Abstract
In the literature, there were studies of Rydberg states of hydrogenic atoms/ions in a high-frequency laser field. It was shown that the motion of the Rydberg electron is analogous to the motion of a satellite around an oblate planet (for a linearly polarized laser field) or around a (fictitious) prolate planet (for a circularly polarized laser field): it exhibits two kinds of precession – one of them is the precession within the orbital plane and another one is the precession of the orbital plane. In this study, we study a helium atom or a helium-like ion with one of the two electrons in a Rydberg state, the system being under a high-frequency laser field. For obtaining analytical results, we use the generalized method of the effective potentials. We find two primary effects of the high-frequency laser field on circular Rydberg states. The first effect is the precession of the orbital plane of the Rydberg electron. We calculate analytically the precession frequency and show that it differs from the case of a hydrogenic atom/ion. In the radiation spectrum, this precession would manifest as satellites separated from the spectral line at the Kepler frequency by multiples of the precession frequency. The second effect is a shift of the energy of the Rydberg electron, also calculated analytically. We find that the absolute value of the shift increases monotonically as the unperturbed binding energy of the Rydberg electron increases. We also find that the shift has a nonmonotonic dependence on the nuclear charge Z: as Z increases, the absolute value of the shift first increases, then reaches a maximum, and then decreases. The nonmonotonic dependence of the laser field-caused energy shift on the nuclear charge is a counterintuitive result.
1 Introduction
The previous studies [1,2,3] focused on hydrogenic atoms/ions in a high-frequency laser field. In particular, ref. [2,3] focused on Rydberg states in the classical description. Ref. [2,3], by “high frequency,” mean that the laser frequency ω is much greater than the Kepler frequency ω K = m e e 4/(n 3 ħ 3) of the highly excited hydrogen atom: ω ≫ ω K. Here, m e and e are the electron mass and charge, respectively; n ≫ 1 is the principal quantum number. In this situation, the laser field and the Rydberg atom can be considered as the fast and slow subsystems, respectively, thus allowing the analytical treatment of the problem. In particular, the authors of ref. [2] generalized Kapitza’s method of effective potentials [4,5].
In ref. [1,2,3], it was revealed that this fundamental problem exhibits a rich physics. When the laser field is linearly polarized or circularly polarized, the system has the axial symmetry, so that the square of the angular momentum M 2 should not be conserved (only its projection M z on the axis of the symmetry is conserved). However, in ref. [1,2,3], it was shown that M 2 is approximately conserved in this situation, so that there is an approximate algebraic symmetry higher than the geometrical symmetry.
In addition, when the laser field is linearly polarized or circularly polarized, the system has celestial analogies. Namely, in the linearly polarized laser field, the motion of the Rydberg electron is analogous to the motion of a satellite around an oblate planet: it exhibits two kinds of precession – one of them is the precession within the orbital plane and another one is the precession of the orbital plane. In the circularly polarized laser field, the motion of the Rydberg electron is analogous to the motion of a satellite around a (fictitious) prolate planet: it also exhibits the same two kinds of precession.
In this article, we study a helium atom or a helium-like ion with one of the two electrons in a Rydberg state, the system being under a high-frequency laser field. For obtaining analytical results, we use the generalized method of the effective potentials from previous studies [2,6] and in the previously published book [7].
Then, we focused at circular Rydberg states.[1] We show that the high-frequency laser field causes the precession of the orbital plane of the Rydberg electron. We calculate analytically the precession frequency and demonstrate that it differs from the case of a hydrogenic atom/ion. In the radiation spectrum, this precession would manifest as satellites separated from the spectral line at the Kepler frequency by multiples of the precession frequency.
We also show that the high-frequency laser field also causes a red shift of the energy of the Rydberg electron. We calculate analytically this energy shift and study its dependence on parameters of the system.
2 New results
We study a He-like atom or ion in a high-frequency laser field. The atom/ion has the inner electron in state 1s and the highly excited (Rydberg) outer electron. The potential Φ of a quasinucleus consisting of the nucleus Z and a spherically symmetric charge distribution corresponding to the inner electron in state 1s is expressed as follows (see, e.g., ref. [39]):
where μ = M n m e/(M n + m e) is the reduced mass of the pair “nucleus Z – electron” (M n is the nuclear mass and m e is the electron mass) and r is the distance from the center of symmetry to the electron. In this study, we use atomic units ħ = e = m e = 1. The atom is subjected to a high-frequency laser field of amplitude F and frequency ω. For Rydberg electrons, a classical or semi-classical treatment is appropriate.
2.1 Linear polarization of the laser field
First, we consider the case of the linear polarization of the laser field. The semi-classical Hamiltonian for the outer (Rydberg) electron in this configuration can be represented in the following form:
where μ 1 = m e(M n + m e)/(M n + 2m e) is the reduced mass of the pair “nucleus Z with the inner electron – outer electron,” the z-axis is in the direction of the laser field F, (r, θ, φ) are the spherical coordinates of the electron, F is the magnitude, and ω is the frequency of the laser field. Both μ and μ 1 are very close to unity: their physical values lie in the range between 0.999864 (for helium, Z = 2) and 0.999998 (for He-like oganesson, Z = 118). For the systems in a high-frequency field, when the frequency of the field is much greater than the highest frequency of the unperturbed system, it is appropriate to use the formalism of the effective potentials [2,4,5,6,7]. As a result, the Hamiltonian H 0 acquires a time-independent term. The zeroth-order effective potential is expressed as follows:
where V = zF, [P, Q] are the Poisson brackets, which is a coordinate-independent energy shift that does not affect the dynamics of the system. The first nonvanishing effect on the dynamics of the system originates from the first-order effective potential:
where we use the notation
so that the effective potential energy of the electron is expressed as follows:
where U is the unperturbed potential energy.
In ref. [2], where the authors studied a hydrogen Rydberg atom in a linearly polarized high-frequency laser field, it was shown that the effective potential energy has the following form:
where θ is the polar angle, that is, the angle between the radius-vector r of the electron and the z-axis chosen along the vector-amplitude F of the laser field. This effective potential energy, which is mathematically equivalent to the potential energy of a satellite around the oblate Earth (see, e.g., ref. [39]), has the following property: in the general case, where the unperturbed electron orbit is elliptical, the orbit undergoes simultaneously two precessions. One is the precession of the ellipse in its plane, and another is the precession of the orbital plane about the vector F. Both of the precession frequencies are of the same order of magnitude and are much smaller than the Kepler frequency of the electron.
In our case, the effective potential energy is generally more complicated. Therefore, we limit ourselves by the situation where the unperturbed orbit of the outer electron is circular. In this situation, the precession in the orbital plane loses its meaning, and we deal only with the precession of the orbital plane.
So, we fix r = const, and only the angle θ remains as the dynamic variable. Then, our effective potential energy can be brought to the form (7) with an additional θ-independent form, as follows:
Introducing the functions:
we can rewrite (4) in the following form:
Therefore, the total energy can be represented in the following form:
where the second term is the energy shift with respect to the unperturbed energy, and the third term is responsible for the precession.
From (9) and (7), we see that the term
is a relative correction to the precession frequency of the orbital plane, and the term
corresponds to an energy shift. Taking into account the factor in the beginning of (9), the energy shift is expressed as follows:
For circular orbits, the energy of the outer electron is E = −(Z − 1)/(2r), and, using (5), we write the energy shift as follows:
Figure 1 shows the dependence of the energy shift on the unperturbed electron energy for Z = 4, F = 1, and ω = 100. The electron Kepler frequency at these energies is expressed as follows:
which is indeed much smaller than the laser frequency.
The energy shift versus the unperturbed electron energy for Z = 4, F = 1, and ω = 100.
It is seen that the shift is zero at the zero unperturbed energy and approaches the limit
The energy shift versus the unperturbed electron energy E for Z = 4, F = 1, and ω = 100 for large negative values of E.
Figure 3 presents the dependence of the energy shift on the nuclear charge Z at F = 1 and ω = 100 for two values of the unperturbed energy: E = −2 (blue, solid line) and E = −5 (red, dashed line). It has a minimum at the point Z m ≈ (1 + (1 + 24|E|)1/2)/4 (in the approximation μ = μ 1 = 1). The nonmonotonic dependence of the energy shift on the nuclear charge is a counterintuitive result.
The energy shift versus the nuclear charge Z for E = −2 (blue, solid line) and E = −5 (red, dashed line) for F = 1, ω = 100.
Figure 4 shows a 3D plot of the dependence of the energy shift on both the nuclear charge and the unperturbed energy for F = 1 and ω = 100. It is seen that the location of the minimum of the energy shift with respect to the nuclear charge indeed moves to higher values of Z as the unperturbed energy increases in the absolute value.
The dependence of the energy shift on Z and E for F = 1, ω = 100.
The motion characterized by the effective potential energy can be expressed as follows:
which is mathematically equivalent to the motion of a satellite around the oblate Earth, and in the latter case, the effective potential (according to ref. [39]) is expressed as follows:
Here M E and m are the Earth and the satellite masses, respectively; R is the equatorial radius of the Earth; and I 2 is a constant characterizing the relative difference between the equatorial and polar diameters of the Earth. The ratio of the precession frequency of the satellite plane Ω to the Kepler frequency ω K is expressed as follows [40]:
where
is the Kepler frequency of the satellite and m and A s are the satellite mass and the major semi-axis of its unperturbed elliptical orbit, respectively. In equation (18), i is the inclination, that is, the angle between the plane of the satellite orbit and the equatorial plane of the Earth. The quantity p is the semi-latus rectum of the unperturbed elliptical orbit.
If in equation (17), we would redefine (i.e., bring into the correspondence)
then the effective potential energy from equation (17) would become identical to equation (16). By substituting |I 2|R 2 = 2γ/e 2 in equation (18), we obtain the corresponding ratio for our case:
where the Kepler frequency ω K is given by equation (15) and the index “0” denotes the unperturbed case.
For circular orbits, one has
so that the ratio from equation (21), corresponding to the unperturbed case, takes the form
where in our case, the quantity γ is (see equation (9)) expressed as follows:
For circular orbits, the dependence of the quantity x from equation (23) on the unperturbed energy is expressed as follows:
In (10), ΔU 1(r) is a relatively small shift of the energy of the electron. The dynamics of the motion beyond the plane of the unperturbed circular orbit is controlled by the following truncated U eff,tr (see (9) and (10)):
where g(x) is given by (8), r is given by (22), and x by (25). Thus, the ratio of the precession frequency to the Kepler frequency of the electron is expressed as follows:
and the relative correction to the precession frequency is expressed as follows:
with g(x) given in (8) and x(E) given in (25).
Figure 5 presents the dependence of the relative correction to the precession frequency of the orbital plane of the Rydberg electron on the electron energy for selected values of Z.
Dependence of the relative correction to the precession frequency of the orbital plane of the Rydberg electron on the electron energy for Z = 4 (blue solid curve), Z = 6 (green dashed curve), and Z = 9 (red dotted curve).
The correction ΔΩ/Ω approaches the limit of 1/(Z − 1) at large negative values of the electron energy.
2.2 Circular polarization of the laser field
Now we consider the case of the circular polarization of the laser field of the amplitude F and frequency ω, the polarization field being perpendicular to the z-axis. The laser field varies as follows:
where e x and e y are the unit vectors along the x- and y-axes (which are perpendicular to the z-axis). The semi-classical Hamiltonian for the outer (Rydberg) electron in this case can be represented as follows:
where H 0 is given in (2). Denoting
where (r, θ, φ) are the spherical coordinates, and using the formalism of the effective potentials [2,6,7], we obtain the zeroth-order effective potential:
and the first-order effective potential:
where x is given in (5). By using the same procedure as in the case of the linear polarization, we can rewrite U 1 in the form as in (9):
It differs from (9) by the sign in the beginning and a different function f 1(x), which is given by
From (34), we find that the energy shift in the case of circular polarization is expressed as follows:
which is twice as much as in the linear case. The relative correction to the precession frequency of the orbital plane is the same as in the linear polarization case (given in (11) and (28)).
3 Conclusions
We studied a helium atom or a helium-like ion with one of the two electrons in a Rydberg state, while the system is subjected to a high-frequency laser field. For obtaining analytical results, we use the generalized method of the effective potentials from the previous works [2,6,7].
Then, we concentrated on circular Rydberg states. We found two primary effects of the high-frequency laser field. The first effect is the precession of the orbital plane of the Rydberg electron. We calculated analytically the precession frequency and showed that it differs from the case of a hydrogenic atom/ion. In the radiation spectrum, this precession would manifest as satellites separated from the spectral line at the Kepler frequency by multiples of the precession frequency.
We also demonstrated that the high-frequency laser field causes a red shift of the energy of the Rydberg electron – the shift that we calculated analytically. We studied its dependence on parameters of the system. We found that the absolute value of the shift increases monotonically as the unperturbed binding energy of the Rydberg electron increases. We also found that the shift has a nonmonotonic dependence on the nuclear charge Z: as Z increases, the absolute value of the shift first increases, then reaches a maximum, and then decreases. The nonmonotonic dependence of the laser field-caused energy shift on the nuclear charge is a counterintuitive result.
Finally, we note that for the interaction of a high-frequency laser field with atoms, the formalism of the effective potentials [4,5,6], employed in this article, has advantages over the formalism developed by Kramers [41] and Henneberger [42] and was later applied by Gavrila and his coauthors to hydrogen atoms [43,44]. The essence of the latter formalism is the transition to the frame oscillating together with an electron in the laser field and then utilizing the time average of the corresponding time-dependent perturbation. However, first, the application of this formalism to hydrogen atoms in the high-frequency laser field, as in the study by Pont and Gavrila [43], misses the hidden (algebraic) symmetry of this system, which is revealed in the previous articles [1,2] via the formalism of the effective potentials; thus, the Kramers–Henneberger formalism lacks the physical insight compared to the formalism of the effective potentials. Second, the formalism of the effective potentials has been developed analytically to the arbitrary order with respect to the laser field [6,7], but this does not seem to be possible for the Kramers–Henneberger formalism.
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Conflict of interest: Eugene Oks is an Editor of Open Physics and was not involved in the review process of this article.
References
[1] Gavrilenko VP, Oks E, Radchik AV. Hydrogen-like atom in the field of high-frequency linearly polarized electromagnetic radiation. Opt Spectrosc. 1985;59:411–2.Search in Google Scholar
[2] Nadezhdin BB, Oks E. Highly excited atoms in linearly polarized high-frequency electromagnetic radiation. Sov Tech Phys Lett. 1986;12:512–3.Search in Google Scholar
[3] Oks E, Davis JE, Uzer T. Rydberg electron dynamics in high-frequency elliptically polarized microwave fields. J Phys B: Mol Opt Phys. 2000;33:207–21.10.1088/0953-4075/33/2/307Search in Google Scholar
[4] Kapitza PL. Dynamic stability of the pendulum when the point of suspension is oscillating. Sov Phys JETP. 1951;21:588–97.Search in Google Scholar
[5] Kapitza PL. A pendulum with oscillating suspension. Uspekhi Fiz Nauk. 1951;44:7–20.10.3367/UFNr.0044.195105b.0007Search in Google Scholar
[6] Nadezhdin BB. Radiatsionnye i relativistskie effekty v atomakh i ionakh [Radiative and relativistic effects in atoms and ions]. Moscow: Scientific Council of the USSR Academy of Sciences on Spectroscopy; 1986. p. 222. RussianSearch in Google Scholar
[7] Oks E. Analytical advances in quantum and celestial mechanics: separating rapid and slow subsystems. Bristol: IOP Publishing; 2020. Appendix A.10.1088/2053-2563/ab3db0Search in Google Scholar
[8] Kryukov N, Oks E. The role of the Debye screening of circular Rydberg states of hydrogenic systems in collinear electric and magnetic fields of arbitrary strengths. Eur Phys J D. 2020;74:125–9.10.1140/epjd/e2020-10142-4Search in Google Scholar
[9] Kryukov N, Oks E. Analytical results for one-electron Rydberg quasimolecules in a high-frequency laser field. Eur Phys J D. 2020;74:82–9.10.1140/epjd/e2020-100414-8Search in Google Scholar
[10] Oks E. Classical analytical solution for Rydberg states of muonic-electronic helium and helium-like atoms. Can J Phys. 2020;98(9):857–61.10.1139/cjp-2019-0483Search in Google Scholar
[11] Kryukov N, Oks E. Classical description of charge exchange involving He-like or Li-like ions in Rydberg states in plasmas. J Phys B: Mol Opt Phys. 2016;49:035002.10.1088/0953-4075/49/3/035002Search in Google Scholar
[12] Kryukov N, Oks E. Analytical solutions for diatomic Rydberg quasimolecules in a laser field. Eur Phys J D. 2014;68:171–7.10.1140/epjd/e2014-40765-7Search in Google Scholar
[13] Kryukov N, Oks E. Muonic-electronic quasi molecules based on a fully stripped multicharged ion. Can J Phys. 2014;92:1405–10.10.1139/cjp-2013-0705Search in Google Scholar
[14] Kryukov N, Oks E. Ionization channel of continuum lowering in plasmas: effects of plasma screening, electric and magnetic fields. J Phys B: At Mol Opt Phys. 2013;46:245701.10.1088/0953-4075/46/24/245701Search in Google Scholar
[15] Kryukov N, Oks E. Muonic-electronic negative hydrogen ion: circular states. Can J Phys. 2013;91:715–21.10.1139/cjp-2013-0077Search in Google Scholar
[16] Kryukov N, Oks E. Electric field induced crossings of energy terms of a Rydberg quasi molecule enhancing charge exchange and ionization. Can J Phys. 2012;90:647–54.10.1139/p2012-059Search in Google Scholar
[17] Tan JN, Brewer SM, Guise ND. Experimental efforts at NIST towards one-electron ions in circular Rydberg states. Phys Scr. 2011;T144:014009.10.1088/0031-8949/2011/T144/014009Search in Google Scholar
[18] Dehesa JS, Lopez-Rosa S, Martinez-Finkelshtein A, Yanez RJ. Information theory of D‐dimensional hydrogenic systems: Application to circular and Rydberg states. Int J Quantum Chem. 2010;110(8):1529–48.10.1002/qua.22244Search in Google Scholar
[19] Nandi T. Experimental confirmation of circular Rydberg states with projectile-like ions. J Phys B: Mol Opt Phys. 2009;42:125201.10.1088/0953-4075/42/12/125201Search in Google Scholar
[20] Jentschura UD, Mohr PJ, Tan JN, Wundt JB. Fundamental constants and tests of theory in Rydberg states of hydrogenlike ions. Phys Rev Lett. 2008;100:160404.10.1103/PhysRevLett.100.160404Search in Google Scholar PubMed
[21] Flannery MR, Oks E. Plasma screening within Rydberg atoms in circular states. Eur Phys J D. 2008;47:27–31.10.1140/epjd/e2008-00032-4Search in Google Scholar
[22] Devoret M, Girvin S, Schoelkopf R. Circuit-QED: How strong can the coupling between a Josephson junction atom and a transmission line resonator be? Ann Phys (Leipz). 2007;16:767–79.10.1002/andp.200710261Search in Google Scholar
[23] Shytov AV, Katsnelson MI, Levitov LS. Atomic collapse and quasi-Rydberg states in graphene. Phys Rev Lett. 2007;99:246802.10.1103/PhysRevLett.99.246802Search in Google Scholar PubMed
[24] Flannery MR, Oks E. Magnetic stabilization of a Rydberg quasimolecule in circular states. Phys Rev A. 2006;73:013405.10.1103/PhysRevA.73.013405Search in Google Scholar
[25] Oks E. Circular Rydberg states of hydrogenlike systems in collinear electric and magnetic fields of arbitrary strengths: an exact analytical classical solution. Eur Phys J D. 2004;28:171–9.10.1140/epjd/e2003-00308-1Search in Google Scholar
[26] Holmlid L. Conditions for forming Rydberg matter: condensation of Rydberg states in the gas phase versus at surfaces. J Phys: Condens Matter. 2002;14:13469–79.10.1088/0953-8984/14/49/305Search in Google Scholar
[27] Oks E. Role of higher multipoles in field-induced continuum lowering in plasmas. Phys Rev E. 2001;63:057401.10.1103/PhysRevE.63.057401Search in Google Scholar PubMed
[28] Oks E. Are crossings of energy levels and charge exchange really quantum phenomena? Phys Rev Lett. 2000;85:2084–7.10.1103/PhysRevLett.85.2084Search in Google Scholar PubMed
[29] Oks E. A purely classical description of crossings of energy levels and spectroscopic signatures of charge exchange. J Phys B: At Mol Opt Phys. 2000;33:3319–30.10.1088/0953-4075/33/17/312Search in Google Scholar
[30] Carlsen H, Goscinski O. Dynamics of particles for circular Rydberg states. Phys Rev A. 1999;59:1063–9.10.1103/PhysRevA.59.1063Search in Google Scholar
[31] Lee E, Farrelly D, Uzer T. A Saturnian atom. Opt Express. 1997;1:221–8.10.1364/OE.1.000221Search in Google Scholar
[32] Germann TC, Herschbach DR, Dunn M, Watson DK. Circular Rydberg states of the hydrogen atom in a magnetic field. Phys Rev Lett. 1995;74:658–61.10.1103/PhysRevLett.74.658Search in Google Scholar PubMed
[33] Cheng CH, Lee CY, Gallagher TF. Production of circular Rydberg states with circularly polarized microwave fields. Phys Rev Lett. 1994;73:3078–81.10.1103/PhysRevLett.73.3078Search in Google Scholar PubMed
[34] Chen L, Cheret M, Roussel F, Spiess G. New scheme for producing circular orbital states in combined RF and static fields. J Phys B: At Mol Opt Phys. 1993;26:L437–L443.10.1088/0953-4075/26/15/002Search in Google Scholar
[35] Cortiñas RG, Favier M, Ravon B, Mehaignerie P, Machu Y, Raimond JM, et al. Laser trapping of circular Rydberg atoms. Phys Rev Lett. 2020;124:123201.10.1103/PhysRevLett.124.123201Search in Google Scholar
[36] Dutta SK, Feldbaum D, Walz-Flannigan A, Guest JR, Raithel G. High-angular-momentum states in cold Rydberg gases. Phys Rev Lett. 2001;86:3993–6.10.1103/PhysRevLett.86.3993Search in Google Scholar
[37] Hulet RG, Hilfer ES, Kleppner D. Inhibited spontaneous emission by a Rydberg atom. Phys Rev Lett. 1985;55:2137–40.10.1103/PhysRevLett.55.2137Search in Google Scholar
[38] Vainberg VM, Popov VS, Sergeev AV. The 1/n expansion for a hydrogen atom in an external field. Sov Phys JETP. 1990;71:470–7.Search in Google Scholar
[39] Nadezhdin BB, Oks EA. Simple quasiclassical model of heliumlike and lithiumlike ions. Opt Spectrosc. 1990;68:12–15.Search in Google Scholar
[40] Beletsky VV. Essays on the motion of celestial bodies. Basel: Birkhäuser/Springer; 2001. Sect. 1.710.1007/978-3-0348-8360-3Search in Google Scholar
[41] Kramers HA. Collected Scientific Papers. Amsterdam: NorthHolland, Amsterdam; 1956. p. 262Search in Google Scholar
[42] Henneberger WC. Perturbation method for atoms in intense light beams. Phys Rev Lett. 1968;21:838–41.10.1103/PhysRevLett.21.838Search in Google Scholar
[43] Pont M, Gavrila M. The levels of atomic hydrogen in intense, high-frequency laser fields. Phys Lett A. 1987;123:469–74.10.1016/0375-9601(87)90346-XSearch in Google Scholar
[44] Pont M, Offerhaus MJ, Gavrila M. Atomic hydrogen in circularly polarized, high-intensity and high frequency laser fields. Z Phys D – Atoms, Molecules Clust. 1988;9:297–306.10.1007/BF01436936Search in Google Scholar
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