Unravelling inter-channel quantum interference in below-threshold nonsequential double ionization with statistical measures

TL;DR

This work addresses interchannel quantum interference in below-threshold NSDI via RESI within the strong-field approximation, deriving analytic phase conditions for interference between excitation channels under arbitrary driving fields. It introduces Earth Mover's Distance–based Equal Mix Metric (EMM) to quantify how equally two channels contribute to the two-electron PMD and identifies three key factors—$E_{ ext{diff}}$, $M_{ ext{diff}}$, and $O_R$—that govern channel mixing. The authors show that channel-exchange interference is typically the most robust interchannel mechanism, while channel-only, channel-temporal, and combined interferences enrich or blur the PMD patterns depending on channel overlap and intensity balance. The framework provides practical guidelines for tuning or suppressing interchannel interference in RESI experiments and offers a scalable approach for extending analyses to more channels in multi-electron strong-field processes.

Abstract

We present a systematic study of interchannel quantum interference in laser-induced nonsequential double ionization (NSDI) within the strong-field approximation. Focusing on the below-threshold intensity regime where the recollision-excitation with subsequent ionization (RESI) pathway dominates, we derive analytical phase conditions governing interference between distinct excitation channels for arbitrary driving fields. To quantify the interplay between channels resulting from a vast number of interfering processes, we introduce statistical metrics based on the Earth Mover's Distance, allowing us to assess the relative weight of each channel's contribution to the two-electron photoelectron momentum distributions (PMDs). We identify key factors that determine whether interchannel interference is appreciable such as comparable channel intensities, strong spatial overlap between the excited-state wavefunctions and the energy difference between contributing channels. We demonstrate that for linearly polarized few-cycle pulses, the typical intrachannel interference features associated with exchange, temporal shifts and combined exchange-temporal interference are retained with interchannel interference. Our findings establish a hierarchy of interference mechanisms in RESI and may provide practical guidelines for enhancing or suppressing interference in different regions of the momentum plane.

Figures (14)

• Figure 1: Schematic representation of the momentum-space regions occupied by the RESI transition amplitudes $M_l, M_u, M_r$ and $M_d$ associated with different events within the pulse [panels (a, a')], and those resulting from their coherent superposition [panel (b)] for multiple channels. The negative (positive) signs in panel (ai) indicate the most probable momenta. The dashed rectangles indicate the momentum constraints that would hold for a monochromatic driving field, while the shaded areas show their counterparts for few-cycle pulses. We have used the same color for $M_l$ and $M_d$, and $M_u$ and $M_r$ to highlight the property $M_l(\mathbf{p_1}, \mathbf{p_2)}= M_d(\mathbf{p_2}, \mathbf{p_1})$ and $M_r(\mathbf{p_1}, \mathbf{p_2)}= M_u(\mathbf{p_2}, \mathbf{p_1})$. We assumed that each subpanel in (a) gives the momentum regions occupied by events within a single cycle, and the events in (a)(i) and (a)(ii) are summed coherently. Overlapping shaded regions indicate that quantum interference may occur. We consider two events separated by a half cycle in each plot in panel (a), which eventually interfere, but one can extend to any number of interfering events. The full coherent photoelectron momentum distribution (i.e. summed over all excitation channels, events and symmetrization) will occupy the full momentum space, and will be the sum of the single-channel PMDs shown in panels (a) and (b). This idea can also be extended for multiple channels.
• Figure 2: Few-cycle pulse associated with the vector potential (\ref{['eq:Apulse']}), with peak intensity $I=1.5 \times 10^{14}\mathrm{W/cm}^2$, wavelength $\lambda=800$ nm ($\omega=0.057$ a.u.), $N=4.3$ and carrier-envelope phases $\phi_1=65^{\circ}$ [panel (a)]. The three main events $p_io_j$ towards the center of the pulse are labeled with their corresponding pair and orbit numbers. The classical ionization and return times associated with the pairs of orbits $p_3$, $p_4$, $p_5$ of the first electron are indicated by arrows, and the most relevant ionization times for the orbits $o_j$ of the second electron, with $(j=4,5,6)$, are signposted by rectangles. The initial numbers chosen for the indices $i,j$ refer to the extremum of the field for which the counting starts. Matching styles and colors have been used for different events $\varepsilon_k=p_io_j$ and the momentum mapping in panel (b), associated with the channel $\mathcal{C}_n$. Panel (b) displaces an analogous mapping for a different channel $\mathcal{C}_m$. To facilitate the interference studies, in this latter panel we have employed different colors, although the sketched PMDs are associated with the same events.
• Figure 3: Temporal shifts between (a) events within a single channel, ($\Delta t"_\varepsilon, \Delta t'_\varepsilon, \Delta t_\varepsilon$) and (b) events in different channels ($\Delta t"_{(C_n, C_m)}, \Delta t'_{(C_n, C_m)}, \Delta t_{(C_n, C_m)}$)
• Figure 4: Schematic representation of different types of pair-wise interchannel interference that may occur for NSDI RESI. Panels (a) and (a') show the locations of the transition amplitudes (and actions) for three different events in the parallel momentum plane, for two different channels. $\varepsilon_2$ occurs in the same cycle as $\varepsilon_1$ and will thus exhibit intracycle interference effects, whilst $\varepsilon_3$ occurs in a different field cycle to $\varepsilon_1$ and so interference between these events will be intercycle. Panel (b) shows processes where only channels are summed over coherently and events and symmetrization are incoherent, thus isolating interference occurring from coherent channels. In panel (c), electron momenta are swapped and channels are summed coherently but intracycle events are not. In panel (d), intracycle events and channels are summed coherently but there is no symmetrization. In panel (e), an event and the symmetrized counterpart of another event in the same cycle are summed over coherently, along with the channels. In panels (b)-(e), the phase differences are denoted by $\alpha_{\mu,\nu}$ where $\mu,\nu=l,d,r,u$ and the subscript $T$ denotes that the event occurs in a different field cycle relative to $\varepsilon_1$. We consider only one event per half-cycle here for simplicity, although one can generalize - see Hashim2025.
• Figure 5: Fully incoherent two-channel momentum distributions for all channels in Table \ref{['tab:channels']}, $\mathcal{P}_{iii} (p_{1\parallel}, p_{2\parallel})$ [given by Table \ref{['tab:possiblesums']}(h)] computed for Argon with a linearly polarized few-cycle pulse with the same parameters as in Fig. \ref{['fig:pulseshape']} and taking the three most dominant events. The axes are indicated with white dashed lines. The excited states for each pair of channels is shown in the top-right corner. For example, $3p4p$ indicates that the $3s\rightarrow3p$ and $3p\rightarrow4p$ transitions have been combined.