Chiral Hole Attosecond Wave-Packet

• The paper demonstrates that chiral hole attosecond wave-packets are coherent electronic deficiencies encoding molecular chirality through ultrafast photoionization.
• It utilizes multiphoton ionization with tailored, rotated polarization pulses to induce quantum interference, yielding PECD values up to 8% and group delays up to 24 as.
• The study employs velocity-map imaging and time-resolved interferometry to map the attosecond dynamics, providing actionable insights into multielectron chiral interactions.

A chiral hole attosecond wave-packet is a time-dependent, coherently evolving electronic deficiency in a molecular cation, formed when a chiral molecule undergoes ultrafast photoionization. This concept describes an attosecond-scale electronic wave-packet whose spatial distribution and handedness encode the molecule's intrinsic chirality and whose temporal evolution provides access to multielectron, chiral-specific dynamics. Chiral hole wave-packets are directly implicated in ultrafast manifestations of photoelectron circular dichroism (PECD), observable even when the ionizing light does not carry helicity, revealing a new mechanism for chiral asymmetry in photoionization driven by molecular—not optical—chirality (Bouskila et al., 13 Dec 2025, Beaulieu et al., 2020).

1. Theoretical Framework and Definition

The chiral hole attosecond wave-packet refers to the missing-electron coherent superposition in the molecular cation following ultrafast photoionization. For a neutral molecule initially in ground state $|\Psi_0\rangle$, the wavefunction after interaction with an ultrashort pulse is approximately:

$$|\Psi(t)\rangle \approx a_0(t) |\Psi_0\rangle + \int d^3k\; a_{\mathbf{k}}(t)\; |\Psi_{\mathbf{k}}^{+}\rangle$$

where $|\Psi_{\mathbf{k}}^{+}\rangle$ comprises the cation and a free electron of momentum $\mathbf{k}$. The hole wave-packet is the projected part onto the surviving cationic basis:

$$|h(t)\rangle = \int d^3k\; a_{\mathbf{k}}(t)\; |\Phi_{\mathbf{k}}^+\rangle \approx \sum_n c_n(t) |\Phi_n^+\rangle$$

with expansion coefficients $c_n(t)$. The resulting electron density

$$\rho_h(\mathbf{r}, t) = \sum_{n,m} c_n^*(t)\, c_m(t)\, \Phi_n^{+*}(\mathbf{r})\, \Phi_m^+(\mathbf{r})$$

evolves on attosecond timescales and, in a chiral molecule, inherently breaks all mirror symmetries. The time-dependent dipole $\mathbf{d}_h(t)$ and pseudoscalar triple product $$\chi_h(t) = \mathbf{d}_h(t)\cdot[\dot{\mathbf{d}}_h(t) \times \ddot{\mathbf{d}}_h(t)]$$ quantify the packet’s three-dimensional handedness (Bouskila et al., 13 Dec 2025).

2. Generation via Attosecond Photoionization

Chiral hole wave-packets are generated through multiphoton or above-threshold ionization (ATI) processes using intense, ultrafast laser pulses. Experimental protocols employ:

• A femtosecond UV pulse (e.g., 400 nm, 40 fs, intensity ≃ $5\times10^{12}\,$W/cm$^2$) with adjustable polarization
• Auxiliary IR pulses (e.g., 800 nm) to enable continuum–continuum transitions and sideband generation

The polarization state of the UV and IR pulses is pivotal. Circularly polarized light creates enantio-sensitive electron emission, while schemes with linearly polarized, temporally delayed, polarization-rotating pulses implement non-helical “storage” of chirality in the evolving hole packet itself (Beaulieu et al., 2020, Bouskila et al., 13 Dec 2025).

Velocimetry of the outgoing electrons employs velocity-map imaging (VMI), yielding full angular-resolved detection of photoelectron wave-packet distributions. The measurement of directional (forward/backward) temporal profiles of electron emission allows one to reconstruct the evolution of the chiral hole wave-packet.

3. Ultrafast Dynamics and Interference Effects

The attosecond evolution of the chiral hole is governed by quantum coherence among ionic states, with its spatial and temporal features manifesting directly in the direction and phase of emitted electron bursts. For probe and pump sequences employing multiple linearly polarized pulses:

$$\mathbf{E}_j(t) = E_0\, \hat{\epsilon}_j\, \cos(\omega (t-t_j) + \phi_j)\, \sin^2\Bigl(\frac{\pi}{T}(t-t_j)\Bigr)$$

the evolution and read-out of the hole packet depend sensitively on the angles between polarization axes $\theta$, inter-pulse delays $\Delta t$, and the carrier–envelope phases.

Photoelectron momentum distributions arise from coherent superpositions of amplitudes from each pulse, leading to interference terms such as:

$$PES(\mathbf{k}) = |A(\mathbf{k})|^2 = \sum_j |A_j|^2 + 2\sum_{j<\ell} |A_j||A_\ell| \cos[\Phi_j(\mathbf{k})-\Phi_\ell(\mathbf{k})]$$

where phase differences $\Delta\Phi_{j\ell} = -\omega(t_\ell-t_j)$ can be attosecond-controlled. The interference gives rise to stark forward/backward asymmetries in observable PECD, unambiguously tracing the signature of chiral hole dynamics.

4. Role in PECD and Chiral Asymmetries

PECD arises as a net asymmetry in the photoelectron angular distribution for randomly oriented chiral molecules, typically requiring nonzero instantaneous optical chirality (helicity). However, with chiral hole attosecond wave-packets, substantial PECD ($\sim8\%$) is produced even when all pulses are linearly polarized and carry zero field chirality at any instant. The key mechanism is:

• The first pulse creates a chiral hole in the ionic manifold, encoding molecular handedness.
• Subsequent, polarization-rotated, phase-synchronized probe pulses induce quantum interference between ionization amplitudes, converting the chiral hole dynamics into observable PECD:

$$PECD(\mathbf{k}) = 2 \frac{PES_R(\mathbf{k}) - PES_S(\mathbf{k})}{PES_R(\mathbf{k}) + PES_S(\mathbf{k})}$$

• The signal is optimized by tuning inter-pulse delay $\Delta t$ and relative polarization $\theta$; the effect is maximal for intermediate angle (e.g., $120^\circ$) and multi-cycle delays, and vanishes for parallel or orthogonal polarization configurations.

This mechanism contrasts fundamentally with CPL-driven PECD, where the field itself carries the handedness (Bouskila et al., 13 Dec 2025).

5. Experimental Reconstruction and Analysis

Time-resolved, angle-resolved photoelectron interferometry combined with VMI enables direct mapping of the attosecond dynamics of the chiral hole. The temporal profile $\psi^{f,b}(t,\theta)$ is reconstructed from the measured spectral amplitude $A^{f,b}(\omega,\theta)$ and phase $\phi^{f,b}(\omega,\theta)$ via Fourier integration:

$$\psi^{f,b}(t,\theta) = \int A^{f,b}(\omega,\theta) e^{-i\omega t + i\phi^{f,b}(\omega,\theta)}\, d\omega$$

The group-delay difference between forward and backward emission,

$$\Delta\tau(\theta) = \frac{\partial}{\partial \omega}[\phi^f(\omega,\theta) - \phi^b(\omega,\theta)]$$

serves as a time-domain fingerprint of chiral dynamical processes, with observed values reaching up to 24 as at certain angles in nonresonant emission (Beaulieu et al., 2020). Resonant autoionizing channels can produce spectrally sharp $\pm\pi$ phase jumps, inducing additional attosecond-scale time shifts and manifesting as pronounced asymmetries in the reconstructed electron wave-packet.

Wigner–Ville distribution analysis of the reconstructed $\psi(t,\theta)$ reveals the time-frequency structure of the outgoing electron wave packets, including destructive interference fringes and the dominance of resonant chiral channels.

6. Quantitative Findings and Optimization Strategies

Numerical simulations and attosecond-resolved experiments yield the following insights:

• PECD values up to $\sim8\%$ are achieved by optimized three-pulse schemes with rotated linear polarizations and delays of approximately three optical cycles at 800 nm.
• Control over inter-pulse delay is the primary parameter affecting the quantum interference and thus the observable PECD. Carrier–envelope phase fluctuations impact the signal by only $3\%$–$5\%$.
• The magnitude and direction of group delay and phase asymmetries are sharply angle- and energy-dependent, reflecting details of the chiral molecular potential and the dynamics of the Dyson orbital and autoionizing resonances.
• Reduction to two-pulse schemes isolates the role of hole creation (pure state preparation via the first pulse) from probing (conversion of chiral hole to PECD via the second, rotated probe pulse).
• In non-resonant regimes, chiral Wigner delays up to 24 as are observed; in resonant cases, phase jumps of $\pm\pi$ over a 0.1 eV window translate into $\sim400$ as forward/backward shifts in the electron emission profile (Beaulieu et al., 2020, Bouskila et al., 13 Dec 2025).

7. Comparison with Traditional Mechanisms and Broader Impact

Traditional PECD relies on circularly polarized light to induce instantaneous, optical-chirality-mediated photoionization. The chiral hole attosecond wave-packet mechanism demonstrates:

• Attosecond chiral asymmetries can arise solely from the time-dependent evolution of the molecular hole, independently of the optical helicity.
• In the absence of field chirality, all observed asymmetries result from the coherent superposition and manipulation of chiral ionic states.
• This delineates a new regime—“hole-wave-packet chirality” as distinct from “field chirality”—enabling novel modalities for ultrafast chiral spectroscopy and control of electron motion.

A plausible implication is that future spectroscopic and control protocols can exploit multi-color, time-delayed, polarization-tailored pulses to sculpt chiral electronic states on attosecond timescales, sensitively probing and manipulating multielectron, chiral-specific dynamics without requiring chiral light (Bouskila et al., 13 Dec 2025).

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Table: Comparison of PECD Mechanisms

Mechanism Type	Origin of Chirality	Key Control Parameter
Standard (CPL-driven) PECD	Optical helicity (light)	Pulse polarization
Chiral hole attosecond wave-packet	Evolving cationic hole packet	Delay/rotation between pulses

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The investigation of chiral hole attosecond wave-packets provides a framework for characterizing and controlling ultrafast electron dynamics in chiral molecules, revealing new fundamental mechanisms for PECD and paving the way for technologically relevant applications in molecular discrimination and coherent control (Beaulieu et al., 2020, Bouskila et al., 13 Dec 2025).