\ \end{array} \right. $
(7) 式中,t′为积分变量。微扰${\mathit{\boldsymbol{\hat W}}{\rm{(}}\mathit{t)}} $可以由系统的偶极矩$ {\mathit{\boldsymbol{\hat d}}}$和X射线激发脉冲电场EX(t)给出,因此:
$ \begin{align} & {{c}_{2}}^{(1)}\left( t \right)=-\frac{\text{i}}{\hbar }\int_{0}^{t}{\text{d}{{t}^{'}}\exp (i\frac{{{E}_{\text{r}}}}{\hbar }{{t}^{'}})\left\langle 2\left| \mathrm{\hat{d}}{{E}_{X}}\text{(}{{\mathsf{t}}^{'}}\text{)} \right|1 \right\rangle }= \\ & -\frac{\text{i}}{\hbar }\int_{0}^{t}{\text{d}{{t}^{'}}\exp (i\frac{{{E}_{\text{r}}}}{\hbar }{{t}^{'}}){{E}_{X}}\text{(}{{\mathsf{t}}^{'}}\text{)}\left\langle 2\left| {\mathrm{\hat{d}}} \right|1 \right\rangle } \\ \end{align} $
(8) 系统的偶极矩阵元$ {\left\langle {2\left| {\mathit{\boldsymbol{\hat d}}} \right|1} \right\rangle }$是未知的,这里假设其取值为常数。因此,将c2(1)(t)的共轭和c1(1)(t)代入(2)式,偶极子振荡可以确定为:
$ \begin{align} & d\text{(}\mathit{t}\text{)∝ }\frac{\text{i}}{\hbar }[\int_{0}^{t}{\text{d}{{t}^{'}}\exp (-i\frac{{{E}_{\text{r}}}}{\hbar }{{t}^{'}})}\mathrm{{E}_{X}}\text{(}{{\mathit{t}}^{'}}\text{)}]\times \\ & \exp (-\frac{\mathit{\Gamma }}{2}t)\exp (i\frac{{{E}_{\text{r}}}}{\hbar }t)+\text{c}.\text{c}. \\ \end{align} $
(9) 与此X射线激发脉冲同时,一束非相干的红外飞秒激光脉冲与该体系作用,飞秒脉冲由于瞬间的斯塔克效应导致激发态能级的变化。在一定延迟时间τ后,另外一束脉冲也与该离子体系作用,再次引起激发态能级的瞬间变化。激光脉冲的强度正比于脉冲的电场强度的模方,系统激发态能级的变化取决于红外激光的强度,亦取决于脉冲的电场强度:
$ \Delta {{\mathit{E}}_{\text{r}}}(t)∝{{I}_{\rm IR}}(t)∝{{\left| {{E}_{\rm IR}}(t) \right|}^{2}} $
(10) 式中,IIR(t)为激光脉冲的强度。此时,偶极子表达式中的Er变为Er(t):
$ {\mathit{E}_{\rm{r}}}(t) = {\mathit{E}_{\rm{r}}} + \Delta {\mathit{E}_{\rm{r}}}(t) $
(11) $ \begin{align} & d\text{(}\mathit{t}\text{)∝}\frac{\text{i}}{\hbar }\left\{ \int_{0}^{t}{\text{d}{{t}^{'}}\exp [-i\frac{{{E}_{\text{r}}}}{\hbar }{{t}^{'}}]}{{E}_{X}}\text{(}{{\mathit{t}}^{'}}\text{)}]\times \right. \\ & \exp (-\frac{\mathit{\Gamma }}{2}t)\exp [i\frac{{{E}_{\text{r}}}}{\hbar }t]+\text{c}.\text{c}. \\ \end{align} $
(12) 双红外激光脉冲的电场可以表示为:
$ \varepsilon (t) = \cos ({\omega _{{\rm{IR}}}}\mathit{t})[{\mathit{E}_{{\rm{IR}}}}(t) + {\mathit{E}_{{\rm{IR}}}}(t - \tau )] $
(13) 式中,ωIR为红外激光脉冲的中心频率,τ为延迟时间。单个激光场的电场为$ {\mathit{E}_{{\rm{IR}}}}(t) = {\varepsilon _0}\sec {\rm{h(}}\gamma t{\rm{)}}$,其中,ε0为激光脉冲的电场振幅,$ \gamma {\rm{ = 2arccosh(}}\sqrt 2 {\rm{)/}}{\mathit{T}_{\rm{d}}}$,Td为脉冲持续时间的半峰全宽。激发脉冲为超短的软X射线脉冲,假设其持续时间为0,那么Be2+离子的吸收光谱由公式$ S(\omega ) ∝{\rm lm}\left\{ {{\rm{FFT[}}\mathit{d}{\rm{(}}\mathit{t}{\rm{)]}}} \right\}$给出,即对系统的偶极子作傅里叶变换取其虚部得系统的吸收光谱[19-22]。
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He和Rb的瞬态吸收光谱研究表明,瞬态吸收光谱的谱线形状的改变对应于系统的动力学过程的演化[17, 23]。实验上,可以通过观察目标系统吸收光谱线型的变化,研究其超快的动力学过程。本文中应用激光诱导相位模型,理论模拟Be2+离子体系中二能级系统的瞬态吸收光谱,研究Be2+离子在双飞秒激光电场中的动力学响应。首先,控制两束红外激光脉冲的时间延迟τ=60fs及脉冲持续时间T=5fs,改变两束红外激光脉冲的强度,研究Be2+的瞬态吸收光谱的变化,如图 2所示。以下所有图中横坐标相对频率为真实能量相对于Be2+离子基态和第一激发态之间能级差的大小,即ΔE=E-Er,其中ΔE代表相对频率,E代表真实能量,Er为Be2+离子基态和第一激发态之间的能级差。一束超短的X射线激光脉冲与离子系统相作用,电子由基态(1s2 1S0)通过单光子吸收被激发到激发态(1s2p1P1),形成一跃迁偶极子d(t)。计算中采用的衰变宽度Γ的取值不仅考虑了系统的自发辐射展宽的因素,同时也考虑了压力展宽[24],多普勒展宽[25]与碰撞展宽[26]的影响。若没有其它电场与该体系作用,则系统的吸收谱线型为对称的洛伦兹型,如图 2a中黑色实线所示。如果引入红外激光脉冲与该离子系统作用,第1束红外激光脉冲与X射线脉冲同时到达,第2束与第1束的时间延迟为60fs。控制第1束红外脉冲的强度为0.94×1012 W/cm2,第2束的强度为0,结果如图 2a中虚线所示。正如OTT在研究He原子的双电子动力学时的发现[17],单电子激发的吸收光谱由洛伦兹型变为法诺型。红外飞秒脉冲与处于激发态的Be2+离子作用,其激发态由于激光电场的作用而产生斯塔克效应。由于激光电场是脉冲式的,所以斯塔克平移存在于脉冲激光的时间宽度内。处于基态的电子会随着激发态平移而发生瞬态运动,结果就表现为离子系统吸收光谱线型的变化而在低频和高频端同时出现了新的频率成分,也就是说电子的动力学过程受到了红外激光场的调控作用,这一过程可以理解为电偶极子的相位受到了红外激光电场的作用而发生改变。相位的改变量取决于红外激光脉冲的电场强度:
$ \Delta \mathit{\Phi } \approx \mathit{ - }\frac{1}{2}{\alpha _{\rm{d}}}\int {{E_{{\rm{IR}}}}^2{\rm{(}}\mathit{t}{\rm{)d}}t} $
(14) 式中,αd=25.7a.u.为1s2p态的有效的动力学极化度。计算表明:实现对偶极子相位由0~π的调控,需要控制相应的红外脉冲强度从0W/cm2~1.87×1012W/cm2变化。为了表述方便,设第1束红外脉冲所引起的相位变化为ΔΦ1,第2束红外脉冲所引起的相位变化为ΔΦ2。控制第1束红外激光脉冲的强度为0,第2束脉冲激光的强度为0.94×1012W/cm2,对应的跃迁偶极子的相位变化为ΔΦ1=0, ΔΦ2=0.5π。吸收谱线变成了更为复杂的“波浪式”结构,出现了更多的频率成分,如图 2a中所示。若同时控制两束红外激光脉冲的强度为0.94×1012W/cm2与Be2+作用,即ΔΦ1=ΔΦ2=0.5π,则吸收谱线型在法诺线型基础上也产生了新的频率成分,如图 2中出现的“波浪式”结构所示,吸收光谱受到的调制作用更为明显。Be2+离子与相干X射线激光脉冲作用,通过单光子吸收跃迁到激发态(1s2p1P1)。由此可见,Be2+离子的吸收光谱线型受到红外激光脉冲的强度和作用时间的双重影响。可以通过非相干红外激光脉冲改变系统偶极子的相位,控制核外内壳层电子的激发动力学。
双激光场调控Be2+离子动力学的研究
The study of ultrafast dynamics of highly changed Be2+ ion in double strong laser fields
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摘要: 为了研究高电荷态Be2+离子在强激光中的超快动力学过程,采用求解系统的密度矩阵的方法,进行了理论分析,获得了Be2+离子系统的光谱响应。结果表明,由于强激光场会导致激发态能级的瞬态斯塔克效应进而引起电子运动状态的改变,对应于Be2+离子系统跃迁偶极子的相位改变量为0.5π;当两束红外激光强度为0.94×1012W/cm2时,Be2+离子吸收光谱向高、低频端延伸,谱线形状由单个孤立的洛伦兹线型转化为“波浪式”结构;改变入射激光场的强度以及激光脉冲的持续时间和相对延迟时间可以控制吸收谱线型。此研究结果说明调控双激光的参量可以实现对高电荷态离子中核外电子运动的控制,也指明了软X射线脉冲整形的一个可行性方案。Abstract: In order to study the ultrafast dynamic process of highly charged state Be2+ ions in a strong laser, the spectral response of Be2+ ions was analyzed by solving the system's density matrix. It is found that the strong laser field causes a change of the electron motion in the excited state due to the transient Stark effect, corresponding to a phase change of 0.5π to the Be2+ system's dipole; the spectral response of Be2+system extends to the high and low frequencies and the line shape is changed from an isolated Lorentz line to a "wave-like" structure with the intensities of both near infrared laser pulses set to be 0.94×1012 W/cm2; the absorption spectral line depends on the intensity of the incident laser field, pulse duration, and relative delay time, which indicates that the movement of electrons outside the nucleus in highly charged ions can be controlled by adjusting the parameters of the pump laser fields. It also find us a possible pulse-shaping scheme for soft X-rays.
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Key words:
- spectroscopy /
- ultrafast dynamic of Be2+ ion /
- phase modulation /
- the dipole response
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