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本文探讨了有限温度下, 飞秒激光脉冲诱导的分子准直现象中内在相干性度量与其准直信号之间的定量关系. 针对超快非共振激光脉冲诱导分子准直问题, 推导了转动体系$l_1 $范数相干性度量($C_{l_1}(\rho)$即密度矩阵ρ中所有非对角元素绝对值之和)与准直幅度(准直度最大值与最小值之差)之比与电场强度之间存在的定量关系, 并通过对CO分子的数值模拟进行了验证. 在此基础上, 进一步给出了这一比值与脉冲强度面积之间的映射关系. 本文的研究结果为实验中探测转动体系的相干性度量提供了新的思路, 并为利用分子转动态进行相干性度量研究奠定了理论基础, 有利于促进超快强场物理与量子信息度量之间的交叉融合.Femtosecond laser-induced excitation of molecular rotational states can lead to phenomena such as alignment and orientation, which fundamentally stem from the coherence between the induced rotational states. In recent years, the quantitative study of coherence in the field of quantum information has received widespread attention. Different kinds of coherence measures have been proposed and investigated. In this work, the quantitative correlation is investigated in detail between the intrinsic coherence measurement and the degree of molecular alignment induced by femtosecond laser pulses at finite temperatures. By examining the molecular alignment induced by ultrafast non-resonant laser pulses, a quantitative relationship is established between the $l_1$ norm coherence measure $C_{l_1}(\rho)$ and the alignment amplitude ${\cal{D}}\langle \cos^2 \theta \rangle$. Here, $C_{l_1}(\rho)$ represents the sum of the absolute values of all off-diagonal elements of the density matrix ρ, ${\cal{D}}\langle \cos^2 \theta \rangle$ represents the difference between the maximum alignment and the minimum alignment. A quadratic relationship $ C_{l_1} = (a + b{\cal{E}}^2_0)\times $$ {\cal{D}}\langle \cos^2 \theta \rangle$ between the the $l_1$ norm coherence measure and ${\cal{D}}\langle \cos^2 \theta \rangle$ with respect to the electric field intensity ${\cal{E}}_0$ is obtained. This relationship is validated through numerical simulations of the CO molecule, and the ratio coefficients a and b for different temperatures are obtained. Furthermore, a mapping relationship between this ratio and the pulse intensity area is established. The findings of this study provide an alternative method for experimentally detecting the coherence measure within femtosecond laser-excited rotational systems, thereby extending the potential applicability of molecular rotational states to the study of the coherence measure in the field of quantum resources. This will facilitate the interdisciplinary integration of ultrafast strong-field physics and quantum information.
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Keywords:
- coherence /
- molecular alignment /
- rotational dynamics /
- ultrafast pulses
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图 1 初始温度$ T = 15\ {\rm{K}}$时, 三种不同电场强度的飞秒激光脉冲诱导的CO分子准直信号, 其中双箭头示意处$ {\cal{D}}(\langle \cos^2 \theta \rangle) $表示准直度变化的最大幅值
Fig. 1. Alignment signal of CO molecule induced by femtosecond pulse of three different field strengths for $ T = 15\ {\rm{K}}$. $ {\cal{D}}(\langle \cos^2 \theta \rangle) $ indicated by the double headed arrow is the maximal change in the amplitude of alignment.
图 2 (a)—(c)不同初始态下$ l_1 $范数低阶相干性度量$ C_{l_1}(\vert \Delta J \vert = 2) $和准直幅度$ {\cal{D}}(\langle \cos^2 \theta \rangle) $随电场强度的变化; (d)—(f)不同初始态下$ C_{l_1}(\vert \Delta J \vert = 2) $与$ {\cal{D}}(\langle \cos^2 \theta \rangle) $比值$ {\cal{R}}_0 $随电场强度的变化. 左列((a)和(d))、中列((b)和(e))和右列((c)和(f))分别对应初始转动态为$ |0, 0\rangle $, $ |2, 0\rangle $和$ |4, 0\rangle $的计算结果
Fig. 2. (a)–(c) Lower order coherence measure $ C_{l_1}(\vert \Delta J \vert = 2) $, amplitude of the degree of alignment with respect to the electric field intensity $ E_0 $ for different initial rotational states, and (d)–(f) their ratios with respect to the field intensity. From left to right, three different initial rotational states$ |0, 0\rangle, |2, 0\rangle, |4, 0\rangle $ are selected
图 3 (a)不同初始转动态$ |J_0, M_0\rangle $和(b)不同初始温度T时, $ l_1 $范数低阶相干性度量$ C_{l_1}(|\Delta J| = 2) $与准直幅度$ {\cal{D}}(\langle \cos^2 \theta \rangle) $之比$ {\cal{R}}_0 $. 图(b)中蓝线为数值模拟计算结果, 红线为不同比例系数按照玻尔兹曼分布加权平均的结果
Fig. 3. Ratio between the lower order coherence measure and the amplitude of the alignment with respect to (a) different initial rotational states and (b) temperatures. The blue line with circles in panel (b) represents the numerical results for $ {\cal{R}}_0 $ and red line is the weighted average of $ {\cal{R}}_0 $ from different initial states.
图 4 (a)—(c)不同初始态下$ l_1 $范数高阶相干性度量$ C_{l_1}(\vert \Delta J \vert = 4) $和准直幅度$ {\cal{D}}(\langle \cos^2 \theta \rangle) $随电场强度的变化; (d)—(f)不同初始态下$ C_{l_1}(\vert \Delta J \vert = 4) $与$ {\cal{D}}(\langle \cos^2 \theta \rangle) $比值$ {\cal{R}}_2 $随电场强度的变化. 空心圆是直接数值计算结果[$ C_{l_1}(\vert \Delta J \vert = 4)/ {\cal{D}}(\langle \cos^2 \theta \rangle) $], 实线是拟合二次函数($ \propto {\cal{E}}_0^2 $)结果. 左列((a)和(d))、中列((b)和(e))和右列((c)和(f))分别对应初始转动态为$ |0, 0\rangle $, $ |2, 0\rangle $和$ |4, 0\rangle $的计算结果
Fig. 4. (a)–(c) Higher order coherence measure $ C_{l_1}(\vert \Delta J \vert = 4) $, amplitude of the degree of alignment with respect to the electric field intensity $ E_0 $ for different initial rotational states, and (d)–(f) their ratios with respect to the field intensity. The empty circle and the solid line correspond to the numerical and fitted results. From left to right, three different initial rotational states$ |0, 0\rangle, |2, 0\rangle, |4, 0\rangle $ are selected
图 5 $ l_1 $范数相干性度量$ C_{l_1} $中高阶部分($ \vert \Delta J \vert = 4 $)与准直幅度$ {\cal{D}}(\langle \cos^2 \theta \rangle) $的比值$ {\cal{R}}_2 $随电场强度以及初始温度的变化
Fig. 5. Ratio between the higher order $ C_{l_1} $ coherence measure and the amplitude of the degree of alignment $ {\cal{D}}(\langle \cos^2 \theta \rangle) $ with respect to the laser intensity and the initial temperature.
图 6 初始温度$ T = 5\ {\rm{K}}$时, $ l_1 $范数相干性度量$ C_{l_1} $随激光脉冲强度面积的变化行为. 实线为利用拟合公式(24)计算所得结果. 空心圆、五角星、三角形是精确数值模拟的三种不同激发脉冲所对应的结果. 插图展示了这三种激发脉冲对应的脉冲包络
Fig. 6. $ C_{l_1} $ coherence measure varying with pulse area for the initial temperature $ T = 5\ {\rm{K}}$. The solid line is the result obtained by calculating Eq. (24). The empty circle, pentagram, triangle are the accurate numerical results induced by three different kinds of laser pulses. The insets are the corresponding pulse envelopes of the pump pulses.
图 7 初始温度(a) $ T = 1\ {\rm{K}}$, (b) $ T = 10\ {\rm{K}}$, (c) $ T = 20\ {\rm{K}}$时, 高阶相干性度量$ C_{l_1}(\vert \Delta J \vert >2) $与准直幅度$ {\cal{D}}(\langle \cos^2\theta \rangle) $之比随着电场强度的变化. 蓝色区域电场强度$ {\cal{E}}_0 \in (0.5, 7.5)\times $$ 10^9 \;{\rm{V}}/{\rm{m}}$是文中讨论所用电场强度范围. 橙色区域电场强度$ {\cal{E}}_0 \in (7.5, 15)\times 10^9 \;{\rm{V}}/{\rm{m}}$是进一步增强电场强度范围
Fig. 7. Changing with the field strength, the ratio of higher-order coherence measure $ C_{l_1}(\vert \Delta J \vert >2) $ to the amplitude of the degree of alignment $ {\cal{D}}(\langle \cos^2\theta \rangle) $ for (a) $ T = 1\ {\rm{K}}$, (b) $ T = 10\ {\rm{K}}$, (c) $ T = 20\ {\rm{K}}$. The blue region corresponds to the range of field strength discussed in the text above, the orange region corresponds to stronger field strengths.
表 1 不同初始温度T时, 相干性度量$ C_{l_1} $与准直幅度$ {\cal{D}}(\langle \cos^2 \theta \rangle) $比值函数的拟合系数. a, b分别是相干性度量$ C_{l_1} $的低阶部分、高阶部分与$ {\cal{D}}(\langle \cos^2 \theta \rangle) $比值函数中的拟合系数. c是用脉冲强度面积表示的$ C_{l_1} $高阶部分与$ {\cal{D}}(\langle \cos^2 \theta \rangle) $比值函数的拟合系数
Table 1. Fit coefficients for the ratio between the $ l_1 $ coherence measure $ C_{l_1} $ and the amplitude of the degree of alignment $ {\cal{D}}(\langle \cos^2 \theta \rangle) $ under different initial temperatures. a represents the coefficients for the lower part of $ C_{l_1} $ which is a constant, b for the higher parts of $ C_{l_1} $ with respect to the electric field intensity and c for the higher parts of $ C_{l_1} $with respect to the pulse area.
T/K 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a 1.68 1.79 1.91 2.01 2.09 2.15 2.21 2.26 2.30 2.33 2.37 2.40 2.42 2.45 2.47 2.49 2.51 2.52 2.54 2.56 b/103 1.04 1.11 1.16 1.19 1.21 1.22 1.22 1.21 1.20 1.19 1.18 1.16 1.15 1.13 1.12 1.10 1.09 1.07 1.05 1.04 c 0.395 0.421 0.441 0.453 0.459 0.462 0.462 0.460 0.456 0.452 0.447 0.442 0.436 0.429 0.423 0.417 0.411 0.405 0.400 0.394 
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