Molecular alignment echoes probe collision-induced rotational-speed changes

We show that the decays with pressure of the alignment echoes induced in N2O-He gas mixtures by two laser pulses with various delays bring detailed information on collision-induced changes of the rotational speed. Measurements and calculations demonstrate that collisions reduce the echo amplitude all the more efficiently when the echo appears late. We quantitatively explain this behavior by the filamentation of the classical rotational phase space induced by the first pulse and the progressive narrowing of the filaments with time. The variation of the echo decay thus reflects the ability of collisions to change the molecules' rotational speed by various amounts, enabling refined tests of models for the dissipation induced by intermolecular forces.


I. INTRODUCTION
In gas media, intermolecular collisions modify the rotational and translational motions of molecules. These processes are of fundamental interest and must be properly modeled for practical applications. Indeed, in gases at equilibrium, molecule-molecule and -atom interactions largely govern the shape of molecular spectra [1] and the energy and mass transports [2,3], for instance. When the system has been perturbed by an external interaction, the return to equilibrium often follows channels enabled by collisions [2].
Many works have studied collisional effects in the frequency domain through their consequences on the shape of light-absorption and -scattering spectra [1,4]. The influences of intermolecular forces in gases have also been investigated in the time domain through the decay of the molecularaxis alignment/orientation induced by an electromagnetic pulse (e.g. [5][6][7][8][9][10][11][12]) and, recently, by using the echo following the excitation of the system by two laser kicks [13,14]. However, these studies provide averaged information with little detail on intermolecular interactions. For instance, the pressure broadening of anisotropic Raman lines or the decay of alignment revivals (which are, through a Fourier transform, equivalent [6]) only tell how fast collisions change the rotational motion, with no information on the respective contributions of dephasing, changing of the rotational speed or of the angular momentum orientation. The "transient" and "permanent" components of alignment signals provide more insight as they tell at which (different) time scales relaxation affects the modulus and orientation of the rotational angular momentum [7,8,10,15]. However, again, no information is provided on how efficient are collisions in modifying the rotational energy by a given amount. The way to overcome these limitations was, up to now, to carry joined time and spectral domains (double resonance) experiments [16,17]. One can, for instance, depopulate a rotational level by a resonant pulse and then probe the evolutions of other levels populations using the time and spectral dependences of the absorption spectrum. Such measurements can provide state-to-state relaxation rates, but they are complicated to carry. Furthermore, disentangling the contributions of the various collisional channels requires the use of a model. This Letter gives the first experimental and theoretical demonstrations that alignment echoes enable detailed investigations of the collisioninduced changes of molecular rotation. These echoes in the alignment of molecular axes after non-adiabatic excitations by two successive nonresonant and linearly-polarized laser pulses were first evidenced in low pressure CO 2 [18]. Some of the molecules are first aligned by a short and intense pulse (P 1 ) thanks to the anisotropy of the molecular polarizability. This results in a peak in the alignment factor 2 cos ( ) ( ) t θ < > , where θ is the angle between the molecule axis and the laser polarization, which vanishes quickly due to dephasing. A second pulse (P 2 ), applied 12 τ later, induces a rephasing creating an echo in the alignment factor at 12 2 t τ = after P 1 (e.g. Fig. 1a).
Further investigations, still carried at pressures for which collisional effects are negligible at the investigated time scales, revealed the existence of fractional, imaginary, and rotated echoes [14,[19][20][21]. The collisional dissipation of echoes was studied later [13,14] τ , of the echo dissipation, showing that the efficiency of collisions in damping the echo significantly varies with time at the early stage of the relaxation process. Associated quantum theoretical calculations evidenced the breakdown of the widely used "secular approximation" at short times.
We here reanalyze the experimental results of Ref. [22] with a completely different approach, using a classical instead of a quantum model. This reveals that the variation of the echo pressureinduced decay with its time of appearance reflects the ability of collisions to change the rotational speed by various amounts. We also show that rotational alignment echoes are some kind of bridge between the quantum and classical worlds.

II. EXPERIMENTS
The experimental set-up, the procedures used for the measurements and their analysis, and the results used below have been presented in Ref. [22]. Two femtosecond laser pulses were applied, separated by various delays 12 τ , and the alignment factor was recorded for N 2 O(4%)+He(96%) gas mixtures at room temperature and several pressures. The variation, with the density d, of the measured density-normalized amplitude 12 ( , ) S d τ (see Fig.  1b) of each echo was fitted by 12 12 0 12 A densitynormalized time constant (in ps.amagat, 1 amagat corresponding to 2.69 10 25 molec/m 3 ) was then defined by 12 12 12 E 0 ( ) 2 ( ) d τ τ = τ τ , which accounts for the fact that the echo appears at 12 2 t = τ after the time origin defined by P 1 .

III. CALCULATIONS
Classical Molecular Dynamics Simulations (CMDS) were carried for N 2 O diluted in He gas, as described in [14,15]. Many molecules and atoms were treated using periodic boundary conditions, nearest neighbors' spheres, and the Verlet algorithm [23]. ( ) . In order to simulate molecules infinitely diluted in He while keeping computer time reasonable, 50% N 2 O + 50% He mixture were treated by only taking N 2 O-He forces into account with the accurate potential of Ref. [24]. The laser pulses characteristics were set to those of the experiments (Gaussian time-envelop with a 100 fs FWHM and peak intensities around 20 TW/cm 2 ) and the N 2 O anisotropic polarizability ∆α =19.8 a.u. [25] was used. The molecules were treated as rigid rotors, and the requantization procedure introduced in [15] was eventually applied, in which the rotational angular speed is changed, at properly chosen times, to match the closest quantum value These requantized CMDS (rCMDS) and the purely classical CMDS enabled to calculate the alignment factor 2 cos ( ) t θ < > for several densities (Fig. 1a). The density-normalized time constant 12 E ( ) τ τ of the echo decay was then determined from the predicted alignments as done with the experimental ones (Sec. II and [22]). Note that the interest of (r)CMDS has been demonstrated for various laserinduced alignment features in different gases [11,12,14,19,20,26,27].

IV. RESULTS
The experimental and predicted values of 12 E ( ) τ τ are displayed in Fig. 2. As pointed out previously [22], the echo decay, which is relatively slow at short delays, becomes faster as the delay increases before a plateau is reached around 12 2 10 τ ≈ ps. This is well reproduced by the rCMDS despite a predicted plateau slightly lower and reached a little later than in the experiments. Concerning the CMDS, they also lead to very satisfactory predictions before about 10 ps but do not predict any plateau. This, which is discussed later, comes from the fact they enable filament widths (and rotational speed changes) to become smaller than the quantum limit / I ℏ .

V. DISCUSSION
Recall that the alignment echo created at t=2τ 12 by laser kicks at t=0 and t=τ 12 is a classical phenomenon [18], contrary to the revivals (see Supplementary Material [28]). It results from the fact that the first pulse, by changing the molecule rotational speed according to its orientation, induces [18] a progressive filamentation of the (ω,θ) rotational phase space. This phenomenon, explained using a 2D model in [18,19], is shown in Fig. 3 and in the Supplementary Movie [29] (both obtained from 3D CMDS for N 2 O gas under collision-free conditions). between their peaks (which very well follows the / t π law deduced from a 2D model [18]) reduce (all with stair-case like profiles not shown on the plot). This observation provides a qualitative explanation for the results in Fig. 2, because it is the progressive removal of the molecules from "their" filament that induces the echo decay, and since it is obvious that this process is all the more rapid when the filaments are narrow. In order to be quantitative, we carried CMDS for N 2 O diluted in He gas at equilibrium (no pulse), and looked at the time evolution of the number of molecules whose angular speed was between 0 / 2 ω − ∆ω and 0 / 2 ω + ∆ω at t=0 and is still within this interval at t. The time constant Loss ( ) τ ∆ω of the decay of this number is shown by the black line in Fig. 5 (obtained for the most probable 12 0 2.3 10 rad/s ω = ). As expected, Loss ( ) τ ∆ω increases with ∆ω , since changing the rotational speed by a larger amount requires more time, because either a strong (and thus little probable) collision or several successive weak collisions are needed. To now cast the results of Fig. 2 into Fig. 5, we need to associate a representative filament width 12 ( ) ∆ω τ to each xaxis coordinate of Fig. 2. For this, and considering that the mean width of the filaments evolves with time (Fig. 4c), it seems reasonable to chose the value for the median time = τ between P 1 and the echo (which is almost identical the mean filament width over this time interval). We thus retained 12 FWHM 12 Fig. 4(c): 12 12 ( ) 1.9/ ∆ω τ = τ , where the factor 2 is introduced in order to include the full filament and not its top half only. Then plotting the measured values from Fig. 2 leads to the open circles in Fig. 5. As can be seen, despite its simplicity, the model (black line) based on the rate of removal of molecules from given rotationalspeed intervals well agrees with the observed results. This quantitatively confirms the above given explanation of the results in Fig. 2 by the time evolution of the filaments widths. Note that the observed plateau of E τ for small values of ∆ω (large delays) is absent in the purely classical calculations of Loss τ (and in the CMDS results of Fig. 2). This is because the CMDS allow filament widths to become smaller than the quantum limit / I ℏ =0.15 10 12 rad/s,. However, a posteriori imposing this limit (i.e. setting Loss Loss ( ) ( ) L τ ∆ω = τ ∆ω for 2 / L I ∆ω ≤ ∆ω = ℏ ) creates a plateau below 0.3 10 12 rad/s in the black curve in Fig. 5 (shown by the dashed curve) making it more consistent with the measurements.
It is finally of interest to relate the present results based on a classical model to those obtained with a quantum approach [22] since, despite the fundamental difference between them, both well predict the observed behavior of the echoes. This connection can be made (see Supplementary Material [28]) by considering that the , involved in the alignment factor, with ρ the density matrix, is represented by those molecules which classically rotate with angular speeds ω such that 2ω is the closest to ℏ . Now, the quantum results show that exchanges between coherences limit the echo decay, a nonsecular effect significant at early times which vanishes with increasing delay and the associated dephasing [22]. The classical equivalent of this is the fact that exchanges between rotational-speed classes limit the echo decay as long as they occur within a given filament, an event all the less probable that the filament width has narrowed with the elapsed time.

VI. CONCLUSION
We have shown, with agreement between theoretical predictions and measurements, that alignment echoes probe the ability of collisions to change the rotational speed by various amounts. This is thanks to the progressive narrowing of the filaments created, in the classical rotational phase space, by the first aligning pulse. With respect to the alignment revivals in the time domain and to the pressure-broadening of absorption lines in the spectral domain, echoes thus give a much more contrasted picture. This unprecedented scrutiny opens renewed perspectives for our understanding of collisional processes and stringent tests of rotational-relaxation models and intermolecular potentials. Rotational alignment echoes being a generic phenomenon, this statement is also valid for very complex/heavy molecules for which quantum calculations are untractable. This further broadens the potential applications of this study.

Supplementary Material for "Molecular alignment echoes probe collision-induced rotational-speed changes"
J

I. Supplementary Note 1: On the nature of the revival and echo phenomena
The time evolution of the density matrix ( ) t ρ of linear molecules subject to a non-resonant laser pulse is, under collision-free conditions, given by: where H 0 is the free rotation Hamiltonian and: with ∆α the anisotropic polarizability, E(t) the envelop of the laser pulse and P 2 a Legendre polynomial. Writing Eq. (S1) in the interaction picture, introducing and assuming a laser pulse applied at t=0 with a negligible duration (sudden approximation) leads, just after the pulse, to: ( 1) where (:::) is a 3J symbol and J and M are the quantum numbers associated to the rotational angular momentum and to its projection along the quantification axis, one has the selection rules ' M M = and ' , 2 J J J = ± . We now make a development of the sudden approximation operators, i.e.: and limit ourselves to the first order. Equation (S4) then becomes: After the pulse, the system evolves freely and one has: (S12) It is obvious that all the oscillating terms of the sum in Eq. (12) rephase, all sine functions being zero, for times t t multiples of rev /(4 ) in the case of CO 2 for which only even J values exist]. This leads, with B(N 2 O)=0.42 cm -1 and B(CO 2 )=0.39 cm -1 , to the so-called "revivals" with (positive and negative) extrema in the alignment factor appearing around multiples of rev 20 t ≈ ps [1,2] and rev 11 t ≈ ps [3,4] ps, respectively. Since these specific times depend on the rotational constant B, the revivals are of purely quantum nature. Equation (S12) shows that the amplitudes of the revivals are, in the weak field limit, proportional to the laser energy, in agreement with measurements and direct calculations of the alignment factor [4,5]. Also note that the time-independent component of the alignment can be obtained using the same approach, but results from the second order terms. The resulting "permanent"-alignment amplitude is thus proportional to the square of the laser energy, a finding also consistent with experiments and independent calculations [4,5].

I.2 Two pulses and the echoes
First note that Eq. (S4) can be rewritten as: , '', ''', '''', ' , ', '', ''', '''', 12 12 12 12 cos ( ) ' cos exp ( 1)( where n C is proportional to the intensity of the n th pulse, and consider the various orders in C 1 and C 2 . The preceding section shows that the contributions to Eq. (S15) which are proportional to 1 C and 2 1 C (resp. to C 2 and 2 2 C ) lead to the revivals and to the permanent alignment generated by the first (resp. second) pulse, respectively. Considering the four terms proportional to 1 2 C C , it is "relatively" easy to show that they cancel out. Let us now look at the contributions proportional to 2 1 2 C C , coming from approximating 1 U or *