Theory of femtosecond strong field ion excitation and subsequent lasing in N$_2^+$

Delayed cavity-free forward lasing at the wavelengths of 391 and 428 nm was observed in recent experiments in air or pure nitrogen pumped with an intense femtosecond laser pulse at wavelength of 800~nm. The mechanism responsible for the lasing is highly controversial. In this article we explain the delayed emission by the presence of long-lived polarizations coupling simultaneously ground state X$^2\Sigma_g^+$ to states A$^2\Pi_u$ and B$^2\Sigma_u^+$ of singly ionized nitrogen molecules N$_2^+$. Ionization of neutral nitrogen molecules in a strong laser field and subsequent ion excitation are described by a system of Bloch equations providing a distribution of ions in the ground and excited states A and B at the end of the laser pulse. The delayed signal amplification at the B-X transition wavelength is described by a system of Maxwell-Bloch equations with polarization coupling maintained by a weak laser post-pulse. Two regimes of signal amplification are identified: a signal of a few ps duration at low gas pressures and a short (sub-picosecond) signal at high gas pressures. The theoretical model compares favorably with experimental results.


I. INTRODUCTION: LASING WITHOUT INVERSION
Several experiments on the interaction of a strong ultra-short laser pulse at 800 nm with molecular nitrogen [1][2][3][4][5][6][7] report on a robust cavity-free lasing in the forward direction at the wavelengths 391 or 428 nm, corresponding to transitions from the excited state B 2 Σ + u to the ground state X 2 Σ + g of N + 2 with vibrational level 0 or 1. It was observed that a femtosecond seed signal at 391 or 428 nm, injected a few ps after the pump pulse, is amplified by twothree orders of magnitude. Furthermore, the lasing emission is delayed from the seed pulse by a few ps. Several explanations for this lasing have been proposed so far. Fast population inversion between B and X due to depletion of X by the pump pulse inducing a transfer of population from X to the intermediate third level A 2 Π u has been discussed in Refs. [5,6].
However, these authors did not offer an explanation for the retarded emission, which is much longer that the pump pulse duration. Increase of the population at the upper level by multiple electron recollisions was suggested in [8]. However, it was shown later that this process itself cannot produce an optical gain needed for lasing [9].
It is known that lasing without population inversion is possible in a V-scheme, which involves a third level resonantly driven by the pump and coupled to the excited state by a quantum interference [10][11][12]. Such a V-scheme, studied in Refs. [13][14][15] and more recently in Ref. [16], can be applied to the interpretation of lasing of nitrogen molecular ions N + 2 driven by a ultrashort laser pulse with peak intensity in the range of a few 10 14 W/cm 2 , but one needs to explain how the third level is driven and what are the conditions for obtaining optical amplification. Recently we have developed a theoretical model that is capable of describing the delayed lasing [17]. The theoretical results are in agreement with experimental data about the temporal shape of the amplified signal and the gas pressure dependence of the gain. The present paper explains in more details this theory: The signal amplification in the B-X transition of nitrogen molecular ions is described by a two-step process: (i) the interaction of a short and intense pump laser pulse at the wavelength 800 nm with a nitrogen gas leads to partial ionization of the nitrogen molecules and partial excitation of the molecular ions to the upper states A and B; (ii) the coherent cross-coupling between excited ion states A and B opens the possibility of retarded signal amplification.
The following condition must be satisfied in order to obtain gain: population at level B must be larger than at level A but smaller than at level X; the coherent cross-coupling between A and B must be maintained by a weak 800 nm post-pulse of a few ps duration.
The paper is organized as follows. Section II addresses the problem of ionization of nitrogen molecules and excitation of the ions by the main laser pulse. While direct ionization into excited ionic states has a low probability, population transfer to these excited states can be quite efficient if the transition frequencies are of the same order of magnitude as the corresponding Rabi frequencies. The calculations show that in the range of intensities around 10 14 W/cm 2 population at level B is always lower than that at level X(0). Nevertheless, as we show in Sec. III, signal amplification after the end of the laser pulse is possible provided that the main laser pulse is followed by a weak coherent post-pulse of a few ps duration, which maintains the A-X polarization. The presence of a post-pulse is consistent with experimental observations [17]. The temporal evolution of populations in these three resonantly coupled levels and the evolution of electromagnetic fields is described by a system of Maxwell-Bloch equations enveloped over the transition frequencies. This long-lived mutual coherence makes the system unstable: it may generate an emission corresponding to the B-X transition or amplify a seed injected at the corresponding wavelength in the absence of population inversion between B and X. Depending on the post-pulse fluence and gas pressure, this amplification may proceed in two regimes: either parametric signal amplification at low pressures or soliton formation at high pressures. Section IV presents an analysis of numerical simulations with a particular emphasis on the dependence of the amplification process on gas pressure and laser post-pulse amplitude. Section V compares our theoretical results with experiments. Section VI presents our conclusions.

II. ION EXCITATION IN A STRONG LASER FIELD
Here we consider the interaction of the main laser pulse at 800 nm with a homogeneous nitrogen gas. The laser pulse intensity, on the order of 10 14 W/cm 2 , is sufficiently strong to create a plasma filament, and it is assumed that it is not appreciably modified by the gas ionization and ion excitation. Therefore, we consider the interaction of a given laser field with a single nitrogen molecule. The physical processes that we are interested in are: (i) ionization of the neutral nitrogen molecule from the neutral ground state to the ionic ground X and excited A and B states and (ii) subsequent transitions during the laser pulse between the ground and excited ionic states. Fig. 1 shows the V-scheme of energy levels of N + 2 containing the ground levels X 2 Σ + g (0,1), the excited level B 2 Σ + u (0) and a series of A 2 Π u (v) levels with different vibrational quantum numbers varying from v = 0 to 3. The resonant couplings considered in what follows Ionization of the nitrogen molecules is described by the PPT model [18,19] with the ionization probability w PPT (U, E)g(θ n ) depending on the ionization potential U , instantaneous laser electric field E(t), and angle θ n between its direction and molecule axis. The angular dependence is interpolated according to [20] by a function g(θ n ) = 0.45 + 0.95 cos 2 θ n + 1.17 cos 4 θ n , which decreases from the maximum value 2.57 for the parallel orientation, θ = 0 • , to 0.45 for the perpendicular orientation, θ = 90 • , with an average value of 1.
The ionization energy of level X(0) U X0 = 15.57 eV is comparable to that of level X(1), U X1 = 15.84 eV, so that both levels have to be considered. As the probability of tunnel ionization decreases exponentially with the transition energy, direct ionization into excited states A and B, separated by the energy gap of 1.5 and 3 eV, respectively, is small, but we account for it for completeness.
The dipolar moments µ ax 0.25 at.u. and µ bx 0.75 at.u. of A-X and B-X transitions [21,22] correspond to the coupling energy µE on the order of a few eV for a laser intensity of 10 14 W/cm 2 . This value is comparable to the energies of transitions. Therefore, several levels can be excited simultaneously irrespectively of the resonance conditions.
Non-resonant excitation of two-and three-level systems in a strong laser field was consid-ered in Refs. [5][6][7]. Here we extend this approach by considering excitation in three states: B(0), A(2) and A (3). Triplet X(0)-A(2)-B(0) with transition energies 1.58 and 3.20 eV, corresponds to the lasing signal observed at 391 nm, another triplet X(1)-A(3)-B(0) with transition energies of 1.55 and 2.93 eV, lying nearby, corresponds to lasing at 428 nm. The scheme of level couplings is shown in Fig. 1.
The temporal evolution of this five-level system in a given laser electric field is described  Equations (A1) -(A11) were solved numerically for a given laser pulse maximum intensity I las and pulse duration t las . The ion state probabilities were evaluated at the end of the laser pulse, t = t las , and averaged over the angle θ n between the laser field and molecular axis by taking into account the angular dependence of the ionization probability g(θ n ) and the polarization couplings. We assume an isotropic distribution of molecules in the gas before the laser pulse arrival.   This general trend presented in Fig. 3 is rather similar for the three pulse duration chosen in panels a, b, and c. Population at level X(0) is always the largest, while an inversion between states occurs at a laser intensity 3 × 10 14 W/cm 2 . The inversion between levels B and A(2) occurs at a laser intensity less than 2×10 14 W/cm 2 , and the inversion point slightly varies with the pulse duration. By increasing the laser intensity beyond 4 × 10 14 W/cm 2 one may also achieve a population inversion between B and X(0) levels, but it is observed only with the shortest pulse duration and it corresponds to an ionization level of more than 40%.
The ionization-excitation model of nitrogen molecules with a short and intense laser pulse presented in this section is quite robust. The calculated populations depend rather weakly on exact values of the frequencies of B-X and A-X transitions and their detuning with respect to the laser frequency. Coupling is strong and it is controlled essentially by the large values of the Rabi frequencies µ a,bx E las / , comparable to the transition frequencies.
Direct ionization to the excited states A(2) and A(3) accounted for in Eqs. (A3) and (A4) makes a relatively small contribution of about 10% to the population in the excited states.
Contribution of the direct ionization to the level B(0) is even smaller, it is less than 1% in the considered cases.
In the next section we investigate the evolution of the excited molecules in a gas after the end of the main laser pulse, assuming that the polarization corresponding to the A-X transition is maintained by a weak laser post-pulse. It is supposed that such a post-pulse cannot affect the population distribution between the levels but it is sufficiently strong and resonant for maintaining one of polarizations d a2x0 or d a3x1 for a time of a few ps, much longer than the main laser pulse duration. Due to the polarization coupling, the laser post-pulse may induces a delayed emission from the level B(0).

A. Maxwell-Bloch equations
Here, we consider the temporal and spatial evolution of the triplet X(0)-A(2)-B(0). One has to account for the possible evolution of the post-pulse in space and in time while propagating through the plasma filament. This implies the use of Maxwell-Bloch equations that account for the evolution of both ion populations and electromagnetic fields in the plasma.
The post-pulse intensity is about four orders of magnitude smaller than the main pulse. It cannot ionize the gas and the coupling to molecular ionic transitions is weak. This allows us to treat each V-triplet independently, to use an envelope approximation for the electric fields and polarization fields and to select only the resonantly coupled levels. Consequently, the electromagnetic field is presented here as a sum of two components operating at the frequencies close to the transitions A(2)-X(0) and B(0)-X(0): Then the Maxwell-Bloch equations for the three level system system read [15]: where θ i is the angle of the ion molecule orientation with respect to the laser polarization, where r = ω bx µ 2 bx /ω ax µ 2 ax is the ratio of characteristic times of evolution of the B-X and A-X transitions andγ ij = γ ij t N are the dimensionless damping rates. For the nitrogen ion this ratio r 18 is quite large because of a large ratio of dipole moments. This implies a much faster temporal evolution of the B-X transition compared to A-X.
It is important to mention that in the absence of damping,γ ij = 0, this system has the propriety of conserving locally in space the following combination of populations: where C is a positive constant which is equal to 1/3 for fully decorrelated and equally partitioned populations. The case of fully correlated system C = 1 is considered in Sec. III B 2.
Another important property is the energy conservation in the system. By time integrating the equations for |e a,b | 2 , one obtains the energy conservation laws for transitions A-X and B-X: Here F a and F b are the energy fluxes of the pump and seed pulses at the end of simulation τ = τ max at the entrance, z = 0, and the exit, z = L, of the system These relations (11) and (12) confirm that the number of emitted photons in A-X and B-X transition is conserved separately and it is equal to the number of ions transferred from the correspondent excited state to the ground state. By taking the difference between the equations for the field intensities one can obtain also the following relation which relates the number of ions transferred from state B to state A through the ground state X to the number of emitted photons in the B-X and A-X transitions. This corresponds to a lasing process without population inversion but has also been viewed as a two-photon stimulated Raman scattering [23]. However, in difference from the conventional, single photon Raman scattering, here the scattered wave is up-shifted in frequency and the energy is provided from the medium. Therefore, in our view, it is better described as a lasing process without population inversion.
Fulfillment of the conservation laws (10), (11) and (12) Before discussing numerical solutions we first present several analytic results that are useful for their interpretation.
B. Analytical solutions to the three level system The system of equations describing the V-scheme has several analytical solutions that are presented in Appendix B for reference. Here, we present two particular solutions that will help us interpret the numerical simulations presented in the next section: the lasing without inversion and the solitons. In both cases we neglect the spontaneous damping for sake of simplicity.

Lasing without population inversion
The system of equations (6) -(9) has a particular solution, discussed by Svidzinsky et al. [16]. It corresponds to an amplification of a signal at a higher frequency ω bx due to a parametric coupling to the pump wave of lower frequency ω ax through the correlated polarization between levels A and B. Here, we assume that there are populations at both excited states, n a0 and n b0 , and that the A-X transition is driven by a pump field e a (τ, z) = e a0 exp(−i∆ω ax τ + iqz) coupled to the polarization p ax (τ, z) = p ax0 exp(−i∆ω ax τ + iqz).
Relations between the amplitudes, p ax0 = 2iqe a0 / sin θ i , the space dephasing parameter q = (n a0 − n x0 ) sin 2 θ i /2∆ω ax and the frequency detuning ∆ω ax follow from the first two equations of the system.
Let us consider seed wave of a small amplitude e b0 at a frequency ω bx corresponding to the B-X transition and investigate the linear response of the system (6) - (9) to this initial perturbation. The perturbed system contains three equations for the field e b and polarizations p bx and p ba : Solving this equation, the signal amplitude can be expressed as an inverse Fourier transform: The integral here is performed in the complex plane ω along the contour going above the singular points, according to the principle of causality. The singular points are solutions of the dispersion equation: which has two solutions ω 1,2 = 1 2 ∆ω ax ∓ 1 2 ∆ω 2 ax + e 2 a0 sin 2 θ i . The first one corresponds to a spontaneous amplification at the B-X transition modified by the presence of the A level. It requires the population inversion n b0 > n x0 as it is shown in Appendix Bb. The second one corresponds to a parametric coupling between A and B levels and requires a softer condition n b0 > n a0 . A convenient way to compute the inverse Fourier transform (14) asymptotically in the limit rzτ cos 2 θ i 1 is to close the integration contour in the lower half plane and to transform it into two circles with the centers in the singular points ω 1,2 .
The radius of each circle is found by equating the amplitudes of the two terms in the exponential. In particular, assuming a sufficiently large detuning |∆ω ax | e a0 | sin θ i |, for the singular point ω 1 ≈ −e 2 a0 sin 2 θ i /4∆ω ax , the circle radius Γ 1 is defined by equation: 2Γ 2 1 τ ≈ rz(n b0 − n x0 ) cos 2 θ i . This singular point corresponds to oscillations decaying in time with the characteristic frequency Γ 1 , if n b0 < n x0 , or to an exponential growth in time, if n b0 > n x0 .
The radius of the contour around the second singular point ω 2 ≈ ∆ω ax +e 2 a0 sin 2 θ i /4∆ω ax is defined by equation: 8Γ 2 2 τ ≈ rz(n b0 − n a0 )e 2 a0 cos 2 θ i sin 2 θ i /∆ω 2 ax . The integral around this singular point corresponds to the modified Bessel function I 1 in the case n b0 > n a0 : In the limit 2Γ 2 τ 1 this expression corresponds to an exponentially growing solution. In physical units expression for the gain factor reads If spontaneous damping is included, it imposes a threshold value of the pump field amplitude for excitation of this instability.

Solitary excitation
The soliton solution is a dynamic structure that propagates along the plasma with a constant velocity. It was described in a two-level system by McCall and Hahn [24] and in more details in Ref. [25]. Considering for example A-X transition alone and neglecting the polarization damping term, system (6) and (8) reduces to the following three equations: Assuming e a to be real and depending on the coordinate and time as ξ = (τ − z/u) sin θ i , this system has two integral relations, |p ax | 2 = 2n a (1 − n a ) and e 2 a = 2n a u, which are the particular case of the more general expressions (10) and (11). The constants in these two relations are chosen assuming that there is no population at the level A before the laser pulse arrival. Then the remaining equation for n a has a soliton solution: , e a = e 0 cosh(wξ) , amplitude. Soliton solutions for a three level system has been constructed in Refs. [26,27] for a particular case of r = 1. However, these are not relevant to our conditions of the large value of r ≈ 18 that makes correlation between B-X and A-X solitons more complicated.
Particular solutions of the V-system described above are found in the numerical analysis presented in the next section.

A. Initial conditions
Before discussing the numerical solutions for the set of equations (6) -(9), we need to consider the role of molecular rotations. The strong laser electric field induces a dipolar moment in a neutral nitrogen molecule and exerts a torque. This leads, after an inertial delay of ∼ 100 fs, to the formation of a coherent rotational wavepacket with a partial alignment of the neutral molecules along the laser field axis. As molecules have a broad discrete distribution in the rotational moments, a coherent rotational wave packet quickly dephases but then experiences spontaneous revivals every half rotation period T rot = 1/(2Bc) [28,29].
(Here, B is the rotation constant equal to 2.0 cm −1 for the neutral molecule, 2.07 cm −1 for level B and 1.93 cm −1 for level X.) The duration of revivals is rather short, it is J 0 times shorter than the revival period, where J 0 µE las t las is the characteristic rotation momentum.
For the parameters of interest in our study J 0 ∼ 10 − 20 and the corresponding revival duration is less than 1 ps. Thus, revivals should not affect significantly the amplification process that proceeds on a longer time scale. Therefore, in our model we assume that the probability of angular distribution of ions, P(θ i ), does not depend on time, and we consider a quasi-classical angular distribution of ions with the average value cos 2 θ i 0.33 corresponding to a non-adiabatic strong short pulse excitation [29,30]. Equations for angle averaged populations, n a,b,x , and corresponding polarizations are given in Appendix C.
In the numerical analysis of our system we consider the B(0)-A(2)-X(0) transition at wavelengths 391.4 and 787.5 nm [17]. The fractions of excited ions and initial values of polarizations are calculated from the system of Bloch equations discussed in Sec. II. As an example, we consider the main laser pulse intensity 2.6 × 10 14 W/cm 2 and duration of 20 fs.
According to Fig. 3, that choice of parameters corresponds to an ionization of 20% and to a situation without population inversion with respect to the ground level X: n a0 = 0.12, n b0 = 0.20 and n a0 = 0.30. However, it satisfies the necessary amplification condition n a0 < n b0 .
We first consider solutions of system (6) -(9) without external fields, e ax = e bx = 0, and with maximum initial values of polarizations: p ij ∼ 1. (The choice of phases has no importance.) The system with these initial conditions is quickly discharged by spontaneously amplifying photons at both transitions. The characteristic de-excitation times of the system, ∆t a ∼ ct 2 N /(2L|n a − n x |) and ∆t b ∼ ct 2 N /(2rL|n b − n x |) are very short, less than 0.1 t N , especially for the B-X transition. This is consistent with the conservation equation (10): any state will terminate with zero polarizations at excited levels and with n x = 1. This situation, however, is not consistent with the delayed emission observed in the experiments. By reducing polarization amplitudes by 100 times or more, one can slow down the spontaneous emission and maintain a large fraction of ions in the excited states. However, we verified that without feeding the polarization p ax with an external pump it is not possible to obtain an efficient coupling between levels A and B and amplification. Therefore, all simulations presented below were conducted with a post-pulse pump applied at t = 0 and decaying exponentially with time.

B. Reference case
As a reference point we take a nitrogen gas pressure p N = 30 mbar, which corresponds to a total ion density n i = 1.2 × 10 17 cm −3 for a 20% ionization for the three considered other seed amplitudes or without seed provided there is small initial polarization p bx0 in the filament. Such an initial polarization p bx0 by field induced recollisions has been attributed as the source for the amplification in the absence of a seed pulse [8,31]. Example of the seed amplification in these conditions is presented in Fig. 4. It shows the intensities of the pump and seed pulses, I a /I N = 1 2 |e a | 2 and at the entrance of the filament, z = 0, and at the exit, z = L, in function of co-propagation time τ . Here I N = 0 cE 2 N = 0.91 GW/cm 2 is the normalization intensity. The pump wave is modulated at the exit with a period of (2 − 3) t N due to the frequency detuning from the A-X transition and partial absorption. The seed pulse of duration 0.2 t N is injected at time 0.5 t N with a duration 0.25 t N . It is amplified more than 25 times in energy and it extracts about 0.8% of the energy initially stored in level B. Parameters are the same as in Fig. 4.
The amplified signal is extended by more than 10 ps and produces at the exit a sequence of pulses of duration of about 2 ps. The amplification is explained by the parametric coupling of levels A and B by the polarization p ba as it is described in Sec. III B 1. By suppressing the corresponding cross-polarization terms in Eqs. (C5) -(C8) one may eliminate completely the amplification. It is also verified that the gain is proportional to the difference of populations between levels B and A, n b0 − n a0 , the filament length, L, and the pump amplitude, e a0 , according to Eq. (16). In particular, no gain is found for equal populations, or when n a0 ≥ n b0 .
The cross-coupling between levels A and B manifests itself also in the spatial and temporal   The increase of signal gain is correlated with an increase of the energy extracted from level B, η b , in the pressure range corresponding to the parametric amplification regime. This is shown in Fig. 6c in per cent. However, the extracted energy is saturated in the pressure range of (3 − 10) p N , and in the soliton regime the extracted energy decreases.
Analysis of the spatio-temporal evolution of the post-pump and seed pulses shown in  in Ref. [17]. Post-pulse duration has to be sufficiently long for enabling seed amplification.
No amplification was observed numerically for pulse duration shorter than 2 − 3 ps. Longer post-pulses favor stronger amplification and at longer seed delays. Seed amplification can be only observed at frequency detuning smaller than the corresponding pump Rabi frequency.

V. COMPARISON WITH EXPERIMENTAL RESULTS
The features of the V-scheme amplification demonstrated in Sec. IV are in agreement with several experimental observations [1-3, 8, 17, 32]. An amplified signal at 391 nm that is delayed from the pump pulse by several ps has been observed in Ref. [3,8]. It displays an increase that is first growing approximately quadratic at low pressures until it reaches a maximum around 30 − 50 mbar, and decreases at higher pressures [8,17]. Similar behavior can be seen in Fig. 6b, curve 2. The measured in these papers gain is on the order of 100, which is also in agreement with Fig. 6b.
Simulations show that the temporal shape of the amplified signal depends sensitively on several parameters such a gas pressure, the length of the gain medium, amplitude, duration of the post pulse and the detuning of its frequency with respect to resonances. Because of a lack of knowledge on these parameters it is difficult make a quantitative comparison Also, it is shown in Ref. [17] that there is good agreement between calculated and measured temporal shapes at different pressures.
An amplification was also observed experimentally without injection of a seed pulse at 391 nm [2,31]. This is consistent with the V-scheme, which predicts amplification even if the initial coherent polarization B-X is on the order of 10 −3 . In this case, the B-X polarization is attributed to electron recollisions during the pump laser pulse duration [31].
Signal amplification has been recently reported in air at normal pressure [32]. The pump pulse wavelength was at 950 nm and the lasing signal occurred at 428 nm. An amplified signal was reported that was delayed by 5.6 ps. According to our model, these conditions correspond to a soliton regime in a B(0)-X(1)-A(2) V-scheme arrangement. The corresponding pulse shapes are shown in Fig. 7f.
Finally, we note that it is also possible to obtain amplification with population inversion between the B and ground X(0) and X(1) states. This, however, requires higher intensities, above ∼ 4 × 10 14 W/cm 2 [33,34]. These conditions can be obtained under tight focusing in thin gas jets [35].

VI. CONCLUSIONS
Our theoretical analysis shows that ionization of nitrogen gas by an intense femtosecond laser pulse at 800 nm is accompanied with a transfer of ionized molecules to higher excited levels due to resonance polarization coupling. This process depends on both the laser intensity and pulse duration. At moderate laser intensities (in the range of 1 − 3 × 10 14 W/cm 2 ), an ionization level of a few percents is reached, and one obtains an inversion between electronic levels B and A but no population inversion between B and X. In this case, a seed amplification can occur due to the coupling between A and B levels in a V-scheme. Two additional conditions must be fulfilled in order for the gain to take place (i) the main laser pulse has to be followed by a post-pulse of a few ps duration and an intensity 4 -5 orders of magnitude smaller than the main pulse; and (ii) the spectrum of the post-pulse has to contain a component sufficiently close to one of the A-X transitions. The large number of rotational and vibrational levels in the excited ion facilitates this resonance condition. A three level V-scheme is sufficient for a description of this process as only one A-X transition closest to the spectral component of the laser post-pulse effectively participates in the coupling.
The seed amplification can be realized in two qualitatively different regimes: three-level parametric coupling or joint soliton propagation. In the former regime that occurs at pressures of less than 100 mbar, the post-pulse needs to be present during the whole process.
This regime is experimentally observed in good agreement with our theory. The soliton regime is realized at higher pressures, where the amplified seed comes out synchronously with the post-pump in the form of a narrow pulse with a delay increasing with pressure.
The soliton regime might have been observed in air at normal pressure.
Finally, at ionization level in excess of 40% our theory shows that it is possible to obtain population inversion between the B and ground X state and to achieve direct amplification of a seed pulse at the B-X transition.
These relations imply that n b − n x n a − n x = |e a | 2 sin 2 θ i |e b | 2 cos 2 θ i = tan 2 ψ.
Taking into account that n a +n b +n x = 1, we find that such a solution may exist for arbitrary field amplitudes: n a = (1 − 2n x ) cos 2 ψ + n x sin 2 ψ, and n b = (1 − 2n x ) sin 2 ψ + n x cos 2 ψ.
This solution implies that the population at the level X is sufficiently low, n x < 0.5. It corresponds to a "dark state" of a three level system, which allows propagation of both electromagnetic waves without absorption. That is, by injecting simultaneously the fields e a and e b one may maintain the population inversion in the states A and B for a long time, assuming that all ions are aligned at the same angle θ i with respect to the electric field.

b. Amplified spontaneous emission
Another known solution corresponds to the exponential amplification of a weak signal in a two level system in the conditions where there exists either a population inversion or a strong polarization. For example, Eqs. (7) and (8) for an isolated B-X transition read: ∂ z e b = r 2 p bx cos θ i , ∂ τ p bx = (n b − n x ) e b cos θ i , ∂ τ n b = − 1 2 Re(p * bx e b ) cos θ i .
Neglecting population variation, n b = n b0 ≈ const, a pair of equations for the electric field and polarization admits either an exponentially growing solution e b ∝ p bx0 exp( 2r(n b0 − n x0 )zτ | cos θ i |), if n b0 > n x0 , or an oscillating solution if n b0 < n x0 . This growing solution may be realized at high pump intensities where a population inversion B-X is created by the pump pulse.
Appendix C: Angle-averaged equations for the V-scheme Equations (6) -(9) are averaged over the ion orientation angle θ i assuming that the probability distribution P(θ i ) is a time independent function with average value cos 2 θ i = 1 2 π 0 cos 2 θ i P(θ i ) sin θ i dθ i .