Coherent control of superradiance from nitrogen ions pumped with femtosecond pulses

Singly ionized nitrogen molecules in ambient air pumped by near-infrared femtosecond laser give rise to superradiant emission. Here we demonstrate coherent control of this superradiance by injecting a pair of resonant seeding pulses inside the nitrogen gas plasma. Strong modulation of the 391.4 nm superradiance with a period of 1.3 fs is observed when the delay between the two seeding pulses are finely tuned, pinpointing the essential role of macroscopic coherence in this lasing process. Based on this time-resolved method, the complex temporal evolution of the macroscopic coherence between two involved energy levels has been experimentally revealed, which is found to last for around 10 picoseconds in the low gas pressure range. These observations provide a new level of control on the"air lasing"based on nitrogen ions, which can find potential applications in optical remote sensing.

Cavity-free lasing of ambient air (nitrogen or oxygen) pumped by ultrafast laser pulses has opened the perspective to perform optical remote sensing with a directional coherent laser beam emitted from the sky to the ground observer [1][2][3][4][5][6][7][8][9][10][11][12]. This is in strong contrast with the traditional Light Detection and Ranging (LIDAR), where the incoherent scattered photons or the fluorescence of the target molecules in the atmosphere excited by a forward-propagating laser was collected by a telescope on the ground. The directionality and coherent nature of the backward propagating air laser beam promises significant improvement of the sensitivity of remote sensing [1][2][3][4]13]. Up to now, both oxygen and nitrogen have been shown to be able to emit lasing radiation under proper pump conditions. The bidirectional near-infrared emission at 845 nm of oxygen atoms has been identified due to population inversion promoted by twophoton dissociation followed by a resonant two photon absorption of the oxygen molecules [1,2,6,14]. In the meantime, the lasing emission of singly ionized nitrogen molecules has been intensely debated concerning its nature and mechanism [3,[7][8][9][15][16][17][18][19][20]. This is partially due to the complexity of this emission process, where the electronic, vibrational, and rotational freedom of the nitrogen molecules are all coupled by the strong laser field [3,15,16,19].
Moreover, it is found that the pump laser wavelength also plays a critical role and different mechanisms can be involved for 800 nm or mid-infrared pump pulses [16,18].
A remarkable observation of the nitrogen ion lasing is that a delayed seeding pulse around 391.4 nm can be significantly amplified, up to a few hundred times, as to its energy [5,[15][16][17] [21,22], inversion of partial rotational quantum levels without entire inversion of the   u B 2 and   g X 2 electronic levels [19]. Recently, it was revealed with time-resolved technique that the amplified 391.4 nm radiation is actually largely lags behind the injected seeding pulse [5,17,23]. More specially, the amplified pulse shows a pressure dependent temporal profile. With increasing nitrogen gas pressure p, both the built-up time d and the pulse width w decrease as p -1 , while the emission peak intensity scales up as p 2 , which are characteristic for superradiance [17,[23][24][25]. Superradiance refers to a cooperative emission process of an ensemble of emitters where a large macroscopic coherence built up, which is normally initialized by spontaneous photons or externally seeding pulse [24,25]. Macroscopic coherence between the two involved energy levels is characteristic for the superradiant radiation process. However, it has not yet been observed directly in the nitrogen-ions lasing experiments.
In this paper, we applied coherent control scheme to the nitrogen ions lasing system by injecting a sequence of seeding pulses at the resonance wavelength (391.4 nm) after the 800 nm pump pulse. In the experiment, it was found that the amplified 391.4nm emission experiences strong intensity modulation with a period of 1.3 fs as the delay between the two seeding pulses were finely tuned. This period corresponds to the transition frequency between the   u B 2 and   g X 2 electronic levels. A long-lived coherence between the two electronic states was revealed since the intensity modulation was observed up to a delay of ~ 10 ps in low gas pressure. We theoretically considered the nitrogen ions as a two-level system, interacting resonantly with the two seeding pulses. It is shown that the relative phase (delay) of the two seeding pulse causes a dynamic modulation on the coherence, which leads to the intensity variation during the superradiant emission process. Based on the modulation contrast, we measured the built-up and decay of the coherence between the  Fig. 2(a)). Time-resolved measurement of this amplified 391.4 nm was performed with a crossed correlation method similar to ref [5,17], where the 391.4 nm pulse was frequency mixed with another weak 800 nm probe pulse in a sum-frequency BBO crystal and the sumfrequency generation signal at 263 nm was measured for varying delay between the 391.4 nm radiation and the weak 800 nm probe pulse. We presented the results in Fig. 2 intensities of the seeding pulse. We note that the amplified 391.4 nm emission lags largely behind the femtosecond seeding pulse, in agreement with the previous reports [17,23]. With increased intensity of the seeding pulse, it was observed that the emission built up more rapidly.
This is expected since the seeding pulse serves as the initial trigger for the formation of the macroscopic coherence during the superradiance process [25,26]. The sharp emission peaks around 8.4, 12.6, and 16.8ps were revealed in our measurement, which corresponds to the quantum alignment of the nitrogen molecular ions [21,27].
Then we injected a pair of seeding pulses inside the nitrogen gas plasma. The spectrum of the emission was recorded as a function of the time delay ss between the two seeding pulses. In this experiment, the delay of the first seeding with respect to the 800 nm pump was fixed to be around 0.3 ps. We present in Fig. 3 the experimental results. In Fig. 3 For the purpose of understanding the fundamental mechanism of this coherent control effect, here we consider a simple model of two-level system, where level 1 is the ground state   g X 2 and level 2 is the excited state   u B 2 . We assume that the system has initial population distribution 11 (0) and 22 (0) as well as the initial coherence 12 (0) at t = 0 after the first 800 nm pump field passes through the medium. We consider the process that the first seeding pulse 1 ( ) + . . interacts with this two-level system. For simplicity, we assume that the center frequency of the seeding field is resonant with the two-level transition, i.e. ω = 12 . Therefore, by defining 12 = 12 , one can write the evolution equation under the rotating-wave approximation: where ℘ is the dipole moment.
The seeding field is ultrashort with a temporal width Δ (from center to nearly zero). It locates at time 1 and the electric field is described as 1 ( − 1 ) ( − 1 ) + . . In the limit of  1 /ℏ ≪ 1, one can integrate Eq. (1) with the first-order approximation, and therefore obtain the expression of the coherence after the passage of the first seeding field passes (t = 1 + Δ): Here 12 gives the amplitude of the coherence built after the first seeding field leaves the medium. The amplitude of the coherence at the moment that the seeding passes is dependent on the population distribution of the medium, the seeding field intensity, and the seeding field profile. Once the coherence is prepared by the first seeding pulse, it starts to grow up due to the optical gain (t) in the system, and the phase of the coherence is oscillating at the frequency , i.e. 12 ( ) = − 12 ( − 1 ) ∫ ′ ( ′) 1 . Before the second seeding pulse comes at t = 2, we assume that its amplitude becomes 12 (| 12 | > | 12 | due to the optical gain of the nitrogen ion system). Next, we consider the second seeding field 2 ( − 2 ) ( − 2 ) + . . coming at = 2 with the same pulse width Δ. The delay between two seed fields is much larger than the pulse width, i.e. 2 − 1 ≫ ∆. We again assume that  2 ∆/ℏ ≪ 1, so one can integrate Eq. (1) and find the coherence after the passage of the second seeding field: where ss = 2 − 1 is the time delay between two probe fields. One can see that the signal is beating at the frequency ω versusss. It is now clear that the strong 1.3 fs modulation of the superradiance (Fig. 3) origins from the coherent interaction of the second seeding pulse with the macroscopic polarization formed by the first seeding pulse, where their relative phase is determined by the delay ss.
The modulation contrast of Fig. 3 can reflect the evolution of the macroscopic polarization in the nitrogen ions after the first seeding pulse. We therefore measured this contrast as a function of the seeding pulse time delay ss. We defined the modulation contrast η as the absolute value of =(Imax-Imin)/(Imax+Imin), where Imax and Imin are maximum and minimum 391.4 nm intensity.
From Eq. (4), it gives = 4| 12 || 12 |/ [| 12 | 2 + | 12 | 2 ]. One therefore can see that the modulation contrast is supposed to be the maximum once | 12 | = | 12 | . In Fig. 4, we presented the results for three different intensity radio between the two seeding pulses. For the case of Is2 = Is1, a monotonous decrease of the modulation contrast was observed. For larger intensity of the second seeding pulse of Is2 = 2Is1 and Is2 = 4Is1, we noticed that the contrast first increases and then decreases. A full understanding of these results will only be possible in the framework of a complete Maxwell-Bloch equation describing the interaction of the resonant seeding fields with the two-level molecular system, as well as the formation process of the superradiance [25,26]. Here we would like to discuss qualitatively the main reasons underlying these observations. We first concentrate on the case where the two seeding pulses had the same energy. After the injection of the first seeding pulse, a seed-induced initial polarization 12 is excited in the nitrogen plasma, with its amplitude depending on the electric field 1 of the seeding pulse and the population difference between the two levels 11( 1 )-22( 1 ). In the presence of optical gain in the system, the seed-induced polarization evolves in the time domain, first grows to a maximum then decreases, with its maximum amplitude and temporal profile depending on the competition of gain, decoherence, and emission of superradiance [25,26]. At time 2, the second seeding was injected in the system. It interacts with the evolving two-level system and provokes an instantaneous polarization change 12 inside the system, presented by the last term in Eq. 3. This net polarization change adds up coherently with the polarization under evolvement from 1 to 2, with their relative phase determined by (1 -2). Strong modulation contrast of the final superradiance intensity is expected when 12 has a comparable amplitude with the instantaneous polarization before the injection of the second seeding pulse 12 . In the case of Is2 = Is1, we expect that 12 is normally less than 12 due to two reasons. First, the polarization after the first seeding pulse is increasing with time and we expect 12 is much larger than the initial polarization 12 . At the same time, the second seeding pulse sees a decreased population difference 11(2)-22(2) compared to that at time 1 and we expect 12 is less than 12 . As a result, the modulation contrast decreases when the time delay ss is increased, presented in Fig. 4. For increased intensity of the second seeding pulse, the polarization change 12 due to its injection can be larger. Therefore, the amplitude of 12 can be comparable to the amplified polarization 12 due to the gain at a proper time delay ss, and hence a maximum modulation contrast can be expected. Obviously, for further increased intensity of the second seeding pulse Is2 = 4Is1, the maximum contrast occurs for longer time delay. Therefore, the evolution of the macroscopic coherence is encoded in the variation of the modulation contrast presented in Fig. 4, while a quantitative understanding will be possible with a complete modeling and numerical simulation.
In conclusion, we demonstrated coherent control of the nitrogen ions "air laser" with a pair of seeding pulses at the resonant wavelength of the   u B 2 to   g X 2 transition. The superradiant emission at 391.4 nm shows a strong intensity modulation with a period determined by the transition frequency when the relative delay (phase) between the two seeding pulses is varied.
With this method, the built-up and decay of the macroscopic coherence between two electronic levels have been experimentally revealed and it has been found to remain for tens of picosecond.
This work highlights the essential role of the macroscopic coherence in the nitrogen ions lasing process and provides a new level of control on the nitrogen ions "air laser", which can be beneficial for its application in remote sensing.

Acknowledgement
The work is supported in part by the National Natural Science Foundation of China (Grants No.