Population inversion versus coherence in N2+ air lasing: discriminating emission mechanisms and building a predictive model for polarization-modulated strong-field pumping

When intense femtosecond laser pulses ionize nitrogen gas, the molecular ions emit mirrorless (cavity-free), directional coherent radiation at 391 nm and 428 nm on the $B^2\Sigma_u^+ \rightarrow X^2\Sigma_g^+$ band — the N2+ air laser. A 2019 study of polarization-modulated pumping (Xie et al., Phys. Rev. A 100, 053419) identified as an open question whether N2+ lasing requires a population inversion in all experimental regimes, or whether coherent mechanisms — superradiant/collective emission or coherence-enabled gain without inversion — can sustain the observed gain. Resolving this question requires quantifying how rotational dynamics, ab initio post-ionization electronic population ratios, and realistic propagation (filamentation, spatial inhomogeneity, broadband wave-plate dispersion) modify the inversion parameter and predicted lasing yields, and establishing systematic experimental discrimination between inversion-mediated stimulated emission and coherence-driven lasing across gas pressures, pulse energies, and polarization-modulation schemes.

Background

Air lasing is cavity-free optical amplification in a gas, produced by the gas constituents themselves when an intense femtosecond laser pulse (typically 800 nm, intensities near 10^14 W/cm^2) ionizes and couples to the molecules; it attracts interest for remote atmospheric sensing because the gain medium forms wherever a focused laser beam passes. In molecular nitrogen the relevant emitter is the singly ionized molecule $\mathrm{N}_2^+$. Its three lowest electronic states — the ground state $X\,^2\Sigma_g^+$, the doubly degenerate $A\,^2\Pi_u$ state, and the excited $B\,^2\Sigma_u^+$ state — form a near-resonant V-system (a three-level scheme in which one common state couples to two others, named for the shape of its level diagram) for 800 nm light: one 800 nm photon is nearly resonant with $X \leftrightarrow A$ transitions (mediated by vibrational levels), and two 800 nm photons are nearly resonant with the $B \leftrightarrow X$ transition whose fluorescence lines at 391 nm ($B(v{=}0) \rightarrow X(v{=}0)$) and 428 nm ($B(v{=}0) \rightarrow X(v{=}1)$) are the observed laser wavelengths (the first negative band system). Backward and forward lasing from strong-field-excited air was reported by Dogariu et al. (Science 331, 442 (2011)), and N2+ lasing in femtosecond filaments (self-guiding intensity channels formed when Kerr self-focusing of the pulse is balanced by the defocusing of the plasma it creates) has been studied intensively since; a 2019 review (Li et al., Chin. Phys. B 28, 114204) states that the mechanism of strong-field-induced N2+ air lasing was at that time the subject of a hot debate.

The debate is the following. In a standard laser, net optical gain requires a population inversion — more population in the upper than the lower state of the lasing transition — to break the reciprocity between stimulated absorption and stimulated emission. Strong-field ionization of $\mathrm{N}_2$ populates the ionic $X$ ground state predominantly, which naively forbids gain on $B \rightarrow X$, yet gain is observed. One school explains this through hidden inversions created by coherent coupling: the pump resonantly couples $X$ and $A$ (and $B$ and $X$) during and after ionization, transferring population out of $X(v{=}0)$ and into $B$, so that a transient inversion between $B$ and individual rovibrational levels of $X$ appears (Ando et al., Phys. Rev. Lett. 123, 203201 (2019); Wang et al., Phys. Rev. A 104, 032823 (2021)). A second school emphasizes coherence-based mechanisms that produce gain without any inversion: lasing without inversion (LWI), in which quantum interference between absorption pathways — here between rotational levels of $X$ coherently prepared by Raman-type cascades — suppresses absorption while emission survives (Richter et al., Optica 7, 1116 (2020); Lytova et al., Phys. Rev. A 102, 013111 (2020)); and superradiance or superfluorescence, in which a coherently or collectively prepared ensemble radiates with an intensity growing faster than linearly in the ion number, mimicking gain. The polarization state of the pump is a key control knob: rotating the ionization-versus-coupling balance by modulating the pulse polarization (e.g., with a multiple-order wave plate, which makes the polarization sweep from circular toward linear within the pulse, or polarization gating, in which the pulse is tailored so that only a short central slice is effectively linearly polarized) can switch the 391 nm response from absorption to lasing. This is the setting of the source paper (Xie et al., Phys. Rev. A 100, 053419 (2019)), which quantified vibrational population transfer between $X$, $A$, and $B$ in polarization-modulated fields and interpreted the lasing through the induced $B$–$X$ inversion (population difference $\Delta n_{B-X}$), while noting that its model neglected molecular rotation, used assumed post-ionization electronic population ratios $X{:}A{:}B$ from separate calculations, and treated propagation effects (intensity clamping — the self-limiting of peak intensity inside a filament by the plasma the beam itself creates — and broadband wave-plate dispersion) only approximately. A complication, established by later work of the same group (Xie et al., Phys. Rev. Research 2, 023329 (2020)), is that rotational coherences — quantum superpositions of rotational states $J$ and $J{+}2$ of $X(v{=}0)$ created by cascaded resonant Raman scattering after ionization — are essential to the observed polarization behavior of the 391 nm P and R branch lines (the two rotational branches of the band, belonging to transitions with rotational quantum number decreased or increased by one), so rotation cannot be treated as a spectator. The mechanistic question thus couples microscopic ion dynamics (quantum coherences, rovibrational populations) with macroscopic propagation (seed amplification, free induction decay — delayed radiation emitted by the coherent polarization the pump leaves behind — and collective buildup of polarization over the ensemble), each of which has been observed separately in the 2019–2025 literature.

The 2019–2025 literature shows that inversion is sufficient and highly beneficial, that inversion-free gain is theoretically possible via rotational quantum beats, and that collective emission and seed amplification coexist with a pressure-dependent crossover; however, no experimental demonstration of inversion-free N2+ lasing and no validated end-to-end regime map exists to date, so the problem remains open.

Problem Statement

Determine the physical mechanism of cavity-free N2+ lasing driven by intense femtosecond laser fields, and specifically whether a population inversion between the $B\,^2\Sigma_u^+$ and $X\,^2\Sigma_g^+$ states is necessary in all experimental regimes, or whether coherent mechanisms — superradiant/collective emission or coherence-enabled gain without inversion (for example, driven by rotational quantum coherences in the $X$ state) — can sustain the observed gain. The common objective has three jointly linked parts: (1) establish experimentally and theoretically which emission mechanism — inversion-mediated stimulated emission, collective (super-radiant/superfluorescent) emission, or coherence-enabled gain without inversion — operates in which regime, as a systematic function of gas pressure, pump intensity/pulse energy, and polarization-modulation scheme (including the multiple-wave-plate and polarization-gating schemes of the source paper); (2) quantify, in a predictive model that includes molecular rotation, ab initio post-ionization electronic population ratios of the $X$, $A$, $B$ states, and realistic propagation (filamentation, spatial inhomogeneity, and the broadband dispersion of real wave plates), how these ingredients modify the population transfer efficiencies, the inversion parameter $\Delta n_{B-X}$, and the predicted lasing yields, including at atmospheric and higher pressures; and (3) on that basis, decide whether inversion-free N2+ lasing is realized in the regimes covered by the published measurements — that is, whether the observed N2+ gain can be accounted for without any transient, rovibrationally resolved $B$–$X$ inversion under the stated conditions (and, if so, identify the conditions) — or else demonstrate that such an inversion, however transient and partial, is required. Completion means a quantitatively predictive account of the gain and emission mechanism whose predictions reproduce the pressure-, intensity-, and polarization-dependent gain and absorption datasets reported in refs 2, 12, 14, and 17 (optionally extended to the further experimental datasets in refs 3, 13, 16), yield falsifiable predictions for regimes not yet measured, and are accompanied by an explicit experimental-design proposal that would discriminate inversion-mediated from coherence-driven gain across the stated experimental parameter space — not merely a plausible scenario for one operating point.

References

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