Rotational Quantum Beat Lasing Without Inversion

In standard lasers, light amplification requires population inversion between an upper and a lower state to break the reciprocity between absorption and stimulated emission. However, in a medium prepared in a specific superposition state, quantum interference may fully suppress absorption while leaving stimulated emission intact, opening the possibility of lasing without inversion. Here we show that lasing without inversion arises naturally during propagation of intense femtosecond laser pulses in air. It is triggered by the combination of molecular ionization and molecular alignment, both unavoidable in intense light fields. The effect could enable inversionless amplification of broadband radiation in many molecular gases, opening unusual opportunities for remote sensing.

A number of different schemes for lasing without inversion have been developed 9, 10 . Taking advantage of destructive interference of different pathways in absorption, they generally strive to maintain a specific phase relationship between the lower-lying states, which carry most of the population. Here we present a scheme that does not follow this tradition. It uses only the natural dynamics of a multi-level quantum system and requires no coherence between the excited and the lower electronic states; effectively, lasing without inversion comes 'for free'. We also show that this mechanism is active in the highly efficient generation of 391 nm radiation during propagation of intense femtosecond laser pulses in air [12][13][14][15][16] , under standard conditions where the process known as "laser filamentation" 17-19 leads to self-guiding of light.
Identifying the mechanism responsible for this effect, commonly referred to as 'air lasing', has been a long-standing puzzle  . The main difficulty in resolving this puzzle stems from the apparent lack of a general physical mechanism capable of generating population inversion for the dominant observed amplification line around 391 nm, for standard filamentation conditions including the clamping of the laser intensity around I ∼ 10 14 W/cm 2 . The 391 nm line corresponds to the transition between the two ground vibrational levels ν , ν = 0 of the electronic states X 2 Σ + g (denoted as X) and B 2 Σ + u (denoted as B) in N + 2 . We believe that our inversionless mechanism provides the key missing component of this puzzle. Compared to the recent proposal in Ref. 35 , we fully account for the vibrational and rotational dynamics of the molecule; our lasing mechanism remains operative even in the absence of initial coherence between the ionic states.

Results and discussion
In its simplest form, our amplification mechanism can be referred to as rotational quantum beat lasing. It is illustrated in Fig.1(b)-(d), and can be easily understood in the time domain. When an intense femtosecond laser pulse interacts with a diatomic molecule, or its ion, it induces molecular alignment 46,47 . This alignment revives periodically after the pulse is gone, Fig.1(b). The revival dynamics demonstrates a sequence of alignment and anti-alignment with the revival period controlled by the rotational constant B 0 . Fig.1(b) shows the evolution of the characteristic alignment measure, cos 2 θ (t), for a nitrogen molecule at room temperature (298 K), after interacting with a 23 fs FWHM, 800 nm pump pulse with a peak intensity of 10 14 W/cm 2 (see Methods). Its minima correspond to anti-alignment, its maxima to maximum alignment with the direction of the linearly polarized, aligning laser field.
Suppose now that similar rotational dynamics is induced in two electronic (vibronic) states, |1 and |2 , with a dipole-allowed electronic transition between them. In diatomics, electronic transitions can be either parallel or perpendicular to the molecular axis; for definiteness, let this transition be parallel. Its probability is then proportional to cos 2 θ, where θ is the angle between the molecular axis and the laser electric field inducing the transition. For a molecule rotating in the lower vibronic state, the probability of parallel absorption |1 → |2 thus depends on cos 2 θ 1 (t) averaged over the rotational dynamics in state |1 , which is controlled by its rotational constant B 1 0 . Conversely, for a molecule rotating in the upper vibronic state |2 , the emission probability |2 → |1 depends on cos 2 θ 2 (t), with its time-dependence controlled by Opportunities for inversionless amplification of a short probe pulse, polarized parallel to the pump and delayed by τ , arise when the |1 state molecules are anti-aligned, so that cos 2 θ 1 (τ ) is minimum and the parallel |1 → |2 absorption is suppressed. This opportunity is further enhanced if the |2 state molecules are aligned at this time, so that cos 2 θ 2 (τ ) is maximum and the parallel |2 → |1 emission is enhanced. This possibility to use molecular alignment for inversionless amplification was previously pointed out by A. K. Popov and V. V. Slabko 48 . The converse is true if the |1 state molecules are aligned when the probe pulse arrives, while the |2 state molecules are anti-aligned, enhancing absorption. As long as B 1 0 and B 2 0 are different, the two rotations will go out of sync, arriving at the point where the lower state is anti-aligned and the upper is still aligned and vice versa (Fig 1c). Overall, temporal windows of gain will be followed by windows of loss, leading to the rotational quantum beats in the time-resolved gain-loss of the short probe pulse.
Figs.1(c,d) give the specific example. Fig.1(c) shows the rotational dynamics in the two ground vibrational levels ν , ν = 0 of the electronic states X and B in N + 2 , induced by the same pump pulse interacting with N 2 , Fig.1(b). The short pump pulse impulsively aligns the neutral N 2 molecules, generating the rotational dynamics in Fig.1(b). It also ionizes some of the N 2 molecules, producing molecular ions predominantly in the ground X, but also in the excited A (A 2 Π u ) and B states, and continues to align them. After the end of the pulse, the ions continue to rotate, reaching maximum alignment at t 90 fs for the X and B states, see Fig.1(c), followed by periodic revivals of alignment and anti-alignment. They correspond to maxima and minima in the rotational ensemble-averaged cos 2 θ X,B (t), where subscripts denote the ionic states (see Methods). These revivals are different due to slight differences in the rotational constants B X,B 0 .
Frequency-integrated absorption at the parallel |X, ν = 0 → |B, ν = 0 transition be- Our theoretical description accounts for laser-induced alignment of the neutral N 2 molecule, its alignment-dependent strong-field ionization into the laser-dressed states of the molecular ion, and full laser-induced electronic, vibrational and rotational dynamics in the ion involving the X, A, B states. Our ab-initio simulations of strong-field ionization use the method of Ref. 49 , and allow us to evaluate the excitation of the different ionic states induced by the recollision of the photoelectron with the parent ion in the X state 23, 24, 33-35, 40 . The role of recollision is gauged by absorbing the electron wavepacket before it is turned around towards the parent ion. Eliminating the recollision reduces the population of the B state by about 1%, compared to the case when the recollison is included. Thus, the recollision can be neglected and the photo-electron can be integrated out. Here, one must account for the electron-ion entanglement. Since the X and B states have opposite parity, the photo-electron wavepackets correlated to them will also carry opposite parity, remaining orthogonal to each other. This eliminates the coherence between the X and B states that could have been produced during strong-field ionization.
Regarding optical X → B excitation by an 800 nm pump, the |X, ν = 0 → |B, ν = 0 transition corresponds to absorption of two photons, which is parity forbidden; noticeable excitation is only generated at very high intensities, I 4 × 10 14 W/cm 2 (the domain of several recent pump-probe experiments but not relevant for standard laser filamentation conditions), when the system is strongly distorted and higher-order multiphoton transitions can occur. In this context, note that strong-field ionization populates not field-free 25, 41, 42, 45, 50 but already polarized (dressed by the field) ionic X, A, and B states. Indeed optical tunneling results from polarization of the many-body wave function of the neutral as one of the polarized electrons leaks through the potential barrier, leaving other electrons polarized. Comparison to ab-initio simulations 49 show that initializing population in the field-free ionic states at the peaks of the instantaneous electric field, where strong-field ionization takes place, generates spurious excitations of the A and B states due to effectively abrupt turn-on of the laser-ion interaction.
Moreover, the initial thermal rotational distribution must be included. Each rotational state in the initial thermal distribution gives rise to a rotational wavepacket induced by the pump. Each wavepacket leads to coherent effects in emission and absorption; these contributions are added incoherently with the corresponding weights.
With these aspects accounted for, our calculations reveal a robust physical mechanism leading to gain at the 391 nm line. In contrast to Ref. 35 , it does not require the coherence between the X and B states, and is based on the mechanism described in Fig.1.
In the frequency domain, the rotational quantum beat mechanism can be understood as follows. The short, intense pump pulse aligning and ionizing the N 2 molecules, generates broad rotational distributions in the X and B states of N + 2 , with coherent population of many adjacent rotational states with similar amplitudes in each electronic state. Fig.2(a) shows the rotational distributions in |X, ν = 0 and |B, ν = 0 for the case of the 800 nm, 23 fs FWHM pump pulse with an intensity of I = 10 14 W/cm 2 and the N 2 molecules initially at room temperature (298 K).
Consider now frequency-resolved transient absorption of a short probe pulse at the dipoleallowed transition |X, J → |B, J + 1 (with the vibrational quantum number equal to zero in both cases). For each initial rotational state of the neutral, rotational coherence generated in the X state of the ion means that the absorption |X, J → |B, J + 1 , stimulated by a broadband probe, is inevitably accompanied by the absorption |X, J + 2 → |B, J + 1 stimulated by the same probe, Fig.2(b). Their interference is governed by the relative phase between the two lower states, φ X (J + 2, J)(t) = ∆E X (J + 2, J)t + φ  Fig.1(c,d). The total gain-loss balance integrated over all frequencies (green) follows the pattern of P X cos 2 θ X − P B cos 2 θ B in Fig.1(d), showing the same gain windows for the inversionless medium (P X > P B ).
lative phase at the start of the field-free evolution when the pump is over. Destructive interference between the |X, J → |B, J + 1 and |X, J + 2 → |B, J + 1 transitions occurs when φ X (J + 2, J)(t) = (2N ± 1)π (N is integer), resulting in suppression of absorption of a short probe pulse arriving at this moment. Conversely, constructive interference enhances absorption when φ X (J + 2, J)(t) = 2N π. This leads to the quantum beat in the time-dependent absorption rate of the short probe pulse with frequency ∆E X (J + 2, J).
Emission at the same frequency, |B, J + 1 → |X, J , is inevitably accompanied by the transition |B, J − 1 → |X, J , Fig.2(b). Their interference is governed by the relative phase As many four-level systems such as the one shown in Fig.2 Fig.1(c). This is not a coincidence. The same phase differences that control the interferences in transient absorption by each (|X, J , |X, J + 2 ) pair, control their contribution to the ensembleaveraged cos 2 θ X measure of the field-free rotations in the X state. The same applies to emission by each (|B, J − 1 , |B, J + 1 ) pair and the ensemble-averaged cos 2 θ B measure. As shown in Methods, integrating the frequency-resolved absorption over all frequencies yields the total transient absorption probability W abs (τ ) ∝ P X cos 2 (θ) X (τ ). Analogous results hold for transient emission.
The green line in Fig.2(d) shows the overall gain-loss balance, integrated over all energies.
As expected, the strongest gain windows coincide with anti-alignment of the X state, cf. Fig.1(c,d), when the gain-loss pattern in Fig.2(c) shows gain across the whole spectrum. If the X and B state molecules would anti-align simultaneously during their field-free evolution, the net gain would vanish.
Results presented in Fig.2 are robust with respect to the pump parameters, and the lasing mechanism described above is completely general. The only necessary ingredients are nonnegligible population in the B state and molecular rotations, both unavoidable during the interaction with an intense laser pulse. For the pump pulse parameters and temperature used for Figs.1,2, our simulations show that net gain sets in already at the ratio P B /P X 10% upon ionization. Fig.3 shows results for two other pump intensities realistic in laser filamentation. The expected "clamping" intensity is I 1 × 10 14 W/cm 2 , but pulse compression during filamentation suggests that short intensity spikes can reach 51, 52 I ∼ 2 × 10 14 W/cm 2 . For 800 nm driver fields with intensities I 2 × 10 14 W/cm 2 , the B state population is not strongly affected by laser-driven excitations from X. However, the higher the intensity, the stronger the depletion of the X state into emerge without population inversion of the medium (P X > P B ) (Gain windows also emerge for the total gain-loss balance integrated over all frequencies, analog to Fig.2d).
In the present mechanism, the gain window travels with the group velocity of the pump pulse, so that amplification occurs only in the forward direction. To achieve lasing in the backward direction, highly desirable for remote sensing, one would greatly benefit from creating population inversion, at least between the rotational states. To this end, a sequence of two well-timed pump pulses can be used to control rotations of X and B states of the ion. Their different rotational periods mean that the second pulse can be timed to simultaneously accelerate rotations of the B-