Fractional Quantum Mechanics Alters How Atoms Release Electrons

Marcelo F. Ciappina, Guangdong Technion, Israel Institute of Technology, and colleagues at Guangdong Provincial Key Laboratory of Materials and Technologies for Energy Conversion have created a new theoretical model explaining atomic ionisation in strong electric fields. Incorporating space-fractional quantum mechanics, which replaces standard kinetic energy with a fractional Laplacian, changes the established scaling law for ionisation probability from the conventional $I_p^{3/2}$ to $I_p^{1+1/α}$. The analytical model, derived within the Ammosov, Delone, Krainov framework, introduces a unique factor dependent on the fractional order, offering a key benchmark for validating nonlocal quantum dynamics and extending strong-field physics beyond conventional approaches.

Fractional quantum mechanics reveals modified power law for ionization potential scaling

Scaling of ionization potential now extends beyond the conventional $I_p^{3/2}$ law to $I_p^{1+1/α}$, a sharp departure achieved through the incorporation of space-fractional quantum mechanics. This represents a significant refinement in understanding how easily an atom loses an electron when subjected to a strong electric field. The conventional $I_p^{3/2}$ scaling law, derived from the Ammosov, Delone, and Krainov (ADK) theory, describes the dependence of the ionization rate on the ionization potential ($I_p$). The new model utilises a fractional Laplacian, effectively ‘smearing out’ particle location and influencing behaviour, a feature inaccessible with standard semiclassical models reliant on quadratic kinetic energy. The fractional Laplacian is a mathematical operator that generalises the standard Laplacian, incorporating long-range interactions and nonlocal effects. This ‘smearing’ arises from the fractional derivative, which considers contributions from a wider range of spatial locations than the standard derivative, thus altering the electron’s probability distribution. A closed-form tunneling exponent demonstrates this deformation of the ionization rate, introducing a $\sin(π/α)$ factor that accounts for nonlocal dispersion, a key characteristic of this fractional approach. The $\sin(π/α)$ factor directly reflects the influence of the fractional order α on the tunneling probability, quantifying the deviation from standard quantum mechanical predictions.

For a given binding potential, ground-state energies and wavefunction distributions demonstrably shift with changes in the fractional order parameter α, further validating this altered scaling. Specifically, the ground-state energy is modified due to the altered kinetic energy operator, and the wavefunction exhibits a different spatial extent. Wavefunctions exhibit heavier momentum-space tails and altered spatial localisation for $α < $2. This indicates a greater probability of finding the electron further from the nucleus, a consequence of the nonlocal nature of the fractional quantum mechanics. Employing both fixed-potential and fixed-ionisation potential protocols, the kinetic operator consistently modifies tunneling dynamics, even when the overall binding energy remains constant. The use of these two protocols ensures that the observed changes are genuinely attributable to the modified kinetic operator and not to alterations in the overall potential. Retuning parameters within a soft-core Coulomb potential maintained a consistent ionization potential across different α values, allowing isolation of the kinetic operator’s impact. A soft-core Coulomb potential avoids the singularity at the origin, simplifying the mathematical analysis while still capturing the essential physics of the atomic potential. This analytical model establishes a key benchmark for validating nonlocal quantum dynamics and extending strong-field physics beyond conventional approaches, providing a reference point for testing the fractional Schrödinger equation in time-dependent simulations. The implications of these findings extend to the development of more accurate models for complex atomic systems and the exploration of novel quantum phenomena, potentially impacting fields like attosecond physics and quantum computing. Understanding the subtle changes in electron behaviour induced by fractional quantum mechanics could lead to improved control over atomic processes.

Refining atomic ionization probability through space-fractional quantum mechanics

The established understanding of atomic ionization, important for technologies like laser development and high-harmonic generation, relies on predicting how electrons escape atoms in electric fields. Accurate modelling of ionization is crucial for optimising laser-based technologies and understanding the fundamental physics of light-matter interactions. This new model offers a refined description of this process, altering the conventional scaling of ionization probability. Its value lies in establishing a key benchmark for future research, despite being built upon a simplified, triangular exit barrier and static fields only. The use of a triangular barrier simplifies the mathematical treatment, allowing for an analytical solution, but it is acknowledged that real atomic potentials are more complex. The current model focuses on static fields to provide a clear understanding of the fundamental effects of fractional quantum mechanics; extending it to time-dependent fields is a natural next step.

Space-fractional quantum mechanics, a mathematical framework altering how electron behaviour is calculated, alters established scaling laws for atomic ionization. This framework provides a more nuanced description of electron behaviour, particularly in situations where nonlocal effects are significant. By modifying the conventional understanding of electron behaviour in electric fields, scientists now have a benchmark for exploring nonlocal quantum dynamics, a departure from standard quantum descriptions. The standard quantum mechanical description assumes that an electron’s behaviour at one point in space is independent of its behaviour at distant points; nonlocal quantum mechanics challenges this assumption. Built upon the established Ammosov-Delone-Krainov framework, the refined model introduces a unique factor dependent on the degree of spatial ‘smearing’, opening questions regarding its application to more complex, time-dependent scenarios and the potential for manipulating electron behaviour in novel ways. The degree of ‘smearing’ is quantified by the fractional order parameter α, which controls the range of nonlocal interactions. Further investigation will focus on extending this approach to include dynamic fields and many-body interactions, potentially unlocking new avenues for controlling atomic processes. Incorporating time-dependent fields will allow for a more realistic simulation of laser-atom interactions, while including many-body interactions will account for the effects of electron-electron correlations. These advancements could lead to the development of new techniques for controlling electron behaviour and manipulating atomic properties.

This research demonstrated that incorporating space-fractional quantum mechanics into models of atomic ionization alters the established scaling laws governing the process. This is significant because it provides a new way to describe electron behaviour, particularly when considering nonlocal effects where an electron’s behaviour is linked to distant points in space. The scientists derived a new tunneling exponent, offering a benchmark for understanding ionization in this framework and validating simulations of the fractional Schrödinger equation. The authors intend to extend this model to include time-dependent fields and many-body interactions to further refine the description of atomic processes.