A Tale of Two Representations: Energy and Time in Photoabsorption

summary

Video file (mp4)

The gist

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.

In short

The essay explores two complementary ways to describe quantum systems: time-dependent and time-independent representations of energy and time. It contrasts these views by examining bound states versus scattering dynamics, using phase shifts to characterize interactions, and introducing concepts like resonances and time delay. The choice between these descriptions depends on the specific technical or experimental context.

Key concepts

Conjugate Pairs
In quantum physics, energy and time are conjugate pairs with a specific complementary relationship. Because of this relationship, a system's state cannot be perfectly described by both simultaneously; one must be chosen over the other to define the physical state.
Phase Shifts
Spectroscopy involves analyzing how electrons interact with atoms. The electron's wave function is analyzed using phase shifts ($\delta$), which depend on energy. These shifts contain all information about the interactions between the electron and its environment, determining the scattering cross-section.
Time Delay ($ au$)
The time delay ($ au = 2\hbar d\delta/dE$) measures how long an electron remains temporarily trapped in a specific configuration before emerging back into the continuum. It is calculated from the energy dependence of phase shifts and reflects residual energy dependencies beyond the first derivative.
Shape Resonances
These resonances occur when an electron experiences a potential barrier that creates a 'two-valley' structure, resulting in a deep inner well and a shallow outer well. These shape resonances arise from the geometry of the potential rather than just being quasi-bound states.

Terminology used across episodes

This episode discusses

The paper

A Tale of Two Representations: Energy and Time in Photoabsorption · Read on arXiv

A. R. P. Rau

Louisiana State University

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.

DOI: 10.1088/1402-4896/ac0aa1

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "A Tale of Two Representations".

Mira: Here's the gist of the paper. 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.

Kai: Conjugate Pairs and State Definition Quantum physics introduces pairs of conjugate quantities bearing a specific complementary relationship, energy and time being one such pair.

Mira: These pairs can be seen as gradients with respect to one another, with Planck’s characterizing our Universe.

Kai: 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.

Paper summary: Kai: So, to recap, the paper "A Tale of Two Representations: Energy and Time in Photoabsorption" argues that energy and time are linked by a complementary relationship in quantum physics, leading to two ways of looking at a system.

Mira: The core thesis is that these two representations—the time-dependent one versus the time-independent one—can both accurately describe the same underlying physics.

Lev: It seems like they're saying that depending on whether you're studying bound states or scattering, you get a different kind of description.

Kai: Exactly. For bound states, time is often background noise because the electron persists forever, so a time-independent view works pretty well there.

Mira: But for scattering and collisions, which involve an electron leaving the system to go somewhere else, you really need to use a representation that evolves with time.

Lev: The paper connects this to how we calculate the S-matrix through phase shifts delta, which are functions of energy E.

Kai: And it uses the ratio of these phase shifts—exp(2i delta)—to get the scattering cross section sigma, which is related to exp(-iE) times some factor involving sin delta <ref:2103.14985#pg2>.

Mira: The paper also introduces the time delay tau equal to two times the derivative of that phase shift with respect to energy, ddelta over dE, as a measure of how long something gets trapped.

Lev: That trapping interpretation is interesting because it suggests a temporary configuration before the electron re-enters the continuum.

Kai: It also covers angular momentum barriers and shape resonances, which are caused by an effective potential that keeps the electron away from the origin.

Mira: These shape resonances are distinct because they arise from the geometry of that potential, not just being trapped in a simple bound state configuration.

Lev: When you look at running this on real hardware, like superconducting qubits or whatever we're working on, you have to figure out how much overhead those time-dependent calculations actually add.

Kai: That’s the engineering reality—figuring out which representation is more convenient and practical for the specific experimental setup we are trying to achieve.

Conclusion: Mira: So, looking at "A Tale of Two Representations: Energy and Time in Photoabsorption," it really boils down to how energy and time descriptions work together.

Kai: The authors show that these two seemingly different descriptions give you complementary insights into photoabsorption phenomena like Cooper minima or angular momentum barrier tunneling.

Mira: They establish that both the time-dependent and time-independent views are valid, but their utility depends entirely on the specific technical questions you're trying to answer in your research.

Lev: For us, it means that when we design quantum error correction schemes or look at pulse sequences, we have to be conscious of whether our model relies more on a time-dependent evolution or a static energy picture.

Kai: Ultimately, the paper is about showing that the choice isn't just academic; it dictates which tools you use for analyzing these complex atomic physics problems.

Mira: It highlights how long-standing concepts in atomic physics are being approached with new tools from experimental capabilities over the past fifty years.

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