Rotational coherence spectroscopy of molecules in helium nanodroplets: Reconciling the time and the frequency domains
Adam S. Chatterley, Lars Christiansen, Constant A. Schouder, Anders V. Jørgensen, Benjamin Shepperson, Igor N. Cherepanov, Giacomo Bighin, Robert E. Zillich, Mikhail Lemeshko, Henrik Stapelfeldt
RRotational coherence spectroscopy of molecules in helium nanodroplets: Reconcilingthe time and the frequency domains
Adam S. Chatterley, ∗ Lars Christiansen, ∗ Constant A. Schouder, Anders V. Jørgensen, Benjamin Shepperson, Igor N. Cherepanov, Giacomo Bighin, Robert E. Zillich, Mikhail Lemeshko, and Henrik Stapelfeldt Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C, Denmark Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria Institute for Theoretical Physics, Johannes Kepler Universit¨at Linz, Altenbergerstraße 69, A-4040 Linz, Austria (Dated: June 5, 2020)Alignment of OCS, CS and I molecules embedded in helium nanodroplets is measured as afunction of time following rotational excitation by a non-resonant, comparatively weak ps laserpulse. The distinct peaks in the power spectra, obtained by Fourier analysis, are used to determinethe rotational, B, and centrifugal distortion, D, constants. For OCS, B and D match the valuesknown from IR spectroscopy. For CS and I , they are the first experimental results reported.The alignment dynamics calculated from the gas-phase rotational Schr¨odinger equation, using theexperimental in-droplet B and D values, agree in detail with the measurement for all three molecules.The rotational spectroscopy technique for molecules in helium droplets introduced here should applyto a range of molecules and complexes. a r X i v : . [ phy s i c s . a t m - c l u s ] J un One of the unique aspects of helium nanodroplets is that molecules embedded in their interior exhibit IR spectrawith rotational fine structure very similar to the case of gas phase molecules. This free-rotation behavior, discoveredfor SF molecules [1], is directly connected to the superfluidity of He droplets because the low density of statesbelow 5 cm − makes coupling between the molecular rotation and the phonons weak [2–4]. The helium droplet does,however, influence the rotational structure of the molecule through a non-superfluid fraction of the surrounding heliumweakly bonding to and co-rotating with the molecule. The result is an effective rotational constant, which is smallerthan that of the isolated molecule. For instance, for the OCS molecule the reduction factor is 2.8 [5].With this knowledge in mind, one might expect that if molecules in helium droplets could be set into rotation bya short laser pulse, the rotational motion measured as a function of time should resemble that of the correspondingisolated molecules although slowed due to the larger moment of inertia of the helium-solvated molecules. Usingnonadiabatic laser-induced alignment techniques, adopted from gas phase molecules, such time-resolved measurementshave been reported [6–8]. The experiments showed that fs or ps pulses can indeed induce rotation of molecules in Hedroplets, leading to transient alignment, i.e. confinement of molecular axes to space-fixed axes [9, 10]. In the case of I molecules, a weak recurrence of the initial alignment maximum, shortly after the laser pulse, was observed about 600ps later and, using the angulon model [11], identified as a rotational revival of the helium dressed molecule [8]. As such,these studies illustrated some similarities to laser-induced rotation of gas-phase molecules but they did not provide aquantitative understanding of the rotational dynamics measured, nor a clear connection to the rotational energy levelstructure of molecules in He droplets established by frequency-resolved spectroscopy and quantum calculations [2–4, 12].Here we explore laser-induced alignment of molecules in helium droplets with the rotational energy kept belowthe roton energy ( ∼ − ) to equal the conditions of frequency-resolved spectroscopy studies and thus test if thefree-rotation behaviour is also observable in time-resolved measurements. We show that the time-dependent degreeof alignment measured can be accurately described by the solution to the time-dependent Schr¨odinger equation foran isolated molecule exposed to the alignment laser pulse, provided the effective B and D constants are employed,and inhomogeneous broadening of rotational energy levels accounted for. Experimentally, OCS, CS and I moleculesare rotationally excited by a non-resonant, picosecond laser pulse and the ensuing degree of alignment recorded bytimed Coulomb explosion. Fourier transformation of the alignment traces recorded reveals well-defined spectral peakscorresponding to the frequencies of a rotational wave packet in a non-rigid linear molecule, and thereby B and D forthe three molecules are determined. This demonstrates that rotational coherence spectroscopy [13, 14] works well forlinear molecules in He droplets and we believe it applies broadly to a variety of molecules and complexes. We showthat D strongly disperses the alignment traces and that the inhomogeneous broadening causes a gradual decay of theoscillatory structure of the alignment degree. The calculated alignment traces enable unambiguous assignment of halfand quarter revivals in the experimental data.The experimental setup and execution is essentially the same as previously reported [8, 15]. Helium dropletsconsisting on average of 6–8,000 He atoms and doped with one OCS, CS or I molecule, are exposed to a linearlypolarized laser pulse centered at 800 nm with a 40 nm (FWHM) bandwidth. Its duration (FWHM) is either 15, 5 or0.45 ps. The purpose of this alignment pulse is to rotationally excite the molecules via the polarizability interaction.After a time, t, the doped droplets are exposed to a 40-fs probe pulse. Its intensity, 6 × W / cm , is sufficientlyhigh to Coulomb explode the molecules. The emission direction of the fragment ions, S + for OCS and CS , andIHe + for I , are recorded by a 2-dimensional detector and the angle, θ , between the ion hit on the detector andthe polarization of the alignment pulse, located in the detector plane, is determined. Hereby (cid:104) cos θ (cid:105) , a standardmeasure for the degree of alignment [16], can be determined, where < .. > means the average over all ion hits, typically2500 recorded for 50,000 laser pulses. The time-dependent alignment traces, i.e. (cid:104) cos θ (cid:105) (t), are obtained byperforming the measurements for a large number of t’s.The black curves in Fig. 1(a1)-(c1) show (cid:104) cos θ (cid:105) as a function of time for OCS, recorded at three differentfluences. In the three cases, a maximum occurring on the trailing edge of the 15 ps alignment pulse is followed by anoscillatory structure. The alignment traces are Fourier transformed to analyze their spectral content. The resultingpower spectra, displayed in Fig. 1(a2)-(c2) contain distinct peaks, just as for rotational wave packets in gas phasemolecules [17–20]. There, the peaks reflect the frequencies of the nonzero matrix elements (cid:104) JM | cos θ | J (cid:48) M (cid:105) , i.e.the coherence (coupling) between state | JM (cid:105) and | J (cid:48) M (cid:105) with J denoting the rotational angular momentum and M its projection on the alignment laser polarization [16]. For a non-rigid linear rotor, the rotational energies are givento the second order by: E rot = BJ ( J + 1) − DJ ( J + 1) , (1) (a1)OCS (a2) c o s D
15 ps 1.1 J/cm (b1) P o w e r ( a r b . un i t s ) (b2) Delay (ps) (c1) Frequency (GHz) (c2) c o s D
15 ps 0.4 J/cm (d1)CS (d2) Delay (ps) (e1) Frequency (GHz) P o w e r ( a r b . un i t s ) (e2) c o s D J/cm (f1)I (f2) Delay (ps) ps 0. J/cm (g1) Frequency (GHz) P o w e r ( a r b . un i t s ) (g2) FIG. 1. Column 1: The time-dependent degree of alignment for OCS, CS and I molecules at different durations and fluencesof the alignment pulse, given on each panel. Black (red) curves: experimental (simulated) results. The intensity profile of thealignment pulses are shown by the shaded grey area. Column 2: The power spectra of the corresponding (cid:104) cos θ (cid:105) traces. Thespectral peaks, highlighted by the colored vertical bands, are assigned as ( J – J + 2) coherences (see text) with J given on topof the panels (blue: even, red: odd). and thus the frequencies corresponding to the dominant ∆ J = J (cid:48) − J = 2 coherences [16], labelled ( J – J + 2), by: ν (J–J+2) = B (4 J + 6)– D (8 J + 36 J + 60 J + 36) . (2)Adopting the gas phase picture, we assign the three prominent peaks in Fig. 1(a2) at 12.8, 20.6 and 27.2 GHz aspertaining to the (0–2), (1–3) and (2–4) coherence, respectively. The colored vertical bands illustrate that the peaksare located at the same positions for the three fluences. The weight of the peaks shifts to higher frequencies as thefluence, F , increases, and for F = 1.4 J/cm , an extra peak appears at 32.3 GHz, which we assign as the (3–5)coherence.Next the central positions of the ( J – J + 2) peaks in Fig. 1(a2)-(c2) are plotted as a function of J in Fig. 2. Thedata points, represented by the blue squares, are fitted using Eq. 2 with B and D as the free parameters. The bestfit, represented by the blue curve in Fig. 2, is obtained for B = 2.18 ± D = 9.5 ± B = 2.19 GHz and D = 11.4 MHz, where D was the average J F r equen cy ( G H z ) OCSB = 2.18 0.06 GHzD = 9.5 1.8 MHz I B = 480 15 MHzD = 450 100 kHzCS B = 730 15 MHzD = 1.2 0.1 MHz
FIG. 2. Central frequencies of the ( J – J + 2) peaks in the power spectra versus J . The full lines represent the best fits usingEq. 2. The B and D constants from the fits are given for each molecule. for the v =0 and v =1 vibrational states in the IR transition [5]. The excellent fitting of Eq. 2 to the peak positionsstrongly indicates that laser-induced rotation of OCS molecules in He droplets is well described by a gas-phase modelemploying the effective B and D constants.To further explore this, we calculated (cid:104) cos θ (cid:105) (t) by solving the time-dependent rotational Schr¨odinger equationfor a linear molecule exposed to the experimental alignment pulse, using the B and D values from the fit. Thecalculations were averaged over the initially populated rotational states, given by a Boltzmann distribution with T =0.37 K, and over the focal volume determined by the measured spot sizes of the alignment ( ω = 30 µ m) and probebeams ( ω = 25 µ m). Also, the effect of inhomogeneous broadening was implemented by a Gaussian distribution ofthe B constants with a FWHM, ∆ B = 90 MHz [21], and a constant B/D ratio.The calculated degree of alignment, shown by the red curves in Fig. 1(a1)-(c1), has a very strong resemblancewith the measured traces and captures in detail most of the oscillatory pattern observed [22]. The good agreementbetween the experimental and calculated time-dependent (cid:104) cos θ (cid:105) corroborates that an effective gas-phase modelaccurately describes the laser-induced rotational dynamics of OCS in He droplets for the fluences and durations ofthe alignment pulses employed. Nevertheless, the observed dynamics appears very different from that of isolatedmolecules. As discussed below, this is due to the much larger D constant for molecules in He droplets compared toisolated molecules (OCS: D He ≈ . × D gas ) [5, 23] and the presence of inhomogeneous broadening.Experiments were also conducted on CS and I molecules. The alignment traces and corresponding power spectra,including the peak assignments, are shown in Fig. 1 panels (d)-(g). The central frequencies are plotted versus J and as for OCS, Eq. 2 provides excellent fits to the experimental results, illustrated by the red (I ) and blue (CS )points/lines in Fig. 2. The B and D values extracted from the best fits are given on the figure. To our knowledge,this is the first experimental determination of B and D for these two molecules in He droplets. In fact, IR and MWdo not apply to I because it lacks a permanent electric dipole moment.For comparison, we obtained B for CS in a helium droplet from a path integral Monte Carlo (PIMC) simulation,using the CS -He interaction of Ref. [24] and the He-He interaction of Ref. [25]. The rotational correlation function [26,27] S (cid:96) ( τ ) = π (cid:96) +1 1 Z (cid:80) m Tr { Y ∗ (cid:96)m (Ω( τ )) Y (cid:96)m (Ω(0)) e − βH } is calculated in imaginary time τ ∈ [0 , β ]. B is obtained byfitting S (cid:96) ( τ ) of a free linear rotor, with B as fit parameter. Unlike a full reconstruction of S (cid:96) ( t ) in real time, thesimple fitting procedure is numerically stable for the statistical errorbars of S (cid:96) ( τ ) achievable for a heavy rotor likeCS . However, the fit is inherently wrong for τ → , β ], i.e. decreasing the temperature T , and extrapolate B to T → T = 0 . . . in clusters of up to N = 150 He atoms and extrapolated the results for B to N → ∞ using a fit linear in N − . With this protocol to remove the bias from the linear rotor fit and from the finite clustersize, the PIMC estimate is B gas /B He = 4 . ± .
05 in large droplets, in good agreement with the experimental valueof 4.5 ± B gas = 3.273 GHz).For I , the experiment shows that B He is reduced by a factor of 2.3 ± B gas . Previous PIMCsimulations for I in a cluster of 150 He atoms gave a reduction factor of 1.7. From our recent experience with CS ,we expect that the reduction factor will also increase for I when the new protocol is applied. If we naively assume c o s D D = 0.0 MHz (a1)
D = 0.0 MHz (a2) c o s D D = 5.0 MHz (b1)
D = 5.0 MHz (b2)
Delay (ps) c o s D D = 9.5 MHz (c1)
Delay (ps)
D = 9.5 MHz (c2)
B = 0 MHz B = 90 MHz
FIG. 3. (cid:104) cos θ (cid:105) as a function of time calculated for OCS (B=2.17 GHz) for three different values of the D constant, without(left column) or with (right column) inhomogeneous broadening included. T = 0.37 K and F = 0.7 J/cm . The yellow andblue bands highlight the position of the half-and full-revival for the D=0 case. the relative correction due to the protocol for I is the same as for CS , the reduction factor would be 2.3, i.e. thesame as the experimental value.As for OCS, we also calculated the time-dependent degree of alignment for I and CS . The results, shown by thered curves in Fig. 1 panels (d1)-(g1) agree very well with the experimental findings [28]. Since the two molecules hadnot been spectroscopically studied in He droplets before, we had to choose a width for the inhomogeneous distributionof the B constant. The best agreement with the measured (cid:104) cos θ (cid:105) (t) was obtained for ∆ B = 50 MHz for CS and40 MHz for I .To elucidate why laser-induced rotational dynamics of molecules in He droplets appears very different from thatof gas phase molecules, we calculated (cid:104) cos θ (cid:105) (t) for three values of the D constant, with or without the effect ofinhomogeneous broadening. The calculations were done for OCS molecules and the experimental 15 ps pulse. When D = 0 and inhomogeneous broadening is neglected, i.e. all molecules have the same B value, Fig. 3(a1) shows that (cid:104) cos θ (cid:105) (t) is periodic with distinct half and full revivals. This case is identical to that of isolated OCS moleculesexcept that the revival period is increased by a factor 2.8 due to the effective B constant. The calculation for D =5.0 MHz, Fig. 3(b1), shows that the centrifugal term introduces an additional oscillatory structure in (cid:104) cos θ (cid:105) (t)and distorts the shape of the revivals. For gas phase molecules, it was already observed and understood that thecentrifugal term modulates the shape of rotational revivals [29, 30] but the influence was moderate [31] and thedifferent revivals remained separated from each other. In Fig. 3(b1), the effect of the centrifugal term is so large that,with the exception of the half-revival, there is essentially no longer distinct, separated revivals. This trend is evenmore pronounced for the calculation with the experimental D value, Fig. 3(c1). The yellow and blue bands providea rigid rotor reference, Fig. 3(a1), for how the centrifugal term distorts and shifts the rotational revivals.The panels in the right column of Fig. 3 show (cid:104) cos θ (cid:105) (t) when inhomogeneous broadening is included by averagingcalculated alignment traces over a 90-MHz-broad Gaussian distribution of B constants. The main influence is a gradualreduction of the amplitude of the oscillations in the alignment traces while preserving the average value of (cid:104) cos θ (cid:105) .This dispersion effect produces alignment traces, which (for D = 9.5 MHz) agrees very well with the experimentalresults for all three fluences studied, see Fig. 1(a1)-(c1). The same is true for the CS results using a 50 MHzdistribution of B constants. Calculations where D is gradually increased from 0 to 1.2 MHz (not shown) identify thevalley-peak structure around t = 200 ps as the quarter revival, see Fig. 1(d1)-(e1). Similarly, for I the oscillatorystructure in the 550-700 ps range I is identified as the half revival, see Fig. 1(f1)-(g1).We note that the average value of the experimental (cid:104) cos θ (cid:105) (t) for I , recorded with the 5 ps pulse, Fig. 1(f1),actually decays over the 1500 ps measured, which is not captured by the calculation. Also, the calculated alignmenttrace shows a transient structure around t = 1100-1200 ps, corresponding to the full revival, which is not observedexperimentally. These two discrepancies between the measurements and simulations indicate that some of the rota-tional states excited by the alignment pulse decay on the time-scale of the measurements, i.e. that the alignmentdynamics is also influenced by lifetime (homogenous) broadening [3, 32]. This is likely also the case for OCS and CS but measurement for longer times would have been needed to observe a similar decay of (cid:104) cos θ (cid:105) (t). For I , electricquadrupole induced coupling between the rotational angular momentum and the nuclear spins may also contribute tothe decay of (cid:104) cos θ (cid:105) (t) [33–35].The good agreement between the measured and calculated (cid:104) cos θ (cid:105) (t) leads us to conclude that, for the relativelyweak laser pulses applied here, the mechanism of non-resonant ps or fs laser-induced rotation of molecules in heliumdroplets is the same as for gas phase molecules. The rotational dynamics does, however, differ significantly fromthat of isolated molecules due to the orders of magnitudes larger centrifugal constant of He-solvated molecules andthe inhomogeneous broadening of the distribution of rotational constants [36–38]. Fourier transformation of themeasured (cid:104) cos θ (cid:105) (t) traces and fitting of the spectral lines to a non-rigid rotor model enabled determination of therotational and centrifugal constants. For OCS, the agreement of our experimental results with the values from IRspectroscopy reconciled frequency resolved spectroscopy and nonadiabatic laser-induced alignment dynamics and inaddition, introduced the latter as a rotational spectroscopy method for molecules in helium droplets. Our methodshould apply to a broad range of molecules [3, 39] and molecular complexes [40, 41]. This includes for instancehomo-dimers of metal atoms [42–44], where neither IR nor microwave spectroscopy can be used.We believe the rotational dynamics reported here is a consequence of superfluidity of the helium droplets. Accordingto the power spectra, shown in Fig. 1, the maximum J and thus rotational energy, E rot ( Eq. 1), are J =5, 1.9 cm − forOCS; J =10, 2.2 cm − for CS ; and J =9, 1.3 cm − for I . In all three cases, E rot is below the roton energy. Thus, thefree molecular rotation we observe is consistent with the weak coupling between molecular rotation and phonons inthis low-energy regime [3]. If stronger alignment pulses are applied, higher rotational states will be excited. Then thefree-rotor description is no longer expected to be valid since the coupling to the phonons (rotons) strongly increasesand the high density of states above the roton energy leads to fast decay [45]. Furthermore, the rotational levelstructure will not continue to be given by Eq. 1, which predicts that E rot starts decreasing when J exceeds a value, J th , e.g. for OCS J th = 10. Time-resolved rotational dynamics measurements may provide an understanding of thisunexplored regime of rotational states.Finally, previous studies have shown that an intense fs laser pulse can deposit so much rotational energy in a moleculethat it transiently decouples from its solvation shell [8]. Such a process will bring the molecule-He system far awayfrom equilibrium. Measuring the rotational dynamics for different delays between the distortion and a subsequentweak alignment pulse will allow to explore how long it takes to restore the equilibrium, gauged by when the alignmenttrace becomes identical to a reference trace recorded without the distortion pulse. Alternatively, the rotational echotechnique may enable real-time characterization of the molecule-helium droplet coupling through measurement of thetime constant for coherence loss of rotational wave packets [46–48].H.S acknowledges support from the European Research Council-AdG (Project No. 320459, DropletControl) andfrom The Villum Foundation through a Villum Investigator grant no. 25886. M.L. acknowledges support by theAustrian Science Fund (FWF), under project No. P29902-N27, and by the European Research Council (ERC) StartingGrant No. 801770 (ANGULON). G.B. acknowledges support from the Austrian Science Fund (FWF), under projectNo. M2461-N27. 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