Control and Preparation of Low Excited States of He Atoms Interacted with Strong Laser Field
1 引言
近年来,随着超强超快激光技术的快速发展,强激光场与原子(或分子)间的相互作用引起了学者们的极大兴趣[1]。当原子处于激光场中时,原子中的束缚电子会通过隧穿电离进入连续态,这些电子在激光场中加速运动,当激光场反向时,一部分电子会携带一定的动能返回到母核附近与母核复合,辐射出高能光子,即高次谐波[2]。高次谐波的波长可以延伸到极紫外和软X射线波段,并且具有很好的时间和空间相干性[3-4],可作为一种优质的桌面型光源应用于许多重要领域,如利用高次谐波对原子分子中的电子轨道[5]和纳米量级的微小物体[6]进行成像,对激光的载波包络相位(CEP)进行测量[7]等。其中,高次谐波重要的应用之一是合成超短的阿秒脉冲[8-11]。
在通常情况下,当激光驱动原子发射高次谐波时,原子处于基态,其电离能较大,导致原子的隧穿电离较弱,故发射的高次谐波强度较小。而激发态原子具有较小的电离能,能获得较大的隧穿电离速率,产生高强度的谐波。因此,学者们提出以原子的基态和激发态的线性组合作为初态来发射高次谐波[12-20]。研究结果表明,利用这种方法产生高次谐波的效率比以原子初态为基态的提高了几个数量级,而且截止能量也会增大。因此,制备具有不同布居数的基态/激发态以及纯粹的激发态原子对于提高高次谐波发射效率具有重要的意义。当原子发射低频高次谐波(电离阈值附近)时,除了辐射出具有奇数阶的高次谐波外,还会出现一些高强度的非整数阶的奇异谐波,而产生这些奇异谐波的正是原子的激发态结构[21-22]。因此,研究原子从基态到激发态的跃迁也有助于理解低能谐波的发射机制[23]。
He原子是一种用于产生高次谐波的重要原子,而关于He原子在激光场中的激发过程的研究鲜有报道。Li等[24]提出了一种依赖角动量量子数的精确的He原子模型势,研究了He原子在激光场中发射高次谐波的性质,得到了与实验一致的结果。本文利用这种精确的原子模型势,研究了He原子从基态到一些低激发态的制备和控制过程。研究结果为实验制备激发态He原子提供了具体的激光参数,同时对利用原子叠加态发射高次谐波的研究具有很好的参考价值。
2 理论方法
由于He原子的初态处于单重态,在线偏振激光驱动的激发过程中总自旋守恒,因此采用Li等[24]构造的He原子单重态的依赖角动量量子数的模型势,该模型势能够精确给出He原子基态和各低激发态的能级:
式中Vl为角动量依赖的模型势;α为He+离子核的偶极极化率;rc为He+的有效原子实半径;N为原子实中的电子数(对于He原子,N=1);r为电子到原子核的距离;W6为核截断函数;S,A1,A2,B1,B2都是依赖角动量量子数的参数;l为角动量量子数。
通过求解含时Schrödinger方程来研究He原子在线偏振激光驱动下的激发性质:
式中Ψ为含时波函数;
为哈密顿算符;r为空间位置矢量;原子与激光场间的相互作用算符
(r,t)=-E(t)·
,其中E(t)为线偏振电场强度,
为空间位置矢量算符。无外场时的体系哈密顿算符为
式中
为球谐函数。
为了求解 (2)式,基于伪谱对空间网格点进行撒点[25],其特点是离原子核近的地方网格点密,离原子核远的地方网格点疏,可在保证计算精度的前提下减少网格点的数量,减少数值计算用时。计算中,空间范围最大值取为200 a.u.(a.u.为原子单位),网格点数为1000,分波数为120。对波函数的时间演化通过分裂算符法[26]完成:
式中Δt为时间步长,ο(·)为三阶无穷小量。t时刻的波函数Ψ(t)在能量空间中的演化时间间隔为Δt/2,在坐标空间中的演化时间间隔为Δt,再到能量空间中演化,得到t+Δt时刻的波函数Ψ(t+Δt),迭代以上步骤可以得到任意时刻的波函数。
含时波函数Ψ(r,t)向原子本征态φn,l(r)的投影为
式中*表示取共轭;n为主量子数。则处在本征态φn,l(r)的原子布居数为
3 结果与讨论
图1所示为He原子的基态和激发态(n≤3)的能级结构及其相应的制备途径,并且给出了利用 (1)式计算得到的单重态能级值以及相关的能级差所对应的光子波长值,其中s、p、d表示能级,标出的能级值为理论计算所得的结果。这些理论计算数值与相关研究[27]结果符合得很好。由于单光子吸收的效率最高,选择该种方式来激发原子。对于2p和3p态,其宇称与基态1s相反,故通过一束单色激光脉冲就可以实现原子布居数向目标态的转移。对于2s、3s和3d态,其宇称与基态相同,故必须先使原子的布居数转移到2p态,然后再利用第二束激光获得各目标态。
图 1. 氦原子低激发态(n≤3)的能级及制备的方案图
Fig. 1. Energy levels and preparation scheme of low excited states (n≤3) of He atoms
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根据图1所示无外场时He原子初态、末态的能级差计算出其单光子跃迁对应的波长值,并将其作为驱动激光脉冲的试探波长。在满足电离较小和脉宽远小于跃迁能级寿命的条件下,选取合适的激光峰值强度和脉宽的初值。通过计算布居数,不断地优化这些激光参数,直到得到所期望的末态原子为止。在参数的选取过程中,激光峰值强度和波长较难确定,这主要是因为在激光场的作用下,原子会产生交流Stark效应,原子的能级会发生移动。实现He原子布居数从基态转移到一些目标态的优化激光脉冲参数如表1所示。
表 1. He原子布居数从基态转移到特定激发态的激光脉冲参数
Table 1. Laser pulse parameters for transferring atomic populations from ground state to specific excited states of He atoms
Initial state | Laser intensity /(W∙cm-2) | Laser wavelength/nm | Pulse duration/cycle | Final state(atomic population) |
---|
1s | 4.50×1013 | 58.1 | 40 | 1s (70%)+2p (30%) | 1s | 8.20×1013 | 58.1 | 40 | 1s (50%)+2p (50%) | 1s | 1.63×1014 | 58.1 | 40 | 1s (20%)+2p (80%) | 1s | 3.20×1014 | 58.1 | 40 | 2p (100%) | 1s | 9.96×1013 | 53.5 | 80 | 1s (50%)+3p (50%) | 1s | 3.80×1014 | 53.5 | 80 | 3p (100%) | 2p | 7.70×109 | 2064.0 | 8 | 2p (50%)+2s (50%) | 2p | 3.10×1010 | 2064.0 | 8 | 2s(100%) | 2p | 2.10×1011 | 715.0 | 24 | 3s (80%)+3d (4%) | 2p | 4.00×1011 | 652.8 | 8 | 3s (7%)+3d (86%) |
|
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图2所示为不同激光峰值强度下基态1s到激发态2p的He原子布居数的转移结果。当激光峰值强度为4.50×1013 W∙cm-2时,可将30%的布居数从基态转移到激发态,如图2(a)所示;当强度为8.20×1013 W∙cm-2时,2p态的布居数增大到50%,得到(|1s>+<2p|)/
的叠加态,如图2(b)所示。进一步增大激光强度至1.63×1014 W∙cm-2时,2p态的布居数增大至80%,如图2(c)所示。使用更高的强度3.20×1014 W∙cm-2时,可以完全将电子从基态转移到2p态,如图2(d)所示。此时,如果激光的持续时间过长,在其作用时间内出现Rabi振荡(周期约为12.6 fs),那么在激光结束时仍会有部分布居数分布在1s态上。
图 2. 不同激光峰值强度下基态1s向2p激发态的He原子布居数的转移。 (a) 4.50×1013 W?cm-2; (b) 8.20×1013 W?cm-2; (c) 1.63×1014 W?cm-2; (d) 3.20×1014 W?cm-2
Fig. 2. Population transfer from 1s ground state to 2p excited state of He atoms under different laser peak intensities. (a) 4.50×1013 W?cm-2; (b) 8.20×1013 W?cm-2; (c) 1.63×1014 W?cm-2; (d) 3.20×1014 W?cm-2
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由图2可知,当激光强度较大时,2p态的布居数会出现振荡,这主要是由电场的振荡引起的。电子的激发、电离都与激光强度有关,故电场振荡时相应的强度也会发生周期性变化,最终导致束缚态上的布居数发生振荡。
采用一束波长为53.5 nm的激光脉冲,通过改变激光峰值强度可以将原子布居数从He原子基态转移到3p态。图3所示为不同激光峰值强度下基态向3p激发态的He原子布居数的转移。
图 3. 不同激光峰值强度下1s基态向3p激发态的He原子布居数的转移。 (a) 9.96×1013 W?cm-2; (b) 3.80×1014 W?cm-2
Fig. 3. Population transfer from 1s ground state to 3p excited state of He atoms under different laser peak intensities. (a) 9.96×1013 W?cm-2; (b) 3.80×1014 W?cm-2
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原子布居数从1s全部转移到2p态后,再使用一束长波激光脉冲(波长为2064 nm)就可以获得布居数可控的2s激发态。当激光强度为7.70×109 W∙cm-2时,可以得到布居数相同的(2s+2p)叠加态,如图4(a)所示。而当强度增强到3.10×1010 W∙cm-2时,则可以产生单纯的2s激发态,如图4(b)所示。
图 4. 不同激光峰值强度下2p态向2s态的He原子布居数的转移。(a) 7.70×109 W?cm-2; (b) 3.10×1010 W?cm-2
Fig. 4. Population transfer from 2p state to 2s state of He atoms under different laser peak intensities. (a) 7.70×109 W?cm-2; (b) 3.10×1010 W?cm-2
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图5所示为由1s制备3s和3d激发态的结果。因为2p态的电离能很小,所以用于激发的第二束激光也会使得一部分电子成为自由电子;另外,由于这两个态的能级非常接近,且激光脉冲具有一定的频宽,用这种方式无法将原子布居数完全转移到3s或3d态。当第二束激光结束时,3s态的最大布居数为0.80,而3d态的则可达到0.86。
图 5. 不同激光峰值强度下1s基态向3s、3d激发态的He原子布居数的转移。 (a) 2.10×1011 W?cm-2; (b) 4.00×1011 W?cm-2
Fig. 5. Population transfers from 1s ground state to 3s and 3p excited states of He atoms under different laser peak intensities. (a) 2.10×1011 W?cm-2; (b) 4.00×1011 W?cm-2
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计算电子密度在激光场中的演化
。图6所示为原子布居数由1s完全转移到2p态的结果,其中z为坐标。从图6可知,电子主要分布在两个区域。在激光场的约前20个周期,电子集中在z=0.5 a.u.附近,这些电子处于基态。从约第20个周期开始,电子跃迁到z=4.6 a.u.附近,由图6(a)可以确定这些电子正好处在2p态。
图 6. (a) He原子1s基态和2p激发态波函数; (b)激光场中电子波包在z轴上随时间的演化;(c)所用到的激光场
Fig. 6. (a) Wavefunctions of 1s ground and 2p excited states of He atoms; (b) time evolution of electron wavepacket along z axis in a laser field; (c) adopted laser field
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4 结论
利用He原子的精确模型势,计算了He原子的基态和单重态的低激发态能级。基于这些精确的能级结构,利用激光脉冲研究了基态向激发态的He原子布居数转移的途径。通过伪谱方法,数值求解了He原子在激光驱动下的含时薛定谔方程,优化了激光脉冲的参数(波长、峰值强度和脉宽),得到了布居数可控的He原子基态和几个低激发态。结果表明,要获得较长寿命的He原子低激发态,采用合适的激光脉冲是一种非常有效的制备手段。对于与基态宇称相反的激发态,可以直接通过单光子吸收将它们制备到2p、3p态上,通过控制激光强度得到具有任意布居数比例的基态与激发态的叠加态,也可以产生完全的激发态。而对于与基态宇称相同的激发态,需先用第一束激光将He原子从基态激发到2p态,再利用第二束激光来制备2s、3s和3d态。利用这种方式可以获得任意布居数的2s态,而3s、3d态的能级非常接近,它们的最高布居数可达到80%左右。所得到的激光参数为获得高强度的高次谐波提供了参考。
致谢 感谢李飞博士的有益讨论。
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