A new method determines the physical bound-state survival amplitude ab(t) at any observation time without prior construction of the full time-dependent propagator or wavefunction. Employing the gauge-equivalent Kramers, Henneberger representation, incorporating the field into contact point motion, it presents an exact amplitude-centred formulation of the problem. The movingcontact dynamics reduces to a closed Volterra equation, with exact endpoint-phase factorization organising complete chronological rescattering history into an ordinary relative-time convolution hierarchy and an exact resolvent. Introducing accumulated bound-state amplitude enables its two-time domain to encompass all relevant dynamical phases.

Analytical derivation of time-dependent bound state probability amplitudes during strong field ionisation An exact nonperturbative solution for a fundamental ionisation model has been developed by careful representation and temporal organization of driven quantum dynamics. Modern theoretical descriptions of strong-field ionisation evolved from pioneering work including that of Keldysh, Perelomov, Popov, Terent’ev (PPT), Ammosov, Delone, and Krainov (ADK) as well as independent contributions by Faisal and Reiss establishing foundations for tunneling, multiphoton, and nonperturbative phenomena. Complementary studies led by Gavrila furthered understanding in atomic stabilisation and high-frequency laser matter interaction.

Considerable progress describes ionization rates, quasienergy spectra, Floquet states, semiclassical trajectories and asymptotic transition amplitudes. Obtaining an explicit analytical finite-time evolution however remains a challenging problem.

Conventional formulations construct the complete time-dependent propagator or wavefunction before projecting onto the desired bound state which necessitates determination of more information than required for just the projected amplitude. This approach focuses on organizing and evaluating this physical bound-state survival amplitude directly using the Kramers, Henneberger representation where influence from an external field transfers into motion of the binding potential. Introducing an accumulated bound-state amplitude reorganizing its two-time structure in terms of relative and centre times reduces chronological rescattering dynamics exactly into an ordinary convolution hierarchy allowing evaluation without first constructing the full propagator.

The work considers a one-dimensional attractive delta-function well subjected to arbitrary dc electric fields retaining essential ingredients while preserving analytical challenges inherent in finite time dynamics. Without an external field, it supports a single localized state which couples with continuum upon application of the dc field providing minimal nontrivial realization for quantum ionization; this model serves as analytically complete testbed enabling development and examination of strategies for driven dynamics at finite times having been used extensively over decades investigating decay, tunneling, resonant transport and solvable problems. Formally exact equations can be established but explicit finite-time expressions remain difficult to obtain often due to methodological approaches that construct propagators before projection.

The Kramers, Henneberger representation transfers influence from an external field into a time-dependent displacement of binding potential converting the problem involving moving contact between interactions; propagation is free without direct driving retaining field effects through trajectory. It provides a framework for strong-field dynamics with renewed interest in formulations of nonperturbative laser matter interaction providing a starting point for amplitude centred formulation. Rather than constructing complete propagator, observable formulates directly using physical bound state survival amplitude reducing dynamics to a closed Volterra equation solved via wavefunction at interaction point and projected onto bound state.

Reorganization renders phase linear when centre time fixed allowing repeated rescattering organized as convolution hierarchy generated by relative time kernel expressing all orders contribution analytically evaluable kernels exact resolvent avoiding full propagator construction; the resulting representation offers finite-time expression consisting explicit contact free term plus all order contact terms separating into direct and repeated scattering requiring no weak field expansion truncation or asymptotic approximation making the delta function model useful foundation general driven quantum dynamic formulations. ## Explicitly solvable ionisation amplitudes via bound state survival probability analysis An explicit finite-time expression for ionisation amplitudes from a one-dimensional delta function potential in an arbitrary dc electric field has been achieved, previously obtaining such expressions necessitated approximations or was simply impossible due to mathematical complexity. Researchers at North Carolina A&T State University obtained a closed form for the field-driven contact-free term utilising the Faddeeva function, a complex analytical function important for evaluating integrals involving Gaussian functions streamlining computations significantly.

All spatial integrations were performed analytically allowing scientists to express direct and repeated scattering terms as explicitly evaluable time integrals without asymptotic approximation. Verification involved confirming agreement with known field-free scenarios alongside independent computer simulations of the same physical system validating accuracy at any dc-field strength and finite observation time; this method centres on determining the probability that an electron remains bound within its initial state simplifying complex quantum dynamics considerably. This new formulation offers complete description of how electrons detach from atoms under intense electric fields, a longstanding problem in atomic physics.

However, current work relies on modelling that atom as an exceptionally simple system: a one-dimensional ‘delta function well’. While allowing for exact analytical solutions, this simplification raises questions about scalability and practical application to more realistic scenarios where electron behaviour is governed by complex multi-dimensional potentials and interactions. The team and North Carolina State University have delivered an exact solution bypassing the need for approximations previously required when calculating how electrons are released from atoms exposed to electric fields; their approach focuses on determining the probability that an electron remains bound within its initial state, the ‘bound-state survival amplitude’, rather than tracking its complete movement over time which significantly simplifies quantum dynamics. This delivers fundamental insights into atomic physics despite relying on a simplified model confining electrons to move along a single line rather than in three dimensions, offering potential avenues for extending these techniques towards more intricate systems with increased computational demands.

The researchers obtained an exact mathematical description of field-induced ionization from a one-dimensional system without using approximations typically required in such calculations. This means they could determine the likelihood of an electron remaining bound to an atom under any electric field strength and at any given moment in time. By focusing on calculating the ‘bound-state survival amplitude’, scientists streamlined complex quantum mechanical computations considerably. The authors note that future work will likely focus on applying this method to more realistic, multi-dimensional atomic models despite current limitations relating to scalability.

👉 More information
🗞 Exact Ionization Amplitudes for a Delta-Function Well in an Arbitrary-Strength DC Electric Field
✍️ Ilki Kim and Gerald J. Iafrate
🧠 ArXiv: https://arxiv.org/abs/2609.16507

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