Abstract

Nitrogen molecules pumped by intense femtosecond laser pulses give rise to coherent emission in the forward direction at a series of wavelengths, coined “air lasing.” We demonstrate the coherent control of these emissions via a pair of seeding pulses at two different frequencies at low pressures, revealing a coherence transfer through vibrational motion. It is found that the injection of a 427.8 nm (391.4 nm) seeding pulse results in its amplification at the expense of the 391.4 nm (427.8 nm) signal, demonstrating a competition between the two spectral components of the emission from the upper level population. Moreover, when the delay between the seeding and pump pulses is finely tuned, the coherent control of both transitions is observed via the coherence transfer. A microscopic molecular relaxation model reproduces these observations, highlighting the crucial role of electronic and vibrational coherences, as well as their coupling, during the lasing action.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Corrections

19 May 2021: A typographical correction was made to the author listing.

1. INTRODUCTION

Ambient air turns into a lasing medium and emits coherent emission in the UV-visible range when it is excited by intense ultrafast laser pulses [110]. This cavity-free lasing action has attracted much attention in the last 10 years due to its potential to generate a remote laser source in the sky, assisted by the long range filamentation driven by powerful femtosecond laser pulses [11]. All three main components of air, i.e., nitrogen, oxygen, and argon, have shown their capacity, under excitation with a properly chosen pump laser wavelength, to generate such a lasing effect [110,12]. Particular attention has been focused on ionic nitrogen molecules, due to the richness of the observed effects and their high concentration in air. It has been demonstrated that lasing action occurs within a few tens of picoseconds in nitrogen ions, and that its intrinsic dynamics can be controlled within femtosecond and attosecond time scales [1318]. The rotational and vibrational degrees of freedom of nitrogen ions enrich and complexify the lasing phenomenon, so that the underlying mechanism of this lasing process is still in debate.

Under excitation by near-infrared 800 nm femtosecond pulses, nitrogen ions give rise to a series of narrow emission lines at 391.4, 427.8, and 471.2 nm in the forward direction. These wavelengths correspond to transitions from the ${{ B }^2}{\Sigma}_{u}^ +$ ($\nu ^\prime = 0$) to the ${{ X }^2}{\Sigma}_{g}^ +$ ($\nu = 0,{1},{2}$) electronic states of the nitrogen ions, where $ \nu $ refers to the vibrational quantum number [2]. Using time-resolved measurements, it has been found that the temporal profile of the 391.4 nm emission presents a characteristic dependence on the density of nitrogen ions, suggesting that the nature of the 391.4 nm emission is superfluorescence, a cooperative emission of many ions enabled by the build-up of a macroscopic electronic coherence [19,20]. Some of the present authors have injected a sequence of two 391.4 nm seeding pulses into nitrogen gas plasma and observed a coherent modulation of the forward 391.4 nm emission [15]. These two observations demonstrate the crucial role of electronic coherence in the lasing process. Moreover, it has also been shown that the rotational coherence of nitrogen ions plays an important role in optical gain, when coherent rotational wave packets are excited non-adiabatically by femtosecond pump pulses [2124]. Concerning vibrational coherence, although its role in this lasing action has been recently revealed by a temporal modulation of 391.4 nm emission [14], it still lacks a complete understanding. This is partially so because the 391.4 and 427.8 nm emission lines are optimized in different pressure ranges so that their interaction may be relatively weak under usual experimental conditions [25,26]. More importantly, while it is well known that the interplay between coherences of different degrees of freedom plays a crucial role in various systems including polyatomic molecules and exciton polaritons [2729], the coupling of vibrational coherence to electronic/rotational coherence in the lasing of molecular cations ${\rm N}_2^ +$ remains elusive due to the weakly created vibrational coherence.

In this work, to clarify the role of vibrational coherence in ${\rm N}_2^ +$ lasing, we inject a pair of resonant seeding pulses at two distinct wavelengths of 391.4 and 427.8 nm in the nitrogen plasma created by an intense 800 nm femtosecond laser pulse. We observe a competition between the two emission pathways originating from the population of upper level ${{ B }^2}{\Sigma}_{u}^ +$ (${\nu ^\prime} = 0$). More importantly, the coherent modulation of the two emissions is achieved by tuning the delay between seeding pulses with, surprisingly, an almost identical temporal period determined by the frequency of the seeding pulse being scanned. This demonstrates electronic coherence transfer via vibrational motion. We interpret these observations using a microscopic model based on the resonant interaction of two seeding pulses with a quantum system consisting of three-level coherent vibrational states. Our findings highlight the role of electronic and vibrational coherence and the inter-conversion between each other, as well as interplay between spectral components of emission from the excited state, associated with the lasing process of nitrogen ions.

2. EXPERIMENTS

In the experiments, we employ a commercial femtosecond laser system that delivers a 35 fs pulse at a central wavelength of 796 nm, with a repetition rate of 1 kHz (Coherent Legend DUO). The maximum pulse energy is about 13 mJ. The output pulses are split into three pulses using two beam splitters, as schematically shown in Fig. 1(a). One of the three pulses serves as the pump pulse, which ionizes nitrogen gas. The other two pulses are transmitted through two 1 mm thick type I beta barium borate (BBO) crystals for generation of the second harmonic. The two BBO crystals are optimized by rotating their azimuthal orientation, to efficiently generate the 391.4 and 427.8 nm seeding pulses. After the two BBO crystals, two narrowband interference filters with central wavelengths of, respectively, 390 and 430 nm are inserted to select the desired spectra of the seeding pulses. The two seeding pulses, with their pulse energy estimated on the order of picojoules, are first recombined with a beam splitter designed for a 400 nm beam. Then, the two seeding pulses are combined with the main 796 nm pulse using a dichroic mirror. The pump ($\sim{1.5}\;{\rm mJ}$), seed 1 (391.4 nm), and seed 2 (427.8 nm) pulses are focused by three individual convex lenses of focal length ${f} = 300\;{\rm mm}$. The spatial overlap of the geometrical focus of the three beams is ensured by slightly adjusting the position of the focal lens. High precision mechanical delay lines are installed for the two seeding pulses. As a result, the relative time delay between the seeding and pump pulses can be individually adjusted with a precision of 10 nm, corresponding to a delay of 30 as. The nitrogen gas pressure is kept at 10 mbar. The forward emission from the nitrogen plasma is properly filtered to eliminate the fundamental pulses around 796 nm, and collected by a convex lens into an optical fiber connected to a spectrometer.

 figure: Fig. 1.

Fig. 1. Schematic experimental setup. The two seeding pulses with central wavelengths around 391 and 428 nm are injected into the nitrogen gas plasma formed by 800 nm pump pulses. The relative delay between seeding and pump pulses ${\tau _{p {\text -} s1}}$ and ${\tau _{p {\text -} s2}}$ can be controlled with precision of 30 as. (b) Energy surfaces of electronic ground and excited states. Solid arrows represent vertical transitions according to Frank–Condon principle, and blue wavy arrows represent nonradiative transitions. (c) Simplified version from (b) for nitrogen cations, where $\Delta \approx 105\;{{\rm cm}^{- 1}}$ denotes the detuning between vertical transitions and the level ${\nu ^\prime} = 0$.

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 figure: Fig. 2.

Fig. 2. (a) Spectra of the two seeding pulses. (b) Injection of the 391.4 nm seeding pulse leads to strong amplification at that wavelength. The seeding pulse itself (red line), the signal produced with only the 800 nm pump pulse (black line), and the amplified signal (blue line) are presented. (c) Injection of the 427.8 nm seeding pulse and corresponding emission. (d) Injection of both 391.4 and 427.8 nm seeding pulses. Delay ${\tau _{p {\text -} s1}} = 300\;{\rm fs}$ and ${\tau _{p {\text -} s2}} = 500\;{\rm fs}$.

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 figure: Fig. 3.

Fig. 3. (a) Variation of the amplified 391.4 and 427.8 nm signal intensities as a function of delay ${\tau _{p {\text -} s1}}$ between the 391.4 nm seeding pulse and the pump pulse. Positive delay means that the 391.4 nm seeding pulse lags behind the 800 nm pump pulse. Delay between pump and 427.8 nm seeding pulse ${\tau _{p {\text -} s2}}$ is fixed at 500 fs. (b) Total energy of 391.4 and 427.8 nm emission as a function of the delay ${\tau _{p {\text -} s1}}$.

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 figure: Fig. 4.

Fig. 4. (a) Transmitted 391.4 and 427.8 nm signal as a function of the delay ${\tau _{p {\text -} s1}}$ being scanned around 0–300 fs. Delay between pump and 427.8 nm seeding pulse ${\tau _{p {\text -} s2}}$ is 130 fs. (b) Theoretical calculation according to Eq. (3). $| {{{\Omega}_1}} |/| {{{\Omega}_2}} | = 1$ and ${\Delta}{\tau _{{p} - {s}2}} \approx 0.82{\pi}$ along the line of the experiments, where two seeding pulses have almost equal intensity and $\Delta = 105\;{{\rm cm}^{- 1}}$. Fitting parameter is $\frac{{| {{{\rho}_{{bc}}}} |}}{{{{\rho}_{{ee}}} - {{\rho}_{{cc}}}}} = - 1.1$ for reproducing the experimental results.

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We first show that the injection of each individual seeding pulse can result in optical amplification at the corresponding wavelengths. The spectrum of the 391.4 and 427.8 nm seeding pulses are presented in Fig. 2(a). With a slight delay of the seeding pulses after the pump pulse, strong amplification at each wavelength is obvious, as shown in Figs. 2(b) and 2(c). In both cases, we observe that the spectral component at 391.4 or 427.8 nm is amplified by two orders of magnitude, in agreement with previous reports [13,20]. We then inject two seeding pulses into the nitrogen gas plasma. The time delay between the pump and the 391.4 nm seeding pulse (${\tau _{p {\text -} s1}}$) is fixed at 300 fs, while the delay between the pump and the 427.8 nm seeding pulse (${\tau _{p {\text -} s2}}$) is fixed at 500 fs. The result of this experiment is presented in Fig. 2(d). Obviously, injection of the 391.4 nm seeding pulse results in strong amplification at 391.4 nm, while a significant reduction of the 427.8 nm amplification is observed as well.

To further examine the amplification effect above, we scan delay ${\tau _{p {\text -} s1}}$ and record the amplified emission intensity at 391.4 and 427.8 nm at the same time, shown in Fig. 3(a). For negative delays (${\tau _{p {\text -} s1}} \lt {0}$), when the 391.4 nm seeding pulse precedes the pump pulse, no amplification occurs at 391.4 nm, and only an amplified 427.8 nm signal is observed. For positive delays, a rapid increase in the 391.4 nm signal was observed for ${\tau _{p {\text -} s1}}$ up to $\sim{500}\;{\rm fs}$, followed by a gradual decrease up to $\sim{5}\;{\rm ps}$. During the same delay interval, a dramatic decrease in the amplified 427.8 nm signal intensity is recorded. Since the 391.4 and 427.8 nm emissions originate from the same electronic upper level, ${{ B }^2}{\Sigma}_{u}^ +$ ($\nu ^\prime = 0$), the competition between the two transitions can be expected. We also measured the total energy of the 391.4 and 427.8 nm with a photodiode, with the result shown in Fig. 3(b). It was observed that the total emitted energy is actually increased in the presence of the second seeding pulse at 390 nm. More interestingly, one can notice that fine oscillations are present for both the 391.4 and 427.8 nm emissions. In Fig. 3, we use a relatively low temporal resolution of 20 fs to examine the mutual influence of the two emissions over a large temporal window of 8 ps.

To resolve the fine oscillations, we next use a better temporal resolution of 30 as. In Fig. 4(a), we present the spectral intensity of 391.4 and 427.8 nm emissions with ${\tau _{p {\text -} s2}}$ fixed at 130 fs and ${\tau _{p {\text -} s1}}$ scanned from 0 to 300 fs. Both the 391.4 and 427.8 nm emissions show clear periodic intensity modulations. For a closer look at these oscillations, we show in Fig. 5 two delay regions with ${\tau _{p {\text -} s1}}\sim {300}\;{\rm fs}$ and ${\tau _{p {\text -} s1}}\sim {480}\;{\rm fs}$. First, the oscillation periods of the two emission lines are found to be nearly identical, about 1.32 fs, equal to the period of the 391.4 nm transition. Second, note that the two signal oscillations can be either anti-phase [Figs. 5(a)–5(c)] or in-phase [Figs. 5(d)–5(f)]. The electronic coherence transfer, as mediated via the vibrational coherence, is indicated, where the 427.8 nm pulse creates the electronic coherence $\langle | e\rangle \langle c |\rangle$ that converts to $\langle | e\rangle \langle b |\rangle$ via the 391.4 nm pulse. The electronic coherence $\langle | e\rangle \langle b |\rangle$ is probed, subsequently yielding the emitted signals. To support this further, we reverse the order of the two seeding pulses in another experiment, with ${\tau _{p {\text -} s1}} = 600\;{\rm fs}$ while scanning ${\tau _{p {\text -} s2}}$. The results are presented in Figs. 5(g)–5(i). Periodic modulations of the emissions at 427.8 and 391.4 nm are observed as well. In this case, the oscillation period of $\sim{1.41}\;{\rm fs}$ corresponds to the 427.8 nm transition. The electronic coherence dynamics coupled to intramolecular vibrational motion will be elaborated later.

 figure: Fig. 5.

Fig. 5. Spectral intensity of emissions around 427.8 and 391.4 nm as a function of time delay ${\tau _{p {\text -} s1}}$ (or ${\tau _{p {\text -} s2}}$) between the pump and the 391.4 nm (or 427.8 nm) seeding pulse. (a)–(f) Delay ${\tau _{p {\text -} s2}}$ fixed at 130 fs: (a)–(c) delay ${\tau _{p {\text -} s1}}$ finely tuned around 300 fs and (d)–(f) delay ${\tau _{p {\text -} s1}}$ finely tuned around 480 fs. (g)–(i) Delay ${\tau _{p {\text -} s2}}$ between the pump and 427.8 nm pulses is finely tuned around 1000 fs when fixing delay ${\tau _{p {\text -} s1}}$ at 600 fs.

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3. DISCUSSION

To understand the experimental results discussed above, we adopt a model of diatomic molecules where the nuclear vibration is coupled the two electronic states [30,31]. The scheme is depicted in Fig. 1(b). For ${\rm N}_2^ +$, there is a notable offset between the minima of the ground and excited electronics energy surfaces. This results in coupling between electronic and nuclear degrees of freedom, which is responsible for the excited-state relaxation and dephasing [31,32]. This can be elaborated by the molecular model Hamiltonian

$$\begin{split}{H_{M}} &= {H_g}({p,q} )| g\rangle \langle g | + {H_e}({p,q} )| e\rangle \langle e |\\&\quad - \left[{{\Omega}(q)| e\rangle \langle g | + {{\Omega}^*}(q)| g\rangle \langle e |} \right]\end{split}$$
under Born–Oppenheimer approximation, where ${H_g}({p,q})$ and ${H_e}({p,q})$ are the nuclear Hamiltonians of the form
$$H({p,q} ) = \frac{{{p^2}}}{2} + {V_g}(q),$$
$${H_e}({p,q} ) = \omega _{{eg}}^0 + \frac{{{p^2}}}{2} + {V_e}({q - d} ),$$
governing the nuclear dynamics. $d$ denotes the offset between the minima of ground energy surface ${V_g}(q)$ and electronic excited energy surface ${V_e}({q - d})$. $\omega _{{eg}}^0$ gives the energy difference between the minima of the two energy surfaces. $p$ and $q$ are the momentum and coordinate of the nucleus, respectively. ${\Omega}(q) =\mu(q)E(t)$, where $\mu (q)$ denotes the electronic transition dipole in the basis of the Frank–Condon principle, and $E(t)$ is the electric field in the pulse. Quantum mechanically, ${H_g}({p,q})$ and ${H_e}({p,q})$ give rise to the vibrational levels $\{{{\nu} = 0,1, \ldots} \}$ for the ground state and $\{{{\nu ^\prime} = 0,1, \ldots} \}$ for the excited state.

The vertical optical transition between the two electronic states promotes the molecules into a superposition of various vibrational levels $\{{{\nu ^\prime} = 0,1, \ldots} \}$ in the electronics excited manifold. Due to the offset $d$, there is a detuning ${\Delta}$ between the vertical optical transition and the energy of the level ${\nu ^\prime} = 0$ of ${\rm N}_2^ +$. Note the vertical transition is resonant with the 391.4 nm pulse; the detuning: ${\Delta} = {{\omega}_1} - {{E}_{{\nu ^\prime} = 0}} \approx 105\;{{\rm cm}^{- 1}} \ll {{\omega}_{{\rm vib}}} = {{E}_{{\nu ^\prime} = 1}} - {{E}_{{\nu ^\prime} = 0}} \approx 2221\;{{\rm cm}^{- 1}}$, and thus the lowest mode $\nu ^\prime = 0$ dominates, while the population of higher vibrational modes can be neglected [33] (see also Supplement 1). We may properly assume that the 391.4 and 427.8 nm pulses induce the ${\nu} = 0 \leftrightarrow {\nu ^\prime} = 0$ and ${\nu} = 1 \leftrightarrow {\nu ^\prime} = 0$ transitions, respectively. We therefore simplify the molecular model given by Eqs. (1) and (2) to the three-level scheme shown in Fig. 1(c), where ${\rm N}_2^ +$ undergoes a near-resonant Raman process with the detuning $ \Delta $. This detuning is crucial for understanding the time-resolved spectrum observed in the experiment. To avoid ambiguity, we hereafter label the levels ${\nu} = 0$, ${\nu} = 1$ in ${{ X }^2}{\Sigma}_{g}^ +$, and ${\nu ^\prime} = 0$ in ${{ B }^2}{\Sigma}_{u}^ +$ as ${b}$, ${c}$, and ${e}$, respectively.

Experimentally detected signals correspond to the frequency-resolved transmission measurement: ${S}({\omega}) = 2{\rm Im}[{{{E}^{*}}({\omega}){P}({\omega})}]$, where ${E}({\omega})$ is the Fourier component of the positive-frequency part of the seeding pulses, and ${P}({\omega})$ is the molecular polarization. We calculate the time-resolved transmission of the 391.4 and 427.8 nm seeds having central frequencies ${{\omega}_1}$, ${{\omega}_2}$, respectively:

$$\begin{split}{S}_1({{{\omega}_1},{{\tau}_{{p} - {s}1}}} ) &= \frac{{2{{| {{{\Omega}_1}} |}^2}| {{{\Omega}_2}} |}}{\gamma}[| {{{\Omega}_2}} |({{\rho _{{ee}}} - {\rho _{{cc}}}} )\\ &\quad - 2| {{{\Omega}_1}} || {{{\rho}_{{bc}}}} |\cos ({\Delta {\tau _{p {\text -} s1}}} )\\&\quad\times{ \cos }({{{{\tilde \omega}}_1}{{\tau}_{{p} - {s}1}} - {{\omega}_2}{{\tau}_{{p} - {s}2}} + \Delta {\tau _{p {\text -} s2}} + \vartheta} )\\{S_2}({{{\omega}_2},{{\tau}_{{p} - {s}1}}} ) &= \frac{{| {{{\Omega}_1}} |{{| {{{\Omega}_2}} |}^2}}}{\gamma}[| {{{\Omega}_1}} |({{\rho _{{ee}}} - {\rho _{{cc}}}} )\cos ({\Delta {\tau _{p {\text -} s2}}} )\\ &\quad - | {{{\Omega}_2}} || {{{\rho}_{{bc}}}} |{ \cos }({{{{\tilde \omega}}_1}{{\tau}_{{p} - {s}1}} - {{\omega}_2}{{\tau}_{{p} - {s}2}} + \vartheta} )],\end{split}$$
where ${\tilde \omega}_{1}=\omega_{1}-\Delta$, and we assume that the pulse duration is much shorter than the timescales of nuclear dynamics and inhomogeneous dephasing. We use 1 and 2 to label the parameters associated with the 391.4 and 427.8 nm seeding pulses, respectively. $|{{\Omega}_1}|$ and $| {{{\Omega}_2}} |$ represent the Rabi frequencies for ${\rm N}_2^ +$ interacting with 391.4 and 427.8 nm pulses, respectively. ${{\Omega}_1} = | {{{\Omega}_1}} |{{e}^{{{{i\alpha}}_1}}}$, ${{\Omega}_2} = | {{{\Omega}_2}} |{{e}^{{{{i\alpha}}_2}}}$, ${{\rho}_{{bb}}}$, ${{\rho}_{{cc}}}$, ${{\rho}_{{ee}}}$, and ${{\rho}_{{bc}}}$ are the populations and vibrational coherence encoded by the 800 nm pump pulse. $\vartheta = {{\alpha}_1} - {{\alpha}_2} + {\beta}$, where ${\beta}$ is the phase of ${{\rho}_{{bc}}}$. ${{\gamma}^{- 1}}$ is the lifetime of the excited state ${\nu ^\prime} = 0$. All details are given in Supplement 1.

Equation (3) shows that the signal amplification is produced jointly by electronic population inversion and electronic coherence, and is further modulated by the initial vibrational coherence. By varying the two delays ${{\tau}_{{p} - {s}1}}$ and ${{\tau}_{{p} - {s}2}}$, Eq. (3) evidences the coherent oscillations with distinct frequencies from the two spectral components of the signal. This reveals the transfer between the electronic coherences. When time delay ${{\tau}_{{p} - {s}1}}$ is much longer than ${\pi}/\Delta$, the ${\pi}$ phase shift modulation can be observed between the signals at frequencies ${{\omega}_1}$ and ${{\omega}_2}$, during every step ${\delta T} = {\pi}/\Delta$. For parameters of the ${\rm N}_2^ +$ experiments, it gives $2{\pi}/{{\tilde \omega}_1} = 1.31\;{\rm fs} $ and $\Delta \approx 105\;{{\rm cm}^{- 1}}$, which leads to ${\delta T} \approx 150\;{\rm fs}$. This is in agreement with our experimental observations shown in Fig. 4(a). The calculated signal versus the ${{\tau}_{{p} - {s}1}}$ delay within 0–300 fs, depicted in Fig. 4(b), shows the oscillation beating measured at 391.4 nm corresponding to the experimental measurement shown in Fig. 4(a). It is worth noting a small discrepancy of $\sim{20}\;{\rm fs}$ in the beating period in Fig. 4(a) (experimental measurement of ${\delta T} \approx {130}\;{\rm fs}$ when zooming in), from the calculated ${\delta T} \approx {150}\;{\rm fs}$ in Fig. 4(b). This can be attributed to the rotation of the ${\rm N}_2^ +$, which has yet to be taken into account in our model. The rotational frequency is typically of the order ${20}\;{{\rm cm}^{- 1}}$ [2123], leading to a correction to the detuning such that ${\Delta} \to {\Delta} + 20\;{{\rm cm}^{- 1}} \approx 125\;{{\rm cm}^{- 1}}$, which subsequently revises ${\delta T} \approx 133\; {\rm fs}$.

 figure: Fig. 6.

Fig. 6. Theoretical calculations for emitted signals around 427.8 and 391.4 nm, according to Eq. (3), as a function of time delay ${\tau _{p {\text -} s1}}$ between the pump and the 391.4 nm seeding pulse. Parameters are the same as in Fig. 5, i.e., ${\tau _{p {\text -} s2}} = 130\;{\rm fs}$: (a) ${\tau _{p {\text -} s1}}$ finely tuned around 300 fs and (b) ${\tau _{p {\text -} s1}}$ finely tuned around 480 fs.

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When the time window for scanning ${{\tau}_{{p} - {s}1}}$ is much narrower than ${\pi}/\Delta$, i.e., scanning by $\pm 10\;{\rm fs}$ in the vicinity of ${{\tau}_{{p} - {s}1}} = 300\; {\rm fs}$ in the experiment, which yields $\cos ({\Delta {{\tau}_{{p} - {s}1}}}) \approx 0.94$, associated with ${\tau _{{p} - {s}2}} \approx 130\; {\rm fs}$, which yields $\Delta {\tau _{{p} - {s}2}} \approx 0.82\pi$, the coherence terms in Eq. (3) show the identical period of oscillations with a ${\pi}$ phase shift between signals ${S}_1$ and ${S}_2$ as presented in Fig. 6(a), in an agreement with our experimental observations in Figs. 5(a)–5(c). When scanning the time delay in the vicinity of ${{\tau}_{{p} - {s}1}} = 480\; {\rm fs}$, which is 180 fs later than that in Figs. 5(a)–5(c), Eq. (3) leads to the identical period of oscillations without a phase shift between signals ${S}_1$ and ${S}_2$ as illustrated in Fig. 6(b). This agrees well with our experimental observations in Figs. 5(d)–5(f).

Finally, we point out that Eq. (3) indicates that population inversion is not necessary for observing signal amplification. To clarify this, it is worth noting that our theoretical analysis demonstrates the third-order nature of the radiative amplification of two coupled spectral components [34]. This process falls into a different parameter regime of amplification compared to a single-mode lasing without inversion [35,36].

4. CONCLUSION

In summary, by injecting two seeding pulses at different wavelengths, we have shown the interplay between spectral components at 391.4 and 427.8 nm of nitrogen ions from the upper level population during the lasing process. Fine tuning of the temporal delay between the seeding and pump pulses reveals simultaneous coherent control of the two emission lines. This modulation period has been determined by the wavelength of the seeding pulse that has been scanned, e.g., 391.4 or 427.8 nm. A theoretical model of molecular relaxation resonantly interacting with laser pulses has been developed to provide a global understanding of the experimental observations.

Our work demonstrates the role of vibrational coherence, reveals its coupling to the electronic coherence dynamics in the lasing of ${\rm N}_2^ +$, and offers new insights for investigating the processes of transient lasing including lasing without inversion.

Funding

National Natural Science Foundation of China (12034013, 11904232, 11934011, 12074124); ARPC-CityU New Research Initiative/Infrastructure Support from Central (9610505); 111 Project (258 B12024); Innovation Program of Shanghai Municipal Education Commission (2017-01-07-00-07-E00007).

Acknowledgment

We acknowledge the instructive discussions with Prof. André Mysyrowicz, Dr. Aurélien Houard, Rostyslav Danylo of ENSTA Paris, and Prof. Vladimir Tikhonchuk of University of Bordeaux. K. E. D. is supported by the Zijiang Endowed Young Scholar Fund, East China Normal University.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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18. Q. Zhang, H. Xie, G. Li, X. Wang, H. Lei, J. Zhao, Z. Chen, J. Yao, Y. Cheng, and Z. Zhao, “Sub-cycle coherent control of ionic dynamics via transient ionization injection,” Commun. Phys. 3, 50 (2020). [CrossRef]  

19. G. Li, C. Jing, B. Zeng, H. Xie, J. Yao, W. Chu, J. Ni, H. Zhang, H. Xu, Y. Cheng, and Z. Xu, “Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization,” Phys. Rev. A 89, 033833 (2014). [CrossRef]  

20. Y. Liu, P. Ding, G. Lambert, A. Houard, V. Tikhonchuk, and A. Mysyrowicz, “Recollision-induced superradiance of ionized nitrogen molecule,” Phys. Rev. Lett. 115, 133203 (2015). [CrossRef]  

21. H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014). [CrossRef]  

22. A. Azarm, P. Corkum, and P. Polynkin, “Optical gain in rotationally excited nitrogen molecular ions,” Phys. Rev. A 96, 051401 (2017). [CrossRef]  

23. M. Richter, M. Lytova, F. Morales, S. Haessler, O. Smirnova, M. Spanner, and M. Ivanov, “Rotational quantum beat lasing without inversion,” Optica 7, 586–592 (2020). [CrossRef]  

24. M. Lytova, M. Richter, F. Morales, O. Smirnova, M. Ivanov, and M. Spanner, “N2+ lasing: gain and absorption in the presence of rotational coherence,” arXiv:2004.04067v1 (2020).

25. Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018). [CrossRef]  

26. B. Xu, J. Yao, Y. Wan, J. Chen, Z. Liu, F. Zhang, W. Chu, and Y. Cheng, “Vibrational Raman scattering from coherently excited molecular ions in a strong laser field,” Opt. Express 27, 18262–18272 (2019). [CrossRef]  

27. A. P. Spencer, W. O. Hutson, and E. Harel, “Quantum coherence selective 2D Raman-2D electronic spectroscopy,” Nat. Commun. 8, 14732 (2017). [CrossRef]  

28. Z. D. Zhang, K. Bennett, V. Chernyak, and S. Mukamel, “Utilizing microcavities to suppress third-order cascades in fifth-order Raman spectra,” J. Phys. Chem. Lett. 8, 3387–3391 (2017). [CrossRef]  

29. D. A. Blank, L. J. Kaufman, and G. R. Fleming, “Fifth-order two-dimensional Raman spectra of CS2 are dominated by third-order cascades,” J. Chem. Phys. 111, 3105–3114 (1999). [CrossRef]  

30. Y. J. Yan and S. Mukamel, “Femtosecond pump-probe spectroscopy of polyatomic molecules in condensed phases,” Phys. Rev. A 41, 6485–6504 (1990). [CrossRef]  

31. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, 1995).

32. Y. J. Yan and S. Mukamel, “Electronic dephasing, vibrational relaxation and solvent friction in molecular nonlinear optical line shapes,” J. Chem. Phys. 89, 5160–5176 (1988). [CrossRef]  

33. G. Herzberg, Molecular Spectra and Molecular Structure (Reitell, 2007), Vol. I.

34. G. S. Agarwal, “Collision-induced coherences in optical physics,” in Advances in Atomic, Molecular, and Optical Physics (1991), Vol. 29, p. 113.

35. M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989). [CrossRef]  

36. A. A. Svidzinsky, L. Yuan, and M. O. Scully, “Transient lasing without inversion,” New J. Phys. 15, 053044 (2013). [CrossRef]  

References

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  1. A. Dogariu, J. B. Michael, M. O. Scully, and R. B. Miles, “High-gain backward lasing in air,” Science 331, 442–445 (2011).
    [Crossref]
  2. J. Yao, B. Zeng, H. Xu, G. Li, W. Chu, J. Ni, H. Zhang, S. L. Chin, Y. Cheng, and Z. Xu, “High-brightness switchable multiwavelength remote laser in air,” Phys. Rev. A 84, 051802 (2011).
    [Crossref]
  3. H. Xu, E. Lötstedt, A. Iwasaki, and K. Yamanouchi, “Sub-10-fs population inversion in N2+ in air lasing through multiple state coupling,” Nat. Commun. 6, 8347 (2015).
    [Crossref]
  4. Y. Liu, Y. Brelet, G. Point, A. Houard, and A. Mysyrowicz, “Self-seeded lasing in ionized air pumped by 800 nm femtosecond laser pulses,” Opt. Express 21, 22791–22798 (2013).
    [Crossref]
  5. A. Laurain, M. Scheller, and P. Polynkin, “Low-threshold bidirectional air lasing,” Phys. Rev. Lett. 113, 253901 (2014).
    [Crossref]
  6. L. Yuan, Y. Liu, J. Yao, and Y. Cheng, “Recent advances in air lasing: a perspective from quantum coherence,” Adv. Quantum Technol. 2, 1900080 (2019).
    [Crossref]
  7. M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
    [Crossref]
  8. D. Kartashov, S. Ališauskas, G. Andriukaitis, A. Pugžlys, M. Shneider, A. Zheltikov, S. L. Chin, and A. Baltuška, “Free-space nitrogen gas laser driven by a femtosecond filament,” Phys. Rev. A 86, 033831 (2012).
    [Crossref]
  9. A. Traverso, R. Sanchez-Gonzalez, L. Yuan, K. Wang, D. V. Voronine, A. M. Zheltikov, Y. Rostovtsev, V. A. Sautenkov, A. V. Sokolov, S. W. North, and M. O. Scully, “Coherence brightened laser source for atmospheric remote sensing,” Proc. Natl. Acad. Sci. USA 109, 15185 (2012).
    [Crossref]
  10. P. Polynkin and Y. Cheng, Air Lasing, Vol. 206 of Springer Series in Optical Sciences (Springer, 2018).
  11. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
    [Crossref]
  12. A. Dogariu and R. B. Miles, “Three-photon femtosecond pumped backwards lasing in argon,” Opt. Express 24, A544–A552 (2016).
    [Crossref]
  13. J. Yao, G. Li, C. Jing, B. Zeng, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, H. Li, and H. Xu, “Remote creation of coherent emissions in air with two-color ultrafast laser pulses,” New J. Phys. 15, 023046 (2013).
    [Crossref]
  14. T. Ando, E. Lötstedt, A. Iwasaki, H. Li, Y. Fu, S. Wang, H. Xu, and K. Yamanouchi, “Rotational, vibrational, and electronic modulations in N2+ lasing at 391 nm: evidence of coherent ${B^2}{\Sigma_u^+}-{X^2}{\Sigma_g^+}-{{A^2}{\Pi_u}}$ coupling,” Phys. Rev. Lett. 123, 203201 (2019).
    [Crossref]
  15. A. Zhang, Q. Liang, M. Lei, L. Yuan, Y. Liu, Z. Fan, X. Zhang, S. Zhuang, C. Wu, Q. Gong, and H. Jiang, “Coherent modulation of superradiance from nitrogen ions pumped with femtosecond pulses,” Opt. Express 27, 12638–12646 (2019).
    [Crossref]
  16. A. Mysyrowicz, R. Danylo, A. Houard, V. Tikhonchuk, X. Zhang, Z. Fan, Q. Liang, S. Zhuang, and L. Yuan, “Lasing without population inversion in N2+,” APL Photon. 4, 110807 (2019).
    [Crossref]
  17. J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
    [Crossref]
  18. Q. Zhang, H. Xie, G. Li, X. Wang, H. Lei, J. Zhao, Z. Chen, J. Yao, Y. Cheng, and Z. Zhao, “Sub-cycle coherent control of ionic dynamics via transient ionization injection,” Commun. Phys. 3, 50 (2020).
    [Crossref]
  19. G. Li, C. Jing, B. Zeng, H. Xie, J. Yao, W. Chu, J. Ni, H. Zhang, H. Xu, Y. Cheng, and Z. Xu, “Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization,” Phys. Rev. A 89, 033833 (2014).
    [Crossref]
  20. Y. Liu, P. Ding, G. Lambert, A. Houard, V. Tikhonchuk, and A. Mysyrowicz, “Recollision-induced superradiance of ionized nitrogen molecule,” Phys. Rev. Lett. 115, 133203 (2015).
    [Crossref]
  21. H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014).
    [Crossref]
  22. A. Azarm, P. Corkum, and P. Polynkin, “Optical gain in rotationally excited nitrogen molecular ions,” Phys. Rev. A 96, 051401 (2017).
    [Crossref]
  23. M. Richter, M. Lytova, F. Morales, S. Haessler, O. Smirnova, M. Spanner, and M. Ivanov, “Rotational quantum beat lasing without inversion,” Optica 7, 586–592 (2020).
    [Crossref]
  24. M. Lytova, M. Richter, F. Morales, O. Smirnova, M. Ivanov, and M. Spanner, “N2+ lasing: gain and absorption in the presence of rotational coherence,” arXiv:2004.04067v1 (2020).
  25. Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
    [Crossref]
  26. B. Xu, J. Yao, Y. Wan, J. Chen, Z. Liu, F. Zhang, W. Chu, and Y. Cheng, “Vibrational Raman scattering from coherently excited molecular ions in a strong laser field,” Opt. Express 27, 18262–18272 (2019).
    [Crossref]
  27. A. P. Spencer, W. O. Hutson, and E. Harel, “Quantum coherence selective 2D Raman-2D electronic spectroscopy,” Nat. Commun. 8, 14732 (2017).
    [Crossref]
  28. Z. D. Zhang, K. Bennett, V. Chernyak, and S. Mukamel, “Utilizing microcavities to suppress third-order cascades in fifth-order Raman spectra,” J. Phys. Chem. Lett. 8, 3387–3391 (2017).
    [Crossref]
  29. D. A. Blank, L. J. Kaufman, and G. R. Fleming, “Fifth-order two-dimensional Raman spectra of CS2 are dominated by third-order cascades,” J. Chem. Phys. 111, 3105–3114 (1999).
    [Crossref]
  30. Y. J. Yan and S. Mukamel, “Femtosecond pump-probe spectroscopy of polyatomic molecules in condensed phases,” Phys. Rev. A 41, 6485–6504 (1990).
    [Crossref]
  31. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, 1995).
  32. Y. J. Yan and S. Mukamel, “Electronic dephasing, vibrational relaxation and solvent friction in molecular nonlinear optical line shapes,” J. Chem. Phys. 89, 5160–5176 (1988).
    [Crossref]
  33. G. Herzberg, Molecular Spectra and Molecular Structure (Reitell, 2007), Vol. I.
  34. G. S. Agarwal, “Collision-induced coherences in optical physics,” in Advances in Atomic, Molecular, and Optical Physics (1991), Vol. 29, p. 113.
  35. M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
    [Crossref]
  36. A. A. Svidzinsky, L. Yuan, and M. O. Scully, “Transient lasing without inversion,” New J. Phys. 15, 053044 (2013).
    [Crossref]

2020 (2)

Q. Zhang, H. Xie, G. Li, X. Wang, H. Lei, J. Zhao, Z. Chen, J. Yao, Y. Cheng, and Z. Zhao, “Sub-cycle coherent control of ionic dynamics via transient ionization injection,” Commun. Phys. 3, 50 (2020).
[Crossref]

M. Richter, M. Lytova, F. Morales, S. Haessler, O. Smirnova, M. Spanner, and M. Ivanov, “Rotational quantum beat lasing without inversion,” Optica 7, 586–592 (2020).
[Crossref]

2019 (6)

B. Xu, J. Yao, Y. Wan, J. Chen, Z. Liu, F. Zhang, W. Chu, and Y. Cheng, “Vibrational Raman scattering from coherently excited molecular ions in a strong laser field,” Opt. Express 27, 18262–18272 (2019).
[Crossref]

T. Ando, E. Lötstedt, A. Iwasaki, H. Li, Y. Fu, S. Wang, H. Xu, and K. Yamanouchi, “Rotational, vibrational, and electronic modulations in N2+ lasing at 391 nm: evidence of coherent ${B^2}{\Sigma_u^+}-{X^2}{\Sigma_g^+}-{{A^2}{\Pi_u}}$ coupling,” Phys. Rev. Lett. 123, 203201 (2019).
[Crossref]

A. Zhang, Q. Liang, M. Lei, L. Yuan, Y. Liu, Z. Fan, X. Zhang, S. Zhuang, C. Wu, Q. Gong, and H. Jiang, “Coherent modulation of superradiance from nitrogen ions pumped with femtosecond pulses,” Opt. Express 27, 12638–12646 (2019).
[Crossref]

A. Mysyrowicz, R. Danylo, A. Houard, V. Tikhonchuk, X. Zhang, Z. Fan, Q. Liang, S. Zhuang, and L. Yuan, “Lasing without population inversion in N2+,” APL Photon. 4, 110807 (2019).
[Crossref]

J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
[Crossref]

L. Yuan, Y. Liu, J. Yao, and Y. Cheng, “Recent advances in air lasing: a perspective from quantum coherence,” Adv. Quantum Technol. 2, 1900080 (2019).
[Crossref]

2018 (2)

M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
[Crossref]

Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
[Crossref]

2017 (3)

A. P. Spencer, W. O. Hutson, and E. Harel, “Quantum coherence selective 2D Raman-2D electronic spectroscopy,” Nat. Commun. 8, 14732 (2017).
[Crossref]

Z. D. Zhang, K. Bennett, V. Chernyak, and S. Mukamel, “Utilizing microcavities to suppress third-order cascades in fifth-order Raman spectra,” J. Phys. Chem. Lett. 8, 3387–3391 (2017).
[Crossref]

A. Azarm, P. Corkum, and P. Polynkin, “Optical gain in rotationally excited nitrogen molecular ions,” Phys. Rev. A 96, 051401 (2017).
[Crossref]

2016 (1)

2015 (2)

H. Xu, E. Lötstedt, A. Iwasaki, and K. Yamanouchi, “Sub-10-fs population inversion in N2+ in air lasing through multiple state coupling,” Nat. Commun. 6, 8347 (2015).
[Crossref]

Y. Liu, P. Ding, G. Lambert, A. Houard, V. Tikhonchuk, and A. Mysyrowicz, “Recollision-induced superradiance of ionized nitrogen molecule,” Phys. Rev. Lett. 115, 133203 (2015).
[Crossref]

2014 (3)

H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014).
[Crossref]

A. Laurain, M. Scheller, and P. Polynkin, “Low-threshold bidirectional air lasing,” Phys. Rev. Lett. 113, 253901 (2014).
[Crossref]

G. Li, C. Jing, B. Zeng, H. Xie, J. Yao, W. Chu, J. Ni, H. Zhang, H. Xu, Y. Cheng, and Z. Xu, “Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization,” Phys. Rev. A 89, 033833 (2014).
[Crossref]

2013 (3)

J. Yao, G. Li, C. Jing, B. Zeng, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, H. Li, and H. Xu, “Remote creation of coherent emissions in air with two-color ultrafast laser pulses,” New J. Phys. 15, 023046 (2013).
[Crossref]

Y. Liu, Y. Brelet, G. Point, A. Houard, and A. Mysyrowicz, “Self-seeded lasing in ionized air pumped by 800 nm femtosecond laser pulses,” Opt. Express 21, 22791–22798 (2013).
[Crossref]

A. A. Svidzinsky, L. Yuan, and M. O. Scully, “Transient lasing without inversion,” New J. Phys. 15, 053044 (2013).
[Crossref]

2012 (2)

D. Kartashov, S. Ališauskas, G. Andriukaitis, A. Pugžlys, M. Shneider, A. Zheltikov, S. L. Chin, and A. Baltuška, “Free-space nitrogen gas laser driven by a femtosecond filament,” Phys. Rev. A 86, 033831 (2012).
[Crossref]

A. Traverso, R. Sanchez-Gonzalez, L. Yuan, K. Wang, D. V. Voronine, A. M. Zheltikov, Y. Rostovtsev, V. A. Sautenkov, A. V. Sokolov, S. W. North, and M. O. Scully, “Coherence brightened laser source for atmospheric remote sensing,” Proc. Natl. Acad. Sci. USA 109, 15185 (2012).
[Crossref]

2011 (2)

A. Dogariu, J. B. Michael, M. O. Scully, and R. B. Miles, “High-gain backward lasing in air,” Science 331, 442–445 (2011).
[Crossref]

J. Yao, B. Zeng, H. Xu, G. Li, W. Chu, J. Ni, H. Zhang, S. L. Chin, Y. Cheng, and Z. Xu, “High-brightness switchable multiwavelength remote laser in air,” Phys. Rev. A 84, 051802 (2011).
[Crossref]

2007 (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

1999 (1)

D. A. Blank, L. J. Kaufman, and G. R. Fleming, “Fifth-order two-dimensional Raman spectra of CS2 are dominated by third-order cascades,” J. Chem. Phys. 111, 3105–3114 (1999).
[Crossref]

1990 (1)

Y. J. Yan and S. Mukamel, “Femtosecond pump-probe spectroscopy of polyatomic molecules in condensed phases,” Phys. Rev. A 41, 6485–6504 (1990).
[Crossref]

1989 (1)

M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
[Crossref]

1988 (1)

Y. J. Yan and S. Mukamel, “Electronic dephasing, vibrational relaxation and solvent friction in molecular nonlinear optical line shapes,” J. Chem. Phys. 89, 5160–5176 (1988).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, “Collision-induced coherences in optical physics,” in Advances in Atomic, Molecular, and Optical Physics (1991), Vol. 29, p. 113.

Ališauskas, S.

D. Kartashov, S. Ališauskas, G. Andriukaitis, A. Pugžlys, M. Shneider, A. Zheltikov, S. L. Chin, and A. Baltuška, “Free-space nitrogen gas laser driven by a femtosecond filament,” Phys. Rev. A 86, 033831 (2012).
[Crossref]

Ando, T.

T. Ando, E. Lötstedt, A. Iwasaki, H. Li, Y. Fu, S. Wang, H. Xu, and K. Yamanouchi, “Rotational, vibrational, and electronic modulations in N2+ lasing at 391 nm: evidence of coherent ${B^2}{\Sigma_u^+}-{X^2}{\Sigma_g^+}-{{A^2}{\Pi_u}}$ coupling,” Phys. Rev. Lett. 123, 203201 (2019).
[Crossref]

Andriukaitis, G.

D. Kartashov, S. Ališauskas, G. Andriukaitis, A. Pugžlys, M. Shneider, A. Zheltikov, S. L. Chin, and A. Baltuška, “Free-space nitrogen gas laser driven by a femtosecond filament,” Phys. Rev. A 86, 033831 (2012).
[Crossref]

Arissian, L.

M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
[Crossref]

Azarm, A.

A. Azarm, P. Corkum, and P. Polynkin, “Optical gain in rotationally excited nitrogen molecular ions,” Phys. Rev. A 96, 051401 (2017).
[Crossref]

Baltuška, A.

D. Kartashov, S. Ališauskas, G. Andriukaitis, A. Pugžlys, M. Shneider, A. Zheltikov, S. L. Chin, and A. Baltuška, “Free-space nitrogen gas laser driven by a femtosecond filament,” Phys. Rev. A 86, 033831 (2012).
[Crossref]

Bennett, K.

Z. D. Zhang, K. Bennett, V. Chernyak, and S. Mukamel, “Utilizing microcavities to suppress third-order cascades in fifth-order Raman spectra,” J. Phys. Chem. Lett. 8, 3387–3391 (2017).
[Crossref]

Blank, D. A.

D. A. Blank, L. J. Kaufman, and G. R. Fleming, “Fifth-order two-dimensional Raman spectra of CS2 are dominated by third-order cascades,” J. Chem. Phys. 111, 3105–3114 (1999).
[Crossref]

Brelet, Y.

Britton, M.

M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
[Crossref]

Brown, G.

M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
[Crossref]

Chen, J.

J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
[Crossref]

B. Xu, J. Yao, Y. Wan, J. Chen, Z. Liu, F. Zhang, W. Chu, and Y. Cheng, “Vibrational Raman scattering from coherently excited molecular ions in a strong laser field,” Opt. Express 27, 18262–18272 (2019).
[Crossref]

Chen, Z.

Q. Zhang, H. Xie, G. Li, X. Wang, H. Lei, J. Zhao, Z. Chen, J. Yao, Y. Cheng, and Z. Zhao, “Sub-cycle coherent control of ionic dynamics via transient ionization injection,” Commun. Phys. 3, 50 (2020).
[Crossref]

Cheng, Y.

Q. Zhang, H. Xie, G. Li, X. Wang, H. Lei, J. Zhao, Z. Chen, J. Yao, Y. Cheng, and Z. Zhao, “Sub-cycle coherent control of ionic dynamics via transient ionization injection,” Commun. Phys. 3, 50 (2020).
[Crossref]

L. Yuan, Y. Liu, J. Yao, and Y. Cheng, “Recent advances in air lasing: a perspective from quantum coherence,” Adv. Quantum Technol. 2, 1900080 (2019).
[Crossref]

B. Xu, J. Yao, Y. Wan, J. Chen, Z. Liu, F. Zhang, W. Chu, and Y. Cheng, “Vibrational Raman scattering from coherently excited molecular ions in a strong laser field,” Opt. Express 27, 18262–18272 (2019).
[Crossref]

H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014).
[Crossref]

G. Li, C. Jing, B. Zeng, H. Xie, J. Yao, W. Chu, J. Ni, H. Zhang, H. Xu, Y. Cheng, and Z. Xu, “Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization,” Phys. Rev. A 89, 033833 (2014).
[Crossref]

J. Yao, B. Zeng, H. Xu, G. Li, W. Chu, J. Ni, H. Zhang, S. L. Chin, Y. Cheng, and Z. Xu, “High-brightness switchable multiwavelength remote laser in air,” Phys. Rev. A 84, 051802 (2011).
[Crossref]

P. Polynkin and Y. Cheng, Air Lasing, Vol. 206 of Springer Series in Optical Sciences (Springer, 2018).

Chernyak, V.

Z. D. Zhang, K. Bennett, V. Chernyak, and S. Mukamel, “Utilizing microcavities to suppress third-order cascades in fifth-order Raman spectra,” J. Phys. Chem. Lett. 8, 3387–3391 (2017).
[Crossref]

Chin, S. L.

D. Kartashov, S. Ališauskas, G. Andriukaitis, A. Pugžlys, M. Shneider, A. Zheltikov, S. L. Chin, and A. Baltuška, “Free-space nitrogen gas laser driven by a femtosecond filament,” Phys. Rev. A 86, 033831 (2012).
[Crossref]

J. Yao, B. Zeng, H. Xu, G. Li, W. Chu, J. Ni, H. Zhang, S. L. Chin, Y. Cheng, and Z. Xu, “High-brightness switchable multiwavelength remote laser in air,” Phys. Rev. A 84, 051802 (2011).
[Crossref]

Chu, W.

J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
[Crossref]

B. Xu, J. Yao, Y. Wan, J. Chen, Z. Liu, F. Zhang, W. Chu, and Y. Cheng, “Vibrational Raman scattering from coherently excited molecular ions in a strong laser field,” Opt. Express 27, 18262–18272 (2019).
[Crossref]

G. Li, C. Jing, B. Zeng, H. Xie, J. Yao, W. Chu, J. Ni, H. Zhang, H. Xu, Y. Cheng, and Z. Xu, “Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization,” Phys. Rev. A 89, 033833 (2014).
[Crossref]

H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014).
[Crossref]

J. Yao, G. Li, C. Jing, B. Zeng, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, H. Li, and H. Xu, “Remote creation of coherent emissions in air with two-color ultrafast laser pulses,” New J. Phys. 15, 023046 (2013).
[Crossref]

J. Yao, B. Zeng, H. Xu, G. Li, W. Chu, J. Ni, H. Zhang, S. L. Chin, Y. Cheng, and Z. Xu, “High-brightness switchable multiwavelength remote laser in air,” Phys. Rev. A 84, 051802 (2011).
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A. Azarm, P. Corkum, and P. Polynkin, “Optical gain in rotationally excited nitrogen molecular ions,” Phys. Rev. A 96, 051401 (2017).
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M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
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A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
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A. Mysyrowicz, R. Danylo, A. Houard, V. Tikhonchuk, X. Zhang, Z. Fan, Q. Liang, S. Zhuang, and L. Yuan, “Lasing without population inversion in N2+,” APL Photon. 4, 110807 (2019).
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Y. Liu, P. Ding, G. Lambert, A. Houard, V. Tikhonchuk, and A. Mysyrowicz, “Recollision-induced superradiance of ionized nitrogen molecule,” Phys. Rev. Lett. 115, 133203 (2015).
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A. Dogariu and R. B. Miles, “Three-photon femtosecond pumped backwards lasing in argon,” Opt. Express 24, A544–A552 (2016).
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A. Dogariu, J. B. Michael, M. O. Scully, and R. B. Miles, “High-gain backward lasing in air,” Science 331, 442–445 (2011).
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A. Mysyrowicz, R. Danylo, A. Houard, V. Tikhonchuk, X. Zhang, Z. Fan, Q. Liang, S. Zhuang, and L. Yuan, “Lasing without population inversion in N2+,” APL Photon. 4, 110807 (2019).
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A. Zhang, Q. Liang, M. Lei, L. Yuan, Y. Liu, Z. Fan, X. Zhang, S. Zhuang, C. Wu, Q. Gong, and H. Jiang, “Coherent modulation of superradiance from nitrogen ions pumped with femtosecond pulses,” Opt. Express 27, 12638–12646 (2019).
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J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
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J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
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D. A. Blank, L. J. Kaufman, and G. R. Fleming, “Fifth-order two-dimensional Raman spectra of CS2 are dominated by third-order cascades,” J. Chem. Phys. 111, 3105–3114 (1999).
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Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
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[Crossref]

Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
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[Crossref]

Y. Liu, P. Ding, G. Lambert, A. Houard, V. Tikhonchuk, and A. Mysyrowicz, “Recollision-induced superradiance of ionized nitrogen molecule,” Phys. Rev. Lett. 115, 133203 (2015).
[Crossref]

Y. Liu, Y. Brelet, G. Point, A. Houard, and A. Mysyrowicz, “Self-seeded lasing in ionized air pumped by 800 nm femtosecond laser pulses,” Opt. Express 21, 22791–22798 (2013).
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A. P. Spencer, W. O. Hutson, and E. Harel, “Quantum coherence selective 2D Raman-2D electronic spectroscopy,” Nat. Commun. 8, 14732 (2017).
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M. Richter, M. Lytova, F. Morales, S. Haessler, O. Smirnova, M. Spanner, and M. Ivanov, “Rotational quantum beat lasing without inversion,” Optica 7, 586–592 (2020).
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T. Ando, E. Lötstedt, A. Iwasaki, H. Li, Y. Fu, S. Wang, H. Xu, and K. Yamanouchi, “Rotational, vibrational, and electronic modulations in N2+ lasing at 391 nm: evidence of coherent ${B^2}{\Sigma_u^+}-{X^2}{\Sigma_g^+}-{{A^2}{\Pi_u}}$ coupling,” Phys. Rev. Lett. 123, 203201 (2019).
[Crossref]

H. Xu, E. Lötstedt, A. Iwasaki, and K. Yamanouchi, “Sub-10-fs population inversion in N2+ in air lasing through multiple state coupling,” Nat. Commun. 6, 8347 (2015).
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A. Zhang, Q. Liang, M. Lei, L. Yuan, Y. Liu, Z. Fan, X. Zhang, S. Zhuang, C. Wu, Q. Gong, and H. Jiang, “Coherent modulation of superradiance from nitrogen ions pumped with femtosecond pulses,” Opt. Express 27, 12638–12646 (2019).
[Crossref]

Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
[Crossref]

Jing, C.

H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014).
[Crossref]

G. Li, C. Jing, B. Zeng, H. Xie, J. Yao, W. Chu, J. Ni, H. Zhang, H. Xu, Y. Cheng, and Z. Xu, “Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization,” Phys. Rev. A 89, 033833 (2014).
[Crossref]

J. Yao, G. Li, C. Jing, B. Zeng, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, H. Li, and H. Xu, “Remote creation of coherent emissions in air with two-color ultrafast laser pulses,” New J. Phys. 15, 023046 (2013).
[Crossref]

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D. Kartashov, S. Ališauskas, G. Andriukaitis, A. Pugžlys, M. Shneider, A. Zheltikov, S. L. Chin, and A. Baltuška, “Free-space nitrogen gas laser driven by a femtosecond filament,” Phys. Rev. A 86, 033831 (2012).
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D. A. Blank, L. J. Kaufman, and G. R. Fleming, “Fifth-order two-dimensional Raman spectra of CS2 are dominated by third-order cascades,” J. Chem. Phys. 111, 3105–3114 (1999).
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Ko, D. H.

M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
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M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
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M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
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Y. Liu, P. Ding, G. Lambert, A. Houard, V. Tikhonchuk, and A. Mysyrowicz, “Recollision-induced superradiance of ionized nitrogen molecule,” Phys. Rev. Lett. 115, 133203 (2015).
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A. Laurain, M. Scheller, and P. Polynkin, “Low-threshold bidirectional air lasing,” Phys. Rev. Lett. 113, 253901 (2014).
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Li, G.

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G. Li, C. Jing, B. Zeng, H. Xie, J. Yao, W. Chu, J. Ni, H. Zhang, H. Xu, Y. Cheng, and Z. Xu, “Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization,” Phys. Rev. A 89, 033833 (2014).
[Crossref]

H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014).
[Crossref]

J. Yao, G. Li, C. Jing, B. Zeng, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, H. Li, and H. Xu, “Remote creation of coherent emissions in air with two-color ultrafast laser pulses,” New J. Phys. 15, 023046 (2013).
[Crossref]

J. Yao, B. Zeng, H. Xu, G. Li, W. Chu, J. Ni, H. Zhang, S. L. Chin, Y. Cheng, and Z. Xu, “High-brightness switchable multiwavelength remote laser in air,” Phys. Rev. A 84, 051802 (2011).
[Crossref]

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T. Ando, E. Lötstedt, A. Iwasaki, H. Li, Y. Fu, S. Wang, H. Xu, and K. Yamanouchi, “Rotational, vibrational, and electronic modulations in N2+ lasing at 391 nm: evidence of coherent ${B^2}{\Sigma_u^+}-{X^2}{\Sigma_g^+}-{{A^2}{\Pi_u}}$ coupling,” Phys. Rev. Lett. 123, 203201 (2019).
[Crossref]

J. Yao, G. Li, C. Jing, B. Zeng, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, H. Li, and H. Xu, “Remote creation of coherent emissions in air with two-color ultrafast laser pulses,” New J. Phys. 15, 023046 (2013).
[Crossref]

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M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
[Crossref]

H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014).
[Crossref]

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A. Zhang, Q. Liang, M. Lei, L. Yuan, Y. Liu, Z. Fan, X. Zhang, S. Zhuang, C. Wu, Q. Gong, and H. Jiang, “Coherent modulation of superradiance from nitrogen ions pumped with femtosecond pulses,” Opt. Express 27, 12638–12646 (2019).
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A. Mysyrowicz, R. Danylo, A. Houard, V. Tikhonchuk, X. Zhang, Z. Fan, Q. Liang, S. Zhuang, and L. Yuan, “Lasing without population inversion in N2+,” APL Photon. 4, 110807 (2019).
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A. Zhang, Q. Liang, M. Lei, L. Yuan, Y. Liu, Z. Fan, X. Zhang, S. Zhuang, C. Wu, Q. Gong, and H. Jiang, “Coherent modulation of superradiance from nitrogen ions pumped with femtosecond pulses,” Opt. Express 27, 12638–12646 (2019).
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[Crossref]

Y. Liu, P. Ding, G. Lambert, A. Houard, V. Tikhonchuk, and A. Mysyrowicz, “Recollision-induced superradiance of ionized nitrogen molecule,” Phys. Rev. Lett. 115, 133203 (2015).
[Crossref]

Y. Liu, Y. Brelet, G. Point, A. Houard, and A. Mysyrowicz, “Self-seeded lasing in ionized air pumped by 800 nm femtosecond laser pulses,” Opt. Express 21, 22791–22798 (2013).
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J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
[Crossref]

B. Xu, J. Yao, Y. Wan, J. Chen, Z. Liu, F. Zhang, W. Chu, and Y. Cheng, “Vibrational Raman scattering from coherently excited molecular ions in a strong laser field,” Opt. Express 27, 18262–18272 (2019).
[Crossref]

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T. Ando, E. Lötstedt, A. Iwasaki, H. Li, Y. Fu, S. Wang, H. Xu, and K. Yamanouchi, “Rotational, vibrational, and electronic modulations in N2+ lasing at 391 nm: evidence of coherent ${B^2}{\Sigma_u^+}-{X^2}{\Sigma_g^+}-{{A^2}{\Pi_u}}$ coupling,” Phys. Rev. Lett. 123, 203201 (2019).
[Crossref]

H. Xu, E. Lötstedt, A. Iwasaki, and K. Yamanouchi, “Sub-10-fs population inversion in N2+ in air lasing through multiple state coupling,” Nat. Commun. 6, 8347 (2015).
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M. Richter, M. Lytova, F. Morales, S. Haessler, O. Smirnova, M. Spanner, and M. Ivanov, “Rotational quantum beat lasing without inversion,” Optica 7, 586–592 (2020).
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M. Lytova, M. Richter, F. Morales, O. Smirnova, M. Ivanov, and M. Spanner, “N2+ lasing: gain and absorption in the presence of rotational coherence,” arXiv:2004.04067v1 (2020).

Miao, Z.

Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
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A. Dogariu, J. B. Michael, M. O. Scully, and R. B. Miles, “High-gain backward lasing in air,” Science 331, 442–445 (2011).
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A. Dogariu and R. B. Miles, “Three-photon femtosecond pumped backwards lasing in argon,” Opt. Express 24, A544–A552 (2016).
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A. Dogariu, J. B. Michael, M. O. Scully, and R. B. Miles, “High-gain backward lasing in air,” Science 331, 442–445 (2011).
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M. Richter, M. Lytova, F. Morales, S. Haessler, O. Smirnova, M. Spanner, and M. Ivanov, “Rotational quantum beat lasing without inversion,” Optica 7, 586–592 (2020).
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M. Lytova, M. Richter, F. Morales, O. Smirnova, M. Ivanov, and M. Spanner, “N2+ lasing: gain and absorption in the presence of rotational coherence,” arXiv:2004.04067v1 (2020).

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A. Mysyrowicz, R. Danylo, A. Houard, V. Tikhonchuk, X. Zhang, Z. Fan, Q. Liang, S. Zhuang, and L. Yuan, “Lasing without population inversion in N2+,” APL Photon. 4, 110807 (2019).
[Crossref]

Y. Liu, P. Ding, G. Lambert, A. Houard, V. Tikhonchuk, and A. Mysyrowicz, “Recollision-induced superradiance of ionized nitrogen molecule,” Phys. Rev. Lett. 115, 133203 (2015).
[Crossref]

Y. Liu, Y. Brelet, G. Point, A. Houard, and A. Mysyrowicz, “Self-seeded lasing in ionized air pumped by 800 nm femtosecond laser pulses,” Opt. Express 21, 22791–22798 (2013).
[Crossref]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[Crossref]

Naumov, A.

M. Britton, P. Laferriere, D. H. Ko, Z. Li, F. Kong, G. Brown, A. Naumov, C. Zhang, L. Arissian, and P. B. Corkum, “Testing the role of recollision in N2+ air lasing,” Phys. Rev. Lett. 120, 133208 (2018).
[Crossref]

Ni, J.

G. Li, C. Jing, B. Zeng, H. Xie, J. Yao, W. Chu, J. Ni, H. Zhang, H. Xu, Y. Cheng, and Z. Xu, “Signature of superradiance from a nitrogen-gas plasma channel produced by strong-field ionization,” Phys. Rev. A 89, 033833 (2014).
[Crossref]

H. Xie, B. Zeng, G. Li, W. Chu, H. Zhang, C. Jing, J. Yao, J. Ni, Z. Wang, Z. Li, and Y. Cheng, “Coupling of N2+ rotational states in an air laser from tunnel-ionized nitrogen molecules,” Phys. Rev. A 90, 042504 (2014).
[Crossref]

J. Yao, G. Li, C. Jing, B. Zeng, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, H. Li, and H. Xu, “Remote creation of coherent emissions in air with two-color ultrafast laser pulses,” New J. Phys. 15, 023046 (2013).
[Crossref]

J. Yao, B. Zeng, H. Xu, G. Li, W. Chu, J. Ni, H. Zhang, S. L. Chin, Y. Cheng, and Z. Xu, “High-brightness switchable multiwavelength remote laser in air,” Phys. Rev. A 84, 051802 (2011).
[Crossref]

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A. Traverso, R. Sanchez-Gonzalez, L. Yuan, K. Wang, D. V. Voronine, A. M. Zheltikov, Y. Rostovtsev, V. A. Sautenkov, A. V. Sokolov, S. W. North, and M. O. Scully, “Coherence brightened laser source for atmospheric remote sensing,” Proc. Natl. Acad. Sci. USA 109, 15185 (2012).
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Polynkin, P.

A. Azarm, P. Corkum, and P. Polynkin, “Optical gain in rotationally excited nitrogen molecular ions,” Phys. Rev. A 96, 051401 (2017).
[Crossref]

A. Laurain, M. Scheller, and P. Polynkin, “Low-threshold bidirectional air lasing,” Phys. Rev. Lett. 113, 253901 (2014).
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D. Kartashov, S. Ališauskas, G. Andriukaitis, A. Pugžlys, M. Shneider, A. Zheltikov, S. L. Chin, and A. Baltuška, “Free-space nitrogen gas laser driven by a femtosecond filament,” Phys. Rev. A 86, 033831 (2012).
[Crossref]

Qiao, L.

J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
[Crossref]

Richter, M.

M. Richter, M. Lytova, F. Morales, S. Haessler, O. Smirnova, M. Spanner, and M. Ivanov, “Rotational quantum beat lasing without inversion,” Optica 7, 586–592 (2020).
[Crossref]

M. Lytova, M. Richter, F. Morales, O. Smirnova, M. Ivanov, and M. Spanner, “N2+ lasing: gain and absorption in the presence of rotational coherence,” arXiv:2004.04067v1 (2020).

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A. Traverso, R. Sanchez-Gonzalez, L. Yuan, K. Wang, D. V. Voronine, A. M. Zheltikov, Y. Rostovtsev, V. A. Sautenkov, A. V. Sokolov, S. W. North, and M. O. Scully, “Coherence brightened laser source for atmospheric remote sensing,” Proc. Natl. Acad. Sci. USA 109, 15185 (2012).
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A. Traverso, R. Sanchez-Gonzalez, L. Yuan, K. Wang, D. V. Voronine, A. M. Zheltikov, Y. Rostovtsev, V. A. Sautenkov, A. V. Sokolov, S. W. North, and M. O. Scully, “Coherence brightened laser source for atmospheric remote sensing,” Proc. Natl. Acad. Sci. USA 109, 15185 (2012).
[Crossref]

Scheller, M.

A. Laurain, M. Scheller, and P. Polynkin, “Low-threshold bidirectional air lasing,” Phys. Rev. Lett. 113, 253901 (2014).
[Crossref]

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Zhang, H.

J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
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J. Yao, B. Zeng, H. Xu, G. Li, W. Chu, J. Ni, H. Zhang, S. L. Chin, Y. Cheng, and Z. Xu, “High-brightness switchable multiwavelength remote laser in air,” Phys. Rev. A 84, 051802 (2011).
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Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
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Q. Zhang, H. Xie, G. Li, X. Wang, H. Lei, J. Zhao, Z. Chen, J. Yao, Y. Cheng, and Z. Zhao, “Sub-cycle coherent control of ionic dynamics via transient ionization injection,” Commun. Phys. 3, 50 (2020).
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Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
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Z. Miao, X. Zhong, L. Zhang, W. Zheng, Y. Gao, Y. Liu, H. Jiang, Q. Gong, and C. Wu, “Stimulated-Raman-scattering-assisted superfluorescence enhancement from ionized nitrogen molecules in 800-nm femtosecond laser fields,” Phys. Rev. A 98, 033402 (2018).
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M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat laser: lasing without inversion and inversion without lasing,” Phys. Rev. Lett. 62, 2813–2816 (1989).
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A. Zhang, Q. Liang, M. Lei, L. Yuan, Y. Liu, Z. Fan, X. Zhang, S. Zhuang, C. Wu, Q. Gong, and H. Jiang, “Coherent modulation of superradiance from nitrogen ions pumped with femtosecond pulses,” Opt. Express 27, 12638–12646 (2019).
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Adv. Quantum Technol. (1)

L. Yuan, Y. Liu, J. Yao, and Y. Cheng, “Recent advances in air lasing: a perspective from quantum coherence,” Adv. Quantum Technol. 2, 1900080 (2019).
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A. Mysyrowicz, R. Danylo, A. Houard, V. Tikhonchuk, X. Zhang, Z. Fan, Q. Liang, S. Zhuang, and L. Yuan, “Lasing without population inversion in N2+,” APL Photon. 4, 110807 (2019).
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Commun. Phys. (1)

Q. Zhang, H. Xie, G. Li, X. Wang, H. Lei, J. Zhao, Z. Chen, J. Yao, Y. Cheng, and Z. Zhao, “Sub-cycle coherent control of ionic dynamics via transient ionization injection,” Commun. Phys. 3, 50 (2020).
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J. Phys. Chem. Lett. (1)

Z. D. Zhang, K. Bennett, V. Chernyak, and S. Mukamel, “Utilizing microcavities to suppress third-order cascades in fifth-order Raman spectra,” J. Phys. Chem. Lett. 8, 3387–3391 (2017).
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Nat. Commun. (2)

A. P. Spencer, W. O. Hutson, and E. Harel, “Quantum coherence selective 2D Raman-2D electronic spectroscopy,” Nat. Commun. 8, 14732 (2017).
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J. Yao, G. Li, C. Jing, B. Zeng, W. Chu, J. Ni, H. Zhang, H. Xie, C. Zhang, H. Li, and H. Xu, “Remote creation of coherent emissions in air with two-color ultrafast laser pulses,” New J. Phys. 15, 023046 (2013).
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Phys. Rev. A (8)

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J. Chen, J. Yao, H. Zhang, Z. Liu, B. Xu, W. Chu, L. Qiao, Z. Wang, J. Fatome, O. Faucher, and C. Wu, “Electronic-coherence-mediated molecular nitrogen-ion lasing in a strong laser field,” Phys. Rev. A 100, 031402 (2019).
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Supplementary Material (1)

NameDescription
» Supplement 1       Additional details of modeling.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. Schematic experimental setup. The two seeding pulses with central wavelengths around 391 and 428 nm are injected into the nitrogen gas plasma formed by 800 nm pump pulses. The relative delay between seeding and pump pulses ${\tau _{p {\text -} s1}}$ and ${\tau _{p {\text -} s2}}$ can be controlled with precision of 30 as. (b) Energy surfaces of electronic ground and excited states. Solid arrows represent vertical transitions according to Frank–Condon principle, and blue wavy arrows represent nonradiative transitions. (c) Simplified version from (b) for nitrogen cations, where $\Delta \approx 105\;{{\rm cm}^{- 1}}$ denotes the detuning between vertical transitions and the level ${\nu ^\prime} = 0$ .
Fig. 2.
Fig. 2. (a) Spectra of the two seeding pulses. (b) Injection of the 391.4 nm seeding pulse leads to strong amplification at that wavelength. The seeding pulse itself (red line), the signal produced with only the 800 nm pump pulse (black line), and the amplified signal (blue line) are presented. (c) Injection of the 427.8 nm seeding pulse and corresponding emission. (d) Injection of both 391.4 and 427.8 nm seeding pulses. Delay ${\tau _{p {\text -} s1}} = 300\;{\rm fs}$ and ${\tau _{p {\text -} s2}} = 500\;{\rm fs}$ .
Fig. 3.
Fig. 3. (a) Variation of the amplified 391.4 and 427.8 nm signal intensities as a function of delay ${\tau _{p {\text -} s1}}$ between the 391.4 nm seeding pulse and the pump pulse. Positive delay means that the 391.4 nm seeding pulse lags behind the 800 nm pump pulse. Delay between pump and 427.8 nm seeding pulse ${\tau _{p {\text -} s2}}$ is fixed at 500 fs. (b) Total energy of 391.4 and 427.8 nm emission as a function of the delay ${\tau _{p {\text -} s1}}$ .
Fig. 4.
Fig. 4. (a) Transmitted 391.4 and 427.8 nm signal as a function of the delay ${\tau _{p {\text -} s1}}$ being scanned around 0–300 fs. Delay between pump and 427.8 nm seeding pulse ${\tau _{p {\text -} s2}}$ is 130 fs. (b) Theoretical calculation according to Eq. (3). $| {{{\Omega}_1}} |/| {{{\Omega}_2}} | = 1$ and ${\Delta}{\tau _{{p} - {s}2}} \approx 0.82{\pi}$ along the line of the experiments, where two seeding pulses have almost equal intensity and $\Delta = 105\;{{\rm cm}^{- 1}}$ . Fitting parameter is $\frac{{| {{{\rho}_{{bc}}}} |}}{{{{\rho}_{{ee}}} - {{\rho}_{{cc}}}}} = - 1.1$ for reproducing the experimental results.
Fig. 5.
Fig. 5. Spectral intensity of emissions around 427.8 and 391.4 nm as a function of time delay ${\tau _{p {\text -} s1}}$ (or ${\tau _{p {\text -} s2}}$ ) between the pump and the 391.4 nm (or 427.8 nm) seeding pulse. (a)–(f) Delay ${\tau _{p {\text -} s2}}$ fixed at 130 fs: (a)–(c) delay ${\tau _{p {\text -} s1}}$ finely tuned around 300 fs and (d)–(f) delay ${\tau _{p {\text -} s1}}$ finely tuned around 480 fs. (g)–(i) Delay ${\tau _{p {\text -} s2}}$ between the pump and 427.8 nm pulses is finely tuned around 1000 fs when fixing delay ${\tau _{p {\text -} s1}}$ at 600 fs.
Fig. 6.
Fig. 6. Theoretical calculations for emitted signals around 427.8 and 391.4 nm, according to Eq. (3), as a function of time delay ${\tau _{p {\text -} s1}}$ between the pump and the 391.4 nm seeding pulse. Parameters are the same as in Fig. 5, i.e.,  ${\tau _{p {\text -} s2}} = 130\;{\rm fs}$ : (a)  ${\tau _{p {\text -} s1}}$ finely tuned around 300 fs and (b)  ${\tau _{p {\text -} s1}}$ finely tuned around 480 fs.

Equations (4)

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H M = H g ( p , q ) | g g | + H e ( p , q ) | e e | [ Ω ( q ) | e g | + Ω ( q ) | g e | ]
H ( p , q ) = p 2 2 + V g ( q ) ,
H e ( p , q ) = ω e g 0 + p 2 2 + V e ( q d ) ,
S 1 ( ω 1 , τ p s 1 ) = 2 | Ω 1 | 2 | Ω 2 | γ [ | Ω 2 | ( ρ e e ρ c c ) 2 | Ω 1 | | ρ b c | cos ( Δ τ p - s 1 ) × cos ( ω ~ 1 τ p s 1 ω 2 τ p s 2 + Δ τ p - s 2 + ϑ ) S 2 ( ω 2 , τ p s 1 ) = | Ω 1 | | Ω 2 | 2 γ [ | Ω 1 | ( ρ e e ρ c c ) cos ( Δ τ p - s 2 ) | Ω 2 | | ρ b c | cos ( ω ~ 1 τ p s 1 ω 2 τ p s 2 + ϑ ) ] ,

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