Mechanism of gain and population inversion on the $B^2\Sigma_u^+\to X^2\Sigma_g^+$ transition of $\mathrm{N_2^+}$ in air lasing

When intense femtosecond near-infrared pulses ionize nitrogen gas, the resulting $\mathrm{N_2^+}$ plasma emits coherent light at 391 nm on the $B^2\Sigma_u^+\to X^2\Sigma_g^+$ transition, a phenomenon known as air lasing. More than a decade after its discovery, the microscopic mechanism that establishes this optical gain — and in particular whether and how a population inversion between the $B$ and $X$ states is produced — remains unresolved. Candidate channels include state-selective tunnel-ionization yields that set the initial ionic populations, post-ionization pump-driven population transfer between the ground state $X$ and the first excited state $A$ (with $A$ acting as a population sink and trapping vibrational population), inversion-free coherence-based (Raman-like) gain among the three lowest states, and propagation-induced gain modulation in filaments. The issue that Britton et al., Phys. Rev. A 102, 053110 (2020), identify as the major unresolved problem in $\mathrm{N_2^+}$ air lasing is the detailed microscopic mechanism behind the 391 nm gain, including whether and how a population inversion between the $B$ and $X$ states is produced.

Background

Air lasing refers to the emission of coherent or amplified radiation directly from the atmosphere (or from nitrogen gas) when an intense femtosecond laser pulse propagating in the gas produces a weakly ionized plasma channel. During femtosecond filamentation — the regime in which Kerr self-focusing balances plasma defocusing so the pulse stays confined over long distances — an 800 nm pump field ionizes $\mathrm{N_2}$ and the molecular cation $\mathrm{N_2^+}$ subsequently emits on the so-called first negative band, the $B^2\Sigma_u^+\to X^2\Sigma_g^+$ electronic transition at 391 nm. The cation's three lowest electronic states are the ground state $X\,{}^2\Sigma_g^+$, the first excited state $A\,{}^2\Pi_u$, and the second excited state $B\,{}^2\Sigma_u^+$. Because the 800 nm pump is near-resonant with certain $X\to A$ vibronic (combined vibrational–electronic) transitions, the ion behaves as a three-level system in which the pump couples $X$ and $A$ while the 391 nm emission connects $B$ and $X$; since $X$ is the common lower level of two upward transitions, this is often drawn as a V-system.

A population inversion — more population in the upper lasing level $B$ than in the lower level $X$ — is the textbook condition for optical gain, and its observation in an ion created by a strong field was surprising, because strong-field (tunnel) ionization, in which the laser field suppresses the Coulomb barrier so strongly that the electron escapes through it, was expected to leave the ion mostly in its electronic ground state. Whether the ionization step itself is state-selective (with unequal yields for producing $\mathrm{N_2^+}$ in $X$, $A$, or $B$), and to what degree the pump field rearranges the populations after ionization, became the central question. One early proposal invoked recollision — the process, fundamental to strong-field science, in which the freed electron is driven back by the oscillating field and can excite the parent ion — but ellipticity tests showed that 391 nm gain survives circular polarization (which suppresses recollision) in a gas jet, ruling recollision out as a significant contributor (Britton et al., 2018). The 800 nm $X\!-\!A$ coupling can launch a vibrational wave packet in the $A$-state potential well; if the Franck–Condon overlap (the overlap of vibrational wave functions governing transition strengths) back to $X$ is poor, population is transiently trapped in $A$, depleting $X$ and helping inversion on $B\to X$ — this is the 'population sink' picture supported by few-cycle-pulse experiments and simulations of Xu et al. (2015).

A competing school holds that no conventional inversion is needed at all: long-lived coherent polarizations that couple $X$ simultaneously to $A$ and $B$ can produce strong gain without population inversion, a molecular analogue of Raman-like or coherence-based gain, reproduced by Maxwell–Bloch simulations (the standard coupled treatment of a classical light field with a quantum density matrix, including propagation) by Mysyrowicz et al. (2019); rotational coherence — coherent rotational wave packets created by the pump — further modulates the gain (Azarm et al., 2017; Xie et al., 2020). Time-resolved measurements show that the 391 nm emission carries rotational, vibrational, and electronic modulation frequencies consistent with coherent $B\!-\!X\!-\!A$ coupling (Ando et al., 2019), and that a second 800 nm pulse can abruptly quench the lasing, consistent with both $X\to A$ population transfer and disruption of the coherence (Danylo et al., 2021). Quantitative rovibronic simulations now reproduce the modulation patterns through rotational population transfer among the three states (Zhang et al., 2021). Direct measurement of the post-ionization state populations by nitrogen K-edge transient absorption (an ultrafast x-ray absorption probe of the valence electronic state) found a remarkably low $A$-state population and nearly equal $X$ and $B$ populations under the measured conditions (Kleine et al., 2022), constraining state-selective-ionization models without settling the question. In parallel, 3D Maxwell–Bloch modeling of cavity-free lasing in filaments (López et al., 2023) and the discovery of additional channels such as bidirectional cascaded superfluorescent lasing (incoherent, cooperative emission build-up) driven by resonant third-harmonic photon exchange from nitrogen to argon in real air (Nie et al., 2024) show that propagation and plasma composition add mechanisms that any complete account must include. Reviews summarize the situation as an unsettled competition between inversion-based and coherence-based gain (Yuan et al., 2019; Fu et al., 2022). Against this background, Britton et al. (2020) performed three-pulse pump–probe/control experiments in a short gas jet (which isolates gain from filament propagation), demonstrated that a weak control pulse acting only on the ion can strengthen, quench, or coherently redirect the emission, and identified as the major unresolved issue the detailed mechanism that builds the inversion, proposing as the obvious next step a pump wavelength chosen so that tunnel ionization — not subsequent $X\!-\!A$ coupling — establishes the initial ionic populations, after which a timed control pulse could manufacture inversion via $A$-state dynamics.

Problem Statement

Determine the detailed microscopic mechanism underlying the optical gain observed on the $B^2\Sigma_u^+\to X^2\Sigma_g^+$ (391 nm) transition of $\mathrm{N_2^+}$ when intense near-infrared femtosecond pulses ionize nitrogen — in particular whether and how a population inversion between the $B$ and $X$ states is produced — the issue identified by Britton et al. (2020) as the major unresolved problem in $\mathrm{N_2^+}$ air lasing, in which the $A\,{}^2\Pi_u$ state seems likely to act as a population sink. Candidate mechanisms proposed in the literature to be discriminated include: (i) the ionic populations fixed at the moment of strong-field tunnel ionization by state-selective ionization yields; (ii) post-ionization pump-driven population transfer between $X\,{}^2\Sigma_g^+$ and $A\,{}^2\Pi_u$, including the action of $A$ as a population sink, vibrational trapping of the $A$-state wave packet, and the rotational decoherence rates of $X$ and $A$; and (iii) the competing inversion-free, coherence-based (Raman-like) gain picture, in which coherent polarizations coupling $B$, $X$, and $A$ produce gain without population inversion (Mysyrowicz et al., 2019; Danylo et al., 2021). Propagation-induced gain modulation and modal selection in filaments (López et al., 2023), which the gas-jet geometry of Britton et al. (2020) deliberately excludes, is a further effect that accounts of the gain in filaments must address. As an obvious next step, Britton et al. (2020) propose an experiment in which the pump wavelength is chosen away from $X\!-\!A$ transitions so that tunnel ionization alone establishes the initial ionic populations, after which an externally timed control pulse could manufacture the inversion through $A$-state dynamics.

References

1. Mathew Britton, Marianna Lytova, Dong Hyuk Ko, Abdulaziz Alqasem, Peng Peng, D. M. Villeneuve, Chunmei Zhang, Ladan Arissian, P. B. Corkum (2020). Control of N2+ air lasing. Physical Review A 102, 053110. DOI: 10.1103/PhysRevA.102.053110, https://doi.org/10.1103/PhysRevA.102.053110 (arXiv:2007.08619, https://arxiv.org/abs/2007.08619)
2. Mathew Britton, Patrick Laferrière, Dong Hyuk Ko, Zhengyan Li, Fanqi Kong, Graham Brown, Andrei Naumov, Chunmei Zhang, Ladan Arissian, P. B. Corkum (2018). Testing the role of recollision in N2+ air lasing. Physical Review Letters 120, 133208. DOI: 10.1103/PhysRevLett.120.133208, https://doi.org/10.1103/PhysRevLett.120.133208
3. Mathew Britton, Marianna Lytova, Patrick Laferrière, Peng Peng, Felipe Morales, Dong Hyuk Ko, Maria Richter, Pavel Polynkin, D. M. Villeneuve, Chunmei Zhang, Misha Ivanov, Michael Spanner, Ladan Arissian, P. B. Corkum (2019). Short- and long-term gain dynamics in N2+ air lasing. Physical Review A 100, 013406. DOI: 10.1103/PhysRevA.100.013406, https://doi.org/10.1103/PhysRevA.100.013406
4. Huailiang Xu, Erik Lötstedt, Atsushi Iwasaki, Kaoru Yamanouchi (2015). Sub-10-fs population inversion in N2+ in air lasing through multiple state coupling. Nature Communications 6, 8347. DOI: 10.1038/ncomms9347, https://doi.org/10.1038/ncomms9347
5. Yi Liu, Yohann Brelet, Guillaume Point, Aurélien Houard, André Mysyrowicz (2013). Self-seeded lasing in ionized air pumped by 800 nm femtosecond laser pulses. Optics Express 21, 22791. DOI: 10.1364/oe.21.022791, https://doi.org/10.1364/oe.21.022791
6. Ali Azarm, Paul Corkum, Pavel Polynkin (2017). Optical gain in rotationally excited nitrogen molecular ions. Physical Review A 96, 051401(R). DOI: 10.1103/PhysRevA.96.051401, https://doi.org/10.1103/PhysRevA.96.051401
7. A. Mysyrowicz, R. Danylo, A. Houard, V. Tikhonchuk, X. Zhang, Z. Fan, Q. Liang, S. Zhuang, L. Yuan, Y. Liu (2019). Lasing without population inversion in N2+. APL Photonics 4, 110807. DOI: 10.1063/1.5116898, https://doi.org/10.1063/1.5116898
8. Toshiaki Ando, Erik Lötstedt, Atsushi Iwasaki, Helong Li, Yao Fu, Siqi Wang, Huailiang Xu, Kaoru Yamanouchi (2019). Rotational, vibrational, and electronic modulations in N2+ lasing at 391 nm: evidence of coherent B2Σu+−X2Σg+−A2Πu coupling. Physical Review Letters 123, 203201. DOI: 10.1103/PhysRevLett.123.203201, https://doi.org/10.1103/PhysRevLett.123.203201
9. Hongqiang Xie, Hongbin Lei, Guihua Li, Qian Zhang, Xiaowei Wang, Jing Zhao, Zhiming Chen, Jinping Yao, Ya Cheng, Zengxiu Zhao (2020). Role of rotational coherence in femtosecond-pulse-driven nitrogen ion lasing. Physical Review Research 2, 023329. DOI: 10.1103/PhysRevResearch.2.023329, https://doi.org/10.1103/PhysRevResearch.2.023329
10. Luqi Yuan, Yi Liu, Jinping Yao, Ya Cheng (2019). Recent advances in air lasing: a perspective from quantum coherence. Advanced Quantum Technologies 2, 1900080. DOI: 10.1002/qute.201900080, https://doi.org/10.1002/qute.201900080
11. Youyuan Zhang, Erik Lötstedt, Toshiaki Ando, Atsushi Iwasaki, Huailiang Xu, Kaoru Yamanouchi (2021). Rotational population transfer through the A2Πu−X2Σg+−B2Σu+ coupling in N2+ lasing. Physical Review A 104, 023107. DOI: 10.1103/PhysRevA.104.023107, https://doi.org/10.1103/PhysRevA.104.023107
12. Jinming Chen, Jinping Yao, Zhihao Zhang, Zhaoxiang Liu, Bo Xu, Yuexin Wan, Fangbo Zhang, Wei Chu, Lingling Qiao, Haisu Zhang, Zhenhua Wang, Ya Cheng (2021). Electronic quantum coherence encoded in temporal structures of N2+ lasing. Physical Review A 103, 033105. DOI: 10.1103/PhysRevA.103.033105, https://doi.org/10.1103/PhysRevA.103.033105
13. Xiang Zhang, Qi Lu, Zhedong Zhang, Zhengquan Fan, Dongjie Zhou, Qingqing Liang, Luqi Yuan, Songlin Zhuang, Konstantin Dorfman, Yi Liu (2021). Coherent control of the multiple wavelength lasing of N2+: coherence transfer and beyond. Optica 8, 668. DOI: 10.1364/OPTICA.417804, https://doi.org/10.1364/OPTICA.417804
14. Rostyslav Danylo, Guillaume Lambert, Yi Liu, Vladimir Tikhonchuk, Aurélien Houard, André Mysyrowicz (2021). Time-resolved study of laser emission in nitrogen gas pumped by two near IR femtosecond laser pulses. Optics Letters 46, 1253. DOI: 10.1364/OL.414863, https://doi.org/10.1364/OL.414863
15. Zhengquan Fan, Xiang Zhang, Qi Lu, Yu Luo, Qingqing Liang, Luqi Yuan, Zhedong Zhang, Yi Liu (2020). Understanding the seeding pulse-induced optical amplification in N2+ pumped by 800 nm femtosecond laser pulses. Photonics 7, 99. DOI: 10.3390/photonics7040099, https://doi.org/10.3390/photonics7040099
16. Carlo Kleine, Marc-Oliver Winghart, Zhuang-Yan Zhang, Maria Richter, Maria Ekimova, Sebastian Eckert, Marc J. J. Vrakking, Erik T. J. Nibbering, Arnaud Rouzée, Edward R. Grant (2022). Electronic state population dynamics upon ultrafast strong field ionization and fragmentation of molecular nitrogen. Physical Review Letters 129, 123002. DOI: 10.1103/PhysRevLett.129.123002, https://doi.org/10.1103/PhysRevLett.129.123002
17. Jingsong Gao, Xiang Zhang, Yang Wang, Yiqi Fang, Qi Lu, Zheng Li, Yi Liu, Chengyin Wu, Qihuang Gong, Yunquan Liu, Hongbing Jiang (2023). Structured air lasing of N2+. Communications Physics 6, 97. DOI: 10.1038/s42005-023-01226-9, https://doi.org/10.1038/s42005-023-01226-9
18. Qi Lu, Xiang Zhang, Santiago López, Haicheng Mei, Liang Xu, Qingqing Liang, Aurélien Houard, Vladimir Tikhonchuk, André Mysyrowicz, Eduardo Oliva, Yi Liu (2023). Spectral splitting of the lasing emission of nitrogen ions pumped by 800-nm femtosecond laser pulses. Optics Letters 48, 664. DOI: 10.1364/ol.478025, https://doi.org/10.1364/ol.478025
19. Santiago López, Alicia García, David Rueda, Eduardo Oliva (2023). 3D modelling of cavity-free lasing in nitrogen plasma filaments. Optics Express 31, 8479. DOI: 10.1364/oe.478100, https://doi.org/10.1364/oe.478100
20. Jingsong Gao, Xiang Zhang, Yang Wang, Jiahao Dong, Mingwei Lei, Yi Liu, Chengyin Wu, Qihuang Gong, Hongbing Jiang (2024). Controlling the polarization of nitrogen ion lasing. The Journal of Physical Chemistry Letters 15, 3805. DOI: 10.1021/acs.jpclett.4c00004, https://doi.org/10.1021/acs.jpclett.4c00004
21. Zan Nie, Noa Nambu, Kenneth A. Marsh, Daniel Matteo, C. Kumar Patel, Chaojie Zhang, Yipeng Wu, Stefanos Carlström, Felipe Morales, Serguei Patchkovskii, Olga Smirnova, Misha Ivanov, Chan Joshi (2024). Bidirectional cascaded superfluorescent lasing in air enabled by resonant third harmonic photon exchange from nitrogen to argon. Physical Review Letters 133, 063201. DOI: 10.1103/PhysRevLett.133.063201, https://doi.org/10.1103/PhysRevLett.133.063201
22. Ping Li, Wenli Yang, Xiaofei Li, Shan Xue, Jinbin Li, Hongchuan Du (2025). Three-photon Rabi oscillations in N2+ lasing at 391 nm. Physical Review A 111, 063501. DOI: 10.1103/PhysRevA.111.063501, https://doi.org/10.1103/PhysRevA.111.063501
23. Xu Lu, Siyi He, Pengyu Xue, Yewei Chen, Ning Zhang, Yajing Wen, Zihan Li, Quanjun Wang, Jinping Yao (2025). Cascaded amplification of air lasing. Science Advances 11, eady7458. DOI: 10.1126/sciadv.ady7458, https://doi.org/10.1126/sciadv.ady7458
24. Yao Fu, Jincheng Cao, Kaoru Yamanouchi, Huailiang Xu (2022). Air-laser-based standoff coherent Raman spectrometer. Ultrafast Science 2022, 9867028. DOI: 10.34133/2022/9867028, https://doi.org/10.34133/2022/9867028
25. Ilya A. Zyatikov, Valery F. Losev (2023). Parameters of 391 nm lasing from molecular nitrogen ions pumped by a 950 nm femtosecond laser pulse. Applied Optics 62, 9133. DOI: 10.1364/ao.508151, https://doi.org/10.1364/ao.508151
26. Yihong Huang, Guihua Li, Runbin Chen, Shuting Wu, Ruixin Che, Hongqiang Xie (2025). Polarization variation of N2+ lasing. Chinese Optics Letters 23, 083201. DOI: 10.3788/col202523.083201, https://doi.org/10.3788/col202523.083201