# Dynamic Instability of Rydberg Atomic Complexes

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}(where n is the principal quantum number), which results in huge dipole moments. This opens up unique opportunities for both the controlled and addressed management of quantum states by external electromagnetic fields [4], and for the creation of long-lived coherent (entangled) states in cold Rydberg media due to the long range dipole-dipole interaction between the medium particles [5,6]. Therefore, the cold Rydberg atoms are considered to be promising objects for solving the problems of quantum information. With their help, the physical carriers of quantum bits [7] can be realized with the simultaneous execution of the basic quantum operations [8].

## 2. Kinetics of Radiative Transitions for Highly Excited Atoms

#### 2.1. Spectral Parameters of an Excited Atom

^{5}, becomes possible due to intense atom-atom interactions. In this case, the principal fraction of the population of excited states is from those with large l values. In the case of a complete l mixing of the Hydrogen atom Rydberg states, the expression for the probabilities ${\langle A\rangle}_{n}$ of radiative decay of the block of n-states is written in the form [31,34]:

#### 2.2. Blocking of the Spectral Transitions. Double Stark (Förster) Resonance

^{−1}[36]. The criterion for the emergence of the FR is simply formulated in terms of a quantum defect of atomic series:

## 3. Rydberg Quasimolecular Complex in the Framework of the Dipole Resonance Mechanism Model

**D**of the quasimolecule ${\mathrm{A}}_{2}^{**}$. The moment

**D**arises in the process of the charge exchange in the system (A + A

^{+}) and it induces an alternating electric field

**E**(t), perturbing the motion of the RE ${e}_{nl}^{-}$ on the Coulomb orbit (see Figure 3). With respect to the ionization channels for the DRI model, the following simplifying assumptions are accepted: (i) both the trajectory of the external electron ${e}_{nl}^{-}$ and relative motion of the ion ${\mathrm{A}}^{+}$ and unexcited atom A are semiclassical (ii) with the initial impact parameter $\rho $; and, (iii) ionization proceeds within a certain region with a given limiting distance ${R}_{ion}$ between colliding atoms, which depends on the type of ion-atom residue and it is a parameter of the theory. The system is traditionally described in the adiabatic approximation using the appropriate potential curves of the Rydberg complex and the molecular ion ${\mathrm{A}}_{2}^{+}$.

^{+}within the quasimolecule ion ${\mathrm{A}}_{2}^{+}$ leads to the splitting $\Delta \left(\mathrm{R}\right)$ (known as “exchange interaction” [45]) of its energy levels and it creates a time-dependent dipole moment

**D**= ${\mathbf{R}}^{\mathrm{cos}(\Delta \left(\mathrm{R}\right)\mathrm{t})/2}$, which oscillates with frequency $\omega =\Delta \left(\mathrm{R}\right)$, i.e., outside the complex ${\mathrm{A}}_{2}^{+}$ an alternating quasimonochromatic microwave electric field

**E**(t) is induced (see Figure 3). We note that, as shown in [49], the effect of the field

**E**(t) on the Rydberg electron ${e}_{nl}^{-}$ is equivalent to its perturbation by an external, spatially uniform field with the frequency ${\omega}_{L}=\Delta \left(\mathrm{R}\right)$, and with polarization along the interatomic axis

**R**. Ionization occurs inside the range of distances ($\mathrm{R}<{\mathrm{R}}_{\mathrm{ion}}$), where the exchange interaction $\omega =\Delta \left(\mathrm{R}\right)$), starting from the threshold value $\Delta ({\mathrm{R}}_{\mathrm{ion}})$, exceeds the binding energy $|{\epsilon}_{nl}|=1/(2{n}^{*2})$ of the ${e}_{nl}^{-}$ electron and, thus, opens the autoionization channel of the quasimolecule complex ${\mathrm{A}}_{2}^{**}$. The probability of ionization per unit time, or the autoionization width of the process, is expressed in terms of the photoionization cross section ${\sigma}_{ph}(nl,\omega )$.

## 4. Rydberg Collisional Complex ${\mathbf{A}}_{\mathbf{2}}^{\mathbf{*}\mathbf{*}}$ in Approximation of Dynamic Chaos

#### 4.1. Nonlinear Dynamic Resonances and the Emergence of Deterministic Chaos

#### 4.2. The Standard Map (SM)

_{1,m}between two adjacent resonance values I

_{1,m}is ΔI

_{1,m}= 1, which gives, for the corresponding energy separation, $\Delta \epsilon =|\omega (I)|\Delta I=|\omega (I)|$ (see Figure 6). Chirikov criterion (11), hence, is reduced to the form

#### 4.3. Conception of Diffusional Ionization

**E**(t), nonlinear dynamic resonances can arise due to the coincidence of the overtone ${k}_{0}{\omega}_{\epsilon}$ of the angular frequency ${\omega}_{\epsilon}$ of motion of RE ${e}_{nl}^{-}$ on the Keplerian orbit with charge-exchange frequency $\Delta \left(\mathrm{R}\right)$ of the internal electron ${e}^{-}$ (see Figure 3). As a result, the motion of the RE becomes unstable, and the RE evolution in the energy space takes the character of random walks along the quasi-intersecting “grid” of potential curves (see Figure 4), which opens the possibility of a kinetic description of the RE dynamics

#### 4.4. Diffusional Ionization of Hydrogen Atom in External Field

**L**in conditions of dynamic chaos development. We take the initial 10P (${n}_{0}=10,{l}_{0}=1$) state of the hydrogen atom in the microwave field with frequency ${\omega}_{L}=3/{10}^{3}$ and amplitude ${E}_{0}$, exceeding its threshold value ${E}_{c}=2/(49{n}_{0}^{4})$ [22]. Two characteristic initial configurations of the vector ${\mathbf{L}}_{0}$ and the Runge-Lenz vector ${\mathbf{A}}_{0}$ (directed along the semiaxis of the unperturbed Keplerian orbit [58]) were chosen, corresponding to the maximal changes in the modulus |

**L|**for cases of two-dimensional (${E}_{0}=8,2{E}_{c}$) and three-dimensional (${E}_{0}=6,5{E}_{c}$) trajectories. Note that, according to the literature data, the range of values of n ≈ 10 corresponds to the range of strong interaction of the dipole field of the cluster ${\mathrm{A}}_{2}^{+}$ with the Rydberg electron ${e}_{nl}^{-}$ (see Figure 3). Also note that the results of numerical calculations in Figure 8 show a significant change of the orbital momentum

**L**in the microwave field under the conditions of development of global chaos, which is in contrast to the main approximation of the authors [22,51], assuming the adiabatic invariance of

**L**.

#### 4.5. Diffusional Ionization of the Rydberg Colisional Complex

#### 4.6. Assotiative Ionization Rate Constants

_{c}of colliding atoms of the same mass M:

#### 4.7. Features of Diffusional Ionization under Conditions of Förster Resonance

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Stebbings, R.F.; Dunning, F.B. (Eds.) Rydberg States of Atoms and Molecules; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Gallagher, T.F. Rydberg Atoms; Cambridge Monographs on Atomic, Molecular and Chemical Physics; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Jones, M.P.A.; Marcassa, L.G.; Shaffer, J.P. Special issue on Rydberg atom physics. J. Phys. B At. Mol. Phys.
**2017**, 50, 060202. [Google Scholar] [CrossRef] - Lim, J.; Lee, H.-G.; Ahn, J. Review of cold Rydberg atoms and their applications. J. Korean Phys. Soc.
**2013**, 63, 867–876. [Google Scholar] [CrossRef] - Hofmann, C.S.; Günter, G.; Schempp, H.; Müller, N.L.; Faber, A.; Busche, H.; Robert-de-Saint-Vincent, M.; Weidemüller, M. An experimental approach for investigating many-body phenomena in Rydberg-interacting quantum systems. Front. Phys.
**2014**, 9, 571–586. [Google Scholar] [CrossRef] - Pillet, P.; Gallagher, T.F. Rydberg atom interactions from 300 K to 300 K. J. Phys. B At. Mol. Opt. Phys.
**2016**, 49, 174003. [Google Scholar] [CrossRef] - Saffman, M.; Walker, T.G.; Mølmer, K. Quantum information with Rydberg atoms. Rev. Mod. Phys.
**2010**, 82, 2313. [Google Scholar] [CrossRef] - Ryabtsev, I.I.; Beterov, I.I.; Tretyakov, D.B.; Entin, V.M.; Yakshina, E.A. Spectroscopy of cold rubidium rydberg atoms for applications in quantum information. Phys.-Uspekhi
**2016**, 59, 196–208. [Google Scholar] [CrossRef] - Marcassa, L.G.; Shaffer, J.P. Interactions in Ultracold Rydberg Gases. In Advances in Atomic, Molecular, and Optical Physics; Arimondo, E., Berman, P.R., Lin, C.C., Eds.; Academic: New York, NY, USA, 2014; Volume 63, pp. 47–133. [Google Scholar]
- Shaffer, J.P.; Rittenhouse, S.T.; Sadeghpour, H.R. Ultracold Rydberg molecules. Nat. Commun.
**2018**, 9, 1965. [Google Scholar] [CrossRef] [PubMed] - Schlagmüller, M.; Liebisch, T.C.; Engel, F.; Kleinbach, K.S.; Böttcher, F.; Hermann, U.; Westphal, K.M.; Gaj, A.; Löw, R.; Hofferberth, S.; et al. Ultracold Chemical Reactions of a Single Rydberg Atom in a Dense Gas. Phys. Rev. X
**2016**, 6, 031020. [Google Scholar] [CrossRef] - Lyon, M.; Rolston, S.L. Ultracold neutral plasmas. Rep. Prog. Phys.
**2017**, 80, 017001. [Google Scholar] [CrossRef] [PubMed] - Klyucharev, A.N.; Vujnović, V. Chemi-ionization in thermal-energy binary collisions of optically excited atoms. Phys. Rep.
**1990**, 185, 55–81. [Google Scholar] [CrossRef] - Graham, W.G.; Fritsch, W.; Hahn, Y.; Tanis, J.A. Recombination of Atomic Ions; Springer Science & Business Media: Berlin, Germany, 2012. [Google Scholar]
- Beterov, I.I.; Tretyakov, D.B.; Ryabtsev, I.I.; Entin, V.M.; Ekers, A.; Bezuglov, N.N. Ionization of rydberg atoms by blackbody radiation. New J. Phys.
**2009**, 11, 013052. [Google Scholar] [CrossRef] - Hahn, Y. Density dependence of molecular autoionization in a cold gas. J. Phys. B At. Mol. Opt. Phys.
**2000**, 33, L655. [Google Scholar] [CrossRef] - Efimov, D.K.; Miculis, K.; Bezuglov, N.N.; Ekers, A. Strong enhancement of Penning ionization for asymmetric atom pairs in cold Rydberg gases: The Tom and Jerry effect. J. Phys. B At. Mol. Opt. Phys.
**2016**, 49, 125302. [Google Scholar] [CrossRef] - Gnedin, Y.N.; Mihajlov, A.A.; Ignjatović, L.J.M.; Sakan, N.M.; Srećković, V.A.; Zakharov, M.Y.; Bezuglov, N.N.; Klycharev, A.N. Rydberg atoms in astrophysics. New Astron. Rev.
**2009**, 53, 259–265. [Google Scholar] [CrossRef] - Buenker, R.J.; Golubkov, G.V.; Golubkov, M.G.; Karpov, I.; Manzheliy, M. Relativity laws for the variation of rates of clocks moving in free space and GPS positioning errors caused by space-weather events. In Global Navigation Satellite System-From Stellar Navigation; Mohamed, A.H., Ed.; In Tech: Berlin, Germany, 2013. [Google Scholar]
- Koch, P.M.; van Leeuwen, K.A.H. The importance of resonances in microwave “ionization” of excited hydrogen atoms. Phys. Rep.
**1995**, 255, 289–403. [Google Scholar] [CrossRef] - Mitchell, K.A.; Handlay, J.P.; Tighe, B.; Flower, A.; Delos, J.B. Analysis of chaos-induced pulse trains in the ionization of hydrogen. Phys. Rev. A
**2004**, 70, 043407. [Google Scholar] [CrossRef] - Krainov, V.P. Ionization of atoms in strong low-frequency electromagnetic field. J. Exp. Theor. Phys.
**2010**, 111, 171–179. [Google Scholar] [CrossRef] - Park, H.; Shuman, E.S.; Gallagher, T.F. Ionization of Rb Rydberg atoms in the attractive ns-np dipole-dipole potential. Phys. Rev. A
**2011**, 84, 052708. [Google Scholar] [CrossRef] - Dashevskaya, E.I.; Litvin, I.; Nikitin, E.E.; Oref, I.; Troe, J. Classical diffusion model of vibrational predissociation of van der Waals complexes Part III. Comparison with quantum calculations. Phys. Chem. Chem. Phys.
**2002**, 4, 3330–3340. [Google Scholar] [CrossRef] - Bezuglov, N.N.; Golubkov, G.V.; Klyucharev, A.N. Ionization of Excited Atoms in Thermal Collisions; The Atmosphere and Ionosphere: Elementary Processes, Discharges and Plasmoids; Springer: New York, NY, USA; London, UK, 2013; Chapter 1; pp. 1–60. [Google Scholar]
- Reichl, L.E. The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations; Springer: New York, NY, USA, 2004. [Google Scholar]
- Zaslavskii, G.M. Physics of Chaos in Hamiltonian Systems, 2nd ed; Imperial College Press: London, UK, 2007. [Google Scholar]
- Bezuglov, N.N.; Golubkov, G.V.; Klyucharev, A.N. Manifestations of “Dynamic Chaos” in Reactions with Participation of Rydberg States; St. Petersburg State University: St. Petersburg, Russia, 2017. (In Russian) [Google Scholar]
- Paris-Mandoki, A.; Gorniaczyk, H.; Tresp, C.; Mirgorodskiy, I.; Hofferberth, S. Tailoring Rydberg interactions via Förster resonances: State combinations, hopping and angular dependence. J. Phys. B
**2016**, 49, 164001. [Google Scholar] [CrossRef] - Gianninas, A.; Dufour, P.; Kilic, M.; Brown, W.R.; Bergeron, P.; Hermes, J.J. Precise atmospheric parameters for the shortest-period binary white dwarfs: Gravitational waves, metals, and pulsations. Astrophys. J.
**2014**, 794, 35–52. [Google Scholar] [CrossRef] - Bezuglov, N.N.; Borisov, E.N.; Verolainen, Y.F. Distribution of the radiative lifetimes over the excited states of atoms and ions. Sov. Phys. Uspekhi
**1991**, 34, 3–29. [Google Scholar] [CrossRef] - Hezel, T.P.; Burkhardt, C.E.; Ciocca, M.; He, L.W.; Leventhal, J.J. Classical view of the properties of Rydberg atoms: Application of the correspondence principle. Am. J. Phys.
**1992**, 60, 329–335. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Quantum Mechanics; Pergamon: Oxford, UK, 1977. [Google Scholar]
- Omidvar, K. Semiclassical formula for the radiative mean lifetime of the excited state of the hydrogenlike atoms. Phys. Rev. A
**1982**, 26, 3053–3061. [Google Scholar] [CrossRef] - Mack, M.; Grimmel, J.; Karlewski, F.; Sárkány, L.; Hattermann, H.; Fortágh, J. All-optical measurement of Rydberg-state lifetimes. Phys. Rev. A
**2015**, 92, 012517. [Google Scholar] [CrossRef] - Tretyakov, D.B.; Beterov, I.I.; Entin, V.M.; Yakshina, E.A.; Ryabtsev, I.I.; Dyubko, S.F.; Alekseev, E.A.; Pogrebnyak, N.L.; Bezuglov, N.N.; Arimondo, E. Effect of photoions on the line shape of the Förster resonance lines and microwave transitions in cold rubidium Rydberg atoms. J. Exp. Theor. Phys.
**2012**, 114, 14–24. [Google Scholar] [CrossRef] - Hund, F. The History of Quantum Theory; Barnes & Noble Books: New York, NY, USA, 1974; Chapter 11. [Google Scholar]
- Bokulich, P.; Bokulich, A. Niels Bohr’s generalization of classical mechanics. Found. Phys.
**2005**, 35, 347–371. [Google Scholar] [CrossRef] - Delone, N.B.; Goreslavsky, S.P.; Krainov, V.P. Dipole matrix elements in the quasi-classical approximation. J. Phys. B
**1994**, 27, 4403. [Google Scholar] [CrossRef] - Arefieff, K.N.; Miculis, N.; Bezuglov, N.N.; Dimitrijević, M.S.; Klyucharev, A.N.; Mihajlov, A.A.; Srećković, V.A. Dynamics Resonances in Atomic States of Astrophysical Relevance. J. Astrophys. Astron.
**2015**, 36, 613–622. [Google Scholar] [CrossRef] - Sommerfeld, A. Atomic Structure and Spectral Lines; Methuen: London, UK, 1934. [Google Scholar]
- Grouzdev, P.F. Atomic and Ionic Spectra in X-ray and Ultraviolet Region’s; Energoatomizdat: Moscow, Russia, 1982. (In Russian) [Google Scholar]
- Zakharov, M.Y.; Bezuglov, N.N.; Klyucharev, A.N.; Matveev, A.A.; Beterov, I.I.; Dulieu, O. Specifics of the stochastic ionization of a Rydberg collision complex with Förster resonance. Russ. J. Phys. Chem. B
**2011**, 5, 537–545. [Google Scholar] [CrossRef] - Fermi, E. Sopra lo spostamento per pressione delle righe elevate delle serie spettrali. Nuovo Cimento
**1934**, 11, 157–166. [Google Scholar] [CrossRef] - Janev, R.K.; Mihajlov, A.A. Resonant ionization in slow-atom-Rydberg-atom collisions. Phys. Rev. A
**1980**, 21, 819–826. [Google Scholar] [CrossRef] - Mihajlov, A.A.; Janev, R.K. Ionisation in atom-Rydberg atom collisions: Ejected electron energy spectra and reaction rate coefficients. J. Phys. B
**1981**, 14, 1639. [Google Scholar] [CrossRef] - Duman, E.L.; Shmatov, I.P. Ionization of highly excited atoms in their own gas. Sov. Phys. JETP
**1980**, 51, 1061–1065. [Google Scholar] - Srećković, V.A.; Dimitrijević, M.S.; Ignjatović, Lj.M.; Bezuglov, N.N.; Klyucharev, A. The Collisional Atomic Processes of Rydberg Hydrogen and Helium Atoms: Astrophysical Relevance. Galaxies
**2018**, 6, 72. [Google Scholar] [CrossRef] - Bezuglov, N.N.; Borodin, V.M.; Kazanskiy, A.K.; Klyucharev, A.N.; Matveev, A.A.; Orlovskii, K.V. Analysis of Fokker-Planck type stochastic equations with variable boundary conditions in an elementary process of collisional ionization. Opt. Spectrosc.
**2001**, 91, 19–26. [Google Scholar] [CrossRef] - Zaslavskij, G.M. Hamiltonian Chaos and Fractional Dynamics; Oxford Univ. Press: Oxford, UK, 2005. [Google Scholar]
- Delone, N.B.; Krainov, V.P.; Shepelyanskii, D.L. Highly-excited atoms in the electromagnetic field. Sov. Phys. Uspekhi
**1983**, 26, 551. [Google Scholar] [CrossRef] - Bezuglov, N.N.; Borodin, V.M.; Ekers, A.; Klyucharev, A.N. A quasi-classical description of the stochastic dynamics of a Rydberg electron in a diatomic quasi-molecular complex. Opt. Spectrosc.
**2002**, 93, 661–669. [Google Scholar] [CrossRef] - Ryabtsev, I.I.; Tretyakov, D.B.; Beterov, I.I.; Bezuglov, N.N.; Miculis, K.; Ekers, A. Collisional and thermal ionization of sodium Rydberg atoms: I. Experiment for nS and nD atoms with n = 8–20. J. Phys. B
**2005**, 38, S17–S35. [Google Scholar] [CrossRef] - Efimov, D.K.; Bezuglov, N.N.; Klyucharev, A.N.; Gnedin, Y.N.; Miculis, K.; Ekers, A. Analysis of light-induced diffusion ionization of a three-dimensional hydrogen atom based on the Floquet technique and split-operator method. Opt. Spectrosc.
**2014**, 117, 8–17. [Google Scholar] [CrossRef] - Chu, S.-I.; Telnov, D.A. Beyond the Floquet theorem: Generalized Floquet formalisms and quasienergy methods for atomic and molecular multiphoton processes in intense laser fields. Phys. Rep.
**2004**, 390, 1–131. [Google Scholar] [CrossRef] - Hairer, E. Numeral Geometric Integration; Universite de Geneve: Geneve, Switzerland, 1999. [Google Scholar]
- Kazansky, A.K.; Bezuglov, N.N.; Molisch, A.F.; Fuso, F.; Allegrini, M. Direct numerical method to solve radiation trapping problems with a Doppler-broadening mechanism for partial frequency redistribution. Phys. Rev. A
**2001**, 64, 022719. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M. Mechanics Course of Theoretical Physics Mechanics; (Nauka, Moscow, 1973); English Tran.; Permagon Press: Oxford, UK; New York, NY, USA; Toronto, ON, Canada, 1976; Volume 1. [Google Scholar]
- Ramsey, N.F. Molecular Beams, 2nd ed.; Clarendon: Oxford, UK, 1989. [Google Scholar]
- Michulis, K.; Beterov, I.I.; Bezuglov, N.N.; Ryabtsev, I.I.; Tretyakov, D.B.; Ekers, A.; Klucharev, A.N. Collisional and thermal ionization of sodium Rydberg atoms: II. Theory for nS, nP and nD states with n= 5–25. J. Phys. B
**2005**, 38, 1811–1831. [Google Scholar] [CrossRef] - Sydoryk, I.; Bezuglov, N.N.; Beterov, I.I.; Miculis, K.; Saks, E.; Janovs, A.; Spels, P.; Ekers, A. Broadening and intensity redistribution in the Na(3p) hyperfine excitation spectra due to optical pumping in the weak excitation limit. Phys. Rev. A
**2008**, 77, 042511. [Google Scholar] [CrossRef] - Kirova, T.; Cinins, A.; Efimov, D.K.; Bruvelis, M.; Miculis, K.; Bezuglov, N.N.; Auzinsh, M.; Ryabtsev, I.I.; Ekers, A. Hyperfine interaction in the Autler-Townes effect: The formation of bright, dark, and chameleon states. Phys. Rev. A
**2017**, 96, 043421. [Google Scholar] [CrossRef] - Porfido, N.; Bezuglov, N.N.; Bruvelis, M.; Shayeganrad, G.; Birindelli, S.; Tantussi, F.; Guerri, I.; Viteau, M.; Fioretti, A.; Ciampini, D.; et al. Nonlinear effects in optical pumping of a cold and slow atomic beam. Phys. Rev. A
**2015**, 92, 043408. [Google Scholar] [CrossRef] - Klyucharev, A.N.; Bezuglov, N.N.; Matveev, A.A.; Mihajlov, A.A.; Ignjatović, L.M.; Dimitrijević, M.S. Rate coefficients for the chemi-ionization processes in sodium- and other alkali-metal geocosmical plasmas. New Astron. Rev.
**2007**, 51, 547–562. [Google Scholar] [CrossRef] - Tantussi, F.; Mangasuli, V.; Porfido, N.; Prescimone, F.; Fuso, F.; Arimondo, E.; Allegrini, M. Towards laser-manipulated deposition for atom-scale technologies. Appl. Surf. Sci.
**2009**, 255, 9665–9670. [Google Scholar] [CrossRef] - Boulmer, J.; Bonanno, R.; Weiner, J. Crossed-beam measurements of absolute rates coefficients in associative ionization collisions between Na*(np) and Na(3s) for 5 ≤ n ≤ 15. J. Phys. B
**1983**, 16, 3015–3024. [Google Scholar] [CrossRef] - Weiner, J.; Boulmer, J. Associative ionization rate constants as a function of quantum numbers n and l in Na*(np) + Na(3s) collisions for 17 ≤ n ≤ 27 and l = 0, l = 1 and l ≥ 2. J. Phys. B
**1986**, 19, 599–609. [Google Scholar] [CrossRef] - Beterov, I.I.; Tretyakov, D.B.; Ryabtsev, I.I.; Bezuglov, N.N.; Miculis, K.; Ekers, A.; Klucharev, A.N. Collisional and thermal ionization of sodium Rydberg atoms III. Experiment and theory for nS and nD states with n = 8–20 in crossed atomic beams. J. Phys. B
**2005**, 38, 4349–4361. [Google Scholar] [CrossRef]

**Figure 1.**Scheme of Rydberg atomic levels, illustrating the effect of the double Stark (Förster) resonance in {l + 1, l} atomic series on blocking of the “long” transitions between remote states.

**Figure 2.**(

**a**) Natural width ${A}_{nl}$ of Rydberg state of s-series (l = 0) with n = 30 as a function of parameter α of Sommerfeld’s atom. (

**b**) The same for the state of p-series (l = 1) with n = 25. The Förster resonance corresponds to the value ${\alpha}_{p,s}$ = 2.81.

**Figure 3.**Scheme of highly excited collisional complex ${\mathrm{A}}_{2}^{**}$ where

**D**—vector of the quasi-molecular ion dipole moment,

**R**—vector of internuclear distance, ${e}_{nl}^{-}$—Rydberg electron, which is shared by the atomic cores of the quasi-molecular ion ${\mathrm{A}}_{2}^{+}$.

**Figure 5.**Schematic of nonlinear dynamic resonances of different $\left\{{k}_{0}{m}_{0}\right\}$ orders for the initial quantum state ${n}_{0}$: (

**a**) single-photon (${m}_{0}=1$) and (

**b**) double-photon (${m}_{0}=2$) resonances.

**Figure 6.**(

**a**) Occurrence of dynamic nonlinear resonances without their widths $\delta \epsilon $ overlapping. (

**b**) Complete overlapping of widths $\delta \epsilon $, corresponding to the formation of the global dynamic chaos. The initial states correspond to the lowest levels ${n}_{0}$ (bold lines).

**Figure 7.**Phase space trajectories of a kicked rotor with Hamiltonian (12) (adopted from [26]).

**Figure 8.**Trajectories of Rydberg electron (frames

**a**,

**c**) and the evolution of its angular momentum

**L**(frames

**b**,

**d**) for the 10P-state (${n}_{0}=10,{l}_{0}=1$) of the hydrogen atom. For two-dimensional (2D) motion of an electron in the {X, Y}-plane (frame

**a**), frame

**b**shows the projection ${L}_{Z}$ of the momentum

**L**on the z-axis, orthogonal to the motion plane {X, Y}. For three-dimensional (3D) motion (frame

**c**), frame

**d**shows |

**L|**.

**Figure 9.**The time evolution of the distribution function $f(n,t)$ of a Rydberg electron ${e}_{nl}^{-}$ in a quasimolecular collisional complex ${\mathrm{A}}_{2}^{**}$ with the impact parameter $\rho =15$ and collision energy $1.9\cdot {10}^{-3}$ a. u. = 600 K. The initial value ${n}_{0}=10$.

**Figure 10.**Distribution function F in a single beam (sb), crossed beams (cb), counter beams (cb’), and gas cell (c).

**Figure 13.**Temporal evolution of binding energy ε of the 13P-state (l = 1) of the Rydberg electron in the Sommerfeld atom in an external microwave field of frequency ω = 1/13

^{3}and amplitude ${E}_{0}=10{E}_{c}$, which exceeds ten times the critical value ${E}_{c}$. The calculations were performed for three values of Sommerfeld parameter: α = 0, 2.81, and 4.5. The arrows indicate the moments of ionization. The occurrence of Förster resonance corresponds to ${\alpha}_{p,s}$ = 2.81.

**Table 1.**Qantum defect ${\mu}_{l}$ for s-, p-series [32] along with $\Delta {\mu}_{P}$ and the factor $1/{\tau}_{P}$ from Equation (1).

Li | Na | K | Rb | Cs | H | |
---|---|---|---|---|---|---|

s | 0.40 | 1.35 | 2.19 | 3.13 | 4.06 | 0 |

p | 0.04 | 0.85 | 1.71 | 2.66 | 3.59 | 0 |

$\Delta {\mu}_{P}$ | 0.36 | 0.50 | 0.48 | 0.47 | 0.47 | 0 |

$10/{\tau}_{P}$ | 0.69 | 0.14 | 0.51 | 0.75 | 0.61 | 10 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dimitrijević, M.S.; Srećković, V.A.; Zalam, A.A.; Bezuglov, N.N.; Klyucharev, A.N.
Dynamic Instability of Rydberg Atomic Complexes. *Atoms* **2019**, *7*, 22.
https://doi.org/10.3390/atoms7010022

**AMA Style**

Dimitrijević MS, Srećković VA, Zalam AA, Bezuglov NN, Klyucharev AN.
Dynamic Instability of Rydberg Atomic Complexes. *Atoms*. 2019; 7(1):22.
https://doi.org/10.3390/atoms7010022

**Chicago/Turabian Style**

Dimitrijević, Milan S., Vladimir A. Srećković, Alaa Abo Zalam, Nikolai N. Bezuglov, and Andrey N. Klyucharev.
2019. "Dynamic Instability of Rydberg Atomic Complexes" *Atoms* 7, no. 1: 22.
https://doi.org/10.3390/atoms7010022