Mapping the Evolution of Optically-Generated Rotational Wavepackets in a Room Temperature Ensemble of D 2
W. A. Bryan, E. M. L. English, J. McKenna, J. Wood, C. R. Calvert, R. Torres, I. C. E. Turcu, J. L. Collier, I. D. Williams, W. R. Newell
aa r X i v : . [ qu a n t - ph ] J un Mapping the Evolution of Optically-Generated Rotational Wavepacketsin a Room Temperature Ensemble of D W. A. Bryan,
1, 2, ∗ E. M. L. English, J. McKenna, J. Wood, C. R. Calvert, R.Torres, I. C. E. Turcu, J. L. Collier, I. D. Williams, † and W. R. Newell ‡ Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK Central Laser Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK Department of Pure and Applied Physics, Queen’s University Belfast, Belfast BT7 1NN, UK Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, UK (Dated: October 24, 2018)A coherent superposition of rotational states in D has been excited by nonresonant ultrafast(12 femtosecond) intense (2 × Wcm − ) 800 nm laser pulses leading to impulsive dynamicalignment. Field-free evolution of this rotational wavepacket has been mapped to high temporalresolution by a time-delayed pulse, initiating rapid double ionization, which is highly sensitive tothe angle of orientation of the molecular axis with respect to the polarization direction, θ . Thedetailed fractional revivals of the neutral D wavepacket as a function of θ and evolution time havebeen observed and modelled theoretically. PACS numbers: 42.50.Hz, 33.80.Gj, 33.80.Wz
The importance of being able to enforce spatial orderon an initially random ensemble of molecules has beenrecognized since the discovery of steric effects in chemi-cal reactions [1]. Traditional ‘brute force’ techniques em-ploying strong DC fields [2, 3] have recently yielded tonew, more subtle and yet more powerful and versatiletechniques using intense laser systems [4, 5]. In partic-ular intense femtosecond pulses have been successfullyused to align molecules along an axis for applicationin areas such as the study of fragmentation dynamics[6, 7], high harmonic generation [8, 9] and the creationof single cycle laser pulses [10]. In this case the align-ment of a molecule (or spatial ordering of an ensembleof molecules) evolves temporally following the creationof a rotational wavepacket by a linearly polarized laserpulse of far shorter duration than the natural period ofrotation.In this Letter, we report on a detailed ultrafast studyof the temporal evolution of such a rotational wavepacketin the neutral deuterium molecule D . The wavepacketis created impulsively by a 12 femtosecond laser pulse,with temporal and angular ordering probed at some latertime with a similar pulse that initiates sequential doubleionization (SI) i.e. D → D +2 → D + + D + [11]. Suchhigh resolution measurements represent the state-of-the-art in ultrafast molecular physics in the high rotationalfrequency limit as dictated by this most fundamental andtheoretically tractable of molecules.It is known that an intense linearly polarized laserpulse interacting with an ensemble of molecules gener-ates a degree of spatial alignment from a random ensem-ble [4, 5]. The only condition for this phenomenon is thatthe molecular polarizability is anisotropic. Thus, the in-duced dipole moment created in the interaction with theelectric field of the laser generates a torque that causesthe axis of maximum polarizability to librate around the polarization vector of the laser field. Quantum mechani-cally, the process is described as a sequence of Rabi-typecycles accompanied by the exchange of two quanta ofangular momentum between the molecule and the laserfield. If the aligning pulse duration is much greater thanthe rotational period, the system evolves adiabaticallyto the the so-called pendular states, which dissipate assoon as the laser pulse passes [12]. However, when thepulse duration is much shorter than the rotational pe-riod, a superposition of rotational states is created thatoutlives the laser pulse, and the phases in the rotationalwavepacket continue to evolve [5]. As a consequence ofthe quantization of the J states, a periodic dephasingand rephasing of the wavepacket occurs in the field-freeregime. At integer and half-integer multiples of (2 Bc ) − (where B is the rotational constant and c is the speed oflight in vacuum) a revival of the rotational wavepacket isexpected, at which point the ensemble exhibits a signifi-cant degree of alignment, with the molecular axes parallelto the polarization vector of the initial pulse. The rela-tive phases of the J states also produces antialignment inthe ensemble, whereby the molecular axes are orientatedto lie on or near the plane normal to the initial pulse po-larization. Alignment and antialignment are well distin-guished temporally. As a consequence of the nuclear spinstatistics, homonuclear systems can also present partialrevivals at 1/4 and 3/4 of the rotational period.Previous observations of impulsive alignment havebeen restricted by the non-availability of laser pulses suf-ficiently shorter than the rotational periods. However,rotational wavepackets have been previously observed inheavy many-electron systems (N [6], O [13, 14], CO [15, 16], CS [16], I [17] etc.), where rotations occur withperiods on the picosecond time scale. The rotational re-vival in D occurs at (2 Bc ) − = 558 fs, requiring an align-ing pulse of the order of tens of femtoseconds in duration. FIG. 1: The experimental configuration used to observe rota-tional wavepackets in D imprinted on the D + rapid sequen-tial ionization (SI) yield (D → D +2 → D + + D + ) as measuredwith a time-of-flight mass spectrometer (TOFMS). The timedelay between the pump and probe pulses is ∆t, and the angle θ between pump and probe polarization directions is variedby rotating the pump polarization. As shown, θ = 0. Dynamic alignment of D using 10 fs pulses has recentlybeen demonstrated by Lee et al [18], however, these ob-servations were of limited temporal resolution and theangular evolution of the rotational wavepacket was notmeasured.In the present work, the spatio-temporal evolution ofthe D rotational wavepacket has been simulated follow-ing the procedure described in [19]. Briefly, a thermal en-semble of rigid rotors is considered, each of which givesrise to a superposition of rotational states, | Ψ J i M i i = P J ≥| M i | F J i J (t) | JM i i in which F J i J ( t ) are the time-dependent complex coefficients (note that M i is con-served in a linearly polarized field). These coefficientsare calculated by numerically solving the time-dependentSchr¨ o dinger equation, and are used to calculate the ther-mal average of the degree of alignment and the angularevolution of the ensemble.Due to the bosonic character of D , the total wavefunc-tion of the molecule must be symmetric under inversion.The electronic ground state is symmetric, and the nucleican form six symmetric (ortho) and three antisymmet-ric (para) nuclear wavefunctions. This restricts the J states populated: para-D occupies only odd J -states,and ortho-D occupies only even J -states. This resultsin a 2 : 1 weighting (ortho-D : para-D ) in a thermalensemble. For a room temperature sample we expect thefollowing initial populations (in parenthesis) for the dif-ferent J states: J = 0 (0.185), 1 (0.208), 2 (0.386), 3(0.112), 4 (0.0899), 5 (0.0128) and 6 (0.00522).The experiment was carried out using 800nm, 0.4 mJ,10 fs pulses at 1 kHz repetition rate. The pulses weresplit by a 4 µ m thick pellicle beamsplitter. A co-linear FIG. 2: (a) False colour representation of the D + rapid se-quential ionization (SI) yield (D → D +2 → D + + D + ) asa function of pump-probe delay (thus wavepacket evolutiontime) ∆ t and ion kinetic energy. (b) Integrated SI yield com-pared to theoretically predicted influence of an impulsivelyexcited rotational wavepacket in D (red). Throughout, θ = π /2, see Fig. 1, and the step in ∆ t = 1 fs. (c) IntegratedSI yield following an 11-point smoothing. The 1/4, 1/2 and3/4 revivals are more clearly observed. Fourier Transformof the theoretical simulation and experimental SI yield areshown in (d) and (e) respectively. The frequency componentspresent in the rotational wavepacket are beats between rota-tional states ∆ J and ∆ J +2. The Fourier amplitudes thusreturn a measure of rotational population. Mach-Zehnder interferometer configuration was used todelay the resulting 70 µ J, 12 fs probe pulse with respectto the 73 µ J, 12 fs pump pulse with 300 attosecond res-olution [20]. A half-wave plate mounted in one arm ofthe interferometer allowed the pump polarization to berotated through an angle θ with respect to the probepolarization, which was fixed parallel to the spectrome-ter axis. In comparison to the core-annulus method [18],the present technique does suffer greater energy loss inthe split pulses. However, this is compensated by thetemporal consistency of the spatial overlap at focus as afunction of time delay over a far greater range. Further-more the present technique allows polarization control.These features are essential to the present study.Following transmission through a fused-silica windowinto an ultra-high vacuum chamber, the pump-probepulses are reflection focused with a silver-coated sphericalmirror capable of supporting the bandwidth of our laserpulses without introducing group-velocity dispersion, asshown in Fig. 1. The laser focus is situated in the sourceregion of a tightly apertured (250 µ m) ion time-of-flightmass spectrometer (TOFMS) [20]. The aperture servestwo purposes: to limit the angular acceptance ( ≤ + ion), and to dramatically limit thefield of view of the spectrometer. By observing only thecentral 250 µ m ‘slice’ of the f /5 focus, the instrument issensitive only to those molecules exposed to a narrow in-tensity range. As a consequence, spatial integration overthe focal volume is obviated, and the behaviour of therotational wavepacket is clearly observed. This results ina remarkably high resolution measurement of the rota-tional wavepacket as compared to earlier observations.Fig. 2(a) shows the measured rapid sequential ioniza-tion D + yield in false colour as a function of pump-probedelay (∆ t , also referred to as evolution time) and D + ionkinetic energy; here, θ = π /2. We are interested in themodulation in the SI yield 0 ≤ ∆ t ≤
800 fs, the signa-ture of the revival of a rotational wavepacket in the D molecule. Importantly, in Fig. 2, the orthogonality ofthe aligning pump pulse to the TOFMS (and probe po-larization) axis results in the initial maximal alignmentof the ensemble producing a minimum in the SI signal asthe majority of the molecules are aligned perpendicularto the detector.The solid circles in Fig. 2(b) represent the exper-imentally obtained high kinetic energy D + yield fromthe rapid SI of D at small internuclear separation for θ = π /2. The half- (280 fs) and full-revivals (560 fs)are clearly apparent in the results of both the integratedSI yield and the theoretical calculation, with excellentagreement observed between experiment and theory. Atsmall delays 0 ≤ ∆ t ≤
100 fs, the discrepancy betweenexperiment and theory is the result of the temporal wingsof the pump and probe pulses overlapping to produce el-liptical/circular polarization depending on ∆ t . When ∆ t ≥ ≤ ∆ t ≤
800 fs, there isevidence of fine structure in the experimentally observedwavepacket dependence, the result of the 1/4, 1/2 and3/4 revivals at respective fractions of (2 Bc ) − . Since theD + yield is recorded every 1 fs, an 11-point smoothingalgorithm applied to the integrated SI yield (Fig. 2b)suppresses statistical fluctuations and hence enhances thefidelity of the fine structure observation without loosingtemporal resolution, as shown in Fig. 2(c).Through the Fourier transform (FT) of the experimen-tal signal (Fig. 2b), we recover the frequency componentspresent in the rotational wavepacket. Fig. 2(d) and Fig.2(e) show histograms of the FT of the calculation andthe experimental data respectively, the peak positionscorresponding to the dominant beat frequencies between FIG. 3: (a) Simulated ‘quantum carpet’ [19] for an ini-tially randomly aligned ensemble of room temperature D molecules. For a particular ∆ t and θ , the false colour-scaleindicates the wavepacket probability with a thermal average.In the range 230 ≤ ∆ t ≤
290 fs, the ensemble is predicted tobe maximally aligned with the pump polarization direction.(b) The integrated rapid SI yield (normalized to the theory)over the same range of ∆ t indicates the theory describes ourresults with impressive accuracy: all the major features arereproduced. rotational states. For a diatomic molecule, ∆ J = 0, ± J in Fig 2 aregoverned by the spacing between rotational levels J and J +2. The peak heights are proportional to the productof the respective quantum amplitudes. The nuclear spinstatistics of D are also evidenced by the domination ofbeats between even J states over odd J states.The beats for J = 0 - 2, 1 - 3 and 2 - 4 occur at 5.5 THz,9.1 THz and 12.8 THz respectively. Rather intriguingly,the recent work of Ergler et al [21], investigating vibra-tional wavepackets in the D +2 molecular ion, similar rota-tional components have been observed, and attributed tothe creation of a rotational wavepacket in the molecularion D +2 . However, the polarizability anisotropy and bondlength change dramatically under ionization from D toD +2 , with corresponing beat frequencies being a factorof two lower than in D . The time-domain and equiv-alent FT frequency domain data described here are inexcellent agreement with the expectation of impulsivelyexcited rotational wavepackets formed in the neutral D molecule.The data presented in Fig. 2 describes only part ofthe rotational dynamics of D as those molecules outsidethe angular acceptance of our detector simply are not de-tected. To fully understand the emergence of rotationalorder, we turn now to predicting and measuring the rapidvariation of the angular distribution of molecules aroundthe first half-revival. Fig. 3(a) is a false-colour repre-sentation of the expected probability distribution as afunction of evolution time ∆ t and θ . When 250 ≤ ∆ t ≤
280 fs, maximal alignment is predicted: the ensem-ble is expected to be preferentially aligned in a narrowdouble-lobed distribution directed along θ = 0, π in this‘quantum carpet’. By rotating θ over the range - π to π , the axis of revival of the rotational wavepacket is ro-tated with respect to our detector axis. The experimen-tal data is shown in Fig. 3(b): the relative D + yield asa function of θ and ∆t directly reflects the thermal av-erage of the rotational wavepacket probability over thethermal average distribution of J . While some statisti-cal scatter is present, the features of the quantum carpetare well resolved. This demonstrates how the molecularaxes align through revival to create order. Up to ≃ do not necessitate supersonic cooling, thusthis demonstration of an ordered molecular ensemble il-lustrates a readily accessible quantum system.The mapping of this ultrafast rotational wavepackethas a number of implications. On the most fundamentallevel, the observation of the quantum carpet in D (con-firmed by our theoretical predictions) illustrates that aroom temperature ensemble of this theoretically impor-tant molecule can be organized into an ordered state.This can allow time- and energy-resolved measurementsof the electronic and nuclear motion with spectroscopicaccuracy. Considerable efforts are underway to perfectultrashort ( ≤ a natural choice) in a hollow waveg-uide with an ultrafast pump pulse, a locally-aligned ro-tational revival propagates along the waveguide at thespeed of light in the medium, and the molecular motioncauses a strong phase modulation due to the refractiveindex change. A second pulse (directly equivalent to ourprobe) injected into the waveguide to coincide with therotational revival is then spectrally broadened and com-pressed through the phase modulation. Importantly, ithas been demonstrated that the time dependent phaseintroduced can be accurately controlled by changing thedelay between the pump and probe pulses.The experiment was carried out at the ASTRA LaserFacility, CCLRC Rutherford Appleton Laboratory, UK,where the assistance of J. M. Smith, E. J. Divall, K. Ertel, O. Chekhlov, C. J. Hooker and S. Hawkes is gratefullyacknowledged. This work was funded by the Engineer-ing and Physical Sciences Research Council (UK). EMLEand JW acknowledge EPSRC studentships, JMcK andCRC wish to acknowledge funding from the Departmentof Employment and Learning (NI). RT acknowledges theSpanish Department of State of Education and Universi-ties, and the European Social Fund. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] R. W. Taft Jr.,
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