Far-from-equilibrium dynamics of angular momentum in a quantum many-particle system
Igor N. Cherepanov, Giacomo Bighin, Lars Christiansen, Anders Vestergaard Jørgensen, Richard Schmidt, Henrik Stapelfeldt, Mikhail Lemeshko
FFar-from-equilibrium dynamics of angular momentumin a quantum many-particle system
Igor N. Cherepanov, ∗ Giacomo Bighin, ∗ Lars Christiansen, Anders VestergaardJørgensen, Richard Schmidt,
3, 4
Henrik Stapelfeldt, † and Mikhail Lemeshko ‡ Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria Department of Chemistry, Aarhus University, 8000 Aarhus C, Denmark Max Planck Institute for Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany Munich Center for Quantum Science and Technology, Schellingstraße 4, 80799 M¨unchen, Germany (Dated: July 9, 2019)We use laser-induced rotation of single molecules embedded in superfluid helium nanodroplets toreveal angular momentum dynamics and transfer in a controlled setting, under far-from-equilibriumconditions. As an unexpected result, we observe pronounced oscillations of time-dependent molec-ular alignment that have no counterpart in gas-phase molecules. Angulon theory reveals that theseoscillations originate from the unique rotational structure of molecules in He droplets and quantum-state-specific transfer of rotational angular momentum to the many-body He environment on pi-cosecond timescales. Our results pave the way to understanding collective effects of macroscopicangular momentum exchange in solid state systems in a bottom-up fashion.
Revealing microscopic dynamics of angular momentumin solids is of key importance for designing molecularmagnets [1], spintronic and nano-magneto-mechanic de-vices [2–5], ultrafast magnetic switches and data regis-ters [6–8], as well as for controlling decoherence in solid-state qubits [9, 10]. Experimentally, being able to fine-tune the relative strength of the spin–orbit, electron–electron, and electron–lattice couplings, would allowone to separate their relative contributions to angularmomentum dynamics in magnetic systems. However,achieving such degree of control is beyond the reach ofmost solid-state experiments. Theoretically, due to thenon-Abelian algebra describing quantum rotations, theproblem of angular momentum dynamics becomes seem-ingly intractable for systems of many particles [11].Here we use a controllable quantum many-body system– isolated molecules trapped in nanodroplets of super-fluid helium – to study out-of-equilibrium angular mo-mentum dynamics. In experiment, we use a picosecondlaser pulse to suddenly align the molecules and therebybring the system far away from equilibrium. As a noveland unexpected result, we observe pronounced oscilla-tions in time-dependent molecular alignment, measuredwith a femtosecond probe pulse, that have no counter-part in gas-phase molecules. Theoretically, we develop afinite-temperature quantum many-body theory based onangulon quasiparticles, which explains the microscopicorigins of this phenomenon.Experimentally, 10-nm-diameter helium droplets, eachdoped with at most one iodine (I ), carbon disulfide(CS ) or carbonyl sulfide (OCS) molecule, are first irra-diated by a 15 ps linearly polarized alignment laser pulse,which excites a superposition of molecular rotationalstates and thus provides the molecules with a tunableamount of rotational angular momentum. This allowsus to explore the energy and lifetime of highly excited angular momentum states, inaccessible through conven-tional infrared and microwave spectroscopies, typicallyrestricted to the linear response regime. After a timedelay, t , the spatial orientation of the molecules is mea-sured by Coulomb explosion with a 40 fs intense probepulse and recording of the emission direction of fragmentions. Hereby, (cid:104) cos θ (cid:105) , the standard measure for thedegree of the molecular alignment is determined, θ be-ing the angle between the alignment pulse polarizationand the projection of the velocity vector of a fragmention on the detector [12].Figure 1 shows (cid:104) cos θ (cid:105) as a function of time forI and CS and various fluences of the alignment pulse, F align . For I , Fig. 1(a), a distinct peak in (cid:104) cos θ (cid:105) isobserved shortly after the alignment pulse at all fluences.This peak (marked by black arrows) grows in magnitudeand shifts to earlier times as F align is increased, consis-tent with previous observations of laser-induced align-ment of molecules in He droplets [13, 14]. The strik-ing new observation is, however, that the distinct oscilla-tions in (cid:104) cos θ (cid:105) with a period of ∼ −
56 ps (red ar-rows in Fig. 1) that follows the prompt alignment peak.The oscillations are already visible at F align = 1 . / cm and get substantially more pronounced at higher flu-ences, reaching a maximum for F align = 7 . / cm . At F align = 14 . / cm the oscillation magnitude slightly de-creases again.A similar effect was observed for CS molecules,Fig. 1(b). Here, the measurements were extended to abroader range of fluences, showing the emergence of os-cillations at around F align = 0 . / cm , their growth inmagnitude with increasing fluence reaching a maximumaround F align = 2 . / cm , and their gradual disappear-ance for F align ≥ . / cm . Strikingly, the period ofthe oscillations ( ∼
46 ps) is essentially constant for the (cid:104) cos θ (cid:105) traces recorded with F align between 2.8 and a r X i v : . [ phy s i c s . a t m - c l u s ] J u l ( c 3 )( c 2 )( c 1 ) ( d 8 )( d 7 )( d 6 )( d 5 )( d 4 )( d 3 )( d 2 )( d 1 ) ( b 9 )( b 8 )( b 7 )( b 6 )( b 5 )( b 4 )( b 3 )( b 2 )( b 1 ) F a l i g n = 0 . 7 J / c m F a l i g n = 0 . 4 J / c m O C S i n h e l i u m d r o p l e t s
C S i n h e l i u m d r o p l e t sI i n g a s p h a s e F a l i g n = 1 . 4 J / c m ( a 1 ) E x p e r i m e n t A n g u l o n t h e o r y L a s e r p u l s e ( a 2 )( a 5 )( a 3 )( a 4 ) t ( p s ) t ( p s ) t ( p s ) t ( p s ) I i n h e l i u m d r o p l e t s
0. Such a transformation excites an infinitenumber of bosons, accounting for a macroscopic defor-mation of the bath by the molecular impurity [17], ontop of which Eq. (2) takes into account additional single-phonon excitations. As a laser creates superpositions ofstates with different L , an appropriate time-dependentvariational ansatz to describe the evolution of the systemas described by the full Hamiltonian, ˆ H + ˆ H laser , is | Ψ i ( t ) (cid:105) = (cid:80) LM | ψ LM,i ( t ) (cid:105) .The finite-temperature Lagrangian is de-fined as a thermal expectation value, L ( t ) = Z − (cid:80) i e − βE i (cid:104) Ψ i ( t ) | i ∂ t − ˆ H − ˆ H laser | Ψ i ( t ) (cid:105) , where E i is the energy of the | i (cid:105) bath state, and Z bos ≡ (cid:80) i exp( − βE i ) is the partition function ac-counting for the finite temperature of the bath [21]. Thefinite temperature of the molecule, on the other hand,is included through the averaging over the statisticalmixture of initial equilibrium configurations, each statebeing weighted by the spin statistics and Boltzmannfactors [12]. Time evolution of the states | Ψ i ( t ) (cid:105) isobtained by numerically solving the Euler-Lagrangeequations corresponding to L ( t ). In addition, in order toaccount for the effect of centrifugal distortion relevantfor high angular momentum states [22–24], we intro-duce a phenomenological term, − D [ L ( L + 1)] , in the FIG. 2. (a) Fourier transform of the (cid:104) cos θ (cid:105) trace forI at F align = 7 . / cm (blue). Vertical red dashed linesdenote the theoretical wave packet frequencies, ν L,L (cid:48) . Inset:theoretically derived rotational energy levels (red dots) com-pared with a B ∗ L ( L +1) interpolation (black line), making therole of centrifugal distortion apparent. (b) Same as (a), butfor CS at F align = 7 . / cm (blue) and 14 . / cm (blackdashes). (c) Same as (a) but for OCS at F align = 2 . / cm . equations of motion, with D the centrifugal distortionconstant [12].The degree of alignment calculated using the angulontheory, shown in Fig. 1(a)-(b) by the red curves, repro-duces the main features observed in the experiment. Atearly times, the theory describes the prompt alignmentpeak for all three molecules. Most important, the theoryreproduces the oscillations observed in the experimentfor I and CS . In line with the experimental findings,the magnitude of these oscillations gradually increaseswith fluence, and then starts decreasing. Persistent os-cillations present for I at F align = 7 . . / cm at long times in the theoretical curves and absent in theexperiment, most likely arise due to the Hilbert spacetruncation enforced by our ansatz [25].However, what is the origin of the oscillations? Asdetailed below, the oscillations are the result of twoconditions caused by the superfluid helium environ-ment: (i) the rotational spectrum of the molecules differsstrongly from that of the gas phase case, and (ii) rota-tional angular momentum of the molecules is transferredto the elementary excitations in the He droplet on pi-cosecond timescales.To get insight into the rotational structure of themolecules we analyze the Fourier transform of the ex-perimentally measured (cid:104) cos θ (cid:105) traces. The blue solid lines in Figs. 2(a) and (b) shows the results for I andCS , at F align = 7 . / cm . In both cases the spectrumis dominated by a single peak, centered at ∼
19 GHz forI and at ∼
22 GHz for CS . These frequencies natu-rally correspond to the oscillation periods of ∼
51 ps inFig. 1(a4) and of ∼
46 ps in Fig. 1(b7). In contrast,for OCS the power spectrum does not contain a singledominant peak. As a result, the oscillations observed in (cid:104) cos θ (cid:105) [12] are less pronounced compared to those forI and CS .The molecule-laser interaction creates a rotationalwave packet, i.e. a coherent superposition of states (eachcomposed of a molecular rotational state dressed byphonons), whose total angular momentum differs by L − L (cid:48) = ± , M = M (cid:48) [19]. Peaks in thepower spectrum of Fig. 2 reflect the coherences betweensuch states. The vertical dashed lines in Fig. 2 markthe frequencies, ν L,L (cid:48) = ( E L − E L (cid:48) ) /h , corresponding tothese coherences, where the rotational energies E L aewobtained from the angulon theory. For I (CS ) seven(four) frequencies cluster around 18 −
20 GHz(21 GHz)and thereby explains the origin of the dominant peak inthe power spectra and the pronounced (cid:104) cos θ (cid:105) oscilla-tions. For OCS the 3-5 4-6 and 5-7 frequencies only liefairly close and, consequently, the (cid:104) cos θ (cid:105) oscillationsbecome less pronounced [12].The reason for the clustering of the frequencies isthe effect of the large centrifugal distortion constant ofmolecules in He droplets [26, 27]. To illustrate this explic-itly, the insets in Fig. 2 compare the calculated rotationalenergies for molecules in He droplets with the centrifugaldistortion constant included (red dots) and not included(black lines). The energies from the calculation includingcentrifugal distortion differs strongly from the pure rigidrotor energies. This stands in stark contrast to the caseof gas phase molecules where the centrifugal term is onlya small perturbation to the rigid rotor structure exceptfor superrotor states accessed by optical centrifuges [28].We have demonstrated that the pronounced (cid:104) cos θ (cid:105) oscillations for I and CS (Fig. 1) results from the pres-ence of a band of equidistant states (insets in Fig. 2).However, why are these states populated in such a ro-bust manner? Notably, why are the oscillations almostidentical in the broad range of intermediate laser flu-ences, Fig. 1(b3-b7), but disappear for the weakest andstrongest pulses, Fig. 1(b1) and (b9)?First of all, a particular feature of the rotational spec-tra shown in Figs. 2(a)-(b) is a large energy gap in thefrequencies of the coherences right after the dominantpeaks at ∼
19 GHz and ∼
22 GHz (For CS2 the nextfrequency after the ∼
22 GHz peak is ν , = 111 GHzoutside the range of Fig. 2(b)). This gap plays an impor-tant role in stabilizing the pronounced oscillations for abroad range of intermediate laser fluences, ranging from F align = 2 . . / cm for I (Fig. 1(a2–a5)), and from F align = 1 . . / cm for CS (Fig. 1(b3–b7)). In the FIG. 3. (a) Theoretical time evolution of (cid:104) cos θ (cid:105) forI neglecting the dynamical transfer of angular momentum(red dashed line) is not able to describe the experimentalobservations (black solid line). (b) Time evolution of themolecular populations, | g LM | , for L = 0 , , ,
10 and M = 0.(c) Total phonon population, (cid:80) λµ | α kλµ | (for the state with L = 2), as a function of time and of the (dimensionless) mo-mentum, ˜ k = k ( m He B ) − / , m He being the mass of a heliumatom. (d) Time evolution of the molecular angular momen-tum, (cid:104) ˆJ (cid:105) ≡ (cid:104) ( ˆL − ˆΛ ) (cid:105) in helium (red) and in the gas phase(blue). Inset: molecular populations | g LM =0 | as a function of L , in helium (red) and in the gas phase (blue), at t = 160 ps. case of a Gaussian alignment pulse, treated as a per-turbation ˆ V ( t ) to first order, the transition probabilitybetween discrete stationary states, L and L (cid:48) , is given by W L,L (cid:48) = exp( − σ ν L,L (cid:48) ) | V L,L (cid:48) | / (cid:126) , where σ is the pulseduration [12]. The exponential factor in W L,L (cid:48) , which,in the spirit of Fermi’s golden rule, we interpret as a dis-crete analogue of the phase space density, strongly sup-presses any transfer of spectral weight beyond the bandof equidistant states, i.e. the gap effectively creates abarrier in angular momentum space. In fact, when F align is increased to 14 . / cm , the spectral weight is eventransferred to lower angular momentum states, as shownin Figs. 2(b) by the black dashed line. This effect is rem-iniscent of bouncing a wave packet against a wall, which,here, promotes population of states with lower energiesand thereby lead to disappearance of the oscillations - see1(b9). Such a behavior is qualitatively different from thecase of gas-phase molecules, where increasing the laserintensity always promotes the population towards higherangular-momentum states [29]. The rotational energy structure of molecules in super-fluid helium is, however, not the only effect responsi-ble for the oscillation dynamics in (cid:104) cos θ (cid:105) . Indeed,Fig. 3(a) shows the alignment simulations for I when thedynamical transfer of angular momentum between themolecule and the many-body helium bath is neglected.(red dashed line). Poor agreement with experimental re-sults (black line) suggests such transfer plays a crucialrole.Let us start by answering the following question: as-suming that we instantaneously switch on the molecule-helium interactions, how long does it take for a moleculeto equilibrate with the helium environment and forman angulon quasiparticle? In Fig. 3(b), we present thetime evolution of molecular rotational state populations, | g LM | , cf. Eq. (2), for I after its instantaneous im-mersion in superfluid helium (in the absence of a laserfield). Crucially, we observe that the equilibrium valueof | g LM | , reached at long times, decreases as the initialangular momentum L increases. Similarly, the equili-bration time is also L -dependent. In other words, theangulon quasiparticle weight, | g LM | , decays during thefirst picoseconds of evolution, and is transferred to thepopulation of phonon amplitudes, α kλn of Eq. (2), re-sulting in a superposition of angular momentum of themolecule and an excitation in helium, see Fig. 3(c). Suchan equilibration time scale, reflecting the time scales ofthe molecule–He interactions, is always on the order oftens of ps, which is comparable to the laser pulse du-ration. This fact has far-reaching consequences: whenthe alignment pulse is on, it pushes a fraction of the ro-tational wave packet to high L -states, thereby increas-ing the respective molecular populations in | g LM | . Atthe same time, the molecule-bath interaction counter-acts by creating a field of excitations in the superfluidhelium around the molecule, thereby decreasing | g LM | .Since these two competing processes happen on the sametimescale their interplay in the first few tens of ps is cru-cial in determining the long-time alignment dynamics.The delicate interplay between laser- and bath-induceddynamics is confirmed by Fig. 3(d), which shows the timeevolution of (cid:104) ˆJ (cid:105) , where ˆJ = ˆL − ˆΛ is the rotationalangular momentum of the molecule alone (disregardingthe angular momentum of the phonon cloud), see Eq. (1),for I at F = 7 . . Already after just a few ps, whenmostly states with low L ’s are populated, the effect ofsuperfluid helium on the angular momentum dynamicsbecomes apparent, as it prevents the rotational angularmomentum of the molecule from increasing as rapidly asit would in the gas phase. The inset of Fig. 3(d), shows | g LM | as a function of L after the pulse for the gas phaseand in helium. One can see that for I in He droplets the L >
15 states are essentially not populated. This furthersupports the ‘barrier’ picture introduced above.Finally, we mention that if the alignment pulse actedfor a period considerably shorter than the molecule-helium interaction timescale, the level structure wouldnot be modified by the dynamical many-body dressing,even at large fluences, thereby making the state cluster-ing of Fig. 2 much less pronounced. In that case the bath-induced dynamics would not have the time to follow thefaster laser-induced dynamics, so that non-equilibratedstates beyond the band of equidistant states could bepopulated, resulting in the absence of oscillations. Thissituation has been observed [14], and can be interpretedas ‘detachment’ of the molecule from the surrounding he-lium shell.Similarly to how the understanding of phonon-mediated superconductivity developed from a descrip-tion of the dressing of single electrons by phonons insolids [30], our results can pave the way, within a bottom-up-approach, to understanding the collective effects ofmacroscopic angular momentum exchange in condensedmatter systems. In contrast to the past studies of angu-lar momentum transfer in collisions between molecularbeams [31, 32], our results shed light on the influence ofa solvent as well as on the timescales of angular momen-tum transfer. Finally, in the spirit of quantum simula-tion in ultracold gases [33], paramagnetic molecules inhelium nanodroplets would provide a controllable modelsystem to study angular momentum dynamics betweenthe electron spin, electron orbital, and lattice degreesof freedom in solids. Such dynamics lies at the core ofthe Einstein-de Haas and Barnett effects [34–36], whosedetailed quantum-mechanical description is still missing,one century after their discovery.The authors acknowledge stimulating discussions withFabian Grusdt, Johan Mentink and Nicol`o Defenu atvarious stages of this work. M.L. acknowledges sup-port by the Austrian Science Fund (FWF), under projectNo. P29902-N27, and by the European Research Coun-cil (ERC) Starting Grant No. 801770 (ANGULON).G.B. acknowledges support from the Austrian ScienceFund (FWF), under project No. M2461-N27. I.C. ac-knowledges the support by the European Union’s Hori-zon 2020 research and innovation programme under theMarie Sk(cid:32)lodowska-Curie Grant Agreement No. 665385.R.S. is supported by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) under Ger-many’s Excellence Strategy – EXC-2111 – 390814868.H.S. acknowledges support from the European ResearchCouncil-AdG (Project No. 320459, DropletControl). ∗ These two authors contributed equally † Corresponding author: [email protected] ‡ Corresponding author: [email protected][1] C. Calero, E. M. Chudnovsky, and D. A. Garanin, Phys.Rev. Lett. , 166603 (2005).[2] A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys. Rev. B , 014430 (2007).[3] M. Matsuo, J. Ieda, E. Saitoh, and S. Maekawa, Phys.Rev. Lett. , 076601 (2011).[4] A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.Rev. Lett. , 167201 (2005).[5] J. Tejada, R. D. Zysler, E. Molins, and E. M. Chud-novsky, Phys. Rev. Lett. , 027202 (2010).[6] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,Phys. Rev. Lett. , 4250 (1996).[7] B. Koopmans, J. J. M. Ruigrok, F. D. Longa, andW. J. M. de Jonge, Phys. Rev. Lett. , 267207 (2005).[8] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod.Phys. , 2731 (2010).[9] L. C. Bassett et al. , Science , 1333 (2014).[10] F. Donati et al. , Science , 318 (2016).[11] D. A. Varshalovich, A. N. Moskalev, and V. K. Kher-sonski, Quantum theory of angular momentum (WorldScientific, Singapore, 1988).[12] See the Supplemental Material for details.[13] D. Pentlehner, J. H. Nielsen, A. Slenczka, K. Mølmer,and H. Stapelfeldt, Phys. Rev. Lett. , 093002 (2013).[14] B. Shepperson et al. , Phys. Rev. Lett. , 203203(2017).[15] R. Schmidt and M. Lemeshko, Phys. Rev. Lett. ,203001 (2015).[16] M. Lemeshko, Phys. Rev. Lett. , 95301 (2017).[17] R. Schmidt and M. Lemeshko, Phys. Rev. X , 011012(2016).[18] M. Lemeshko and R. Schmidt, Molecular impurities in-teracting with a many-particle environment: from ultra-cold gases to helium nanodroplets, in Cold Chemistry:Molecular Scattering and Reactivity Near Absolute Zero ,edited by O. Dulieu and A. Osterwalder, (Royal Societyof Chemistry, 2017); arXiv:1703.06753[19] M. Lemeshko, R. V. Krems, J. M. Doyle, and S. Kais,Mol. Phys. , 1648 (2013).[20] P. Kramer and M. Saraceno,
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EXPERIMENTAL SETUP AND METHODS
Details of the experimental setup were described in Ref. [1] and, thus, only a brief description of the relevant detailsare given here. Helium droplets are produced using a continuous helium droplet source with stagnation conditionsof 25 bar and 14 K for I and OCS and 16 K for CS , giving ∼
10 nm diameter helium droplets [2]. The beam of Hedroplets exit the source chamber through a skimmer with a 1 mm diameter opening and enters a pickup cell containinga vapor of either CS , OCS or I molecules. The partial pressure of the molecular vapor was kept sufficiently low toensure the pickup of at most one molecule per droplet. Hereafter, the doped droplets pass through a liquid nitrogentrap that captures the majority of the effusive molecules that are not picked up by the droplets. In order to furtherreduce the contribution from effusive molecules the doped droplets pass through a second skimmer with a 2 mmdiameter opening followed by a second liquid nitrogen trap. Finally, the doped droplets enter a velocity map imaging(VMI) spectrometer placed in the middle of the target chamber. Here, the droplet beam is crossed perpendicularlyby two collinear, focused, pulsed laser beams, both with a central wavelength of 800 nmThe pulses in the first beam are used to induce alignment. These pulses have a duration of 15 ps (the measuredtemporal intensity profile is shown by the shaded blue shape in each panel in Fig. 1 in the main article) and aGaussian spotsize, ω = 30 µ m. The pulses, termed probe pulses, in the second beam, sent at time t with respectto the center of the alignment pulses, are used to measure the spatial orientation of the molecules. This occurs byCoulomb explosion of the molecules and recording of the direction of the fragment ions recoiling along the internuclearaxis of their parent molecule (IHe + for I , S + for CS and OCS). These pulses have a duration of 40 fs, spotsize, ω = 22 µ m, and a peak intensity 8 × W / cm .The VMI spectrometer projects the ions produced by the probe pulse onto a 2-dimensional detector. The anglebetween the position of an ion hit and the polarization direction of the alignment beam, contained in the detectorplane, is denoted θ . The degree of alignment is characterized by (cid:104) cos θ (cid:105) , a standard measure used in manyprevious works [3]. The 2-dimensional ion images are recorded at a larger number of delays between the alignmentand the probe pulse. Hereby the time-dependent (cid:104) cos θ (cid:105) curves, displayed in Fig. 1 in the main article, areobtained. THEORETICAL FRAMEWORK: OUT-OF-EQUILIBRIUM DYNAMICS OF ANGULONS IN THESTRONG COUPLING REGIME
Our theoretical approach is based on the angulon quasiparticle [4], a quantum rotor interacting with many-particlebosonic bath. Originally derived to describe a molecule immersed in a dilute BEC, the theory has been extended todescribe phenomenologically molecules trapped inside much denser and strongly-interacting solvents, such as superfluid He nanodroplets. In particular, angulon theory showed a good agreement in describing experimental observations inhelium droplets, namely, renormalization of rotational constants [5], impulsive molecular alignment [1, 6] and selectivebroadening of spectral lines [7]. Heavy molecules (with rotational constant B (cid:46) − ) such as I , CS , and OCSconsidered in the present paper are known to strongly interact with surrounding helium [5]. This is mostly due tothe fact that the kinetic energy of molecular rotation is comparable to the potential energy of the molecule-heliuminteraction, as opposed to light molecules. Accordingly, we make use of the strong coupling angulon theory developedin Ref. [8], accounting for an infinite number of helium excitations. The angulon Hamiltonian
We start from the Hamiltonian defined in the molecular (body-fixed) coordinate system, co-rotating with themolecule, describing a linear molecule interacting with a bosonic bathˆ H = B ( ˆL − ˆΛ ) + (cid:88) kλµ ω k ˆ b † kλµ ˆ b kλµ + (cid:88) kλ V λ ( k ) (cid:0) ˆ b † kλ + ˆ b kλ (cid:1) . (S1)Here we used the notation (cid:80) k = (cid:82) dk , set (cid:126) ≡ B = (cid:126) / (2 I ) is the gas phase rotational constant of the molecule,with I is the molecular moment of inertia, and ˆL the total angular-momentum operator acting in the frame co-rotatingwith the molecule. Note that the ˆL operator acts on symmetric top states, since the linear-rotor molecule molecule isturned into an effective symmetric top by dressing of the boson field [8]. Moreover ˆΛ = (cid:88) kλµν ˆ b † kλµ σ λµν ˆ b kλν (S2)is the angular-momentum operator for the bosonic helium bath, whose excitations are described by the creation(annihilation) operators, ˆ b † kλµ (ˆ b kλµ ), respectively. Furthermore, σ λ = { σ λ − , σ λ , σ λ +1 } denotes the vector of the angularmomentum matrices fulfilling the SO (3) algebra in the representation of angular momentum λ . For convenience, thecreation and annihilation operators are cast in the angular momentum basis [9], with k the magnitude of linearmomentum, λ angular momentum of the boson, and µ the angular momentum projection onto the laboratory z -axis. Finally, ω k gives the empirical dispersion relation of superfluid helium [10]. The details of the molecule-heliuminteraction are encoded in the potential V λ ( k ) V λ ( k ) = (cid:115) n k πmω k (cid:90) drr f λ ( r ) j λ ( kr ) (S3)where n is the particle density of helium atoms, m is the mass of a He atom, and j λ ( kr ) are the spherical Besselfunctions. The form factor f λ ( r ) determines the components of the expansion of the molecule-He potential energysurface (PES) into the spherical harmonics. Such a treatment is fully justified only for a low value of helium density.However, we previously demonstrated [5–7] that fine details of the molecule-helium potential are irrelevant, and theproblem can be approached from a phenomenological perspective by scaling the coupling constants, V λ ( k ), accordingto the particle density in helium. For the sake of simplicity we choose Gaussian form-factors f λ = u λ (2 π ) − / exp − r r λ (S4)as model potentials. Here r λ , the interaction range, is set to the distance of the global minimum in the molecularPES, whereas u λ , the interaction strength, is fixed phenomenologically to reproduce known properties of the molecule-helium interaction, More details on the model parameters will be given in a subsequent Section. The Hamiltonian (S1)can be diagonalized in the limit of the slowly rotating molecule B → H = ˆ U − ˆ H ˆ U (S5)where ˆ U = exp (cid:34)(cid:88) kλ V λ ( k ) ω k + Bλ ( λ + 1) (ˆ b † kλ − ˆ b kλ ) (cid:35) . (S6)After the transformation, the bosonic vacuum | (cid:105) bos becomes the ground state of the Hamiltonian (S5). It is alsoworth noting that for a given total angular momentum state | LM (cid:105) , the ground state of the Hamiltonian (S1), | ψ LM (cid:105) = ˆ U | (cid:105) bos | LM (cid:105) mol , involves an infinite number of bath excitations and describes the macroscopic deformationof bosons caused by molecular rotation. In a simple picture, one can regard this state as a shell of bosons co-rotatingalong with the molecule. In the case of a helium nanodroplet playing a role of the many-body bath, such a deformationis known as a nonsuperfluid helium solvation shell [11]. In the absence of external fields, the system wavefunction canbe described by the following time-dependent variational Ansatz | ψ LM,i ( t ) (cid:105) = ˆ U (cid:18) g LM ( t ) | LM (cid:105) mol + (cid:88) kλn α LMkλn ( t ) | LM n (cid:105) mol ˆ b † kλn (cid:19) | i (cid:105) bos . (S7)where | i (cid:105) represents the many-body states of the bath, L and M are constants of motion corresponding to thetotal angular momentum of the system and its projection onto the laboratory z -axis, n defines the projection of thetotal angular momentum onto the molecular axis, g LM ( t ) and α kλn ( t ) are time-dependent variational coefficients. Adetailed derivation of the Hamiltonian and its properties can be found in Refs. [8, 9]. The Lagrangian
The Lagrangian of the system in the absence of the laser is L = 1 Z bos (cid:88) i e − βE i (cid:104) ψ LM,i ( t ) | i∂ t − ˆ H | ψ LM,i ( t ) (cid:105) (S8)the index i running over the energy eigenstates of the bosonic Fock space, the | i (cid:105) state having E i energy, Z bos ≡ (cid:80) i e − βE i being the partition function for the bosonic bath. Here and below, the notation (cid:68) f (ˆ b † , ˆ b ) (cid:69) T ≡ Z bos (cid:88) i e − βE i (cid:104) i | f (ˆ b † , ˆ b ) | i (cid:105) (S9)indicates a finite-temperature bosonic expectation value. For example, one sees immediately that (cid:68) ˆ b kλn ˆ b † k (cid:48) λ (cid:48) n (cid:48) (cid:69) T = δ ( k − k (cid:48) ) δ λλ (cid:48) δ nn (cid:48) (1 + f BE ( ω k )) f BE ( x ) = 1 e βx − f BE , β ≡ ( k B T ) − . Generalizations to an arbitrary number ofbosonic operators are readily found by means of Wick’s theorem. Substituting the Hamiltonian (S1) into Eq. (S8)we obtain L = ig ∗ LM ( t ) ˙ g LM ( t ) + i (cid:88) kλn α ∗ LMkλn ( t ) ˙ α LMkλn ( t )(1 + f BE ( ω k )) + L A + L B + L C (S11)where L A = − BL ( L + 1) | g LM ( t ) | , (S12) L B = − B (cid:88) kλn g ∗ LM ( t ) α LMkλn ( t ) V λ ( k ) W λ ( k ) (cid:112) λ ( λ + 1) L ( L + 1) δ n ± ++ 2 B (cid:88) kλ λ ( λ + 1) α LMkλ ( t ) g ∗ LM ( t ) V λ ( k ) W λ ( k ) (1 + f BE ( ω k )) f BE ( ω k ) , (S13)and L C = − (cid:88) kλn BL ( L + 1) | α LMkλn ( t ) | (1 + f BE ( ω k )) − (cid:88) kλn (cid:20) ω k + Bλ ( λ + 1) (cid:21) | α LMkλn ( t ) | (1 + f BE ( ω k )) + − B (cid:88) kk (cid:48) λλ (cid:48) n α ∗ LMkλn ( t ) α LMk (cid:48) λ (cid:48) n ( t ) V λ ( k ) W λ ( k ) V λ (cid:48) ( k (cid:48) ) W λ (cid:48) ( k (cid:48) ) (cid:112) λ ( λ + 1) (cid:112) λ (cid:48) ( λ (cid:48) + 1) δ n ± (1 + f BE ( ω k ))(1 + f BE ( ω k (cid:48) ))++ 2 B (cid:88) kλnν α LMkλn ( t ) α ∗ LMkλν ( t ) η Lνn σ λνn (1 + f BE ( ω k )) . (S14)Note that W λ ( k ) = ω k + Bλ ( λ + 1), and the angular momentum coupling term is given by η Lnν σ λnν = n δ nν + 12 (cid:112) λ ( λ + 1) − ν ( ν + 1) (cid:112) L ( L + 1) − ν ( ν + 1) δ nν +1 ++ 12 (cid:112) λ ( λ + 1) − ν ( ν − (cid:112) L ( L + 1) − ν ( ν − δ nν − (S15) The equations of motion
The equations of motion are given by ddt ∂ L ∂ ˙ x i − ∂ L ∂x i = 0 (S16)where x i = { g LM , α LMkλn } . By substituting the Lagrangian (S11) into Eq. (S16) we obtain the system of integro-differential equations for g LM ( t ) and α LMkλn ( t ) i ˙ g LM ( t ) = (cid:20) BL ( L + 1) − D [ L ( L + 1)] (cid:21) g LM ( t )++ 2 B (cid:88) kλn V λ ( k ) W λ ( k ) (cid:112) λ ( λ + 1) L ( L + 1) δ n ± α LMkλn ( t )(1 + f BE ( ω k ))++ 2 B (cid:88) kλ λ ( λ + 1) α LMkλ ( t ) V λ ( k ) W λ ( k ) (1 + f BE ( ω k )) f BE ( ω k ) (S17a) i ˙ α LMkλn ( t ) = (cid:20) BL ( L + 1) − D [ L ( L + 1)] + ( Bλ ( λ + 1) + ω k )(1 + f BE ( ω k )) (cid:21) α LMkλn ( t )++ B V λ ( k ) W λ ( k ) (cid:112) λ ( λ + 1) (cid:88) k (cid:48) λ (cid:48) V λ (cid:48) ( k (cid:48) ) W λ (cid:48) ( k (cid:48) ) (cid:112) λ (cid:48) ( λ (cid:48) + 1) δ n ± α LMk (cid:48) λ (cid:48) n ( t )(1 + f BE ( ω k (cid:48) ))++ B V λ ( k ) W λ ( k ) (cid:112) λ ( λ + 1) L ( L + 1) δ n ± g LM ( t ) + 2 Bλ ( λ + 1) g LM ( t ) V λ ( k ) W λ ( k ) δ n f BE ( ω k )+ − B (cid:88) ν η Lnν σ λnν α LMkλν ( t )(1 + f BE ( ω k )) (S17b)Here we introduced a phemenological term − D [ L ( L + 1)] , with D the centrifugal distortion constant, accounting fornon-rigidity of the system being in the highly-excited total angular momentum states, acting up to a cutoff L max .For molecules trapped in helium droplets D is four orders of magnitude larger than for the gas phase [12–14]. Thecentrifugal correction to the spectrum becomes non-negligible for L (cid:38) MODELLING THE INTERACTION OF A MOLECULE WITH A LASER PULSE‘Laser part’ of the Lagrangian
Since the laser will create a superposition of states with different L , an appropriate wavefunction for the systemdescribed by the full Hamiltonian given in the main text, including the laser-molecule interaction, is | Ψ i ( t ) (cid:105) = (cid:88) LM | ψ LM,i ( t ) (cid:105) . (S18)The interaction of the molecule with a linearly polarized far-off-resonant laser is modelled, in the laboratory frame ofreference, by the following term [15] ˆ H laser = −
14 ∆ αE ( t ) cos ˆ θ . (S19)One can use the unitary transformation ˆ S introduced in Ref. [8] to express it in the molecular frame of referencewhere the angulon Hamiltonian (S1) and the variational Ansatz (S18) are defined, obtaining ˆ H laser = ˆ S − ˆ H laser ˆ S .Then, the molecular-laser interaction will enter the Lagrangian of the system as the additional term L laser = − Z bos (cid:88) i e − βE i (cid:104) Ψ i ( t ) | ˆ H laser | Ψ i ( t ) (cid:105) , (S20)which yields the ‘laser Lagrangian’ L laser = 14 ∆ αE (cid:90) dΩ cos θ Z bos (cid:88) i e − βE i (cid:104) Ψ i ( t ) | Ω (cid:105) (cid:104) Ω | Ψ i ( t ) (cid:105) . (S21)The selection rules for the Clebsch-Gordan coefficients imply that M = M (cid:48) , as expected since the electric field isconserving the z -component of angular momentum. This means that the time evolution does not mix states withdifferent values of M . After a straightforward calculation one gets that the time-evolution on a manifold with defined M is described by the following Lagrangian L laser = 16 ∆ αE ( (cid:88) LL (cid:48) g ∗ LM g L (cid:48) M (cid:114) L + 12 L (cid:48) + 1 C L (cid:48) M LM C L (cid:48) L ++ (cid:88) LL (cid:48) (cid:88) kλn α LM ∗ kλn ( t ) α L (cid:48) Mkλn ( t )(1 + f BE ( ω k )) (cid:114) L + 12 L (cid:48) + 1 C L (cid:48) M LM C L (cid:48) n Ln ++ 12 (cid:88) L g ∗ LM g LM + 12 (cid:88) L (cid:88) kλn α LM ∗ kλn ( t ) α LMkλn ( t )(1 + f BE ( ω k ))) (S22)via the corresponding Euler-Lagrange equations of motion. ‘Laser part’ of the equations of motion In order to simplify the notation let us define C ≡
16 ∆ αE Q LL (cid:48) MN ≡ (cid:114) L + 12 L (cid:48) + 1 C L (cid:48) M LM C L (cid:48) N LN (S23)so that L laser = C ( (cid:88) LL (cid:48) g ∗ LM g L (cid:48) M Q LL (cid:48) M + (cid:88) LL (cid:48) (cid:88) kλn α LM ∗ kλn ( t ) α L (cid:48) Mkλn ( t )(1 + f BE ( ω k )) Q LL (cid:48) Mn ++ 12 (cid:88) L g ∗ LM g LM + 12 (cid:88) L (cid:88) kλn α LM ∗ kλn ( t ) α LMkλn ( t )(1 + f BE ( ω k ))) (S24)Then ∂ L laser ∂g ∗ LM = C (cid:88) L (cid:48) Q LL (cid:48) M g L (cid:48) M + C g LM (S25)and ∂ L laser ∂α LM ∗ kλn = C (cid:88) L (cid:48) ≥ Q LL (cid:48) Mn α L (cid:48) Mkλn (1 + f BE ( ω k )) + C α LMkλn (1 + f BE ( ω k )) (S26)to be added to the equations of motion of Eq. (S17). It is important noting that, due to the selection rules imposedby Q LL (cid:48) Mn , the summation on L (cid:48) of Eq. (S25) and Eq. (S26) extends only on L (cid:48) = L − , . . . , L + 2. Also in Eq.(S26) one needs to introduce the condition L (cid:48) ≥ α LMkλn vanish for L ≤ CALCULATION OF THE PROJECTED COSINE, cos θ D The experimentally-measured molecular alignment is defined in terms of the operators describing the molecularangles ˆ θ and ˆ φ as cos ˆ θ D = cos ˆ θ cos ˆ θ + sin ˆ θ sin ˆ φ . (S27)In order to calculate the matrix elements of this operator in the angular momentum basis (cid:104) j (cid:48) m (cid:48) | cos ˆ θ D | jm (cid:105) weexpand it as follows cos ˆ θ D = (cid:88) λµ f λµ Y λµ (ˆ θ, ˆ φ ) (S28)and clearly the inverse expansion, giving the coefficients f λµ in terms of cos θ D , is f λµ = (cid:90) dΩ Y λµ (Ω) cos θ D . (S29)Here the idea is that, rather than dealing with the cumbersome expectation value in Eq. (S27), one can calculate aninfinite series of expectation values of the type (cid:104) j (cid:48) m (cid:48) | Y λµ (ˆ θ, ˆ φ ) | j (cid:48) m (cid:48) (cid:105) – inserting the expansion in Eq. (S28) – which willbe effectively limited by an angular momentum cutoff. The calculation of the expectation value (cid:104) j (cid:48) m (cid:48) | cos ˆ θ D | jm (cid:105) is then simplified to (cid:104) j (cid:48) m (cid:48) | cos ˆ θ D | jm (cid:105) = (cid:88) λµ f λµ (cid:104) j (cid:48) m (cid:48) | Y λµ (ˆ θ, ˆ φ ) | jm (cid:105) == (cid:88) λµ f λµ ( − m (cid:48) (cid:114) (2 j + 1)(2 j (cid:48) + 1)(2 λ + 1)4 π (cid:18) j j (cid:48) λ (cid:19) (cid:18) j j (cid:48) λm − m (cid:48) µ (cid:19) . (S30)We still have to calculate f λµ through Eq. (S27). Since, as already noted, the laser does not break the symmetry forrotations around the z -axis, one will always have to deal with the m = m (cid:48) case, which implies µ = 0 in Eq. (S30).Then the coefficients f λ are given by f λ = (cid:114) λ + 14 π (cid:90) +1 − d x (cid:90) π d φ P λ ( x ) x x + (1 − x ) sin φ . (S31)By noting that (cid:82) π φx +(1 − x ) sin φ = π | x | one gets f λ = 4 π (cid:114) λ + 14 π (cid:90) d xP λ ( x ) P ( x ) if λ is even (S32)whereas for odd λ one sees that f λ = 0 due to symmetry arguments. The integral in Eq. (S32) can be evaluated inclosed a form [16] – note that it is not the orthogonality relation for Legendre polynomials – and it finally leads tothe following closed form for f λ , f λ = π (cid:113) λ +14 π ( − λ λ !2 λ ( λ − λ +2)( λ !) λ even0 λ odd (S33) ENSEMBLE AND FOCAL AVERAGING
After calculating the time-evolution of the wavefunction | Ψ i ( t ) (cid:105) by solving the Euler-Lagrange equations of motion,one may want to calculate expectation values for different observables. For concreteness’ sake let us consider a certainsolution to the equations of motion | Ψ i ( t ) (cid:105) L ,M obtained starting from an initial purely-molecular wavefunction | Ψ i ( t = 0) (cid:105) = | LM (cid:105) , that will be dynamically dressed; let us also consider a generic operator ˆ O . Taking theexpectation value of ˆ O following Ref. [17], one accounts for the finite temperature of the bosonic environment as (cid:104) ˆ O (cid:105) L ,M ( t ) = 1 Z bos (cid:88) i e − βE i (cid:104) Ψ i ( t ) | ˆ O | Ψ( t ) (cid:105) L ,M . (S34)However, the energy scale corresponding to the rotational constant for all the three molecules we consider is consid-erably smaller than the temperature of the bath T bath = 0 . P LM = e − βB ∗ L ( L +1) Z (S35)with Z = (cid:80) LM P LM , B ∗ being the effective rotational constant as derived in Ref. [5], β = ( k B T ) − . In additionto this, one also has to take into account the fact, generally, the even and odd states will have different relativeabundances. To this extent, the occupation probabilities of Eq. (S35) need to be calculated separately for the evenand odd components, as P even LM = e − βB ∗ L ( L +1) Z even Z even = (cid:88) L even ,M P even LM (S36)and similarly for P odd LM and Z odd . Practically, both partition functions can be calculated up to some angular momentumcutoff L max , for the molecules we consider in the present paper a cutoff L max = 8 which suffices to always includemore than 99% of the ensemble. Going back to the expectation value of the operator ˆ O , we define the averages overthe even and odd components as (cid:104) ˆ O (cid:105) even ( t ) = (cid:88) L ,M L is even P even L ,M (cid:104) ˆ O (cid:105) L ,M ( t ) (S37)and (cid:104) ˆ O (cid:105) odd ( t ) = (cid:88) L ,M L is odd P odd L ,M (cid:104) ˆ O (cid:105) L ,M ( t ) . (S38)Introducing the abundances of the even and odd states, let us call them A even and A odd , respectively, one gets: (cid:104) ˆ O (cid:105) ( t ) = A even (cid:104) ˆ O (cid:105) even ( t ) + A odd (cid:104) ˆ O (cid:105) odd ( t ) A even + A odd (S39)which corresponds to the expectation value of the operator ˆ O taking into account finite-temperature effects for thebath and for the molecule. Finally, we performed a weighted average of five different peak intensities defined by themeasured spot sizes of the alignment and the probe pulse laser beams. TRANSITION PROBABILITY UNDER A GAUSSIAN TIME-DEPENDENT PERTURBATION
Here we derive the transition probability W L,L (cid:48) given in the main text. Let us consider a time-dependent pertur-bation whose time-dependence is Gaussianˆ V ( t ) = ˆ V √ πσ exp( − t / (2 σ )) . (S40)The probability of transition from an initial stationary state | i (cid:105) to a final stationary state | f (cid:105) under the action of ˆ V ( t )is given by [18] W fi = 1 (cid:126) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) + ∞−∞ V fi ( t ) e i ω fi t d t (cid:12)(cid:12)(cid:12)(cid:12) (S41)where ω fi ≡ ( E f − E i ) / (cid:126) and V fi ( t ) is a matrix element of ˆ V ( t ). After carrying out the integral over d t one has W fi = | V fi | (cid:126) exp (cid:0) − σ ω (cid:1) (S42)and using L (cid:48) , L as initial (final) states, respectively, one has the expression for W L,L (cid:48) given in the main text.
PARAMETERS OF THE MODEL
For most molecules f λ =2 ( r ) is the dominant anisotropic term in the molecule-helium PES expansion (S3) [19–24].Therefore, in order to simplify the final equations of motion (S17), one can keep only the many-body excitationswith λ = 2. Such an approach has been successfully employed for predicting renormalization of rotational constantsleading to a good quantative agreement with experimental data [5]. All parameters used for simulations of moleculardynamics are listed in Table S1. The constant accounting for centrifugal correction, D , is fixed such as to reproduceexperimentally the measured value of D ∗ for OCS and calculated by the following empirical relationship for I andCS [25]. D ∗ = 0 . B ∗ . (S43)the formula being valid for B ∗ and D ∗ measured in cm − . We set r , the parameter characterizing the range of themolecule-helium interaction, using the position of the global minimum of the molecule-He PES as a reference. Thestrength of interaction, u , was chosen as to reproduce the renormalization of rotational constants in helium droplet, B ∗ /B . The particle density of helium atoms in the center of the droplet and their temperature are 0.022 ˚A − and0 .
38 K, respectively [2]. The mass of a He atom is m = 4 .
03 amu. B (GHz) B ∗ /B D ∗ (GHz) u (THz) r (˚A) ∆ α (˚A ) Spin abundance(even:odd)I . · − . · − . · − [11] 0.50 3.8 [23] 4.7 [32] 1:1TABLE S1. Parameters used in the simulations. TIME-DEPENDENT ALIGNMENT FOR OCS MOLECULES
Here we present the experimental and theoretical results for the time-dependent alignment of OCS, analogous tothat of Fig. 1 of the main text. Figure S1 shows (cid:104) cos θ (cid:105) as a function of time for OCS and a series of differentfluences of the alignment pulse, F align . Here, the oscillations are observed at intermediate fluences, most clearlyat F align = 2 . / cm . Compared to I and CS (Fig. 1 of the main text), however, the oscillations are much lesspronounced and disappear around F align = 5 . / cm . One can see that the results of the angulon quasiparticle theory(red solid line) reproduce all the main features observed in experiment: the peak of prompt alignment (black arrows),weak oscillations (red arrows), as well as the revival structure (blue arrows). [1] B. Shepperson et al. , J. Chem. Phys. , 013946 (2017).[2] J. P. Toennies and A. F. Vilesov, Ang. Chem. Int. Ed. , 2622 (2004).[3] A. A. Søndergaard, B. Shepperson, and H. Stapelfeldt, J. Chem. Phys. , 013905 (2017).[4] M. Lemeshko and R. Schmidt, Phys. Rev. Lett , 203001 (2015).[5] M. Lemeshko, Phys. Rev. Lett. , 95301 (2017).[6] B. Shepperson et al. , Phys. Rev. Lett. , 203203 (2017).[7] I. N. Cherepanov and M. Lemeshko, Phys. Rev. Materials , 35602 (2017).[8] R. Schmidt and M. Lemeshko, Phys. Rev. X , 11012 (2016).[9] M. Lemeshko and R. Schmidt, Molecular impurities interacting with a many-particle environment: from ultracold gasesto helium nanodroplets, in Cold Chemistry: Molecular Scattering and Reactivity Near Absolute Zero , edited by O. Dulieuand A. Osterwalder, (Royal Society of Chemistry, 2017); arXiv:1703.06753[10] R. J. Donnelly and C. F. Barenghi, J. Phys. Chem. Ref. Data , 1217 (1998).[11] S. Grebenev et al. , J. Chem. Phys. , 4485 (2000).[12] K. K. Lehmann, J. Chem. Phys. , 4643 (2001).[13] J. Harms, M. Hartmann, J. P. Toennies, A. F. Vilesov, and B. Sartakov, J. Mol. Spectrosc. , 204 (1997).[14] M. Hartmann, R. E. Miller, J. P. Toennies, and A. Vilesov, Phys. Rev. Lett. , 1566 (1995).[15] M. Lemeshko, R. Krems, J. Doyle, and S. Kais, Mol. Phys. , 1648 (2013).[16] W. E. Byerly, An Elementary Treatise on Fourier’s Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, withApplications to Problems in Mathematical Physics
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E x p e r i m e n t A n g u l o n t h e o r y L a s e r p u l s e F a l i g n = 0 . 7 J / c m F a l i g n = 0 . 4 J / c m O C S i n h e l i u m d r o p l e t s C S i n h e l i u m d r o p l e t sI i n g a s p h a s e F a l i g n = 1 . 4 J / c m I i n h e l i u m d r o p l e t s( a 1 )( a 5 )( a 3 )( a 2 )( a 4 ) t ( p s ) F a l i g n = 1 . 4 J / c m t ( p s ) t ( p s ) t ( p s )