Angular Distributions and Polarization of Fluorescence in an XUV Pump–XUV Probe Scheme

Abstract

This work provides theoretical calculations of fluorescence angular distribution and polarization within an XUV pump–XUV probe scheme designed for determining ultra-short lifetimes of highly charged heavy ions. The initial pumping leads to a non-zero alignment in the excited levels. After the probing stage, the anisotropies in angular distribution and polarization of subsequent fluorescence are significantly enhanced due to the existence of a previous alignment. Furthermore, two-photon sequential excitation from a ground state with zero angular momentum to a level with angular momentum one by two aligned linearly polarized photon beams is strictly prohibited by the selection rules and may be used as a diagnostic tool to determine beam misalignment. The present approach is based on the density matrix and statistical tensor framework. We provide the analytical form for the alignment parameters caused by successive photoexcitation either with linearly polarized photon beams, or with unpolarized photons. The analytical results can generally be used to compute angular distribution asymmetry parameters and linear polarization of subsequent fluorescence for a large array of atomic systems used in pump–probe experiments.

1. Introduction

Quantum systems are generally examined by measuring the properties of emitted radiation following different excitation mechanisms. Anisotropies in astrophysical or fusion plasmas have been diagnosed by measuring the polarization of dielectronic recombination satellite lines in highly charged heavy ions [1,2,3]. Quantum electrodynamical effects such as Breit interaction have been investigated by monitoring the angular distribution and linear polarization of dielectronic recombination radiation [4,5,6]. Similarly, the angular distribution and polarization of emitted radiation following electron-impact excitation and dielectronic recombination on highly charged heavy ions can predict interference effects between electric dipole and magnetic quadrupole transitions [7]. Coherent excitation of closely lying resonances [8] may lead to significant depolarization effects of the subsequent emission lines [9,10], relating the polarization of radiation to the energy splitting of levels.

The ultra-short lifetime of Be-like ions can be determined by measuring the XUV fluorescence obtained following a pump–probe scheme with a controllable delay between pulses [11]. The latter can be obtained by high-order harmonic generation (HHG) in noble gases [12,13,14,15,16] and selecting the harmonics optimal for the successive resonant excitation of specific highly charged heavy ions. Although the polarization of harmonics is mainly dependent on the driving field, significant depolarization can occur during the transport of pulses [17,18,19]. The effects of pulse polarization on the excitation process and subsequent fluorescence are well established [8,20,21]. However, to the best of the authors’ knowledge, the exact descriptions of alignment parameters corresponding to excited levels obtained by successive linearly polarized or even unpolarized pulses are not available in the literature.

This paper is structured as follows: Section 2 presents the density matrix and statistical tensor approach to modeling primary and secondary fluorescence in the pump–probe experiment [11] and provides the analytical expressions for the alignment parameters of the excited levels. It is worth mentioning that three descriptive proofs relevant to the calculation are provided in Appendix A, Appendix B and Appendix C. Section 3 provides explicit values for the alignment, anisotropy, asymmetry parameters and linear polarization of primary and secondary fluorescence for individual transition types. Here, the effects of excited levels’ alignment prior to the probing stage are shown to drastically enhance the anisotropies in the angular distribution and linear polarization of secondary fluorescence photons. Section 4 provides the concluding remarks.

2. Theoretical Background

This paper follows the XUV pump–XUV probe scheme described by Rothhradt et al. [11] based on the generation and selection of two high-order harmonics [12,13,14] with a controllable delay between them that can act on an ensemble of trapped Be-like ions in a storage ring. The explicit scheme given in Figure 1 comprises the photoexcitation from ground $1s^{2}2s^2$ ${}^{1}S_0^e$ to an excited $1s^{2}2snp$ ${}^{1}P_1^o$ ($2\le n\le 5$) state (pumping stage), followed by a longer time period of spontaneous emission (temporal delay stage), photoexcitation to levels of $J^{\pi }=0,1,2^e$ symmetry (probing stage), and secondary fluorescence emission (final stage). By measuring the secondary fluorescence at different time delays, one would be able to record the population at the $1s^{2}2snp$ ${}^{1}P_1^o$ level ($2\le n\le 5$), obtain the decay curve as a plot against time, and determine the level lifetime.

It is worth mentioning that for the entire Be-like isoelectronic sequence, the $1s^{2}2snp$ ${}^{1}P_1^o$ ($2\le n\le 5$) levels are generally well separated energetically from their neighboring states allowing a better experimental selection during the pumping stage. The density matrix description only requires the energy, total angular momentum J, and parity $\pi$ of levels; therefore, we avoid writing the spurious $LSJ$ terms for all the de-excited levels or for the ones obtained after the probing stage. Furthermore, these levels’ positioning may change with respect to individual $1s^{2}2snp$ ${}^{1}P_1^o$ ($2\le n\le 5$) levels depending on the element Z.

2.1. Initial System Description

An ensemble of ground-state ions with uniformly distributed magnetic quantum numbers is described by the following density matrix [8]:

$$\begin{array}{c}\hfill {\widehat{\rho }}_{gs}={\displaystyle \frac{1}{2J_i+1}}\sum _{M_i=-J_i}^{J_i}\left|{\alpha }_i{J}_i{M}_i\rangle \langle {\alpha }_i{J}_i{M}_i\right|\end{array}$$

(1)

The pumping pulse can be viewed as an ensemble of unpolarized photons with energy $\hslash {\omega }_1$ along ${\widehat{k}}_1$ direction defined by their density matrix operator [8]:

$$\begin{array}{c}\hfill {\widehat{\rho }}_{{\gamma }_1}={\displaystyle \frac{1}{2}}\sum _{{\lambda }_1=\pm 1}\left|{\overrightarrow{k}}_1{\lambda }_1\rangle \langle {\overrightarrow{k}}_1{\lambda }_1\right|\end{array}$$

(2)

The initial system $\widehat{\rho }\left(0\right)={\widehat{\rho }}_{gs}\otimes {\widehat{\rho }}_{{\gamma }_1}$ is a direct sum between the two ensembles. By applying the evolution operator $\widehat{\mathcal{U}}\left(t\right)$, the coupled ensembles evolve as follows:

$$\begin{array}{c}\hfill \widehat{\rho }\left(t\right)={\displaystyle \frac{1}{2(2J_i+1)}}\sum _{{\lambda }_1=\pm 1}\sum _{M_i=-J_i}^{J_i}\widehat{\mathcal{U}}\left(t\right)\left|{\alpha }_i{J}_i{M}_i;{\overrightarrow{k}}_1{\lambda }_1\rangle \langle {\alpha }_i{J}_i{M}_i;{\overrightarrow{k}}_1{\lambda }_1\right|{\widehat{\mathcal{U}}}^{†}\left(t\right)\end{array}$$

(3)

We will limit the time evolution operator to the first order of the Dyson series [8,22]:

$$\begin{array}{c}\hfill \widehat{\mathcal{U}}\left(t\right)\approx \widehat{I}-{\displaystyle \frac{i}{\hslash }}\underset{0}{\overset{t}{\int }}{\widehat{\mathcal{U}}}_0^{†}\left(t^{\prime }\right){\widehat{H}}_{int}{\widehat{\mathcal{U}}}_0\left(t^{\prime }\right)d{t}^{\prime }\end{array}$$

(4)

where ${\widehat{\mathcal{U}}}_0\left(t\right)$ is the time evolution operator [8] in the absence of light-matter coupling term:

$$\begin{array}{c}\hfill {\widehat{\mathcal{U}}}_0\left|\alpha J_{\alpha }{M}_{\alpha };\overrightarrow{k}\lambda \right.〉=e^{-i{E}_{\alpha }t/\hslash }{e}^{-{\Gamma }_{\alpha }t/2}{e}^{-i{\omega }_{k}t}\left|\alpha J_{\alpha }{M}_{\alpha };\overrightarrow{k}\lambda \right.〉\end{array}$$

(5)

where $E_{\alpha }$ and ${\Gamma }_{\alpha }$ are the energy and total radiative decay rate of level $\alpha J_{\alpha }{M}_{\alpha }$, while $\hslash {\omega }_k$ is the energy of the $\overrightarrow{k}\lambda$ photon. At a later time t, the new ensemble $\widehat{\rho }\left(t\right)$ contains ions in the ground and excited states with or without photons, respectively. Depending on the particular subsystem of interest, the reduced density matrix is determined via a suitable trace, over either the ionic or photonic basis states.

2.2. Pumping Stage (Rapid Photoexcitation)

The ensemble of excited ions can be described by the spherical statistical tensors:

$$\begin{array}{c}〈{\widehat{T}}_{kq}^{†}({\alpha }_b^{\prime }{J}_b^{\prime },{\alpha }_b{J}_b)〉={\displaystyle \frac{\sqrt{2k+1}}{2(2J_i+1)}}\sum _{M_b^{\prime }=-J_b^{\prime }}^{J_b^{\prime }}\sum _{M_b=-J_b}^{J_b}{(-1)}^{J_b^{\prime }-M_b^{\prime }}\left(\begin{array}{ccc}{J}_b^{\prime }& J_b& k\\ M_b^{\prime }& -M_b& -q\end{array}\right){\mathcal{C}}_{ii}^{b^{\prime }b}(t_1,{\omega }_1)\hfill \\ \hfill \times \sum _{{\lambda }_1=\pm 1}\sum _{M_i=-J_i}^{J_i}〈{\alpha }_b^{\prime }{J}_b^{\prime }{M}_b^{\prime }|{\widehat{H}}_{int}|{\alpha }_i{J}_i{M}_i;{\overrightarrow{k}}_1{\lambda }_1\rangle \langle {\alpha }_i{J}_i{M}_i;{\overrightarrow{k}}_1{\lambda }_1\left|H_{int}\right|{\alpha }_b{J}_b{M}_b〉\end{array}$$

(6)

where $b,b^{\prime }$ are indexes for excited levels, the term on the right in the first line is a Wigner $3j$ symbol, and ${\mathcal{C}}_{ii}^{b^{\prime }b}(t_1,{\omega }_1)$ is a term containing the time integrals (see Appendix A). In the present work, we exclusively use the electric dipole interaction for laser–ion coupling within the non-relativistic limit. The matrix elements are expressed in length gauge [22]. The Wigner–Eckart theorem [8] is used to factorize the elements into geometrical (angular) and dynamical (radial) contributions using Wigner $3j$ symbols and reduced matrix elements, respectively. It is assumed that the narrow laser pulses cannot coherently excite a multitude of neighboring levels due to large energy spacing; thus, only diagonal statistical tensors are non-zero:

$$\begin{array}{c}〈{\widehat{T}}_{kq}^{†}\left({\alpha }_b{J}_b\right)〉={\displaystyle \frac{\pi e^2{\hslash }^2{\omega }_{bi}^2}{(2J_i+1)\hslash {\omega }_{1}V}}{\left|〈{\alpha }_b{J}_b\Vert r^1\Vert {\alpha }_i{J}_i〉\right|}^2{\mathcal{C}}_{ii}^{bb}(t_1,{\omega }_1) \hspace{1em} \hfill \\ \hfill \times \left\{\begin{array}{ccc}1& 1& k\\ J_b& J_b& J_i\end{array}\right\}\sum _{{\lambda }_1=\pm 1}{(-1)}^{{\lambda }_1+k-J_i-J_b}C\begin{array}{ccc}1& 1& k\\ {\lambda }_1& -{\lambda }_1& -q\end{array}\end{array}$$

(7)

where ${〈{\alpha }_i{J}_i\Vert r^1\Vert {\alpha }_b{J}_b〉}^{*}={(-1)}^{J_i-J_b}〈{\alpha }_b{J}_b\Vert r^1\Vert {\alpha }_i{J}_i〉$. The Clebsch–Gordan coefficient requires $q=0$, and the sum over ${\lambda }_1$ only allows $k=0,2$ statistical tensors. The reduced density matrix corresponding to the ensemble of excited ions is given as follows:

$$\begin{array}{c}\hfill {\widehat{\rho }}_b=\sum _{k=0,2}〈{\widehat{T}}_{k0}^{†}\left({\alpha }_b{J}_b\right)〉{\widehat{T}}_{k0}\left({\alpha }_b{J}_b\right)\end{array}$$

(8)

2.3. Delay Stage (Fluorescence)

During the controllable delay $T_d$ stage, the excited ions undergo spontaneous emission due to the interaction with vacuum electromagnetic fluctuations. By taking the trace over all possible de-excited levels ${\alpha }_r{J}_r{M}_r$ of the evolved ${\widehat{\rho }}_b\left(t\right)$ ionic ensemble, one extracts the fluorescence reduced density matrix:

$$\begin{array}{c}\hfill 〈\overrightarrow{k}\lambda \left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}{\lambda }^{\prime }〉=\sum _{{\alpha }_r}\sum _{J_r}\sum _{M_r=-J_r}^{J_r}〈{\alpha }_r{J}_r{M}_r;\overrightarrow{k}\lambda \left|{\widehat{\rho }}_b\left(t\right)\right|{\alpha }_r{J}_r{M}_r;\overrightarrow{k}{\lambda }^{\prime }〉\end{array}$$

(9)

Substituting all quantities leads to the following:

$$\begin{array}{c}〈\overrightarrow{k}\lambda \left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}{\lambda }^{\prime }〉=\sum _{k=0,2}〈{\widehat{T}}_{k0}^{†}\left({\alpha }_b{J}_b\right)〉\sum _{{\alpha }_r}\sum _{J_r}\sum _{M_r=-J_r}^{J_r}\sum _{M_b=-J_b}^{J_b}{(-1)}^{J_b-M_b}\left(\begin{array}{ccc}{J}_b& J_b& k\\ M_b& -M_b& 0\end{array}\right)\hfill \\ \hfill \times \sqrt{2k+1}〈{\alpha }_r{J}_r{M}_r;\overrightarrow{k}\lambda |\widehat{\mathcal{U}}\left(t\right)|{\alpha }_b{J}_b{M}_b\rangle \langle {\alpha }_b{J}_b{M}_b\left|{\widehat{\mathcal{U}}}^{†}\left(t\right)\right|{\alpha }_r{J}_r{M}_r;\overrightarrow{k}{\lambda }^{\prime }〉\end{array}$$

(10)

The fluorescence may be emitted along any available direction $\widehat{k}$; therefore, we must project the polarization vector ${\overrightarrow{e}}_{\overrightarrow{k}\lambda }$ onto the quantization axis of the ions using Wigner-D rotation matrices. The electric dipole elements in the length gauge are given as follows:

$$\begin{array}{c}〈{\alpha }_r{J}_r{M}_r;\overrightarrow{k}\lambda \left|{\widehat{H}}_{int}\right|{\alpha }_b{J}_b{M}_b〉={(-1)}^{\lambda }ie\hslash {\omega }_{rb}\sqrt{{\displaystyle \frac{2\pi }{\hslash {\omega }_{k}V}}}\sum _{\sigma =-1}^1{\mathcal{D}}_{\sigma ,-\lambda }^{\left(1\right)}\left(\widehat{k}\right)〈{\alpha }_r{J}_r{M}_r\left|r_{\sigma }^1\right|{\alpha }_b{J}_b{M}_b〉\hfill \\ \hfill ={(-1)}^{\lambda }ie\hslash {\omega }_{rb}\sqrt{{\displaystyle \frac{2\pi }{\hslash {\omega }_{k}V}}}\sum _{\sigma =-1}^1{\mathcal{D}}_{\sigma ,-\lambda }^{\left(1\right)}\left(\widehat{k}\right){(-1)}^{J_r-M_r}\left(\begin{array}{ccc}{J}_r& 1& J_b\\ -M_r& \sigma & M_b\end{array}\right)〈{\alpha }_r{J}_r\Vert r^1\Vert {\alpha }_b{J}_b〉\end{array}$$

(11)

Using $〈{\alpha }_b{J}_b\Vert r^1\Vert {\alpha }_r{J}_r〉={(-1)}^{J_b-J_r}〈{\alpha }_r{J}_r\Vert r^1\Vert {\alpha }_b{J}_b〉$ and performing the angular momentum coupling, we obtain the following:

$$\begin{array}{c}〈\overrightarrow{k}\lambda \left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}{\lambda }^{\prime }〉=\sum _{k=0,2}〈{\widehat{T}}_{k0}^{†}\left({\alpha }_b{J}_b\right)〉\sum _{{\alpha }_r{J}_r}{\displaystyle \frac{2\pi e^2{\hslash }^2{\omega }_{br}^2}{\hslash {\omega }_{k}V}}{\left|\langle {\alpha }_r{J}_r\Vert r^1\Vert {\alpha }_{b}1\rangle \right|}^2{\mathcal{C}}_{bb}^{rr}(T_d,{\omega }_k)\hfill \\ \hfill \times {(-1)}^{J_r+J_b+\lambda }\left\{\begin{array}{ccc}1& 1& k\\ J_b& J_b& J_r\end{array}\right\}C\begin{array}{ccc}1& 1& k\\ -\lambda & {\lambda }^{\prime }& {\lambda }^{\prime }-\lambda \end{array}{\mathcal{D}}_{0,{\lambda }^{\prime }-\lambda }^{\left(k\right)}\left(\widehat{k}\right)\end{array}$$

(12)

The angle and energy-integrated emission probability are obtained by tracing the fluorescence-reduced density matrix on all available photon modes [8,22]:

$$\begin{array}{c}\hfill \overline{\overline{\mathcal{P}}}=\sum _{\overrightarrow{k}}\sum _{\lambda =\pm 1}〈\overrightarrow{k}\lambda \left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}\lambda 〉={\displaystyle \frac{V}{{\left(2\pi c\right)}^3}}\int d{\Omega }_k\int {\omega }_k^{2}d{\omega }_k\sum _{\lambda =\pm 1}〈\overrightarrow{k}\lambda \left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}\lambda 〉\end{array}$$

(13)

The fluorescence angular distribution at different photon energies is then given by:

$$\begin{array}{c}\hfill \mathcal{P}({\omega }_k,{\theta }_k)={\displaystyle \frac{\overline{\mathcal{P}}\left({\omega }_k\right)}{4\pi }}\left[1+{\beta }_2\left({\omega }_k\right)P_2(cos{\theta }_k)\right]\end{array}$$

(14)

where $\overline{\mathcal{P}}\left({\omega }_k\right)$ is the angle integrated emission probability of photons with $\hslash {\omega }_k$ energy. The alignment parameter characterizing the ensemble of excited ions is given as follows:

$$\begin{array}{c}\hfill {\mathcal{A}}_2\left({\alpha }_b{J}_b\right)={\displaystyle \frac{〈{\widehat{T}}_{20}^{†}\left({\alpha }_b{J}_b\right)〉}{〈{\widehat{T}}_{00}^{†}\left({\alpha }_b{J}_b\right)〉}}=\sqrt{{\displaystyle \frac{3(2J_b+1)}{2}}}{(-1)}^{J_b+J_i-1}\left\{\begin{array}{ccc}1& 1& 2\\ J_b& J_b& J_i\end{array}\right\}\end{array}$$

(15)

The asymmetry parameter present in the angular distribution becomes:

$$\begin{array}{c}\hfill {\beta }_2\left({\omega }_k\right)={\mathcal{A}}_2\left({\alpha }_b{J}_b\right){\alpha }_2\left({\omega }_k\right)\end{array}$$

(16)

Here, ${\alpha }_2$ represents an intrinsic anisotropy parameter:

$$\begin{array}{c}\hfill {\alpha }_2\left({\omega }_k\right)=\sqrt{{\displaystyle \frac{3(2J_b+1)}{2}}}{\displaystyle \frac{{\displaystyle \sum _{{\alpha }_r{J}_r}}{\omega }_{br}^2{\left|d_{rb}\right|}^2{\mathcal{C}}_{bb}^{rr}(T_d,{\omega }_k){(-1)}^{J_b+J_r+1}\left\{\begin{array}{ccc}1& 1& 2\\ J_b& J_b& J_r\end{array}\right\}}{{\displaystyle \sum _{{\alpha }_r{J}_r}}{\omega }_{br}^2{\left|d_{rb}\right|}^2{\mathcal{C}}_{bb}^{rr}(T_d,{\omega }_k)}}\end{array}$$

(17)

where $d_{rb}=〈{\alpha }_r{J}_r\Vert r^1\Vert {\alpha }_b{J}_b〉$ is the reduced dipole element. For an individual transition line, the intrinsic anisotropy parameter becomes

$$\begin{array}{c}\hfill {\alpha }_2=\sqrt{{\displaystyle \frac{3(2J_b+1)}{2}}}{(-1)}^{J_b+J_r+1}\left\{\begin{array}{ccc}1& 1& 2\\ J_b& J_b& J_r\end{array}\right\}\end{array}$$

(18)

The linear polarization can be extracted from the correlation between photon density matrix and Stokes parameters:

$$\begin{array}{c}\hfill {\mathbb{P}}_1({\omega }_k,{\theta }_k)=-{\displaystyle \frac{〈\overrightarrow{k}-1\left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}+1〉+〈\overrightarrow{k}+1\left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}-1〉}{〈\overrightarrow{k}+1\left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}+1〉+〈\overrightarrow{k}-1\left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}-1〉}}\end{array}$$

(19)

The direct calculation and usage of explicit form for Clebsch–Gordan coefficients yields the following:

$$\begin{array}{c}\hfill {\mathbb{P}}_1({\omega }_k,{\theta }_k)=-{\displaystyle \frac{3{\mathcal{A}}_2\left({\alpha }_b{J}_b\right){\alpha }_2\left({\omega }_k\right){sin}^2{\theta }_k}{2-{\mathcal{A}}_2\left({\alpha }_b{J}_b\right){\alpha }_2\left({\omega }_k\right)(1-3{cos}^2{\theta }_k)}}\end{array}$$

(20)

It is worth mentioning that except for ${\mathcal{A}}_2\left({\alpha }_b{J}_b\right){\alpha }_2\left({\omega }_k\right)=-1$ or 0, where the linear polarization is independent of polar angle ${\theta }_k$, the maximum (absolute) value is reached at ${\theta }_k={\displaystyle \frac{\pi }{2}}$. The linear polarization will, thus, be given at ${\theta }_k={\displaystyle \frac{\pi }{2}}$ values in the tables.

2.4. Probing Stage (Rapid Photoexcitation)

The remaining b-excited ions are the ones that did not couple to the electromagnetic vacuum fluctuations and evolved through the unperturbed ${\widehat{\mathcal{U}}}_0$ operator. After this delay period, an unpolarized probing pulse of energy $\hslash {\omega }_2$ along the ${\widehat{k}}_2\Vert {\widehat{k}}_1$ direction (parallel to the pumping pulse) is coupled to the remaining excited ions ensemble, leading to the following:

$$\begin{array}{c}{\widehat{\rho }}_d={\widehat{\mathcal{U}}}_0\left(T_d\right){\widehat{\rho }}_b{\widehat{\mathcal{U}}}_0^{†}\left(T_d\right)\otimes {\widehat{\rho }}_{{\gamma }_2}={\displaystyle \frac{e^{-{\Gamma }_b{T}_d}}{2}}\sum _{{\lambda }_2=\pm 1}\sum _{k=0,2}〈{\widehat{T}}_{k0}^{†}\left({\alpha }_b{J}_b\right)〉\sqrt{2k+1} \hspace{1em} \hfill \\ \hfill \times \sum _{M_b=-J_b}^{J_b}{(-1)}^{J_b-M_b}\left(\begin{array}{ccc}{J}_b& J_b& k\\ M_b& -M_b& 0\end{array}\right)\left|{\alpha }_b{J}_b{M}_b;{\overrightarrow{k}}_2{\lambda }_2\rangle \langle {\alpha }_b{J}_b{M}_b;{\overrightarrow{k}}_2{\lambda }_2\right|\end{array}$$

(21)

The statistical ensemble of newly excited states at a time t is then given as follows:

$$\begin{array}{c}〈{\widehat{T}}_{KQ}^{†}({\alpha }_f^{\prime }{J}_f^{\prime },{\alpha }_f{J}_f)〉=\sum _{M_f^{\prime },M_f}{(-1)}^{J_f^{\prime }-M_f^{\prime }}\left(\begin{array}{ccc}{J}_f^{\prime }& J_f& K\\ M_f^{\prime }& -M_f& -Q\end{array}\right)\sqrt{2K+1} \hspace{1em} \hfill \\ \hfill \times 〈{\alpha }_f^{\prime }{J}_f^{\prime }{M}_f^{\prime }\left|\widehat{\mathcal{U}}\left(t\right){\widehat{\rho }}_d{\widehat{\mathcal{U}}}^{†}\left(t\right)\right|{\alpha }_f{J}_f{M}_f〉\end{array}$$

(22)

The explicit calculations are presented in Appendix B, leading to the following statistical tensors:

$$\begin{array}{c}〈{\widehat{T}}_{K0}^{†}({\alpha }_f^{\prime }{J}_f^{\prime },{\alpha }_f{J}_f)〉=e^{-{\Gamma }_b{T}_d}{\displaystyle \frac{\pi {\hslash }^2{\omega }_{f^{\prime }b}{\omega }_{fb}}{\hslash {\omega }_{2}V}}\sum _{k=0,2}〈{\widehat{T}}_{k0}^{†}\left({\alpha }_b{J}_b\right)〉\sqrt{[K,k]}{\mathcal{C}}_{bb}^{f^{\prime }f}(t_2,{\omega }_2)\sum _{{\lambda }_2=\pm 1}\hfill \\ \times \sum _{k_{{\gamma }_2}=Max\left(\right|K-k|,0)}^{Min(K+k,2)}(2k_{{\gamma }_2}+1)\left(\begin{array}{ccc}K& k& k_{{\gamma }_2}\\ 0& 0& 0\end{array}\right)\left(\begin{array}{ccc}{k}_{{\gamma }_2}& 1& 1\\ 0& {\lambda }_2& -{\lambda }_2\end{array}\right)\left\{\begin{array}{ccc}{J}_f& 1& J_b\\ J_f^{\prime }& 1& J_b\\ K& k_{{\gamma }_2}& k\end{array}\right\}\\ \hfill \times 〈{\alpha }_f^{\prime }{J}_f^{\prime }\Vert r^1\Vert {\alpha }_b{J}_b〉{〈{\alpha }_f{J}_f\Vert r^1\Vert {\alpha }_b{J}_b〉}^{*}\end{array}$$

(23)

Due to summation over ${\lambda }_2=\pm 1$, the statistical tensor rank of the probing photon ensemble $k_{{\gamma }_2}$ can only take the values 0 and 2. Using Wigner $3j$ and $6j$ symbol selection rules, we are left with the following values for the K rank of the final statistical tensors:

$$\begin{array}{ccccccc}K& 0& 2& 2& 0& 2& 4\\ k& 0& 0& 2& 2& 2& 2\\ k_{{\gamma }_2}& 0& 2& 0& 2& 2& 2\end{array}$$

The first two columns are the isotropic contributions from $k=0$. We notice four new contributions arising from the existence of an alignment in the excited states $k=2$ obtained during the pumping stage. The photon density matrix is obtained in Appendix B, and the $K=4$ rank tensor bears no effect due to the Clebsch–Gordon sum rule, leading to $\Delta \left(11K\right)$ triangular relation (see Equation B.3.56 in ref. [23]) since we used the dipole approximation:

$$\begin{array}{c}〈\overrightarrow{k}{\lambda }^{\prime }\left|{\widehat{\rho }}_{\gamma }\right|\overrightarrow{k}\lambda 〉=\sum _{K=0,2}\sum _{\begin{array}{c}{\alpha }_f^{\prime }{J}_f^{\prime }\\ {\alpha }_f{J}_f\\ {\alpha }_r{J}_r\end{array}}〈{\widehat{T}}_{K0}^{†}({\alpha }_f^{\prime }{J}_f^{\prime },{\alpha }_f{J}_f)〉{\displaystyle \frac{2\pi e^2{\hslash }^2{\omega }_{f^{\prime }r}{\omega }_{fr}}{\hslash \omega V}}{\mathcal{C}}_{f^{\prime }f}^{rr}(\omega ,t){(-1)}^{{\lambda }^{\prime }+J_r+J_f}\hfill \\ \hfill \times 〈{\alpha }_r{J}_r\Vert r^1\Vert {\alpha }_f^{\prime }{J}_f^{\prime }〉{〈{\alpha }_r{J}_r\Vert r^1\Vert {\alpha }_f{J}_f〉}^{*}C\begin{array}{ccc}1& 1& K\\ -{\lambda }^{\prime }& \lambda & \lambda -{\lambda }^{\prime }\end{array}{\mathcal{D}}_{0,\lambda -{\lambda }^{\prime }}^{\left(K\right)}\left(\widehat{k}\right)\left\{\begin{array}{ccc}1& 1& K\\ J_f^{\prime }& J_f& J_r\end{array}\right\}\end{array}$$

(24)

In the case of incoherent excitation, only a set of ${\alpha }_f{J}_f$ is populated. In such a case, the asymmetry and linear polarization terms maintain their same dependence on alignment and intrinsic anisotropy parameters as in Equations (16) and (20), respectively. However, the latter terms change. The structure of $K=0,2$ statistical tensors is different compared to that of Equation (7) due to the non-zero intermediate alignment. The $K=2$ alignment parameter describing the final excited states is then given by the following:

$$\begin{array}{c}{\mathcal{A}}_2\left({\alpha }_f{J}_f\right)={\displaystyle \frac{〈{\widehat{T}}_{20}^{†}\left({\alpha }_f{J}_f\right)〉}{〈{\widehat{T}}_{00}^{†}\left({\alpha }_f{J}_f\right)〉}}=\left\{\sqrt{{\displaystyle \frac{3(2J_f+1)}{2}}}{(-1)}^{J_b+J_f+1}\left\{\begin{array}{ccc}1& 1& 2\\ J_f& J_f& J_b\end{array}\right\}+\right.\hfill \\ +\left.{\mathcal{A}}_2\left({\alpha }_b{J}_b\right)\sqrt{(2J_b+1)(2J_f+1)}\left[{(-1)}^{J_f+J_b+1}\left\{\begin{array}{ccc}{J}_b& J_b& 2\\ J_f& J_f& 1\end{array}\right\}-5\sqrt{{\displaystyle \frac{3}{7}}}\left\{\begin{array}{ccc}{J}_f& 1& J_b\\ J_f& 1& J_b\\ 2& 2& 2\end{array}\right\}\right]\right\}\\ \times {\left[1+{\mathcal{A}}_2\left({\alpha }_b{J}_b\right)\sqrt{{\displaystyle \frac{3(2J_b+1)}{2}}}{(-1)}^{J_b+J_f+1}\left\{\begin{array}{ccc}1& 1& 2\\ J_b& J_b& J_f\end{array}\right\}\right]}^{-1}\hfill \end{array}$$

(25)

In the absence of intermediate alignment, the ${\mathcal{A}}_2\left({\alpha }_f{J}_f\right)$ parameter becomes similar to (15):

$$\begin{array}{c}\hfill \underset{{\mathcal{A}}_2\left({\alpha }_b{J}_b\right)\to 0}{lim}{\mathcal{A}}_2\left({\alpha }_f{J}_f\right)=\sqrt{{\displaystyle \frac{3(2J_f+1)}{2}}}{(-1)}^{J_b+J_f+1}\left\{\begin{array}{ccc}1& 1& 2\\ J_f& J_f& J_b\end{array}\right\}\end{array}$$

(26)

Furthermore, in addition to unity, the denominator contains a product between ${\mathcal{A}}_2\left({\alpha }_b{J}_b\right)$ that resulted from $J_i\to J_b$ and a similar term that would have resulted from $J_f\to J_b$. Time-reversing the successive excitation processes $(J_f\to J_b\to J_i)$ would just invert the product factors, not affecting the denominator. It is worth mentioning that the unity from denominator is actually the result of triangular rule $\Delta (J_b,J_f,1)$, but because we assume the electric dipole transitions only then this rule is automatically satisfied.

2.5. Linearly Polarized Beams

Similar calculations are also performed for the case of two linearly polarized pulses in Appendix C, where the direction of quantization is chosen along the polarization axis, leading to similar statistical tensor ranks with projection zero. The labels $UP$ or $LP$ are now used for the cases where both pulses are either unpolarized or linearly polarized. The anisotropy parameters ${\alpha }_2$ remain unchanged, while the alignment parameters for excited ions obtained during the pumping stage differ by a multiplication factor:

$$\begin{array}{c}\hfill {\mathcal{A}}_2^{LP}\left({\alpha }_b{J}_b\right)=-\sqrt{2}{\mathcal{A}}_2^{UP}\left({\alpha }_b{J}_b\right)=-\sqrt{6(2J_b+1)}{(-1)}^{J_b+J_i-1}\left\{\begin{array}{ccc}1& 1& 2\\ J_b& J_b& J_i\end{array}\right\}\end{array}$$

(27)

Moreover, the alignment of excited ions obtained during the probing stage becomes

$$\begin{array}{c}{\mathcal{A}}_2^{LP}\left({\alpha }_f{J}_f\right)={\displaystyle \frac{〈{\widehat{T}}_{20}^{†}\left({\alpha }_f{J}_f\right)〉}{〈{\widehat{T}}_{00}^{†}\left({\alpha }_f{J}_f\right)〉}}=\left\{-\sqrt{6(2J_f+1)}{(-1)}^{J_b+J_f+1}\left\{\begin{array}{ccc}1& 1& 2\\ J_f& J_f& J_b\end{array}\right\}+\right.\hfill \\ +\left.{\mathcal{A}}_2^{LP}\left({\alpha }_b{J}_b\right)\sqrt{(2J_b+1)(2J_f+1)}\left[{(-1)}^{J_f+J_b+1}\left\{\begin{array}{ccc}{J}_b& J_b& 2\\ J_f& J_f& 1\end{array}\right\}+10\sqrt{{\displaystyle \frac{3}{7}}}\left\{\begin{array}{ccc}{J}_f& 1& J_b\\ J_f& 1& J_b\\ 2& 2& 2\end{array}\right\}\right]\right\}\\ \times {\left[1-{\mathcal{A}}_2^{LP}\left({\alpha }_b{J}_b\right)\sqrt{6(2J_b+1)}{(-1)}^{J_b+J_f+1}\left\{\begin{array}{ccc}1& 1& 2\\ J_b& J_b& J_f\end{array}\right\}\right]}^{-1}\hfill \end{array}$$

(28)

Similar to the previous case, in the absence of intermediate alignment, one obtains

$$\begin{array}{c}\hfill \underset{{\mathcal{A}}_2^{LP}\left({\alpha }_b{J}_b\right)\to 0}{lim}{\mathcal{A}}_2^{LP}\left({\alpha }_f{J}_f\right)=-\sqrt{6(2J_f+1)}{(-1)}^{J_b+J_f+1}\left\{\begin{array}{ccc}1& 1& 2\\ J_f& J_f& J_b\end{array}\right\}\end{array}$$

(29)

which would correspond to the alignment formed by excitation from a state $J_b$ with no alignment to a state $J_f$ by linearly polarized light photons.

With these adjustments to alignment parameters, one can readily determine the angular asymmetry coefficients (see Equation (16)) and polarization (see Equation (20)) of fluorescence from excited levels obtained by pumping the system with linearly polarized light.

3. Results and Discussion

The large energy separation of $1s^{2}2snp$ ${}^{1}P_1^o$ levels ($2\le n\le 5$) in highly charged heavy ions only allows incoherent excitation from the ground state to occur; hence, the non-zero statistical tensors are necessarily diagonal. Taking $J_i=0$ corresponding to the ground state $1s^{2}2s^2$ ${}^{1}S_0^e$ and $J_b=1$ corresponding to the excited level, we obtain an alignment parameter ${\mathcal{A}}_2^{UP}\left({\alpha }_{b}1\right)={\displaystyle \frac{1}{\sqrt{2}}}$ for unpolarized pumping beam and ${\mathcal{A}}_2^{LP}\left({\alpha }_{b}1\right)=-\sqrt{2}$ for linearly polarized pumping beam. The asymmetry and linear polarization parameters are provided for individual transitions, as seen in

Table 1. Alignment ${\mathcal{A}}_2^{UP}\left({\alpha }_b{J}_b\right)$, anisotropy ${\alpha }_2$, asymmetry ${\beta }_2^{UP}$ and linear polarization ${\mathbb{P}}_1^{UP}(\pi /2)$ parameters describing the primary fluorescence of $J^{\pi }=1^o$ levels excited from the 0^e ground state by absorption of unpolarized photons.

${\mathit{J}}_{\mathit{i}}^{\mathit{e}}$	${\mathit{J}}_{\mathit{b}}^{\mathit{o}}$	${\mathcal{A}}_2^{\mathit{UP}}\left({\mathit{\alpha }}_{\mathit{b}}{\mathit{J}}_{\mathit{b}}\right)$	${\mathit{J}}_{\mathit{r}}^{\mathit{e}}$	${\mathit{\alpha }}_2$	${\mathit{\beta }}_2^{\mathit{UP}}$	${\mathbb{P}}_1^{\mathit{UP}}(\mathit{\pi }/2)$
0	1	${\displaystyle \frac{1}{\sqrt{2}}}$	0	${\displaystyle \frac{1}{\sqrt{2}}}$	${\displaystyle \frac{1}{2}}$	$-1$
0	1	${\displaystyle \frac{1}{\sqrt{2}}}$	1	$-{\displaystyle \frac{1}{2\sqrt{2}}}$	$-{\displaystyle \frac{1}{4}}$	${\displaystyle \frac{1}{3}}$
0	1	${\displaystyle \frac{1}{\sqrt{2}}}$	2	${\displaystyle \frac{1}{10\sqrt{2}}}$	${\displaystyle \frac{1}{20}}$	$-{\displaystyle \frac{1}{13}}$

and

Table 2. Alignment ${\mathcal{A}}_2^{LP}\left({\alpha }_b{J}_b\right)$, anisotropy ${\alpha }_2$, asymmetry ${\beta }_2^{LP}$ and linear polarization ${\mathbb{P}}_1^{LP}(\pi /2)$ parameters describing the primary fluorescence of $J^{\pi }=1^o$ levels excited from the 0^e ground state by absorption of linearly polarized photons.

${\mathit{J}}_{\mathit{i}}^{\mathit{e}}$	${\mathit{J}}_{\mathit{b}}^{\mathit{o}}$	${\mathcal{A}}_2^{\mathit{LP}}\left({\mathit{\alpha }}_{\mathit{b}}{\mathit{J}}_{\mathit{b}}\right)$	${\mathit{J}}_{\mathit{r}}^{\mathit{e}}$	${\mathit{\alpha }}_2$	${\mathit{\beta }}_2^{\mathit{LP}}$	${\mathbb{P}}_1^{\mathit{LP}}(\mathit{\pi }/2)$
0	1	$-\sqrt{2}$	0	${\displaystyle \frac{1}{\sqrt{2}}}$	$-1$	1
0	1	$-\sqrt{2}$	1	$-{\displaystyle \frac{1}{2\sqrt{2}}}$	${\displaystyle \frac{1}{2}}$	$-1$
0	1	$-\sqrt{2}$	2	${\displaystyle \frac{1}{10\sqrt{2}}}$	$-{\displaystyle \frac{1}{10}}$	${\displaystyle \frac{1}{7}}$

for unpolarized and linearly polarized pumping beams, respectively.

During the probing stage, photoexcitation populates levels with $J^{\pi }=0,1,2^e$ symmetry, which then further decay to levels of $0,1,2,3^o$ symmetries. The secondary fluorescence occurring after the probing stage is then characterized by the asymmetry and linear polarization parameters in

Table 3. Alignment ${\mathcal{A}}_2^{UP}\left({\alpha }_f{J}_f\right)$, anisotropy ${\alpha }_2$, asymmetry ${\beta }_2^{UP}$, and linear polarization ${\mathbb{P}}_1^{UP}(\pi /2)$ parameters describing the subsequent fluorescence of $J^{\pi }=0,1,2^e$ levels excited from $1s^{2}2snp$ ${}^{1}P_1^o$ by unpolarized photons.

${\mathit{J}}_{\mathit{b}}^{\mathit{o}}$	${\mathit{J}}_{\mathit{f}}^{\mathit{e}}$	${\mathcal{A}}_2^{\mathit{UP}}\left({\mathit{\alpha }}_{\mathit{f}}{\mathit{J}}_{\mathit{f}}\right)$	${\mathit{J}}_{\mathit{r}}^{\mathit{o}}$	${\mathit{\alpha }}_2$	${\mathit{\beta }}_2^{\mathit{UP}}$	${\mathbb{P}}_1^{\mathit{UP}}(\mathit{\pi }/2)$
1	0	0	1	0	0	0
1	1	$-\sqrt{2}$	0	${\displaystyle \frac{1}{\sqrt{2}}}$	-1	1
1	1	$-\sqrt{2}$	1	$-{\displaystyle \frac{1}{2\sqrt{2}}}$	${\displaystyle \frac{1}{2}}$	$-1$
1	1	$-\sqrt{2}$	2	${\displaystyle \frac{1}{10\sqrt{2}}}$	$-{\displaystyle \frac{1}{10}}$	${\displaystyle \frac{1}{7}}$
1	2	${\displaystyle \frac{5}{7}}\sqrt{{\displaystyle \frac{10}{7}}}$	1	${\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{7}{10}}}$	${\displaystyle \frac{5}{14}}$	$-{\displaystyle \frac{15}{23}}$
1	2	${\displaystyle \frac{5}{7}}\sqrt{{\displaystyle \frac{10}{7}}}$	2	$-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{7}{10}}}$	$-{\displaystyle \frac{5}{14}}$	${\displaystyle \frac{5}{11}}$
1	2	${\displaystyle \frac{5}{7}}\sqrt{{\displaystyle \frac{10}{7}}}$	3	${\displaystyle \frac{1}{\sqrt{70}}}$	${\displaystyle \frac{5}{49}}$	$-{\displaystyle \frac{5}{31}}$

and

Table 4. Alignment ${\mathcal{A}}_2^{LP}\left({\alpha }_f{J}_f\right)$, anisotropy ${\alpha }_2$, asymmetry ${\beta }_2^{LP}$, and linear polarization ${\mathbb{P}}_1^{LP}(\pi /2)$ parameters describing the subsequent fluorescence of $J^{\pi }=0,1,2^e$ levels excited from $1s^{2}2snp$ ${}^{1}P_1^o$ by linearly polarized photons.

${\mathit{J}}_{\mathit{b}}^{\mathit{o}}$	${\mathit{J}}_{\mathit{f}}^{\mathit{e}}$	${\mathcal{A}}_2^{\mathit{LP}}\left({\mathit{\alpha }}_{\mathit{f}}{\mathit{J}}_{\mathit{f}}\right)$	${\mathit{J}}_{\mathit{r}}^{\mathit{o}}$	${\mathit{\alpha }}_2$	${\mathit{\beta }}_2^{\mathit{LP}}$	${\mathbb{P}}_1^{\mathit{LP}}(\mathit{\pi }/2)$
1	0	0	1	0	0	0
1	1	Undefined 1	0	${\displaystyle \frac{1}{\sqrt{2}}}$	Undefined 1	Undefined 1
1	1	Undefined 1	1	$-{\displaystyle \frac{1}{2\sqrt{2}}}$	Undefined 1	Undefined 1
1	1	Undefined 1	2	${\displaystyle \frac{1}{10\sqrt{2}}}$	Undefined 1	Undefined 1
1	2	$-\sqrt{{\displaystyle \frac{10}{7}}}$	1	${\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{7}{10}}}$	$-{\displaystyle \frac{1}{2}}$	${\displaystyle \frac{3}{5}}$
1	2	$-\sqrt{{\displaystyle \frac{10}{7}}}$	2	$-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{7}{10}}}$	${\displaystyle \frac{1}{2}}$	$-1$
1	2	$-\sqrt{{\displaystyle \frac{10}{7}}}$	3	${\displaystyle \frac{1}{\sqrt{70}}}$	$-{\displaystyle \frac{1}{7}}$	${\displaystyle \frac{1}{5}}$

¹ The “Undefined” label accounts for null statistical tensors corresponding to states forbidden by selection rules.

for unpolarized and linearly polarized beams, respectively.

Interestingly, the selection rules do not allow for a two-photon sequential excitation with beams of similar linear polarization of the $J_i^{\pi }=0^e$ ground state to the $J_f^{\pi }=1^e$ level. This can be understood in terms of angular momentum coupling. With the quantization axis along the direction of photon polarization, the pumping pulse forms the excited ions of $J_b^{\pi }=1^o$ symmetry with zero angular momentum projection, since both the ground state and the spherical photon state described in Equation (A25) have zero angular momentum projection. Further absorption of a similar probing photon would not be able to excite the ions to $J_f^{\pi }=1^e$ since the corresponding Clebsch–Gordan coupling coefficient $C_{000}^{111}$ for this transition is null. As a result, the statistical tensors for $J_f=1$ are all null; thus, the corresponding alignment parameter remains undefined.

In practice, two-photon sequential excitation by similar linearly polarized beams can occur from a ground state $J_i^{\pi }=0^e$ to $J_f^{\pi }=1^e$ level provided their polarization axes are not perfectly aligned. For example, the $2s^2 {}^{1}S_0^e\to 2s2p {}^{1}P_1^o\to 2p3p {}^{1}P_1^e$ sequential excitation in Be-like $CIII$, where $1s^2$ is closed, should not be possible if the polarization axes of the two photon beams are perfectly aligned. Assuming the angle between the two axes is small, the population of ions excited to a level with $J_f^{\pi }=1^e$ symmetry, or $2p3p {}^{1}P_1^e$ in the above example, becomes roughly proportional to the square value of that angle expressed in radians. It is worth mentioning that the selection rules only prohibit this two-photon excitation if one considers a non-zero intermediate alignment. Fluorescence from such a level would be proof that the pump and probe pulses are misaligned, which may be useful in an experimental setting.

The angular and linear polarization distributions for subsequent fluorescence obtained after the system absorbs two delayed unpolarized beam pulses are plotted in Figure 2 and Figure 3, respectively.

The sign of asymmetry parameter is directly connected with the type of spheroid representing the angular distribution, namely oblate for $\beta <0$ and prolate for $\beta >0$ with respect to the $\widehat{z}$ axis, which points upward in the graphs of Figure 2. In the case of fluorescence from $J_b=0$, no anisotropies (${\alpha }_2=0$) in the angular distribution or polarization are registered; therefore, they have been omitted from both tables and graphs.

It is worth mentioning that the $J_f=1\to J_r=0$ transition corresponds to a fully oblate angular distribution, where the asymmetry parameter attains its minimum possible value ${\beta }_2=-1$. Additionally, the fluorescence is $100\%$ polarized since the emitted photon fully carries out the angular momentum of the excited state as the lower level has zero angular momentum. For this reason, the polarization of this transition type is not angular-dependent, a feature that is also held by $J_f=0\to J_r=1$, where the polarization is zero regardless of the emitted photon direction.

Neglecting the intermediate excited states’ alignment parameter would significantly underestimate the angular anisotropies, as all asymmetry parameters would be reduced in absolute value while still maintaining their sign. Similarly, the polarization is vastly reduced compared to the case where intermediate alignment is non-zero.

4. Conclusions

The alignment of excited levels obtained within a pump–probe scheme with linearly polarized or unpolarized pulses was investigated using the density matrix and statistical tensor approach. The analytical form of the alignment was deduced by performing angular momentum algebra calculations. Asymmetry parameters together with the polarization of subsequent fluorescence are provided as a result. The analytical forms for the alignments obtained by linearly polarized (Equation (25)) or unpolarized (Equation (28)) beams can be adapted to any type of atomic system and only require the angular momentum of the initial $J_i$, intermediate $J_b$, and final $J_f$ excited levels. It was found that the presence of an intermediate alignment drastically enhances anisotropies in angular distributions (Equation (16)) and maximizes the absolute value of linear polarization (20). These results may provide useful insights into the diagnostics of future pump–probe experiments.