Two-colour coherent control of nuclear and electron dynamics in photoionization of molecular hydrogen with FEL pulses

This study demonstrates that two-colour ($\omega$-$2\omega$) coherent control using seeded FERMI free-electron laser pulses can selectively steer the coupled electron-nuclear dynamics in molecular hydrogen photoionization, revealing how relative phases between ionization pathways depend on final vibrational states and are influenced by autoionizing resonances.

The Gist

Imagine you are trying to steer a tiny, chaotic boat (a hydrogen molecule) through a stormy sea using two different types of waves. One wave is a single, powerful surge, and the other is a pair of smaller, rhythmic ripples. If you time these waves perfectly, they can either cancel each other out or combine to push the boat in a specific direction.

This paper describes a groundbreaking experiment where scientists did exactly that, but with light instead of water and molecules instead of boats. Here is the story of how they used a special "two-color" laser to control the dance of electrons and nuclei inside a hydrogen molecule.

The Setup: The "Twin-Color" Flashlight

Usually, lasers are like a single-color flashlight. But at a massive facility called FERMI (a Free-Electron Laser), scientists created a unique "twin-color" beam.

• Color 1 (The Ripples): A lower-energy light wave (let's call it $\omega$).
• Color 2 (The Surge): A higher-energy light wave (let's call it $2\omega$), which is exactly twice the frequency of the first.

Think of these two lights as two different keys trying to unlock the same door (ionize the molecule).

1. The Direct Key ($2\omega$): A single, high-energy photon hits the molecule and knocks an electron out immediately. This is like kicking a door open with one strong shove.
2. The Two-Step Key ($\omega$): Two lower-energy photons hit the molecule in quick succession. The first one knocks the molecule into a temporary, excited "waiting room" state, and the second one kicks the electron out. This is like knocking once to get the door's attention, then pushing it open.

The Magic: The Quantum Interference

Here is the cool part: In the quantum world, particles can act like waves. When both "keys" (the one-step and two-step paths) are used at the same time, the electron doesn't just take one path; it takes both simultaneously.

These two paths interfere with each other, just like ripples in a pond.

• If the waves line up perfectly, they boost the electron's chance of flying out in a specific direction.
• If they are out of sync, they cancel each other out in that direction.

The scientists could control this "sync" by slightly delaying one of the light beams. By shifting the timing by a tiny fraction of a second (attoseconds, which is a billionth of a billionth of a second), they could make the electron spray out in a specific pattern, effectively steering the molecule's reaction.

The Twist: The Molecule is a Dancing Duo

Most previous experiments did this with single atoms (like Neon), which are like simple, round balls. But molecules are different. A hydrogen molecule ($H_2$) is like two dancers holding hands.

• The Electron: The dancer spinning fast.
• The Nuclei (Protons): The dancers holding hands, moving back and forth (vibrating).

In this experiment, the scientists didn't just watch the electron spin; they watched how the two dancers moved together.

• They tuned their laser to hit a specific "note" (vibrational level) of the molecule.
• They found that the direction the electron flew depended heavily on where the two protons were holding hands at the exact moment the second photon hit.

The Discovery: A Map of the Invisible

By measuring the angles at which the electrons flew out, the scientists created a "map" of the molecule's internal dance.

• They saw sudden "jumps" in the electron's behavior.
• These jumps told them that the electron was interacting with a complex, hidden "ghost state" (an autoionizing state) inside the molecule.
• Essentially, they used the interference pattern to take a snapshot of the molecule's nuclear wavefunction—a map of where the protons were likely to be found.

Why This Matters

Think of this as learning to conduct an orchestra where the musicians are moving at the speed of light.

• Before: Scientists could see the notes being played (the electrons) but couldn't control the rhythm of the musicians (the nuclei).
• Now: They have a "remote control" that can steer the reaction. By adjusting the phase (timing) of the light, they can force the molecule to break apart or react in a way that was previously impossible.

The Bottom Line

This paper proves that we can use ultra-fast, multi-colored laser pulses to act as a steering wheel for chemical reactions. We can now control not just the electrons, but the heavy nuclei too, opening the door to designing new chemical reactions with atomic precision. It's like being able to tell a molecule exactly how to dance, step-by-step, before it even hits the floor.

Technical Summary

1. Problem Statement

Controlling chemical reactions on the timescale of electron and nuclear motion (attoseconds to femtoseconds) is a major goal in physical chemistry. While coherent control schemes using $\omega$-$n\omega$ (fundamental and harmonic) light fields have been successfully applied to atomic systems to steer electron dynamics, extending this to molecular systems remains challenging due to:

• Additional Degrees of Freedom: Molecules possess rovibrational structures and nuclear motion, complicating the interference patterns.
• Symmetry Breaking: The lack of spherical symmetry in molecules leads to complex partial wave contributions and orientation-dependent dynamics.
• Spectral Congestion: Broadband pulses often excite multiple vibrational and electronic states simultaneously, obscuring specific dynamical pathways.

The specific challenge addressed here is the ability to disentangle and control coupled electron-nuclear dynamics in a molecule using a two-color coherent control scheme, specifically retrieving the relative phases between one-photon and two-photon ionization pathways.

2. Methodology

The study employs a combined experimental and theoretical approach using the seeded Free-Electron Laser (FEL) FERMI at the Low Density Matter (LDM) end-station.

• Experimental Setup:

  • Target: Molecular hydrogen ($H_2$) in its ground state ($X\ ^1\Sigma_g^+, v=0$).
  • Light Source: A bichromatic VUV pulse generated by tuning the first five undulators to the fundamental frequency ($\hbar\omega = 12.1$ eV) and the sixth undulator to the second harmonic ($\hbar2\omega = 24.2$ eV).
  • Coherent Control: A phase shifter between the undulators controls the relative optical phase ($\phi$) between the $\omega$ and $2\omega$ fields with a resolution of ~17 attoseconds.
  • Detection: Photoelectron Velocity Map Imaging (VMI) records angle-resolved photoelectron spectra (PADs) as a function of the phase shift $\phi$ and electron kinetic energy $\epsilon$.
  • Resonance: The $\omega$ photon energy is tuned to resonantly excite the specific intermediate vibronic state $H_2(B\ ^1\Sigma_u^+, v'=6)$, enabling Resonance-Enhanced Multiphoton Ionization (REMPI).

• Theoretical Framework:

  • Calculations were performed using second-order time-dependent perturbation theory.
  • The model accounts for randomly oriented molecules, summing over all accessible singlet states ($\Sigma, \Pi, \Delta$ symmetries) and partial waves.
  • It includes the coupling between electronic and nuclear motions, specifically treating autoionizing states ($1\Sigma_g^+$ and $1\Pi_g$) and the mapping of the intermediate nuclear wavefunction to final ionic states.
  • Theoretical spectra were convoluted with a Gaussian function (110 meV width) to match experimental energy resolution.

3. Key Contributions

• First Molecular $\omega$-$2\omega$ Coherent Control: This work represents the first successful implementation of the $\omega$-$2\omega$ coherent control scheme in a molecular system, moving beyond previous atomic studies (Ne, He).
• Vibrational Selectivity: By exploiting the narrow bandwidth of seeded FEL pulses, the authors achieved selective excitation of a single vibrational level ($v'=6$) in the intermediate neutral state, effectively eliminating vibrational congestion.
• Phase Retrieval: The study successfully extracted the relative phase ($\Delta\zeta$) between the one-photon ionization (OPI, $2\omega$) and two-photon ionization (TPI, $\omega$) amplitudes as a function of photoelectron energy and emission angle.
• Decoupling Dynamics: The methodology allows for the separation of electronic dynamics (influenced by autoionizing states) from nuclear dynamics (Franck-Condon factors and wavefunction mapping).

4. Results

• Interference Signatures: The Photoelectron Angular Distributions (PADs) exhibited clear asymmetry relative to the polarization axis, characterized by the odd-rank anisotropy parameters $\beta_1$ and $\beta_3$. These parameters oscillated sinusoidally with the relative optical phase $\phi$.
• Phase Jumps: The relative phases $\eta_1(\epsilon)$ and $\eta_3(\epsilon)$ (derived from $\beta_1$ and $\beta_3$) showed a monotonic increase with electron energy but featured pronounced phase jumps (of order $\pi$ or $\pi/2$) at specific energies corresponding to final vibrational states ($v_f$) of the $H_2^+$ cation.
  • Jumps were observed at $\epsilon \approx 6.55$ eV ($v_f \approx 11$), $6.75$ eV ($v_f \approx 10$), and around $7.0-7.2$ eV ($v_f \approx 8, 7$).
• Mechanism of Jumps:
  • Electronic Origin: The jumps near $7$ eV were attributed to the changing composition of the continuum partial waves ($\sigma_g$ vs. $\pi_g$) and the influence of autoionizing states ($1\Sigma_g^+$ and $1\Pi_g$).
  • Nuclear Origin: The other jumps were assigned to the mapping of the intermediate nuclear wavefunction ($v'=6$ of the $B$ state) onto the final vibrational states of the cation. The interference is sensitive to the internuclear distance ($R$) at which ionization occurs, which varies depending on the final $v_f$.
• Dominant Pathways: Analysis revealed that the interference is dominated by a single partial wave in each pathway: the perpendicular OPI path ($l=1$, $p\pi_u$) and the parallel TPI path ($l=2$, $d\sigma_g$). This simplification holds true except in a narrow energy region ($7.0-7.25$ eV) where multiple partial waves contribute.
• Agreement: There was excellent quantitative agreement between the experimental data and the ab initio theoretical calculations, validating the model of coupled electron-nuclear dynamics.

5. Significance

• Fundamental Insight: The study demonstrates that $\omega$-$2\omega$ coherent control can be used as a "molecular interferometer" to probe the nuclear wavefunction of a selected intermediate excited state with sub-femtosecond time resolution.
• Control Paradigm: It establishes a new paradigm for steering chemical reactions by manipulating the relative phase of light fields to selectively enhance or suppress specific vibrational pathways in the ionization process.
• Future Applications: The work paves the way for:
  • Molecular Frame Measurements: Future experiments aligning molecules could provide orientation-resolved dynamics.
  • Pump-Probe Schemes: Utilizing shorter pulses (few-fs or attosecond) to observe electron-nuclear dynamics in the absence of the external field.
  • Complex Molecules: Extending these techniques to larger polyatomic molecules to control complex chemical reaction pathways.

In summary, this paper provides a benchmark demonstration of how seeded FELs can be used to achieve unprecedented control and observation of coupled electron-nuclear dynamics in molecules, bridging the gap between atomic coherent control and complex molecular chemistry.