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Stark–Zeeman and Blokhintsev Spectra of Rydberg Atoms

  • ATOMS, MOLECULES, OPTICS
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Abstract

The problem of Rydberg atomic spectra in crossed electric F and magnetic B fields (combined Stark–Zeeman effects), as well as an oscillating electric field are under consideration. The main problem is the great array of radiative transitions between the Rydberg atomic states. The different versions of semiclassical approaches are applied for the solution of the problem. New approximate selection rules are established to simplify the matrix elements of radiative transitions, making it possible to reduce the complex expressions of Gordon’s hypergeometric series to the trivial trigonometric functions. Simple analytical formulas for the dipole matrix elements of a Rydberg hydrogen-like atom in external crossed electric and magnetic fields for Hnα are obtained. The specific calculations are presented for n = 10 to 9 transition both for parallel and perpendicular orientations of FB fields. The result of Blokhintsev for a single Stark component in oscillating field is generalized for the array of radiative transitions between Rydberg atomic states. Such spectra with large values of modulation index are compared with static ones.

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Correspondence to A. Yu. Letunov.

Appendices

APPENDIX A

1.1 X-Matrix Element in Terms of Jacobi Polynomials

There is a connection between the hypergeomtric function and the Jacobi polynomials:

$$\begin{gathered} F\left( { - k, - k - \alpha ,1 + \beta , - {{{\cot }}^{2}}\frac{\varphi }{2}} \right) = \frac{{k!\Gamma (1 + \beta )}}{{k + 1 + \beta }} \\ \times {{\left( { - \frac{1}{{{{{\sin }}^{2}}\frac{\varphi }{2}}}} \right)}^{k}}P_{k}^{{(\alpha ,\beta )}}(\cos \varphi ), \\ \end{gathered} $$
(55)

where \(P_{k}^{{(\alpha ,\beta )}}\)(z) is a Jacobi polynomial and Γ(z) is the gamma function. Using formula (55), we can obtain the result for Gordon’s X-matrix element in terms of Jacobi polynomials:

$$X_{{{{n}_{1}},{{n}_{2}},m - 1}}^{{{{{\bar {n}}}_{1}}{{{\bar {n}}}_{2}},m}} = {{\Pi }_{x}}[\Upsilon _{{{{x}_{1}}}}^{J} - \Upsilon _{{{{x}_{2}}}}^{J}],$$
(56)

where

$$\begin{gathered} {{\Pi }_{x}} = {{( - 1)}^{{{{{\bar {n}}}_{1}} + {{{\bar {n}}}_{2}}}}}\frac{{{{a}_{0}}}}{{4{{{((m - 1)!)}}^{2}}}} \\ \times \sqrt {\frac{{({{n}_{1}} + m)!({{n}_{2}} + m)!({{{\bar {n}}}_{1}} + m - 1)({{{\bar {n}}}_{2}} + m - 1)!}}{{{{n}_{1}}!{{n}_{2}}!{{{\bar {n}}}_{1}}!{{{\bar {n}}}_{2}}!}}} \\ \times {{\left( {\frac{{4n\bar {n}}}{{{{{(n - \bar {n})}}^{2}}}}} \right)}^{{m + 1}}}{{\left( {\frac{{n - \bar {n}}}{{n + \bar {n}}}} \right)}^{{n + \bar {n}}}}, \\ \end{gathered} $$
$$\begin{gathered} \Upsilon _{{{{x}_{1}}}}^{J} = \frac{{{{N}_{1}}!{{N}_{2}}!{{{[(m - 1)!]}}^{2}}}}{{({{N}_{1}} + m)!({{N}_{2}} + m)}}{{\left( { - \frac{1}{{{{{\sin }}^{2}}\frac{\varphi }{2}}}} \right)}^{{{{N}_{1}} + {{N}_{2}}}}} \\ \, \times P_{{{{N}_{1}}}}^{{({{\Delta }_{1}},m - 1)}}(\cos \varphi )P_{{{{N}_{2}}}}^{{({{\Delta }_{2}},m - 1)}}(\cos \varphi ), \\ \end{gathered} $$
$$\begin{gathered} \Upsilon _{{{{x}_{2}}}}^{J} = {{\left( {\frac{{\Delta n}}{{n + \bar {n}}}} \right)}^{2}}\frac{{{{{\bar {N}}}_{1}}!{{{\bar {N}}}_{2}}!{{{[(m - 1)!]}}^{2}}}}{{({{{\bar {N}}}_{1}} + m - 1)!({{{\bar {N}}}_{2}} + m - 1)!}} \\ \, \times P_{{{{{\bar {N}}}_{1}}}}^{{({{{\bar {\Delta }}}_{1}},m - 1)}}(\cos \varphi )P_{{{{{\bar {N}}}_{2}}}}^{{({{{\bar {\Delta }}}_{2}},m - 1)}}(\cos \varphi ), \\ \end{gathered} $$
$$\begin{gathered} {{\Delta }_{1}} = {\text{|}}{{n}_{1}} - {{{\bar {n}}}_{1}}{\text{|}},\quad {{\Delta }_{2}} = {\text{|}}{{n}_{2}} - {{{\bar {n}}}_{2}}{\text{|}}, \\ {{{\bar {\Delta }}}_{1}} = {\text{|}}{{n}_{1}} - {{{\bar {n}}}_{1}} - 1{\text{|}},\quad {{{\bar {\Delta }}}_{2}} = {\text{|}}{{n}_{2}} - {{{\bar {n}}}_{2}} - 1{\text{|}}, \\ \Delta = n - \bar {n}, \\ \end{gathered} $$
$${{N}_{1}} = \min ({{n}_{1}},{{\bar {n}}_{1}}),\quad {{N}_{2}} = \min ({{n}_{2}},{{\bar {n}}_{2}}),$$
$${{\bar {N}}_{1}} = \min ({{n}_{1}} + 1,{{\bar {n}}_{1}}),\quad {{\bar {N}}_{2}} = \min ({{n}_{2}} + 1,{{\bar {n}}_{2}}).$$

In this expression, the following relations were taken into account:

$$\begin{gathered} {{\cot }^{2}}\frac{\varphi }{2} = \frac{{4n\bar {n}}}{{{{\Delta }^{2}}}},\quad \sin \frac{\varphi }{2} = \frac{\Delta }{{n + \bar {n}}}, \\ \cos \varphi = \frac{{4n\bar {n} - {{\Delta }^{2}}}}{{{{{(n + \bar {n})}}^{2}}}}. \\ \end{gathered} $$
(57)

Now, it is easy to show for the nα series that in the case of n ≪ Δn. expression (55) coincides with the classical Born result. We should use the asymptotic forms for Jacobi polynomials.

$$\begin{gathered} P_{k}^{{(\alpha ,\beta )}}(\cos \varphi ) \simeq \frac{{\Gamma (k + \alpha + 1)}}{{k!{{U}^{\alpha }}}} \\ \times \sqrt {\frac{\varphi }{{\sin \varphi }}} \frac{{{{J}_{\alpha }}(U\varphi )}}{{{{{\left( {\sin \frac{\varphi }{2}} \right)}}^{\alpha }}{{{\left( {\cos \frac{\varphi }{2}} \right)}}^{\beta }}}},\quad \varphi \ll 1, \\ \end{gathered} $$
(58)

where U2 = \(A_{0}^{2}\)\(\left( {\frac{{{{\beta }^{2}}}}{4}} \right.\) + \(\frac{{{{\alpha }^{2}}}}{{12}}\)\(\left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}\frac{1}{{12}}} \right)\) and A0 = k + \(\frac{{\alpha + \beta + 1}}{2}\).

Then we have to simplify U in the case of n ≫ 1:

$$\begin{gathered} {{U}_{1}} = \sqrt {{{{\left( {{{N}_{1}} + \frac{{{{\Delta }_{1}}}}{2}} \right)}}^{2}} - \frac{{{{{(m - 1)}}^{2}}}}{2} + \frac{{\Delta _{1}^{2}}}{{12}} - \frac{1}{{12}}} \\ = \sqrt {\left( {{{N}_{1}} + \frac{{m + {{\Delta }_{1}}}}{2} + \frac{{m - 1}}{2}} \right)\left( {{{N}_{1}} + \frac{{m + {{\Delta }_{1}}}}{2} - \frac{{m - 1}}{2}} \right) + \frac{{\Delta _{1}^{2} - 1}}{{12}}} \\ \end{gathered} $$

If we take into account the fact that Δ1 ~ 1, we get

$${{U}_{1}} \simeq \sqrt {{{N}_{1}}({{N}_{1}} + m)} $$

Similarly, it can be done for U1 = U3, U2 = U4.

After the substitution of expression (57) into (55) and simple mathematical transformations, we can obtain the classical Born formula for the X-matrix element (16).

APPENDIX B

1.1 Selection Rules for σ Component for H nα Series

Let us consider system (32) with +1 in the second equation:

$$\left\{ \begin{gathered} ({{i}_{2}} - {{i}_{1}}) - ({{{\bar {i}}}_{2}} - {{{\bar {i}}}_{1}}) = 0 \hfill \\ {\text{|}}{{i}_{2}} + {{i}_{1}}{\text{|}} = {\text{|}}{{{\bar {i}}}_{2}} + {{{\bar {i}}}_{1}}{\text{|}} + 1. \hfill \\ \end{gathered} \right.$$

(a) i2 + i1 and \({{\bar {i}}_{2}}\) + \({{\bar {i}}_{1}}\) ≥ 0:

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} + \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} + \frac{1}{2}. \hfill \\ \end{gathered} \right.$$

(b) i2 + i1 ≤ 0, \({{\bar {i}}_{2}}\) + \({{\bar {i}}_{1}}\) ≥ 0. These transitions are forbidden by the selection rule for m.

(c) i2 + i1 ≥ 0, \({{\bar {i}}_{2}}\) + \({{\bar {i}}_{1}}\) ≤ 0. There is only one transition that satisfies the selection rule for m: 0 → –1. It may be included in the next item (d).

(d) i2 + i1 and \({{\bar {i}}_{2}}\) + \({{\bar {i}}_{1}}\) ≤ 0:

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} - \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} - \frac{1}{2}. \hfill \\ \end{gathered} \right.$$

Doing the same manipulations, we can obtain the result for system (32) with –1:

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} - \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} - \frac{1}{2}, \hfill \\ \end{gathered} \right.$$

when i2 + i1 and \({{\bar {i}}_{2}}\) + \({{\bar {i}}_{1}}\) ≥ 0, and

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} + \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} + \frac{1}{2}, \hfill \\ \end{gathered} \right.$$

when i2 + i1 and \({{\bar {i}}_{2}}\) + \({{\bar {i}}_{1}}\) ≤ 0.

We can combine different cases in the following way:

$$\left\{ \begin{gathered} {{i}_{2}} = {{{\bar {i}}}_{2}} \pm \frac{1}{2} \hfill \\ {{i}_{1}} = {{{\bar {i}}}_{1}} \pm \frac{1}{2}. \hfill \\ \end{gathered} \right.$$
(59)

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Letunov, A.Y., Lisitsa, V.S. Stark–Zeeman and Blokhintsev Spectra of Rydberg Atoms. J. Exp. Theor. Phys. 131, 696–706 (2020). https://doi.org/10.1134/S106377612010012X

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