Relativistic calculations of angular dependent photoemission time delay

This study utilizes the dipole relativistic random phase approximation to investigate the angular dependence of photoemission time delays for valence $np_{3/2}$ and $np_{1/2}$ subshells in Ar, Kr, and Xe, revealing strong anisotropy near Cooper minima and significant spin-orbit effects near the ionization threshold.

The Gist

Imagine you are watching a magic show where a magician (a laser) hits a crystal ball (an atom) with a flash of light, causing a tiny marble (an electron) to pop out. For a long time, scientists thought this happened instantly. But recently, they discovered that there is actually a tiny, tiny delay—a fraction of a billionth of a billionth of a second—between the flash and the marble flying out.

This paper is about measuring that delay with extreme precision, but with a twist: it depends on which way the marble flies and how heavy the crystal ball is.

Here is a breakdown of the paper's findings using simple analogies:

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

Imagine the electron trying to leave the atom is like a car trying to exit a highway. Usually, there are two lanes it can take: a "fast lane" (the dominant path) and a "slow lane" (a weaker path).

• The Cooper Minimum: Sometimes, the fast lane gets a massive construction zone (a "Cooper minimum"). The car gets stuck, and the exit time slows down dramatically.
• The Twist: When the fast lane is blocked, the car is forced to use the slow lane. But here's the kicker: the time it takes to exit depends heavily on which direction the car is trying to leave.
  • If the car tries to exit straight ahead (aligned with the laser), the delay is huge.
  • If it tries to exit at an angle, the delay changes completely.
• The Finding: The authors found that for lighter atoms like Argon, this directional delay is very strong. It's like a traffic jam that only happens if you try to turn left, but not if you turn right.

2. The "Heavy Suit" Effect (Relativity and Spin)

Now, imagine the marble isn't just a marble; it's a spinning top. In the world of heavy atoms (like Krypton and Xenon), the electron is moving so fast that it starts to feel the effects of Einstein's relativity. It's like the electron is wearing a heavy, spinning suit of armor.

• Spin-Orbit Splitting: Because the electron is spinning and moving fast, the "suit" splits into two slightly different versions: one where the spin helps the motion, and one where it fights the motion.
• The Result: Near the edge of the atom (the threshold), these two versions of the electron take different amounts of time to escape. It's like two twins running a race; one is running with the wind, and the other is running against it. They finish at different times, and this paper calculates exactly how much time separates them.

3. The "Ghostly Push" (Coulomb-Laser Coupling)

There is one more invisible force at play. When the electron leaves, the atom it left behind is now positively charged (like a magnet). It tries to pull the electron back. At the same time, the laser is still pushing it forward.

• The Tug-of-War: This creates a "tug-of-war" between the atom pulling back and the laser pushing forward.
• The Paper's Insight: The authors had to account for this invisible tug-of-war. They found that for lighter atoms, this tug-of-war makes the electron take longer to get away near the starting line. For heavier atoms, the electron is already moving so fast that this tug-of-war matters less.

The Big Picture: Why Does This Matter?

The scientists used a super-advanced computer model (called RRPA) to simulate these events for three noble gases: Argon (light), Krypton (medium), and Xenon (heavy).

• For Argon: Their model matched perfectly with non-relativistic (simpler) models. It confirmed that the "direction" of the electron matters a lot when the main exit path is blocked.
• For Krypton and Xenon: Here, the "heavy suit" (relativity) kicks in. The time delay splits based on the electron's spin. This is something simpler models couldn't predict.

In summary: This paper is like a high-speed camera study of electrons escaping atoms. It tells us that time isn't just a number; it's a shape. The time an electron takes to leave depends on where it's going (angle), how heavy the atom is (relativity), and which spin direction it has. By understanding these tiny delays, we are learning how to control matter at the fastest speeds imaginable, which is crucial for future technologies like ultra-fast computers and advanced medical imaging.

Technical Summary

1. Problem Statement

The paper addresses the theoretical understanding of attosecond time delays in laser-driven atomic ionization, specifically focusing on two complex aspects often treated separately or approximated in non-relativistic models:

• Angular Anisotropy: The variation of photoemission time delay relative to the polarization direction of the incident light. This is particularly significant near Cooper minima (energies where the photoionization cross-section drops due to destructive interference between partial waves).
• Spin-Orbit (SO) Splitting: The influence of relativistic effects, specifically the splitting of valence subshells ($np_{1/2}$ and $np_{3/2}$), on time delays. This is crucial for heavier atoms where relativistic effects are non-negligible.
• Threshold Behavior: The behavior of time delays near the ionization threshold, where Coulomb singularities and Coulomb-Laser Coupling (CLC) corrections dominate.

The authors aim to resolve discrepancies between non-relativistic models and experimental data for noble gases (Ar, Kr, Xe) by incorporating full relativistic effects and channel interference.

2. Methodology

The authors employ the Dipole Relativistic Random Phase Approximation (RRPA).

• Formalism: They utilize the multichannel RRPA formalism developed by Johnson and Lin. This approach treats the atom as a system of coupled channels, accounting for electron-electron correlations and relativistic effects simultaneously.
• Transition Amplitudes: The photoionization amplitude is calculated for transitions from ground states to excited continuum states. The formalism explicitly includes the interference of all spin-orbit coupled photoionization channels (e.g., $np_{1/2} \to \epsilon s_{1/2}, \epsilon d_{3/2}$ and $np_{3/2} \to \epsilon s_{1/2}, \epsilon d_{3/2}, \epsilon d_{5/2}$).
• Time Delay Definition:
  • Wigner Time Delay ($\tau_W$): Defined as the energy derivative of the scattering phase ($\tau = d\eta/dE$).
  • Angular Resolution: The authors derive expressions for the time delay as a function of the emission angle $\theta$ relative to the laser polarization. They calculate spin-averaged time delays for specific magnetic quantum numbers ($m$) and sum them to get the total angular-dependent delay.
  • CLC Correction: For near-threshold comparisons with two-color (XUV+IR) experiments, they add the Coulomb-Laser Coupling (CLC) correction ($\tau_{CLC}$) to the Wigner delay to obtain the total atomic time delay ($\tau_a = \tau_W + \tau_{CLC}$).
• Target Systems: Calculations were performed for Argon (Ar), Krypton (Kr), and Xenon (Xe). Neon (Ne) was excluded as it lacks a Cooper minimum in its photoionization cross-section.

3. Key Contributions

• Full Relativistic Interference: Unlike previous studies that often evaluated only the dominant channel, this work includes the full interference of all spin-orbit coupled channels, providing a more rigorous description of the time delay.
• Angular Dependence Mapping: The paper provides a comprehensive mapping of time delay anisotropy across different emission angles ($\theta = 0^\circ$ and $45^\circ$) and photon energies, specifically near Cooper minima.
• Validation and Calibration: The model is validated against non-relativistic RPAE (Random Phase Approximation with Exchange) calculations and experimental data for Argon, establishing the reliability of the RRPA approach before applying it to heavier atoms.
• Resolution of Threshold Discrepancies: The authors successfully explain the sign and magnitude of the time delay difference between $np_{1/2}$ and $np_{3/2}$ subshells near the ionization threshold, reconciling theoretical predictions with recent experimental observations.

4. Key Results

A. Argon (Ar) - Validation

• Agreement: For Ar (a light atom with weak relativistic effects), the RRPA results for the $3p$ subshell agree very well with non-relativistic RPAE calculations and experimental data (RABBITT experiments).
• Anisotropy: A strong angular anisotropy is observed near the Cooper minimum. The time delay differs significantly between $\theta = 0^\circ$ (polarization axis) and $\theta = 45^\circ$.
• Spin-Orbit Splitting: The difference between $3p_{1/2}$ and $3p_{3/2}$ time delays is negligible, confirming that relativistic spin-orbit effects are minor for Ar in this energy range.

B. Krypton (Kr) and Xenon (Xe) - Relativistic Effects

• Spin-Orbit Splitting: As the atomic number increases, the splitting between the time delays of the $np_{1/2}$ and $np_{3/2}$ subshells becomes clearly visible.
• Cooper Minimum Behavior:
  • The Cooper minimum in the time delay becomes shallower as the emission angle moves away from the polarization axis ($\theta = 0^\circ \to 45^\circ$).
  • This is due to the kinematic node of the $f$-wave (generated in two-color experiments) and the increasing contribution of weaker channels ($\epsilon s$) which compete with the dominant $\epsilon d$ channel.
  • The angular dependence is less pronounced in Kr and Xe compared to Ar, but the relativistic splitting is the dominant new feature.
• Resonances: In Xe, the calculations resolve rapid oscillations in the time delay caused by autoionization resonances ($5s^1np$ and $4d^9np$ series), which are less resolved in non-relativistic models.

C. Near-Threshold Region

• Coulomb Singularity: Near the ionization threshold, the Wigner time delay diverges to positive infinity due to the Coulomb singularity.
• CLC Correction: The CLC correction diverges to negative infinity. The total atomic delay is the sum of these two competing terms.
• Subshell Difference:
  • The $np_{1/2}$ subshell has a higher ionization potential than $np_{3/2}$. Consequently, at the same photon energy, electrons from $np_{1/2}$ have lower kinetic energy and are more strongly affected by the Coulomb singularity.
  • Kr: The net time delay difference ($\tau_{3/2} - \tau_{1/2}$) remains negative or close to zero near the threshold.
  • Xe: Due to lower ionization thresholds, photoelectrons have higher kinetic energy, reducing the Coulomb effect. The net time delay difference becomes positive near the threshold.
  • These findings align perfectly with recent experimental data (Huppert et al., 2015).

5. Significance

• Theoretical Benchmark: This work establishes the RRPA as a robust tool for predicting angular-dependent time delays in heavy atoms, where non-relativistic approximations fail.
• Experimental Interpretation: The results provide a critical theoretical framework for interpreting attosecond streaking and RABBITT experiments, particularly those involving heavy noble gases where spin-orbit coupling cannot be ignored.
• Physical Insight: The paper clarifies the interplay between angular anisotropy (driven by channel competition) and relativistic splitting (driven by spin-orbit coupling), showing that while anisotropy dominates near Cooper minima, relativistic splitting dominates near ionization thresholds.
• Resolution of Controversies: By including the CLC correction and full channel interference, the authors resolve previous discrepancies between theory and experiment regarding the sign of the time delay difference in Xe near the threshold.