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.
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.
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]):
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:
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:
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:
Therefore, the total energy can be represented in the following form:
From (9) and (7), we see that the term
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:
It is seen that the shift is zero at the zero unperturbed energy and approaches the limit
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.
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 motion characterized by the effective potential energy can be expressed as follows:
If in equation (17), we would redefine (i.e., bring into the correspondence)
For circular orbits, one has
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)):
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.
The correction ΔΩ/Ω approaches the limit of 1/(Z − 1) at large negative values of the electron energy.
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:
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:
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.
Conflict of interest: Eugene Oks is an Editor of Open Physics and was not involved in the review process of this article.
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© 2021 Nikolay Kryukov and Eugene Oks, published by De Gruyter
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