1. Introduction

It has been predicted that nonspreading quantum mechanical wave packets moving on classical trajectories can be constructed for hydrogenic electrons, if they are correctly supported by a resonant radiation field. These wave packets [1, 2, 3, 4] travel along circular trajectories and are dressed Floquet states [5, 6, 7] of hydrogen. They are interesting as a context to study quantum chaos [8] and as a tool for nonperturbative quantum control, for example for controllable adiabatic transfer of population from one hydrogenic eigenstate to another [9]. We have previously developed both classical and quantum theories of these circular wave packets [1],[10], and present here the extension to non-spreading packets on non-circular orbits. Because the trajectory equations for these packets enjoy a remarkably close mathematical analogy with the equations for the Trojan asteroids of the solar system, they are frequently referred to as Trojan wave packets.

We approach the nonspreading wave packets as wave functions corresponding to the first KAM resonance island of hydrogen perturbed by a resonant field. The basic idea of this method can be outlined as follows. The phase space of a classical nonlinear resonance is locally that of a one-dimensional pendulum, with some less prominent motion in other dimensions [11]. This means that the quantum mechanical eigenfunctions of such a system must be approximately those of a one-dimensional pendulum [12, 13]. Since the ground state wave function of a pendulum has a near-Gaussian shape, the same will be true for a system with a nonlinear resonance.

Nonlinear resonance theory in classical hydrogen has been widely discussed during the last two decades in connection with stochastic ionization [14]. In this paper we use a version of this theory, employing uniform semiclassical quantization rules. This allows us to predict the existence of objects that were not explicitly discussed earlier -nonspreading localized wave packets adiabatically connected to hydrogenic eigenstates with arbitrary quantum numbers n, l. Trojan wave packets are a particular case for circular states, l = n - 1.

2. Classical Resonance Theory

For simplicity, we consider 2D hydrogen, which is almost equivalent to the 3D case when the electron angular momentum is perpendicular to the plane of polarization of the incident radiation. We consider an atom dressed in a circularly polarized (CP) field with its frequency equal to the Kepler frequency ω = 1/$n_0^3$, where n ₀ is the principal quantum number.

The first pair of the action-angle variables of this system comprises the action l = p_φ and the corresponding angle φ -the angle of periapse of the ellipse of electron motion. The second pair is the action n = p_φ + 1/2π ∮p_rdr and the corresponding angle θ - the mean anomaly of the orbit, denoting an electron coordinate on the ellipse.

According to the third Kepler law, θ is proportional to the surface covered by electron radius-vector during electron motion along the orbit. In action-angle variables, the Hamiltonian is 𝛨 ₀ = -l/2n ² and the equations of motion have the form

In a CP field of frequency ω and field strength ε, the Hamiltonian takes the form

Starting from Eq. (2), and expressing x and y in action-angle variables [14] one can make a transition to the appropriate rotating frame in the standard way with the help of the generating function F = ñ(θ + φ - ωt) + l̃φ so that the new coordinates are

The new Hamiltonian is

where η_k (n,l) and ζ_k (n,l) are expansion coefficients dependent only on n and l [14]. The unperturbed Hamiltonian can be expanded near some n ₀ up to the second order, namely

We can exploit the resonance condition, ω = 1/$n_0^3$, and average the Hamiltonian over fast oscillations at frequency ω. After the averaging, only the term with k = 1 will remain for consideration. The Hamiltonian then takes the form

3. Mathieu Wave Packets

The uniform semiclassical quantization of 2D hydrogen replaces the action n by the operator -i∂/∂θ+1/2,l by -i∂/∂φ [16, 17, 18, 19, 20, 21], and applies periodic boundary conditions on the wavefunction. In the frame given by Eq.(3), corresponding replacements are made: ñ → -i∂/∂θ̃ +1/2, and l̃ → -i∂/∂φ̃ -1/2. The stationary Schrödinger equation 𝛨ψ = Eψ then takes the form

The boundary conditions in this case imply that the solution must be 2π-periodic in both θ̃ and φ̃, and this defines the set of possible “dressed” energy values E^dr as well as the corresponding dressed eigenfunctions ψ^dr .

An unperturbed hydrogenic eigenstate with quantum numbers n ₀ and l in (θ̃, φ̃)-representation has the form exp[i L φ̃] exp[i N θ̃], where N and L are integers [16, 17, 18]. According to uniform semiclassical quantization and Eq.(3), N = n ₀ - 1/2, L = l - n ₀ + 1/2. By seeking a dressed solution in the form ${\psi }_{\mathit{\text{NL}}}^{\mathit{\text{dr}}}$ = exp[i L φ̃]exp[i N θ̃] g(θ̃), we get the Mathieu equation for the function g(θ̃):

where $E_N^0$ = - 1/2(N + 1/2)² - ω)(N + 1/2), is the exact eigenenergy of unperturbed 2D hydrogen in the frame (3), with N = 0,1, 2,… [23].

 

Fig. 1. (left) Coefficient η ₁ + ζ ₁ in equation (9). (right) Energy spectrum in the rotating frame. Color dashed lines are the predictions of the Mathieu theory.

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Eq. (8) is analogous to Eq. (6) of Ref. [10] which describes circular Rydberg states in a resonant CP microwave field. The only difference between these two equations is that in Eq. (8) we have the term [η ₁(n,l) + ζ ₁(n,l)] whereas for a circular state [η ₁ + ζ ₁] = n ². The total dependence of [η ₁ + ζ ₁] on l/n is shown in Fig. 1. This figure indicates that the larger the ratio l/n of an initial state the larger effect a dressing resonant CP field has on this state. From Eq. (8) one can immediately derive the dressed eigenfunctions [22] as:

where M(θ̃) is a π-periodic Mathieu function of the argument (θ̃- π)/2.

We are particularly interested in a localized state that is adiabatically connected with the hydrogenic eigenstate ψ ⁰ _n0l for which exact resonance is fulfilled. In the rotating frame such a localized state has the form const * e -^iLφ̃ e^iNθ̃ e ₀[(θ̃ - π)/2] where e ₀ is the zero-order even π-periodic Mathieu function . In the laboratory frame this state is expressed is

Thus the dressed eigenstate ${\psi }_{\mathit{\text{nl}}}^{\mathit{\text{dr}}}$ is just the corresponding undressed eigenstate ${\psi }_{\mathit{\text{nl}}}^0$ modulated in angle as θ + φ - ωt.

Recall that under a strong perturbation the function e ₀ can be approximated by a Gaussian [22]. In addition, recall that in classical mechanics the combination θ + φ can be approximated by ϕ - sin θ (2e - e ³/4 + …) + … where ϕ is the usual polar angle, and e = (1 - l ²/n ²)^1/2 is the eccentricity of the orbit [15]. With this approximation, and using the Gaussian approximation of the Mathieu function, the dressed wave function can be rewriten in the form

where

The first exponential term on the right-hand side of Eq. (11) implies that the dressed state is an angularly localized wave packet rotating with the frequency ω. The second exponential term is responsible for radial localization of the state near the points where sin θ = 0, that is, for localization near turning points of clasical motion. The third term is responsible for an additional radius-dependent angular redistribution at radii at which sin θ ≠ 0. Thus a dressed state at exact resonance is an angular wave packet that is strongly modulated in radius and rotates around the nucleus with the field axis.

The energy of this state can be approximated as [22]

Eq.(13) holds for any reference frame if by $E_n^0$ one understands the zero-field energy of the state ${\psi }_{\mathit{\text{nol}}}^0$. Thus Eqs.(12) and (13) coincide with those of Ref. [10] with εn ² changed to ε[η ₁ + ζ ₁]. The energies of the states adjacent to the resonant one are determined by the other stability lines of Mathieu equation (8), and the dressed wavefunctions are the corresponding π-periodic Mathieu functions of θ + φ - ωt.

4. Numerical Calculations

In order to check these “pendulum” predictions, we made two different numerical calculations. In the first one, we solved the Schrödinger equation in the frame rotating with the field. In that frame, the Hamiltonian has the form

 

Fig. 2. (a) Probability distribution of the field-free 2D state n = 20, l = 14. (b) Probability distribution of the field-dressed state obtained with the help of matrix elements of 3D aligned states. The gauge in the left lower corner of left plot indicates the distance equal to 100 atomic units.

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We solved the 3D stationary Schrödinger equation with Hamiltonian (14), using a truncated basis of aligned undressed 3D hydrogenic eigenstates (states with l = m), which is approximately equivalent to considering two-dimensional hydrogen [23]. Fig. 1 (right) shows the energy spectrum calculated as a function of electric field for the frequency ω = 1/n ³, for the state n = 20, l = m = 14 and adjacent states versus predictions of the Mathieu theory. The coincidence of the calculated spectrum with the theoretical pedictions confirms that the 2D theory describes well-aligned states of 3D hydrogen.

Fig. 2 shows the field-free state n = 20, l = m = 14, and the dressed state obtained by numerical calculation for the field strength ε = 0.02/$n_0^4$ (red point on Fig. 1). The Floquet state in the laboratory frame is the state given in Fig. 2 rotating with the frequency ω. On Fig. 2, one can see that the dressed state is the initial state localized both angularly and radially, as predicted by the theory.

In a second calculation, we solved the time-dependent Schrödinger equation in two dimensions, using the split-operator method. We took the 2D hydrogenic eigenstate ${\psi }_{\mathit{\text{nl}}}^0$ with n ₀ = 20 and l ₀ = 14 as the initial state. A CP field with frequency ω = 1/$n_0^3$ was switched on adiabatically during thirty optical cycles according to the time dependence ε = ε ₀ e ^0.2(t-30) until t = 30 when the value ε ₀ = 0.02/$n_0^4$ was reached. After the turn-on was complete we monitored an additional ten cycles of evolution with the amplitude held constant at ε ₀.

 

Fig. 3. Two snapshots of time evolution of the dressed state n = 20, l = 14 obtained by ab initio calculation. The gauge in the left lower corner of left plot indicates the distance equal to 100 atomic units.

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Fig. 3 shows the initial state ${\psi }_{2014}^0$ and the dressed state ${\psi }_{2014}^{\mathit{\text{dr}}}$ at two different moments during the 35th cycle. One can see a resemblance between this state and the one given in Fig. 2.

5. Conclusions

To conclude, we have extended the treatment of nonspreading Rydberg wave packets in hydrogen, dressed in a resonant microwave field, to the case of states with a significant deviation from circularity. We predicted and tested numerically the predictions of the Mathieu theory for angularly localized wave packets adiabatically connected to hydrogen eigenstates with definite spherical quantum numbers. The cause of localization is the nonlinear resonance in hydrogen in a radiation field.

We believe that the states investigated in this paper have potential significance for nonperturbative state control and preparation in Rydberg atoms. It has been shown that by slowly changing the driving frequency, one can transfer different circular states into each other through Trojan wave packets [9]. We suppose that the states described here can be used for adiabatic nonperturbative control in a wider class of hydrogenic states. This is a topic to be investigated in the future.