Time delay in valence shell photoionization of noble gas atoms

This paper employs the non-relativistic random phase approximation with exchange to calculate and analyze valence shell photoionization time delays for noble gas atoms (Ne, Ar, Kr, and Xe) across a 200 eV energy range, validating the results against experimental data to reveal fundamental atomic physics insights accessible through attosecond measurements.

The Gist

The Big Picture: The Atomic "Escape Artist" Race

Imagine an atom (like Neon, Argon, or Xenon) as a crowded dance floor. The electrons are the dancers, spinning in specific groups (shells) around a central DJ (the nucleus).

Sometimes, a flash of light (a photon) hits the dance floor. This light is so energetic that it kicks one of the dancers out of the club entirely. This is called photoionization.

For a long time, scientists thought this happened instantly. If you shine a light, the electron pops out immediately. But recent experiments using ultra-fast "stop-motion cameras" (called attosecond streaking) revealed something surprising: The electrons don't all leave at the exact same time.

Some electrons take a tiny fraction of a second longer to escape than others. This tiny delay is measured in attoseconds (one attosecond is to a second what a second is to the age of the universe!).

The Mystery: Why the Delay?

The author of this paper, A. S. Kheifets, wanted to figure out why these delays happen and how long they actually are. He focused on the "valence shell" electrons—the outermost dancers who are the easiest to kick out.

He looked at four noble gases: Neon, Argon, Krypton, and Xenon. These are like a family of atoms getting heavier and more complex as you go down the list.

The Two Theories: The Soloist vs. The Crowd

To calculate these delays, the author used two different ways of thinking about the electrons:

1. The "Soloist" Model (Hartree-Fock): Imagine each dancer is an independent person. They only care about the DJ and their own path out the door. They don't really notice the other dancers. This is the "Independent Electron" model.
2. The "Crowd" Model (RPA): Imagine the dancers are all holding hands. If one gets pushed, they pull on their neighbors. They react to each other. This is the Random Phase Approximation with Exchange (RPA). It accounts for the fact that electrons are messy and interact with one another.

The Key Findings: What the Paper Discovered

The author ran massive computer simulations to see how long it takes an electron to escape under both models. Here is what he found, using some metaphors:

1. The "Cooper Minimum" (The Traffic Jam)

In some atoms (like Argon and Xenon), there is a specific energy level where the electron's path out gets weird. It's like a highway where the lanes suddenly merge or disappear.

• The Soloist Model says: "The electron just slows down a little bit."
• The Crowd Model says: "Whoa! Because the electrons are holding hands, when one hits this traffic jam, it causes a massive ripple effect. The electron gets stuck for a moment, then suddenly bursts out."
• The Result: Near these "traffic jams" (called Cooper minima), the delay time explodes. Instead of a few attoseconds, the delay can jump to hundreds of attoseconds. It's the difference between a car slowing down for a red light and a car getting stuck in a massive pile-up.

2. The "Heavy" Atoms are Messier

As the atoms get heavier (from Neon to Xenon), the electrons are more crowded.

• In Neon (the lightest), the "Soloist" model works okay. The delays are small, and the "Crowd" model doesn't change things much.
• In Xenon (the heaviest), the "Crowd" model is essential. The electrons are so tightly packed that they act like a single, chaotic fluid. The delays become huge and unpredictable if you ignore how they interact.

3. The "Who Left First?" Debate

Scientists had been arguing about which electron leaves first: the one from the inner shell or the outer shell?

• In Argon, experiments suggested the outer electron left later than the inner one.
• The author's "Crowd" model (RPA) showed that the interactions between shells actually flip the script. Depending on the energy of the light, the inner electron might actually be the one that gets delayed, or vice versa. It's like a race where the runners keep tripping over each other, changing the order of who finishes first.

The Problem: Theory vs. Reality

Here is the twist at the end of the story.
The author's calculations (using the "Crowd" model) are very sophisticated. They match the experimental data for how many electrons get kicked out (the cross-section) very well.

However, when it comes to the time delay, the author's numbers are still about half of what the experiments measured.

• Experiment says: "The delay is 21 attoseconds."
• Author's Math says: "The delay is only about 12 attoseconds."

Even after adding all the known corrections (like the tug-of-war between the electron and the laser field), the math still doesn't quite match the stopwatch.

The Conclusion: A Work in Progress

The paper concludes that while we have a great understanding of how electrons interact (the "Crowd" model is a huge improvement over the "Soloist" model), there is still a missing piece of the puzzle.

It's like having a perfect map of a city, but when you try to time a drive from Point A to Point B, your GPS says 10 minutes, but the car actually takes 20. We know the roads (the physics), but there's a hidden traffic factor we haven't figured out yet.

In short:

• Electrons don't leave atoms instantly; they take a tiny, measurable pause.
• This pause gets huge when electrons interact with each other (the "Crowd" effect).
• We can predict the pause pretty well, but our predictions are still shorter than what we see in real life.
• This mystery drives the field of attosecond physics forward, pushing scientists to build better models and faster cameras to understand the fundamental rules of the universe.

Technical Summary

1. Problem Statement

The paper addresses the discrepancy between theoretical calculations and experimental measurements of attosecond time delays in atomic photoionization.

• Context: Recent experiments using attosecond streaking and two-photon sideband (RABBITT) interference have measured relative time delays between photoelectrons emitted from different sub-shells (e.g., $2s$ vs. $2p$ in Neon, $3s$ vs. $3p$ in Argon).
• The Discrepancy:
  • In Neon (106 eV), experiments measured a relative delay of $21 \pm 5$ attoseconds (as) between $2p$ and $2s$ emission. Theoretical Hartree-Fock (HF) calculations predicted only $\sim 6$ as, and even with Random Phase Approximation with Exchange (RPAE) corrections, the gap remained large.
  • In Argon (34–40 eV), experiments near the $3s$ Cooper minimum showed conflicting results regarding the sign of the delay between $3s$ and $3p$ emission. Early theories based on independent electron models failed to predict the correct ordering or magnitude.
• Core Question: Can a systematic theoretical investigation using advanced many-body methods (RPAE) resolve these discrepancies and explain the underlying physics of time delays across the noble gas series (Ne, Ar, Kr, Xe)?

2. Methodology

The author employs a non-relativistic Random Phase Approximation with Exchange (RPAE) to calculate photoionization matrix elements and derive time delays.

• Theoretical Framework:
  • Independent Electron Model (HF): Used as a baseline. It accounts for Coulomb interactions indirectly via a self-consistent potential but neglects dynamic inter-shell correlations.
  • RPAE Model: An extension of HF that explicitly accounts for inter-shell correlations. It solves a system of integral equations where the dipole matrix element includes contributions from virtual electron-hole pair excitations in neighboring sub-shells (both time-forward and time-reverse processes).
• Calculation of Time Delay:
  • The Wigner time delay ($\tau_W$), representing the atomic contribution, is defined as the energy derivative of the complex phase of the photoionization amplitude:
    $$\tau_W = \frac{d}{dE} \arg(f(E))$$
  • The total measured delay in experiments includes corrections for continuum-continuum transitions ($\tau_{CC}$) in interferometry or Coulomb-laser coupling ($\tau_{CLC}$) in streaking. The paper focuses on isolating and calculating the intrinsic atomic $\tau_W$.
• Scope: Calculations were performed for valence shells of Ne, Ar, Kr, and Xe from their ionization thresholds up to 200 eV.
• Validation: The computational technique was validated by comparing calculated partial photoionization cross-sections and angular asymmetry parameters ($\beta$) against extensive experimental data.

3. Key Contributions

• Systematic Mapping: The first comprehensive theoretical mapping of Wigner time delays across the entire noble gas series (Ne to Xe) over a wide photon energy range.
• Resolution of Sign Reversal in Argon: The paper demonstrates that inter-shell correlation is the critical factor determining the sign of the relative time delay between $3s$ and $3p$ sub-shells in Argon. While HF predicts one ordering, RPAE predicts the reverse, aligning better with specific experimental observations near the Cooper minimum.
• Identification of Phase Jumps: The study highlights that $\pi$-phase jumps occurring at Cooper minima (where the photoionization cross-section drops to a minimum) are the primary drivers of extreme time delays (hundreds of attoseconds).
• Clarification of Coulomb vs. Correlation Effects: The work disentangles the effects of the long-range Coulomb potential (which dominates at low energies) from short-range many-electron correlations (which dominate near Cooper minima and shape resonances).

4. Key Results

A. Neon (Ne)

• Cross-sections: RPAE improves agreement with experiment for $2s$ and $2p$ cross-sections compared to HF.
• Time Delay: The calculated relative delay between $2p$ and $2s$ at 106 eV is $\sim 11.9$ as (after adding $\tau_{CLC}$ corrections). This is significantly closer to the experimental $21 \pm 5$ as than HF, but still leaves a gap. The $2s$ delay is negative relative to $2p$ at this energy.

B. Argon (Ar)

• Cooper Minimum: The $3s$ cross-section exhibits a deep Cooper minimum driven by inter-shell correlation with $3p$.
• Phase Jump: At the Cooper minimum, the phase of the $3s \to \epsilon p$ amplitude jumps by $\pi$.
• Time Delay Reversal:
  • HF Prediction: $3p$ emission is delayed relative to $3s$.
  • RPAE Prediction: The correlation causes a massive delay in the $3s$ channel (hundreds of as), reversing the order so that $3s$ is delayed relative to $3p$.
  • This reversal explains the conflicting experimental reports and highlights the sensitivity of time delay measurements to correlation effects near Cooper minima.

C. Krypton (Kr) and Xenon (Xe)

• Kr: Similar to Ar, the $4s$ sub-shell shows a deep minimum and a large time delay peak ($\sim 300$ as) due to correlation with $4p$. The $3d$ sub-shell is dominated by a shape resonance (intra-shell effect) and is well-described by HF.
• Xe:
  • The $4d$ sub-shell exhibits a "giant resonance" followed by a Cooper minimum.
  • Strong Inter-shell Coupling: The behavior of the $4d$ resonance is replicated in the $5s$ and $5p$ cross-sections due to strong inter-shell interaction.
  • Extreme Delays: The $5s$ sub-shell shows a time delay of nearly 300 as near the Cooper minimum (below 30 eV) due to the merging of the phase jump with the Coulomb singularity.
  • Negative Delays: Both $5s$ and $5p$ show large negative delays near 150 eV due to correlation with the $4d$ shell.

5. Significance and Conclusion

• Fundamental Physics: The paper establishes that attosecond time delay measurements are a powerful probe of many-electron correlations and elastic scattering phases. The time delay is not just a trivial delay but contains rich phase information about the atomic potential and electron dynamics.
• Theoretical vs. Experimental Gap: Despite the improved accuracy of RPAE, a discrepancy remains between the calculated Wigner delays and experimental values (even after adding $\tau_{CC}$ or $\tau_{CLC}$ corrections).
  • The author suggests that while the accuracy of cross-sections is high, the accuracy of group delay calculations is harder to quantify (estimated within 10% for lighter atoms, potentially larger for Xe).
  • Relativistic effects are deemed negligible for the phase in these systems.
• Future Outlook: The unresolved discrepancy suggests that either the current corrections (CLC/CC) are insufficient, or there are subtle many-body effects not fully captured by the RPAE model. The work underscores the need for further refinement in both experimental techniques and theoretical models to fully understand the attosecond dynamics of photoionization.

In summary, this paper provides a rigorous theoretical framework demonstrating that inter-shell correlations are the dominant factor shaping photoionization time delays in noble gases, particularly near Cooper minima, and that these delays can reach hundreds of attoseconds, far exceeding simple independent-electron predictions.