Rotational Echoes: Rephasing of Centrifugal Distortion in Laser

Letter pubs.acs.org/JPCL

Rotational Echoes: Rephasing of Centrifugal Distortion in LaserInduced Molecular Alignment Dina Rosenberg,†,‡,⊥ Ran Damari,†,‡,⊥ Shimshon Kallush,§,∥ and Sharly Fleischer*,†,‡ †

Raymond and Beverly Sackler Faculty of Exact Sciences, School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel Tel-Aviv University Center for Light-Matter Interaction, Tel Aviv 6997801, Israel § Department of Physics and Optical Engineering, ORT Braude College, P.O. Box 78, Karmiel 21982, Israel ∥ The Fritz Haber Research Center and The Institute of Chemistry, The Hebrew University, Jerusalem 91904, Israel ‡

S Supporting Information *

ABSTRACT: We study and demonstrate the rephasing property of the echo response in a multilevel rotational system of iodomethane via long time-resolved optical birefringence measurements. The strong centrifugal distortion of iodomethane is utilized as a dephasing mechanism imprinted on the echo signal and is shown to rephase throughout its evolution. The dependence of the echo signal amplitude on the driving pulses’ intensities is theoretically and experimentally explored. The analogy to Hahn’s spin echoes is discussed, and a quantum-mechanical version of Hahn’s track runners is provided for the case of multilevel rotational system. ince its first demonstration by E. L. Hahn in NMR1 in 1950 and its adoption in optics2 in 1964, the phenomenon of echoes has been realized and utilized in a wide range of imaging and spectroscopic applications such as 2D electronic,3−5 vibrational,6−9 and rotational spectroscopy.10−13 Their ability to distinguish between homogeneous and inhomogeneous broadening has made echo experiments highly important in modern nonlinear spectroscopy and imaging.14 Whereas echo experiments are typically treated in the quantum-mechanical framework of molecular degrees of freedom, the underlying physics of the echo phenomenon was beautifully described using classical physics and experimentally demonstrated in a series of works recently.10−12 In this work we experimentally demonstrate the rephasing property of the echo phenomenon in a multilevel rotational system. We utilize the strong centrifugal distortion of iodomethane (CH3I) as a rotational dephasing mechanism and monitor the field-free evolution of the echo signal as it rephases and dephases as a function of time. The paper is organized as follows: Section I is a brief introduction to laserinduced rotational dynamics with an emphasis on the dephasing due to centrifugal distortion. Section II focuses on the similarities and dissimilarities of the echo response in a multilevel rotational system and two-level spin system. Section III describes the main experimental results of rotational echoes induced by two time-delayed laser pulses showing the rephasing and dephasing echo dynamics. In Section IV, we provide a quantum-mechanical version of Hahn’s famous track runners for the case of rotational echoes. Section I. Laser-induced molecular alignment in the gas phase is a vastly-researched field in the last three decades. Since the

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early works of Felker, Baskin, and Zewail,15,16 rotational coherence spectroscopy has been used to obtain highly accurate rotational coefficients of gas-phase molecules.17−23 In parallel to the advances in high-resolution spectroscopy, laser-induced molecular alignment emerged as a key player in ultrafast molecular dynamics, providing means to control the angular distribution of molecules in the gas phase and under field-free conditions. On this front, laser-aligned molecular ensembles have yielded a rich variety of applications ranging from molecular-phase modulators24,25 and ultrashort pulse compression26 to controlled molecular ionization27 and high harmonic generation (HHG)28−30 and have even been suggested for quantum information processing.31,32 (For recent review articles, see refs 33−36 and references therein.) Alignment and orientation of the molecular ensemble associate the molecular frame and the laboratory frame, an essential component in ultrafast molecular dynamics such as timeresolved ultrafast X-ray diffraction, electron diffraction, and HHG spectroscopy.37−41 The physics of molecular alignment has been thoroughly explored and is provided here in brief for the sake of clarity: A femtosecond (∼100 fs) pulse with linear polarization imparts torque to molecules via their anisotropic polarizability Δα = α∥ − α⊥ (difference in polarizability components parallel and perpendicular to the molecular axis). As a result, the molecules rotate toward the polarization direction of the pulse (taken as Received: August 22, 2017 Accepted: September 26, 2017 Published: September 26, 2017 5128

DOI: 10.1021/acs.jpclett.7b02215 J. Phys. Chem. Lett. 2017, 8, 5128−5135

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The Journal of Physical Chemistry Letters the z axis) and their angular distribution peaks along the z direction shortly after the excitation, following which the molecules dephase due to their field-free rotation. The Hamiltonian that describes the rotational dynamics is given 2 2 by Ĥ = L̂ /2I + V̂ (t ) with the first term, L̂ /2I, as the field1 free Hamiltonian and V̂ (t ) = − 4 Δα |E(t )|2 cos2 θ as the fieldmolecule interaction term. L̂ is the angular momentum operator, I is the moment of inertia, |E(t)|2 is the pulse intensity envelope, and θ is the angle between the rotor axis and the pulse polarization (z axis). The quantization of angular momentum and corresponding rotational energy spectrum EJ,m = hBcJ(J + 1) (J is the rotational quantum number and B is the rotational coefficient of the molecule) manifests in periodic dynamics termed “quantum rotational revivals”.42,43 Thus, following its rotational excitation, the molecular ensemble manifests periodic recurrences of the anisotropic angular distribution, observed as a series of alignment events separated by the rotational revival period Trev = (2Bc)−1, where B is in units of cm−1 and c is the speed of light. However, because of their finite rigidity, real molecules are amenable to the centrifugal force that acts to stretch them with the increase in angular momentum (increasing J), and hence a correction to the level energy, known as “centrifugal distortion” (CDN) was added such that EJ,m = hBcJ(J + 1) − hDcJ2(J + 1)2. D is the centrifugal coefficient (in cm−1), with typical B/D in the range of 105 to 107. Experimental Section. The alignment dynamics of CH3I gas was measured using the weak-field polarization detection technique.44,45 A 100 fs pulse is split to form an optical pump beam (λ = 800 nm) and a weak probe pulse (λ = 400 nm, via SHG in a BBO crystal). The pump beam is passed through a Michelson interferometer with computer-controlled delay stage on one of its arms to yield two collinear pump pulses with a controlled delay apart (P1 and P2). The pump and probe beams, polarized at 45°, are recombined by a dichroic mirror and are collinearly focused into the sample cell. The diameter of the pumps beam is reduced (to 4 mm at the surface of the f = 150 mm focusing lens) by an iris to ensure good overlap with the probe beam waist at the center of the sample cell. The pump pulses’ energies are selectively varied in the range of 1−20 μJ/pulse by a λ/2 waveplate situated before the Michelson interferometer and a λ/4 on one of the interferometer arms. Both pump pulses pass through a polarizer at the output of the interferometer, with few microjoules for each pump pulse, set to avoid strong-field ionization. The anisotropic angular distribution of the sample manifests as optical birefringence and polarization changes of the probe pulse that are recorded by a polarizer and a photodiode using lock-in detection. (See the Supporting Information, Section SI.1 for detailed schematics). The observed signal is given by [ΔI/I](t) ∝ [⟨cos2θ⟩(t)1/3], where θ is the molecular axis angle relative to the polarization angle of the pump beams.46−48 Figure 1 depicts the time-resolved alignment signal of CH3I (20 Torr, ambient temperature 295 K) following excitation by a single laser pulse (at t = 0). CH3I is a prolate symmetric-top with a moment of inertia about the C−I axis (3.26 amu·A2) much smaller than its two other axes (67.4 amu·A2).49 The ΔK = 0 selection rule for the Raman excitation, and the low K values (relative to J values) at the ambient temperature of the experiment (295 K) result in a qualitatively similar rotational dynamics to that of linear molecules.13,46,50,51 (See SI.2 for numerical simulations of the

Figure 1. Time-resolved birefringence measurement of CH3I at the pressure of 20 Torr. The revival period of CH3I is Trev = 66.7 ps (B = 0.25 cm−1) with modulations separated by Trev/2. Strong centrifugal distortion of CH3I results in “signal smearing” transforming the nearly single oscillation observed at the first revival (1Trev) to multiple oscillations at larger delays (see enlarged signals at 2, 3, and 4Trev).

alignment dynamics of CH3I, simulated as a linear rotor versus symmetric top with excellent agreement.) Thus to simplify the theoretical treatment and analytic expressions of the CDNinduced phases, CH3I is treated as a linear rotor with B = 0.25 cm−1, and D = 2.1 × 10−7 cm−1 throughout this work. Following its rotational excitation at t = 0, the ensemble shows transient recurrences of the alignment, separated by Trev/ 2. CDN manifests as the “smearing” of the alignment signals with gradually increasing number of oscillations and has been explored in details previously (e.g., in refs 19 and 20). Next, we provide a qualitative analysis of the CDN ramifications to the alignment signal, starting from a general expression of the rotational wavepacket |ψJ ⟩ = ∑J cJ |J ⟩ formed via the interaction with the short laser pulse. (cJ are the expansion coefficients of the eigenstates |J⟩.) Because cos2 θ serves as the operator for both the excitation and the alignment observable, coupling rotational states with Δm = 0 (m is the magnetic quantum number), we restrict the discussion to the principal angular momentum quantum number J without compromising the underlying physics of CDN. Once created,ψJ evolves under the 2 field-free Hamiltonian Ĥ = L̂ /2I, such that |ψJ ⟩(t ) =

∑ cJ ·e−iHt̑ /ℏ|J ⟩ = ∑ cJ ·e−iE t /ℏ|J ⟩ J

J

J

with EJ = hBcJ(J + 1) − hDcJ (J + 1)2. The degree of alignment is given by ⟨ψJ|cos2 θ|ψJ′⟩ with selection rules J′ = J (time independent population signal) and J′ = J ± 2, where the latter governs the transient alignment signals that are of interest in this work. Thus the timedependent alignment is a summation of oscillatory terms 2

S(t ) = ⟨ψJ(t )|cos2 θ|ψJ(t )⟩ =

∑ FJ ·exp[−i(EJ+ 2 − EJ )t /ℏ] + c.c (1)

J

with (complex) amplitudes FJ ≡ cJ*cJ+2⟨J|cos θ|J + 2⟩ (at the end of the pulse t ≈ 100 fs) and frequencies defined by the rotational energy differences EJ+2 − EJ = 2B(2J + 3) − 4D(2J3 + 9J2 + 15J + 9). We note that the ensemble consists of numerous different rotational states populated prior to excitation; that is, following the interaction with the short laser pulse, each and every initially populated state |Ji⟩ is described by a rotational wavepacket |ψJ⟩ and gives rise to the alignment signal S(t) of expression 1. The observable ⟨⟨cos2 θ⟩⟩ is thus a weighted 2

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Figure 2. Echo signal dependence on pulse energy P1 and P2 for CS2, 22 Torr at 295 K (ambient temperature). (a) Simulated echo signal induced by two pulses (P1, P2) at a fixed delay apart, Δτ = 9 ps. The amplitude of the Echo signal Secho is measured by the peak-to-peak difference (marked by Secho). (b) Simulated Secho as a function of P1 power (P2 fixed). (c) Simulated Secho as a function of P2 (P1 fixed). (d) Experimental Secho versus P1 (P2 fixed at 2.5 μJ). (e) Experimental Secho versus P2 (P1 fixed at 3.5 μJ). The pump beams were reduced by an iris to 4 mm, and were focused in the sample cell by an f = 150 mm lens. Pulse duration was ∼100 fs (fwhm).

order dispersion (∝J3) that breaks the time-symmetry of the signal and manifests in multiple oscillations toward the positive time direction.53 At the ambient temperature of our experiment, the most populated state of CH3I is J ≈ 20, and thus the leading term is the cubic phase term, resulting in multiple oscillations toward the positive time axis, readily observed in Figure 1. We consider the CDN as a dephasing mechanism that is imprinted on both the fundamental alignment signals (as in Figure 1) and the alignment echoes (discussed in Section III). Section II. Two-Level Spin Echo versus Multilevel Rotational Echo. A two-level spin system such as magnetic spin-half in NMR is typically described by the Bloch sphere,54 starting from the magnetization in the positive z direction. In a typical echo experiment, an RF pulse (at t = 0) rotates the spin vectors by 90° to the x axis of the Bloch sphere; that is, it creates a coherent superposition of the two spin levels (termed “a π/2 pulse”). Because of inhomogeneities in the magnetic field and chemical environment, different spins rotate at different velocities (frequencies) and disperse in the xy plane, which corresponds to dephasing, manifested as a decay of the free induction signal amplitude with time. At t = Δτ, a second RF pulse is applied to invert the magnetization vectors (rotates the spins by 180°, termed “π pulse”), and the spins start rephasing. Following another Δτ of free propagation, that is, at t = 2Δτ,

average of the alignment signals emanating from all of the different initial states and is accounted for in our density matrix simulations. Rearrangement of expression 1 yields S(t ) =

∑ FJ ·  exp[−i 4πBc(2J + 3)t ] ·  J

φ1

exp[i8πDc(2J 3 + 9J 2 + 15J + 9)t ] + c .c  φ2

(2)

At the vicinity of integer revival times t = n·Trev = n/2Bc {n ∈ integers} corresponding to φ1 = 1, the alignment signal is governed by φ2 and takes the form S(n· Trev) =

∑ FJ exp[i 4πn(D/B)(2J 3 + 9J 2 + 15J + 9)] + c.c J

(3)

With each revival period, the alignment signal accumulates a phase across its frequency components containing a global phase (∝J0), a linear phase (∝J) that introduces a slight shift in time with each revival, a quadratic dispersion (∝J2) that increases the signal duration and introduces a gradient in its oscillation frequency with time (positive chirp),52 and a third5130


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higher orders that are beyond the scope of the paper. In addition, our numerical simulations and additional experimental results show that while Secho depends on the intensities of both pulses Secho ∝ P1, sin2(a·P2), the optimal pulse energies (intensities) are not coupled; that is, there exists a P2 value that maximizes the obtainable echo amplitude, regardless of P1 energy. However, the optimal P2 value does depend on Δτ, and so does the echo amplitude (in contrast with two-level systems). These observations are in agreement with the previous work of Karras et al.12 Section III. Rephasing of the CDN: Experimental Results. Before we proceed to the experimental demonstration of the rephasing character of rotational echoes, it is important to decipher between “revival” signals and “rephasing” echo signals. In our experiments, an ultrashort laser pulse (with a bandwidth of ∼13 nm fwhm centered at 800 nm) induces molecular rotations via a nonresonant stimulated Raman process (Figure 3a). A pair of

the spin vectors come back into phase (“rephased”) and a free induction signal is observed once again, however, with a reversed phase-time evolution. A combination of 90° and 180° for the first and second pulses, respectively, induces the maximal echo signal amplitude, regardless of the delay, Δτ, between the pulses. However, two RF pulses with general flip angles α1 and α2, respectively, produce an echo signal with amplitude Secho ∝ sin(α1) sin2(α2/2),55 and thus for small flip angles (sin(α) ≈ α) the echo amplitude increases linearly with the first pulse (P1) and quadratically with the second pulse (P2) amplitudes. In a multilevel system, such as the molecular rotor, characterization of the pulses’ areas or flip angles is not possible because ultrashort pulses practically couple all of the accessible rotational levels of the molecule via multiple interactions with each pulse, transferring populations to higher rotational states and creating higher order coherences. While the echo response of classical rotors can be described analytically10,12, quantum-mechanical rotors require numerical approaches for simulating their echo dynamics. Thus, to characterize the rotational echo dependence on the excitation pulses’ intensities and the delay between them, we simulated the rotational dynamics of molecules modeled as quantummechanical rigid rotors subject to two, time-delayed optical pulses with varying intensities. The simulations were performed by numerically propagating the Liouville−Von Neumann equation ∂ρ/∂t = −(i/ℏ)[Ĥ ,ρ] as in our recent works46,56 with Ĥ = L̂ 2/2I + V̂ (t). Here the interaction term V̂ (t) 1 =− 4 Δα |E(t )|2 cos2 θ consists of two pulses with Gaussian intensity envelopes, with Δτ delay apart |E(t)|2 = I1 exp[−t2/σ2] + I2 exp[−(t − Δτ)2/σ2] and fwhm = 2 ln 2 σ = 100 fs. Figure 2 shows the simulated and experimental results of rotational echoes induced in a 22 Torr carbonyl-sulfide (CS2) gas sample at 295 K (with rotational constant B = 0.109 cm−1). CS2 was chosen for its high polarizability anisotropy (Δα = 9.5 Å3)20 and correspondingly efficient interaction with the excitation pulses, avoiding the use of strong fields and their associated unwanted responses such as ionization and plasma formation. In Figure 2 we extract the dependence of the echo signal amplitude on the first (P1) and second (P2) pulses’ energies from numerical simulations (Figure 2b,c) and experimental measurements (Figure 2d,e) of the CS2 gas. The echo signal amplitude, Secho, is quantified by its peak-to-peak signal height, as marked by the red double-sided arrow in Figure 2a. Each data point represents Secho obtained with specific P1 and P2 energies. To extract the dependence of Secho on the intensity of the first pulse, we performed a series of numerical simulations where P2 was fixed and P1 was varied (Figure 2b) and vice versa 2 Sym (Figure 2c). We found SSym echo ∝ P1 and Secho ∝ sin (aP2) (where a is a free parameter). Figure 2d,e depicts the experimental results of Secho dependence on P1 and P2 (given in μJ) respectively. By fitting the experimental Secho versus P1 to a 0.95 power law, we obtained SExp echo ∝ P1 , in good agreement with the numerical results. The experimental data in Figure 2e were fitted by b·sin2(aP2), also in good agreement with the numerically predicted dependence. We note that the sinesquared dependence on P2 is valid for a limited region of P2 energies (or intensities when considering the beam and focusing parameters given in the caption of Figure 2), as shown in Figure 2e. Further increase in P2 leads to additional oscillations of Secho but with diminished amplitudes and reversal of the echo signal phases due to molecule−field interactions of

Figure 3. Double-sided Feynman diagrams of the multiple Raman excitation pathways leading to rephasing echo alignment signals. (a) Sketch of two-photon stimulated Raman transitions. (Blue and red photons are contained within the ultrashort pulse bandwidth.) (b−d) Different Feynman pathways that induce rephasing coherences.

photons (blue and red arrows, contained in the pulse spectrum) induces rotational coherences between states |J⟩⟨J + 2| (pulse P1 in Figure 3b−d). Once created, these coherences accumulate phase under field-free evolution with frequencies ωJ,J+2 = 4πBc(2J + 3) (see eq 2). It is the spacing between the rotational levels of quantum rotors (the transition frequencies between rotational states |J⟩ and |J + 2⟩), forming an harmonic series, that dictates the periodic recurrences (“revivals”) when all of the coherences come back in phase (as discussed in Section I). “Rephasing” echo signals, on the contrary, are induced via three stimulated Raman processes provided by two time-delayed laser pulses. Figure 3b−d describes different interaction pathways in Liouville space using double-sided Feynman diagrams starting from an initial population |J⟩⟨J| in rotational level J. In all three diagrams, the first pulse (P1) creates the |J⟩⟨J + 2| coherences that evolves for Δτ. At t = Δτ the second pulse (P2) is applied and interacts with the existing coherences twice (hence the quadratic dependence of the echo signal on the second pulse (P2) intensity): The first Raman interaction of P2 (marked as P(1) 2 ) with the |J⟩⟨J + 2| coherences creates populations (|J + 2⟩⟨J + 2| in the pathways described by Figure 3b,d and |J⟩⟨J| in the pathway of Figure 3c), and the second Raman interaction (marked as P(2) 2 ) creates the rephasing coherences (|J + 2⟩⟨J| , | J⟩⟨J − 2| , and |J + 4⟩⟨J + 2| in Figure 3b−d, respectively). The phase accumulated by |J⟩⟨J + 2| during the delay Δτ between the two pulses is exactly subtracted during the evolution of |J + 2⟩⟨J| for Δτ after the application of P2 and gives rise to a rephased echo signal. Next, we present the key experimental results of this work. We conducted time-resolved birefringence measurements induced by two time-delayed ultrashort laser pulses in a gas 5131


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Figure 4. Time-resolved optical birefringence of CH3I gas (30 Torr, 295 K) induced by two pump pulses (P1 and P2) delayed by (a) 9.8 ps and (b) 140.9 ps (P1 is applied at t = −281.8 ps, not shown in the scan). t = 0 is centered at the rephased echo signal in both scans. Dashed red insets show the enlarged echo signals as they rephase at t < 0 and dephase at t > 0. Dashed green insets show the enlarged nonrephasing revivals of P1.

In Figure 4b we perform the same measurement as in Figure 4a, with only one difference: The delay between the pulses is set to Δτ = 140.9 ps. Here, P1 is applied at t = −281.8 ps (not shown) and P2 is shown on the left-most side (at t = −140.9 ps); that is, the echo signal is expected at t = 0 (−281.8 ps + 2Δτ). Indeed, a completely rephased echo signal is observed at t = 0 (enlarged “echo 0” inset), free of CDN effects. However, the echo 0 signal is preceded by several echo signals at t = −133.7 ps (enlarged inset “echo −2”), t = −100.3 ps (marked “echo (− 3 )”), t = −66.8 (enlarged inset “echo −1), and t =

of CH3I molecules. The strong CDN of CH3I serves as a dephasing mechanism that allows us to monitor the evolution of the echo signal phase as a function of time. Figure 4a shows the alignment signal obtained with a short (Δτ = 9.8 ps) delay between the pulses (marked by P1, P2). Each pulse initiates rotational dynamics that manifest in transient birefringence (2) every 1/2Trev, marked by nT(1) rev or nTrev for the nth revival induced by P1 or P2, respectively. At integer (n = 1, 2, ...) revival times the birefringence signal evolve from negative (antialignment) to positive (alignment) direction (see signals marked (1) (2) (2) 1T(1) rev , 2Trev , 1Trev , 2Trev for the first two revival signals in Figure 4a and 1Trev in Figure 1). The first echo response (dashed red inset marked “0”) is observed at a delay of Δτ = 9.8 ps after P2 with a reversed phase-time with respect to that of integer revivals, that is, evolving from alignment to antialignment. This reversed phasetime evolution is induced by P2, which imparts a π-phase shift to the rotational coherences previously induced by P1 (turning | J⟩⟨J + 2| coherences to |J + 2⟩⟨J|, as shown in Figure 3b). This is equivalent to reversing the direction of time in eq 1. A perfect π-pulse (as in a two-level system) would have resulted in an echo signal that is a perfect inversion (phase and amplitude) of the revival signal induced by P1 alone (1Trev in Figure 1 P1induced revivals. However, in the multilevel rotor system this is impossible, as discussed above. Thus P2 only partially rephases the rotational coherences’ amplitudes and results in a diminished echo signal amplitude. Correspondingly, the amplitudes of the P1-induced revivals are also diminished (see 1 (1) (1) T , 1Trev , ... in Figure 4a) but not completely eliminated 2 rev and form the “nonrephasing” signals.57 Like all other rotational responses, the echo signals are also governed by the revival phenomenon and experience increasing dephasing due to CDN with each revival period, as observed for the enlarged first and second echo revivals in the dashed red insets marked (+1) and (+2), respectively. Half-integer revivals of the echo are also marked in the Figure by echo(+ 1 ) and echo(+ 3 ). 2

2

−33.4 ps (marked “echo (− 1 )”), with an appreciable phase 2

distortion due to CDN. Different from the recently reported (1) − Δτ, the “imaginary echoes”11 observed at t = nTrev “preceding echoes” are observed at integer and half-integer revival periods (Trev = 66.7 ps) before the fully rephased echo signal (“echo 0”) and are equivalent to propagating echo 0 in negative time. The reversed propagation manifests by an inverse CDN effect on the preceding signals, as is readily observed by comparing echo−2 to echo+2 or echo −1 to echo +1 with perfectly inverted CDN phase accumulation. Half and full integer revivals of P1 and of P2 are also marked in Figure 4b (2) by nT(1) rev or nTrev . To summarize the above discussion, P1 induces rotational coherences that experience CDN dephasing during their fieldfree evolution. At t = Δτ, P2 is applied and inverses the direction of phase accumulation by the rotational coherences; that is, the CDN phase that was accumulated during t = 0 → Δτ is gradually removed during t = Δτ → 2Δτ, and an echo signal, free of CDN phase effects, is observed at t = 2Δτ, referred to as “Rephased Echo Signal”. At t > 2Δτ the CDN phase that continues to accumulate manifests in multiple oscillations and “smearing” of the echo signal but in the reversed direction. For simulated alignment dynamics like those shown in Figure 4b, see the Supporting Information, Section SI.3.

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The Journal of Physical Chemistry Letters Section IV. Pictorial Representation of Rotational Echoes. In what follows, we provide a pictorial representation of the quantum-mechanical rotational echoes discussed above, inspired by the famous Physics Today cover figure of Hahn’s runners,58 drawing an analogy between spin echoes and runners on a circular track. While in Hahn’s case the runners progress at various velocities in some continuous range of velocities, in our case their velocities are (very close to being) integer multiples of the slowest runner’s velocity, that is, quantized. Each runner represents the phase of a rotational coherence |J⟩⟨J + 2| as it evolves with time, and their deviation from perfect commensurability (stepping out of line as time progresses) represents the CDN discussed before. Gray Silhouettes Represent the Rephasing Coherences (Echo Signals). In snapshot (a) of Figure 5, all of the runners are perfectly placed at the starting line as the race begins. Shortly

after (snapshot (b)) the runners are completely dispersed along the track because they run at very different velocities. However, owing to the quantization of their velocities, all of the runners meet for a short instance at the starting line (snapshot (c)), representing the first revival period. Up to this point, each of the runners completed an (almost) integer number of laps. The deviations from perfectly integer number of laps arise from the slight incommensurability of the runners velocities (due to CDN); therefore, the runners are not perfectly aligned at the starting line, and a small drift is observed (marked by the curved gray line). (d) is the same as (b) only for times between the first and second revivals (1Trev < t < 2Trev). The drift of the runners from the straight starting line increases with each revival, as shown for the clockwise runners (e,h). When a second shot is fired (corresponding to the application of the second laser pulse) at t = 2Trev+ Δτ in snapshot (f), each runner splits into two. One continues in the forward clockwise direction (black silhouettes) and the other in the reversed direction (gray silhouettes), resulting in two groups of runners: clockwise (black) and counterclockwise (gray). In snapshot (g), at t = 2Trev+ 2Δτ, the gray runners are back at the starting line; this is the alignment echo signal, with a larger drift between the runners compared to the second revival (e). Note that because the two groups run in opposite directions, the observer sees the fronts of the black runners and the rears of the gray runners as each group accumulates at the starting line. This corresponds to a time-reversed signal phase of the rephasing gray runners. The black, nonrephasing signals are also shown, and their drift from straight line keeps increasing with each revival (snapshot (h)), while for the gray runners, this drift decreases (snapshot (i)). At exactly twice the delay between the pulses, namely, at t = 4Trev+ 2Δτ, the gray runners are perfectly aligned at the starting line, reaching complete rephasing (snapshot (j)). As the race continues, the drift of the gray runners increases again but this time in the opposite direction (snapshots (k,l)). To conclude, rotational echoes were experimentally induced by two time-delayed ultrashort laser pulses in a gas of CH3I molecules, demonstrating strong centrifugal distortion. The results provide clear indication of the rephasing nature of the echo, starting from phase-distorted signal (due to CDN) that evolves through distortion-free (“rephased signal”) and to phase-distortion in the reversed direction. We’ve shown that the echo responses of two-level and multilevel system share common features such as the delay at which the echo is observed, the rephasing nature of their dynamics, and, to some extent, their dependence on the driving pulses’ intensities. However, they differ by their dependence on Δτ (no dependence in a two-level system) as well as by the echo behavior at increased P2 intensities. Unlike rotational revival signals that are observed at specific times depending on the molecular moment(s) of inertia, echo responses can be induced “on demand” (by setting the delay between pulses) and are applicable to characterization of fast decay processes in dense gas samples where revivals cannot be measured or for exploring large molecular entities with long revival periods that are beyond the reach of time-resolved ultrafast spectroscopy. In addition to their spectroscopic importance, rotational echoes can be utilized to judiciously control the space-time angular distribution of gas media, for example, to enable quasi-phasematched harmonic generation.

Figure 5. Quantum-mechanical analogue of Hahn’s runners.58 (a−e) Snapshots at selected times before the second shot (pulse) is fired 0 < t < Δτ. (f−l) Snapshots at selected times after the second pulse t > Δτ. Black silhouettes: Nonrephasing coherences. 5133


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b02215. SI.1. Sketch of the experimental setup. SI.2. Simulated CH3I dynamics: Asymmetric top versus linear rotor model. SI.3. Simulation of the rephasing echo signal. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Sharly Fleischer: 0000-0003-0213-2165 Author Contributions ⊥

D.R. and R.D. contributed equally to the work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This work is supported by several funding agencies: S.F. acknowledges support of the Israel Science Foundation (ISF) grant no.1065/14, the Marie Curie CIG grant no. 631628, ISF grant no. 2797/11 (INREP - Israel National Research Center for Electrochemical Propulsion), and the Wolfson Foundation grant no. PR/ec/20419. S.K. acknowledges support of the Binational Science Foundation grant no. 2012021 and by CMST COST Action CM1405 MOLIM.

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Rotational Echoes: Rephasing of Centrifugal Distortion in Laser

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