A Tale of Two Representations: Energy and Time in Photoabsorption

The gist The essay discusses two alternative representations of energy and time, a time-dependent and a time-independent one, both capable of embracing the same physics.

Conjugate Pairs and State Definition

Quantum physics introduces pairs of conjugate quantities bearing a specific complementary relationship, energy and time being one such pair. These pairs can be seen as gradients with respect to one another, with Planck’s characterizing our Universe. In quantum mechanics, the state of a system cannot simultaneously be prescribed in terms of both members of the conjugate pair because sets a limitation on such simultaneous prescription of entities whose commutator is non-zero. Only mutually commuting entities can be fixed simultaneously and used to define physical states, which points to choosing either one but not both of a non-commuting pair.

Bound States vs. Scattering Dynamics

The text contrasts the view of bound states in t-independent terms with collisions or scattering handled as evolving in time. Stationary states of fixed energy played a central role in the development of quantum physics, and Bohr’s description accounted for both atomic sizes and energies. In such a state, time becomes irrelevant as the electron persists over all time, occurring as a background in a generally ignorable pure phase factor of the wave function. Conversely, collisions or scattering are handled another way as evolving in time.

Phase Shifts and Cross Sections

Spectroscopy, involving bound states and transitions, is best viewed on the same footing as scattering involving asymptotic states with the electron reaching infinity. The electron’s asymptotic wave function is given by a superposition of regular and irregular solutions, leading to phase shifts δ that are functions of E. These phase shifts contain all effects of interactions, and the scattering cross-section is determined by the S-matrix ratio exp(2iδ).

Resonances and Time Delay

A rapid climb through π over some energy interval Γ around Er denotes a “resonance” of width Γ at that energy Er, where time may be invoked through /Γ as a “lifetime” of a state at that energy. The Wigner-Eisenbud-Smith “time delay” τ = 2 dδ/dE is introduced to analyze the energy dependence of phase shifts, reflecting residual energy dependences beyond the first derivative. This time delay can be interpreted as a temporary trapping into that configuration before the electron again emerges back in the continuum.

Angular Momentum Barriers and Shape Resonances

The angular momentum barrier experienced by the electron is related to Wigner threshold laws, which involve an effective potential that keeps the electron away from the origin. In complex atoms, this barrier can lead to a “two-valley” potential with an inner deep attractive well and a shallow outer well separated by an intervening positive barrier. This structure can result in “shape” resonances, which lie above threshold (E > 0) and arise from the shape of the potential.

Technological Influence

The choice of representation is influenced by what is technologically convenient or indeed possible, as direct measurements of short times were outside the reach of experiment in the first decades. The advent of very short laser pulses and streak cameras has only recently permitted such direct access to the very short times involved.

Acknowledgments

The perspectives one develops of a field owe much to several discussions with colleagues over many years. The author notes that this article in honor of his contributions to the field is also offered as a tribute to others with whom he discussed from those first months and extending over decades but who have now, sadly, passed away.

The paper explores the complementary nature of energy and time representations in quantum physics, highlighting how these two seemingly different descriptions provide distinct yet unified insights into phenomena like photoabsorption. The discussion delves into the relationship between bound states and scattering processes, utilizing phase shifts to characterize interactions, resonances defined by energy structure, and the introduction of time delay as a measure of state evolution. Furthermore, the paper connects these concepts to physical barriers like angular momentum barriers and illustrates how technological advancements have enabled direct measurement of short timescales relevant to these quantum dynamics. The core argument is that both representations are valid tools for understanding atomic physics, with the choice between them depending on the specific technical or experimental context.

How it works

The paper contrasts two primary ways of describing physical systems: a time-dependent and a time-independent one. In the context of quantum states, the state cannot be prescribed in terms of both members of the conjugate pair because ~ sets a limitation on such simultaneous prescription. This necessitates choosing either energy or time, but not both, for a non-commuting pair.

The analysis of scattering involves constructing the full wave function by superposing different solutions of the radial equation with appropriate outgoing (+) or ingoing (-) wave boundary conditions. The ratio of coefficients exp(±iδ), namely exp(2iδ), is identified as the S-matrix, and its modulus-square yields the scattering cross-section.

Resonances are characterized by a rapid climb through π over some energy interval Γ around Er, which can be interpreted as a quasi-bound state embedded in a continuum. The resonance cross section is described by the Beutler-Fano profile, which is asymmetric about Er and reduces to a symmetric Breit-Wigner form for large q.

The time delay τ = 2 dδ/dE provides an energy-dependent measure of the phase shift, reflecting residual energy dependences beyond the first derivative. This time delay can be interpreted as a temporary trapping into a configuration before the electron emerges back in the continuum.

Angular momentum barriers are related to an effective potential that keeps the electron away from the origin, which involves a k2 + 1 dependence in the phase shift that forms the basis of Wigner threshold laws. These barriers can lead to “shape” resonances, distinguishing them from quasi-bound state or “Feshbach” resonances.

The paper emphasizes that the choice of representation is often dictated by technological feasibility, as direct measurements of short times were previously impossible until the advent of attosecond measurement technology. The integration of time-dependent Schrödinger equation with very small time steps via supercomputers allows for meaningful direct integration without being restricted to time-independent calculations.

The concepts discussed, such as the Wigner threshold laws and the interpretation of resonances, are grounded in established spectroscopic techniques and theoretical models derived from decades of research in atomic physics. The paper concludes by noting that alternative descriptions exist outside physics, such as those used in astronomy and biology to manage vastly different time scales.

The study is grounded in established spectroscopic techniques and theoretical models derived from decades of research in atomic physics. The paper concludes by noting that alternative descriptions exist outside physics, such as those used in astronomy and biology to manage vastly different time scales. This comprehensive analysis underscores the deep interplay between fundamental quantum concepts and the practical constraints imposed by experimental capabilities.

The study is grounded in established spectroscopic techniques and theoretical models derived from decades of research in atomic physics.

1. textbf Rewriting Time-Dependent Simulations for Scattering Events: The Time Delay Metric. This allows AI models to predict not just energy-dependent cross sections, but also incorporate a time delay, defined as τ = 2 dδ/dE (Eq. 6), providing a measure of the temporary trapping into that configuration mentioned in the text when analyzing resonances like Cooper minima.

2. textbf Developing Fano Resonance Prediction via Time Domain Analysis. AI systems can be trained to analyze the time domain following a resonance, using measurements that probe the phase of the photoionization amplitude with free electron lasers, as this allows probing Fano resonances in the time domain, following their very build up [41–44] in time.

3. textbf Implementing Angular Momentum Barrier Modeling via Effective Potentials. AI can utilize the concept of a two-valley potential described as an inner deep attractive well and a shallow outer well separated by an intervening positive barrier, which is crucial for understanding how the angular momentum barrier suppresses electron escape in complex atoms.

4. textbf Enhancing Photoionization Prediction with Shape Resonance Identification. Models can be improved to distinguish between different resonance types by analyzing the energy dependence of cross sections, specifically identifying shape resonances (section 4.5 of [11]) to distinguish them from quasi-bound state or 'Feshbach' resonances discussed earlier [33, 34].

5. textbf Creating High-Resolution Spectral Analysis via Energy Derivative Mapping. AI can leverage the mapping provided by the decomposition in Eq. (4) to map energy values from smaller than Er to larger into a variation of δb from 0 to π as the cotangent goes from −∞ to ∞, enabling precise identification of resonance widths Γ and lifetimes.

This essay is based on a talk at Advances in Atomic, Molecular, and Optical Sciences 2020 (AAMOS20) in a symposium honoring Prof. S. T. Manson's decades-long contribution to photoabsorption studies. Quantum physics introduced into physics pairs of conjugate quantities bearing a specific complementary relationship, energy and time being one such pair. This gives rise to two alternative representations, a time-dependent and a time-independent one, seemingly very different but both capable of embracing the same physics. They give complementary descriptions and insight, with technical questions, theoretical and experimental, determining which may be the more convenient and practicable at any juncture. Two recent topics, Cooper minima in photoabsorption in Cl- and Ar, and angular-momentum barrier tunneling of f photoelectrons from Se in WSe 2, provide illustrative examples, also of the role that technological developments over the past five decades played in our approach to and understanding of phenomena.