Twin-Capture Rydberg State Excitation Enhanced with Few-Cycle Laser Pulses

Jing Zhao; Jinlei Liu; Xiaowei Wang; Zengxiu Zhao

doi:10.1088/0256-307X/41/1/013201

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Twin-Capture Rydberg State Excitation Enhanced with Few-Cycle Laser Pulses

1.
Department of Physics, National University of Defense Technology, Changsha 410073, China
2.
Hunan Key Laboratory of Extreme Matter and Applications, National University of Defense Technology, Changsha 410073, China

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Corresponding author:
Zengxiu Zhao, E-mail: zhaozengxiu@nudt.edu.cn

Received Date: October 22, 2023

Published Date: November 20, 2023

Abstract

Abstract

Quantum excitation is usually regarded as a transient process occurring instantaneously, leaving the underlying physics shrouded in mystery. Recent research shows that Rydberg-state excitation with ultrashort laser pulses can be investigated and manipulated with state-of-the-art few-cycle pulses. We theoretically find that the efficiency of Rydberg state excitation can be enhanced with a short laser pulse and modulated by varying the laser intensities. We also uncover new facets of the excitation dynamics, including the launching of an electron wave packet through strong-field ionization, the re-entry of the electron into the atomic potential and the crucial step where the electron makes a U-turn, resulting in twin captures into Rydberg orbitals. By tuning the laser intensity, we show that the excitation of the Rydberg state can be coherently controlled on a sub-optical-cycle timescale. Our work paves the way toward ultrafast control and coherent manipulation of Rydberg states, thus benefiting Rydberg-state-based quantum technology.

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Due to their significant dipole–dipole interactions and enduring stability, Rydberg atoms have emerged as a promising avenue in the realm of detecting microwave electric fields, high-speed terahertz (THz) imaging and quantum information applications.^[1–4] Conventionally, population of Rydberg states involves a sequential single-photon resonant transition through a predetermined pathway using a series of narrow-band laser fields.^[5] In contrast, coherent Rydberg wave packets, which consist of a superposition of multiple excited states, can be efficiently generated and persist under intense laser pulses.^[6] Achieving coherent excitation and manipulation of Rydberg states depends on a thorough understanding of the mechanisms involved in strong-field excitation, facilitated by advanced ultrafast laser techniques.

For atoms exposed to strong laser fields, the excitation of Rydberg wave packets has long been predicted.^[7] However, in the tunneling regime, valence electrons were previously believed to undergo ionization and have little opportunity to settle on the excited states. Tunneling ionization can be well-described theoretically using Keldysh theory^[8] and can be easily investigated experimentally through charged particles or coherent emissions, such as terahertz waves^[9] and high-order harmonics,^[10] resulting from the rescattering of liberated electrons by the ionic core. It was later discovered that Rydberg states played roles in both the phase-matching^[11] and emitting processes of high-order harmonic generation (HHG).^[12] Additionally, Rydberg states were found to lead to two kinds of independent emission of HHG: extreme-ultraviolet free induced decay (XFID) and excited state HHG.^[13] Notably, bright XFID emission was observed^[14,15] in excited atoms and the XFID intensity strongly depended on the ellipticity and carrier-envelope phase of the laser pulses.^[16]

With increasing experimental observations, more insight has been gained into the buildup dynamics of Rydberg states under strong laser fields. A stabilization mechanism through destructive interference transitions or the formation of Kramers–Henneberger states^[17,18] was proposed to explain the survival of Rydberg states under strong laser fields. Channel closing induced by the ponderomotive shift of the ionization potential was identified to be responsible for the oscillations in the intensity dependence of excitation probabilities.^[19–22] As a complement to the rescattering model, frustrated tunneling ionization^[14,23] provided a more intuitive picture, where the liberated electron can be recaptured by the parent ion driven by the laser field. Regardless of which mechanism dominates, the excitation of Rydberg states can be considered as interference between wave packets created in every half laser cycle.^[24,25] Therefore, the phases, including the transition phase and propagation phase, play crucial roles. The electron capture occurs every half cycle and its coherent superposition determines the final occupation of Rydberg states, depending on constructive or destructive interferences. However, how and when the Rydberg state is populated determines the phase of amplitude and, consequently, the final population of Rydberg states after the laser field, which remains unexplored.

In this work, we conducted numerical investigations on the coherent development of excited states in neon atoms driven by intense femtosecond pulses with varying pulse durations and intensities. We found that the excitation probability of Rydberg states is more efficient with shorter laser pulses and can be modulated by varying the laser intensities. Within the framework of strong-field approximation, we propose an analytical model to confirm that the freed electron after tunnel ionization surfs over the atomic potential and the Rydberg state is dominantly populated twice when the electron makes a U-turn within one optical cycle. By varying the peak intensity of the pulse, we observed modulations in the excitation probabilities of Rydberg states, which can be attributed to the intensity-dependent interference of the two capture events during the U-turn.

Numerical simulations were performed with the time-dependent Schrödinger equation (TDSE) describing the interaction of a one-dimensional model neon atom with an intense laser pulse (atomic units are used unless stated otherwise),

$i \frac{\partial ψ (x, t)}{\partial t} = [\frac{1}{2} \frac{d^{2}}{d x^{2}} - x E (t) + V (x)] ψ (x, t),$

(1)

where the soft-core potential is chosen as

$V (x) = - 1 / \sqrt{x^{2} + α}$ with α = 0.65 to reproduce the ionization potential of the real neon atom. The electric field of the laser pulse is E(t) = E₀f(t)cos(ωt+φ) with a Gaussian envelope f(t) and carrier-envelope phase (CEP) φ, and ω denotes the frequency of 800 nm. The calculations were performed with Gaussian laser pulses with different peak intensities and pulse durations. The populations of excited states (n = 2–20) as a function of driving laser intensity are illustrated in Fig. 1(a). The excitation probability exhibits pronounced oscillations with increasing laser intensity, which was observed due to the fact that the production of Rydberg states is associated with the occurrence of channel closing.^{[19,20,22,26]} The higher excited states are easier to be coherently populated, in agreement with a previous study.^[27] The energy of the high excited states is shifted approximately by the intensity-dependent ponderomotive energy of the electron U_p = I/4ω². For the states with specific parity, if the energy levels of the excited states are shifted by the energy of two photons, the channel is open again and provides the condition for the channel closing as ΔU_p = 2ω. For laser wavelength centered at 800 nm, the oscillation period of the excitation probability for a single excited state is about 0.5 × 10¹⁴ W/cm², shown clearly with the populations of Rydberg states with n = 8 in Fig. 1(c) and n = 9 in Fig. 1(d). The state-resolved population measurement is difficult in the experiment. The oscillation period for the total excitation yields of 0.25 × 10¹⁴ W/cm² has been observed experimentally by measuring the intensity dependence of the excited atoms.^[28] Moreover, we present the derivative of the Rydberg state phase after the laser pulse with respect to the laser intensity, as shown by the green lines in Figs. 1(c) and 1(d). As the resonant excitation channel is closed, the phase of the Rydberg state changes dramatically. When the resonant excitation channel is open, i.e., the energy difference between the ground state and the laser-dressed Rydberg state equals multiple photon energies, the phase of the Rydberg state varies smoothly, where phase matching is easier to achieve and it leads to efficient excitation.

Fig. 1. TDSE calculations for the coherent excitation probabilities of Rydberg states with 5 fs laser pulse and CEP of φ = 0. Results of excited states (n = 2–20) are shown in (a), and n = 8, 9 are shown in (b), with the laser intensity from 2 × 10¹⁴ W/cm² to 2.5 × 10¹⁵ W/cm². Phase variation dependence on the laser intensity for the Rydberg states of (c) n = 8 and (d) n = 9.

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Interestingly, except for the fast oscillations, it can be seen from Fig. 1(b) that a much slower oscillation appears as a function of the laser intensity, which depends on the quantum number. To highlight the slow oscillation appearing in the intensity dependence, populations of the Rydberg states are averaged over the laser intensities. The populations of Rydberg state n = 8 using laser pulses with different pulse duration are shown in Fig. 2. For few-cycle pulses, the population reached a maximum above 10¹⁵ W/cm² and was followed by an oscillation dependence on the laser intensity. As the pulse duration increases, the total population decreased and the peak position shifted to a smaller intensity. Different Rydberg state populations show similar oscillation structures and pulse duration dependences, and the slight difference originates from their orbital properties.

Fig. 2. Intensity-averaged populations of Rydberg state n = 8 as functions of the laser intensity and pulse duration.

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As will be discussed in the following, the excitation behavior of the Rydberg states can be explained by the interference of multiple recapture events for each ionization time within the recollision picture. Considering a neon atom exposed to the strong field, an electron freed at time t_i will have the probability to revisit the Rydberg state by acceleration of the laser field. Electrons revisiting at different times can be coherently recaptured into the Rydberg states. Interference between the two adjacent recapture times within a half laser cycle contributes to the low oscillation structure dependence on the laser intensity observed in the experiments. The recapture probability for each ionization time t_i can be calculated based on the strong-field approximation,

$a (t_{i}) = \sum_{m} a_{g} (t_{i}) \sqrt{W (t_{i})} e^{i ϕ_{m}} a_{r} (t_{m}),$

(2)

where W(t_i) is the ionization rate given by the Ammosov–Delone–Krainov theory^[29] and the ground-state depletion is taken into account using

$a_{g} (t) = \exp [- \int_{- \infty}^{t} W (t)]$ . The phase accumulated by the electron from the ionization time t_i to the m-th recapture time t_m is

$ϕ_{m} = \int_{t_{i}}^{t_{m}} (E_{n} + \frac{A^{2} (τ)}{2}) d τ$ , with the energy of the Rydberg state E_n and the vector potential of the laser field A(t). The phase describes the Rydberg electron evolving with instantaneous energy with a laser-induced ac-Stark shift. The electron motion after ionization is calculated classically with zero initial velocity and position. Once the electron wave packet passes through the spatial range of the Rydberg state, the recapture occurs with the probability a_r(t). The total amplitude for the Rydberg excitation is obtained by the coherent sum over all ionization events,

$a_{total} (t_{f}) = \int_{0}^{t_{f}} e^{- i ϕ (t_{f}, t^{''})} a (t_{i}) d t_{i},$

(3)

with additional phase accumulation

$ϕ (t_{f}, t^{''}) = \int_{t^{''}}^{t_{f}} (E_{n} + A^{2} / 2) d t$ acquired by the Rydberg electron from the capture time t″ to the end of the laser pulse at time t_f. For a slowly varying pulse envelope, we can write the final amplitude of occupying the Rydberg state as a sum over contributions from every half cycle (assume that the pulse duration is N/2 times laser period),

$a_{total} (t_{f}) = \sum_{n = 1}^{N} e^{i n (π - ϕ_{0})} \int_{0}^{T / 2} a (t_{i}) d t_{i},$

(4)

where

$ϕ_{0} = \int_{0}^{T / 2} (E_{n} + \frac{A^{2}}{2}) d t = (E_{n} + U_{p}) \frac{T}{2}$ denotes the accumulated phase during each half cycle. Contributions from subsequent half cycles lead to the π phase difference because the electrons are recaptured in opposite directions of the same Rydberg state. Finally, the occupation probability can be written as

$\begin{array}{l} | a_{total} (t_{f}) |^{2} = & \frac{\sin^{2} [N (π - ϕ_{0}) / 2]}{\sin^{2} [(π - ϕ_{0}) / 2]} | \int_{0}^{T / 2} a (t_{i}) d t_{i} |^{2} \\ \propto \frac{\sin^{2} [N (π - ϕ_{0}) / 2]}{\sin^{2} [(π - ϕ_{0}) / 2]} {[1 + \cos (Δ ϕ)]}^{2} . \end{array}$

(5)

From the first term in Eq. (5), it can be seen that, when multiphoton resonance occurs, that is, ϕ₀ = mπ, the ac-Stark shifted energy of the Rydberg state equals multiple photon energy E_n + U_p = mω. For a state with specified parity, the occupation probability peaks are separated by ΔU_p = 2ω, corresponding to the channel-closing effects observed with a fine adjustment of the laser intensity discussed in previous work. The second term arising from the intra-cycle phase accumulation Δϕ is attributed to the slow modulation observed in Fig. 2.

By analyzing the electron trajectories within each half laser cycle, we find that electrons born around the peaks of the electric field would have a typical ‘surfing’-like trajectory, where the electron turns around in the spatial range of the Rydberg state such that it can be recaptured coherently into the Rydberg state twice within a half laser cycle as indicated by t₂ and t₃ in Fig. 3. Here, we consider only the first surfing recapture event due to the quantum dispersion of the electron wave packet. The interference effect in the surfing trajectory contributes to the oscillatory structure in the laser intensity dependence of the Rydberg state population. Let us consider an electron born at the peak of the electric field during the rising edge of the laser pulse. The electron trajectory is calculated classically by neglecting the Coulomb force of the ion. The time interval between these two capture times, Δt = t₃ – t₂, in the surfing trajectory can be obtained by cos(ωΔt/2) = R₀ωα/E₀, with R₀ being the radius of the Rydberg orbital. Here, E₀ is the field amplitude of the laser pulse and α is the slope of the electric field envelope at the ionization time, which is inversely proportional to the pulse duration. The phase difference of the twin-capture event is approximated to be $Δ ϕ \approx (E_{n} + \frac{A^{2}}{2}) Δ t$ . Constructive interference appears when Δϕ = 2nπ, leading to an effective Rydberg state population. Destructive interference occurs when Δϕ = (2n+1)π, decreasing the coherent excitation of the Rydberg state. Similar intra-cycle interference has been observed in above-threshold ionization spectra^[30] and in the photoelectron holography^[31,32] arising from a pair of electron trajectories released within one laser cycle.

Fig. 3. The twin-capture model of Rydberg state excitation. Illustration of electron wave packet launch, reentry and twin events of capture into Rydberg orbitals at times t₂ and t₃. Electron wave packets are partially captured into the Rydberg orbital during each intersection. However, only those wave packets captured at times t₂ and t₃ can be in phase due to the smaller phase accumulation during the quick U-turn.

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To verify our proposed model, the coherent excitation probability calculated with Eq. (4) is shown in Fig. 4, considering only the contribution from ionization within one half cycle before the peak intensity of the pulse envelope. It can be seen that both the oscillation period and the optimal intensity decrease as the pulse duration increases, which agrees quite well with the TDSE simulations. The interference structure is determined by the phase difference that accumulates during the time interval Δt between the twin capture events, which is proportional to the laser intensity and the pulse duration. Therefore, the oscillation period is inversely proportional to the pulse duration. As the capture dynamics repeats at every half cycle, the coherent superposition of the Rydberg state also leads to the intra-cycle interference effect. However, for a few-cycle pulse with high intensity, the phase difference between adjacent half cycles can be as large as 38π due to ∼ 60 eV ponderomotive energy at 1 × 10¹⁵ W/cm² pulse intensity, which makes the amplitudes out of phase. Therefore, the populating efficiency decreases as the pulse duration increases, confirmed by the TDSE simulations in Fig. 2.

Fig. 4. The twin-capture process reproduces the TDSE intensity dependence. The solid and dashed line in the bottom plane indicates the optimal intensity and the poorest intensity respectively.

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In conclusion, the reported data demonstrate that the Rydberg population can be enhanced by one order of magnitude when the pulse duration is reduced to nearly a single-cycle laser pulse. Furthermore, the Rydberg population can be significantly modulated by changing the intensity of the driving pulses. The underlying physics is attributed to the crucial step where the returned electron makes a U-turn in the vicinity of Rydberg orbital, leading to constructively interfered twin-capture events. Our work advances new means of coherent Rydberg states excitation by tailoring laser waveforms, which benefits future Rydberg-state-based precision measurements and quantum detection.

Acknowledgments: This work was supported by the National Key Research and Development Program of China (Grant No. 2019YFA0307703), the National Natural Science Foundation of China (Grant Nos. 12234020, 11874066, 12274461, and 11974426), and the Science and Technology Innovation Program of Hunan Province (Grant No. 2022RC1193).

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Twin-Capture Rydberg State Excitation Enhanced with Few-Cycle Laser Pulses

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