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Purpose : The interaction of intense, ultrashort laser pulses with atoms gives rise to a rich tapestry of non-perturbative phenomena, encoded within the final-state photoelectron momentum distribution (PMD). A particularly enigmatic feature, often observed in the multiphoton ionization regime (Keldysh parameter $\gamma \gtrsim 1$), is a complex, fan-like interference pattern in the near-threshold, low-energy region of the PMD. The physical origin of this structure has been the subject of extensive debate, with proposed mechanisms ranging from multipath interference in the Coulomb field to complex sub-barrier dynamics. This work aims to provide a physical explanation for this phenomenon. We hypothesize and demonstrate that this fan-like structure is not a mere consequence of Coulomb focusing but serves as a direct and sensitive signature of non-adiabatic dynamics occurring as the electron tunnels through the laser-dressed atomic potential barrier. Our goal is to unambiguously isolate the key physical ingredients responsible for shaping this quantum interference.
Methodology : To achieve this, we employ a synergistic three-pronged approach that combines experiment, exact numerical simulation, and a sophisticated theoretical model.
1. Experiment : We performed velocity-map imaging measurements on argon atoms ionized by a 798 nm, 35 fs laser pulse at a peak intensity of $6.3 \times 10^{13}$ W/cm$^2$. This provides the experimental result, clearly revealing the low-energy fan-like interference pattern.
2. Quantum Benchmark : We solved the time-dependent Schrödinger equation (TDSE) within the single-active-electron (SAE) approximation, using a well-established model potential for argon that accurately reproduces its ionization potential and ground-state properties. After performing a focal-volume average to simulate experimental conditions, the TDSE results show excellent qualitative agreement with the measurements, establishing the TDSE as a reliable quantum benchmark for our investigation.
3. Semiclassical Model (CTMC-p) : The core of our analysis relies on a custom-developed semiclassical trajectory model based on the Feynman path-integral formulation. In this framework, ionization is a two-step process: (i) an electron tunnels through the potential barrier at an initial time $t_0$ and position $\mathbf{r}_0$, and (ii) it propagates classically in the combined laser and ionic fields according to Newton's equations. Crucially, each trajectory is endowed with a quantum phase accumulated along its path, $\Phi_k$, allowing for the coherent summation of all trajectories ending with the same final momentum, $M_j = \sum_k e^{i\Phi_k}$. Our model incorporates two critical physical effects beyond standard treatments:
Non-Adiabatic Tunneling : We introduce a non-zero initial longitudinal momentum, $v_{0\parallel} = -A(t_0)(\sqrt{1+\gamma_{\text{eff}}^2}-1)$, acquired by the electron at the tunnel exit. This term accounts for the non-instantaneous nature of the tunneling process, a key non-adiabatic effect.
Core Polarization : We include an induced dipole potential, $U_{\text{ID}} = -\alpha^I \mathbf{E}(t) \cdot \mathbf{r}/r^3$, to model the dynamic polarization of the Ar$^+$ ionic core, a multi-electron effect.
By selectively including or excluding these effects, we can unambiguously isolate their respective contributions to the final PMD.
Results : Our central finding is that the non-adiabatic initial longitudinal momentum is the decisive factor for correctly describing the near-threshold interference. This is powerfully illustrated in Figure 6. The benchmark TDSE calculation [Fig. 6(a)] for a single intensity of $5 \times 10^{13}$ W/cm$^2$ ($\gamma \approx 1.6$) reveals a distinct 6-lobe interference pattern. A conventional semiclassical simulation based on the quasi-static tunneling approximation (i.e., setting $v_{0\parallel}=0$) qualitatively fails, predicting an incorrect 8-lobe structure [Fig. 6(c)]. However, upon including the non-zero initial longitudinal momentum ($v_{0\parallel} \neq 0$), our non-adiabatic semiclassical model quantitatively reproduces the correct 6-lobe structure in perfect agreement with the TDSE benchmark [Fig. 6(b)].
To understand the underlying physics, we performed a quantum-orbit decomposition. This analysis reveals that the overall fan-like structure arises from the interference of multiple trajectory types, including 'direct' (Category I), 'forward-scattered' (Category II), and 'glory-scattered' (Category III) orbits. While the full structure results from the collective interference of these paths, we have pinpointed the origin of the lobe-count correction. The initial longitudinal momentum contributes a phase term, $\Delta\Phi_{\text{initial}} \approx -\mathbf{v}_{0\parallel} \cdot \mathbf{r}_0$, to the total accumulated action. We found that the relative phase between the 'direct' and 'glory' trajectories is exquisitely sensitive to this term due to their vastly different paths and birth conditions. It is this specific and dramatic change in the I-III interference channel that ultimately corrects the topology of the entire pattern, reducing the lobe count from 8 to 6. In contrast, other interference pairs, such as the holographic pair II-III, are largely robust against this effect as their nearly identical birth conditions cause the initial phase term to cancel in their relative phase. In parallel, our simulations show that the ionic core polarization has a negligible effect on this low-energy structure but is essential for accurately describing higher-energy rescattering features by smoothing unphysical caustics caused by a pure Coulomb potential.
Conclusion : We have unequivocally demonstrated that the near-threshold fan-like interference pattern in the multiphoton regime is a direct manifestation of non-adiabatic dynamics during tunneling, specifically the acquisition of a longitudinal momentum component by the electron during its finite-time passage under the potential barrier. Our findings not only provide a clear, intuitive, and orbit-based physical picture for this complex quantum phenomenon but also highlight the predictive power of semiclassical methods when crucial non-adiabatic effects are properly incorporated. This understanding lays a foundation for future investigations, including the extension of this model to more complex molecular systems and its application in retrieving attosecond electron dynamics from holographic interference patterns.-
Keywords:
- Non-adiabatic tunneling dynamics /
- Quantum orbit interference /
- Semiclassical method /
- Near-threshold photoelectron spectra
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