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Collapse of Xe polarized atomic states in magnetic fields

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Abstract

Ionization of two-photon excited states \(5p^{\mathrm {5}}(^{\mathrm {2}}P_{\mathrm {3/2}})6\hbox {p}[^{\mathrm {3}}/_{\mathrm {2}}\), \(^{\mathrm {5}}/_{\mathrm {2}}]_{\mathrm {2}}\), \(\hbox {M}=2\) (jl-coupling) of xenon atoms by circularly polarized probe light was studied experimentally in a supersonic beam. The observed photoionization signals revealed oscillation structure due to the Larmor precession of atomic states in an external magnetic field. We derived analytical formulas for the photoelectron current and explained the diversity in the structure of the detected oscillations in terms of the principal lines among multiplet components of optical transitions. The obtained numerical data demonstrate collapse and revival (beating) behavior of the photocurrent due to nonlinearity of Zeeman shifts in the presence of the Paschen–Back effect. Our results indicate the possibility of implementing Doppler-free spectroscopy involving bound-free transitions.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All relevant data are in the paper.].

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Acknowledgements

This work was supported by the Latvian Science Council Grant No. lzp-2019/1-0280 and by the Russian Science Foundation under the Grant No. 18-12-00313 in the part regarding the theoretical analysis of photoionization signals observed in experiments with Xe atoms. The equipment of the Resource Center “Physical Methods of Surface Investigation” of the St. Petersburg State University was used in experiments. We thank Professor M. Auzinsh for useful discussions on issues related to our work.

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Correspondence to M. S. Dimitrijević.

Appendix A: Temporal dependence of polarization moments \(\rho _{q=0}^{\kappa } \)

Appendix A: Temporal dependence of polarization moments \(\rho _{q=0}^{\kappa } \)

If we restrict ourselves to the effects of linear Zeeman shifts, we can independently study the evolution of the polarization moments of atoms \(\rho _{q}^{\kappa } (t)\,(q=-\kappa ,-\kappa =1,\ldots ,\kappa )\) arising from the Larmor precession of the atomic angular momentum \(\mathbf {{J}}\). The \(\rho _{0}^{0} \) moment describes excited states population and is not affected by a magnetic field \(\mathbf {{B}}\), i.e. \(\rho _{0}^{0} \) does not depend on t.

In the “excitation coordinate system” (quantization z-axis), three components \(\rho _{q}^{1} (t)\) correspond to spherical components \(J_{0,\pm } \) of the vector \(\mathbf {{J}}(t)\) [1, 4] which is initially oriented along the unit vector \(\mathbf {{e}}_{z} \) (see discussion regarding Eq. (8)). In other words \(\rho _{\pm }^{1} (t=0)=0\) while \(\Lambda \rho _{0}^{1} (t=0)\equiv J_{0} =J_{z} =\vert \mathbf {{J}}\vert \) (\(\Lambda \) is some constant depending on the excited states parameters). The vector \(\mathbf {{J}}\) precession around the magnetic field (x-axis) results in \(\Lambda \rho _{0}^{1} (t)\equiv J_{z} (t)=\vert \mathbf {{J}}\vert \cos (\omega _{L} t)\) (compare with Eq. (4)).

In the case of the alignment moments (\(\kappa =2)\), let’s construct two matrix operators \(\hat{{T}}(t)\) and \(\hat{{F}}\). The first one \(T_{q{q}'} (t)\) is proportional to the product of the vector \(\mathbf {{J}}(t)\) spherical components \(T_{q{q}'} =J_{q} J_{{q}'} \), while the second immutable matrix \(F_{q{q}'} \) is formed by the product of the spherical components of the unit vector \(\mathbf {{e}}_{z} \): \(F_{q{q}'} =\delta _{q0} \delta _{{q}'0} \), where \(\delta \) is the Kronecker delta function. In the next step, we consider the trace of the matrix product:

$$\begin{aligned} tr\left( \hat{{T}}(t)\hat{{F}}\right) =\sum \limits _{q{q}'} {T_{q{q}'} F_{{q}'q} } = \left| {J_{z} (t)} \right| ^{2}=\vert \mathbf {{J}}\vert ^{2}\cos ^{2}\left( \omega _{L} t\right) \nonumber \\ \end{aligned}$$
(A1)

On the other hand, the trace can be viewed as the scalar product of the involved matrices [9], i.e. the trace may be written in terms of their irreducible tensor operators [1, 9]:

$$\begin{aligned} tr\left( \hat{{T}}(t)\hat{{F}}\right)= & {} \sum \limits _{\kappa ,q} {T_{q}^{\kappa } } F_{-q}^{\kappa } (-1)^{q}\nonumber \\= & {} \sum \limits _{\kappa =0,2} {T_{0}^{\kappa } } F_{0}^{\kappa } = \vert \mathbf {{J}}\vert ^{2}/3+T_{0}^{\kappa =2} (t)F_{0}^{\kappa =2} ,\nonumber \\ \end{aligned}$$
(A2)

where we account for the following equalities: \(T_{0}^{\kappa =0} =\vert \mathbf {{J}}\vert ^{2}/\sqrt{3} \), \(F_{0}^{\kappa =0} =1/\sqrt{3} \), \(F_{q}^{\kappa =1} \equiv 0\), \(F_{q\ne 0}^{\kappa =0} \equiv 0\) [1]. The comparison of Eq. (A1) with Eq. (A2) yields the temporal dependence

$$\begin{aligned} T_{0}^{\kappa =2} (t)F_{0}^{\kappa =2} =\vert \mathbf {{J}}\vert ^{2}(\cos ^{2}(\omega _{L} t)-1/3) \end{aligned}$$
(A3)

presented in Eq. (4), for the atomic polarization moment \(\rho _{0}^{\kappa =2} \sim T_{0}^{\kappa =2} \).

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Viktorov, E.A., Dimitrijević, M.S., Srećković, V.A. et al. Collapse of Xe polarized atomic states in magnetic fields. Eur. Phys. J. D 75, 13 (2021). https://doi.org/10.1140/epjd/s10053-020-00029-9

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  • DOI: https://doi.org/10.1140/epjd/s10053-020-00029-9

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