Laser picoscopy of valence electrons in solids

High-Order Harmonic Generation in Photoexcited Three-Dimensional Dirac Semimetals

High-order harmonic generation (HHG) in condensed matter is highly important for potential applications in various fields, such as materials characterization, all-optical switches, and coherent light source generation. Linking HHG to the properties or dynamic processes of materials is essential for realizing these applications. Here, a bridge has been built between HHG and the transient properties of materials through the engineering of interband polarization in a photoexcited three-dimensional Dirac semimetal (3D-DSM). It has been found that HHG can be efficiently manipulated by the electronic relaxation dynamics of 3D-DSM on an ultrafast time scale of several hundred femtoseconds. Furthermore, time-resolved HHG (tr-HHG) has been demonstrated to be a powerful spectroscopy method for tracking electron relaxation dynamics, enabling the identification of electron thermalization and electron-phonon coupling processes and the quantitative extraction of electron-phonon coupling strength. This demonstration provides insights into the active control of HHG and measurements of the electron dynamics.

Coulomb-induced emission time shifts in high-order harmonic generation from H2

Accurate emission times of high-order harmonic generation (HHG) are vital for high-precision ultrafast detection in attosecond science, but a quantitative analysis of Coulomb effects on this time is absent in the molecular HHG. Here, we investigate the Coulomb-induced emission-time shift in HHG of H2+ with two different internuclear distances R, where the times obtained via the Gabor transform of numerical data from solving the time-dependent Schrödinger equation are used as simulation experiment results. Based on the molecular strong-field approximation, we develop a trajectory-resolved classical model that takes into account the molecular two-center structure. By selecting appropriate electron trajectories and including Coulomb interactions, the classical trajectory method can reproduce Gabor emission times well. This consistence reveals that Coulomb tails cause an emission-time shift of ∼35 as at the R = 2.0 a.u. case and of ∼40-60 as at the R = 2.6 a.u. case under the present laser parameters when compared to the Coulomb-free quantum-orbit model. Our results are of significance to probe the attosecond dynamics via two-center interference.

Ultrafast Electron-Electron Scattering in Metallic Phase of 2H-NbSe_{2} Probed by High Harmonic Generation

Electron-electron scattering on the order of a few to tens of femtoseconds plays a crucial role in the ultrafast electron dynamics of conventional metals. When mid-infrared light is used for driving and the period of light field is comparable to the scattering time in metals, unique light-driven states and nonlinear optical responses associated with the scattering process are expected to occur. Here, we use high-harmonics spectroscopy to investigate the effect of electron-electron scattering on the electron dynamics in thin film 2H-NbSe_{2} driven by a mid-infrared field. We observed odd-order high harmonics up to 9th order as well as a broadband emission from hot electrons in the energy range from 1.5 to 4.0 eV. The electron-electron scattering time in 2H-NbSe_{2} was estimated from the broadband emission to be almost the same as the period of the mid-infrared light field. A comparison between experimental results and a numerical calculation reveals that competition and cooperation between the driving and scattering enhances the nonperturbative behavior of high harmonics in metals, causing a highly nonequilibrium electronic state corresponding to several thousand Kelvin.

Tracking electron motion within and outside of Floquet bands from attosecond pulse trains in time-resolved ARPES

Floquet engineering has recently emerged as a technique for controlling material properties with light. Floquet phases can be probed with time- and angle-resolved photoelectron spectroscopy (Tr-ARPES), providing direct access to the laser-dressed electronic bands. Applications of Tr-ARPES to date focused on observing the Floquet-Bloch bands themselves, and their build-up and dephasing on sub-laser-cycle timescales. However, momentum and energy resolved sub-laser-cycle dynamics between Floquet bands have not been analyzed. Given that Floquet theory strictly applies in time-periodic conditions, the notion of resolving sub-laser-cycle dynamics between Floquet states seems contradictory-it requires probe pulse durations below a laser cycle that inherently cannot discern the time-periodic nature of the light-matter system. Here we propose to employ attosecond pulse train probes with the same temporal periodicity as the Floquet-dressing pump pulse, allowing both attosecond sub-laser-cycle resolution and a proper projection of Tr-ARPES spectra on the Floquet-Bloch bands. We formulate and employ this approach inab-initiocalculations in light-driven graphene. Our calculations predict significant sub-laser-cycle dynamics occurring within the Floquet phase with the majority of electrons moving within and in-between Floquet bands, and a small portion residing and moving outside of them in what we denote as 'non-Floquet' bands. We establish that non-Floquet bands arise from the pump laser envelope that induces non-adiabatic electronic excitations during the pulse turn-on and turn-off. By performing calculations in systems with poly-chromatic pumps we also show that Floquet states are not formed on a sub-laser-cycle level. This work indicates that the Floquet-Bloch states are generally not a complete basis set for sub-laser-cycle dynamics in steady-state phases of matter.

High-Order Harmonic Generation in Solids: The Role of Intraband Transitions in Extreme Nonlinear Optics

High-order harmonic generation (HHG) in gases is frequently used nowadays to produce attosecond pulses and coherent radiation in the visible-to-soft X-ray spectral range. HHG in solids is a natural extension of the idea of HHG in gases, and its first observation about ten years ago opened the door to investigations on attosecond electron dynamics in solids and the development of solid-state attosecond light sources. The common process in both types of HHG is nonlinear photocarrier generation, and thus, transitions between different bands (interband transitions) are always important for HHG. As well, in the case of solids, the transitions within a band (intraband transitions) also need to be considered, because efficient carrier acceleration is possible due to them. This Perspective focuses on experimental findings that show how intraband transitions can be controlled because such an understanding will be essential in the development of unique optoelectronics that can operate at petahertz frequencies.

Solid-state high harmonic spectroscopy for all-optical band structure probing of high-pressure quantum states

High pressure has triggered various novel states/properties in condensed matter, as the most representative and dramatic example being near-room-temperature superconductivity in highly pressured hydrides (~200 GPa). However, the mechanism of superconductivity is not confirmed, due to the lacking of effective approach to probe the electronic band structure under such high pressures. Here, we theoretically propose that the band structure and electron-phonon coupling (EPC) of high-pressure quantum states can be probed by solid-state high harmonic generation (sHHG). This strategy is investigated in high-pressure Im-3m H3S by the state-of-the-art first-principles time-dependent density-functional theory simulations, where the sHHG is revealed to be strongly dependent on the electronic structures and EPC. The dispersion of multiple bands near the Fermi level is effectively retrieved along different momentum directions. Our study provides unique insights into the potential all-optical route for band structure and EPC probing of high-pressure quantum states, which is expected to be helpful for the experimental exploration of high-pressure superconductivity in the future.

Observation of interband Berry phase in laser-driven crystals

Ever since its discovery1, the notion of the Berry phase has permeated all branches of physics and plays an important part in a variety of quantum phenomena2. However, so far all its realizations have been based on a continuous evolution of the quantum state, following a cyclic path. Here we introduce and demonstrate a conceptually new manifestation of the Berry phase in light-driven crystals, in which the electronic wavefunction accumulates a geometric phase during a discrete evolution between different bands, while preserving the coherence of the process. We experimentally reveal this phase by using a strong laser field to engineer an internal interferometer, induced during less than one cycle of the driving field, which maps the phase onto the emission of higher-order harmonics. Our work provides an opportunity for the study of geometric phases, leading to a variety of observations in light-driven topological phenomena and attosecond solid-state physics.

Microscopic nonlinear optical response: Analysis and calculations with the Floquet-Bloch formalism

We analyze microscopic nonlinear optical response of periodic structures within the Floquet-Bloch formalism. The analysis is focused on the real-space distributions of optically induced charge and electron current density within the unit cell of a crystal. We demonstrate that the time-reversal symmetry of a crystal determines the phases of the temporal oscillations of these distributions. We further analyze their spatial symmetries and connection to macroscopic optical response. We illustrate our study with ab initio calculations that combine density functional theory with the Floquet-Bloch formalism. The calculations provide time-dependent optically induced charge distributions and electron current densities within the unit cells of a crystal with inversion symmetry MgO and a crystal without inversion symmetry GaAs in response to a strong-field excitation. The real-space, microscopic view on nonlinear optical response provides insightful information about the strong field-matter interaction.

Mapping Light-Dressed Floquet Bands by Highly Nonlinear Optical Excitations and Valley Polarization

Ultrafast nonlinear optical phenomena in solids have been attracting a great deal of interest as novel methodologies for the femtosecond spectroscopy of electron dynamics and control of the properties of materials. Here, we theoretically investigate strong-field nonlinear optical transitions in a prototypical two-dimensional material, hBN, and show that the k-resolved conduction band charge occupation patterns induced by an elliptically polarized laser can be understood in a multiphoton resonant picture, but, remarkably, only if using the Floquet light-dressed states instead of the undressed matter states. Our work demonstrates that Floquet dressing affects ultrafast charge dynamics and photoexcitation even from a single pump pulse and establishes a direct measurable signature for band dressing in nonlinear optical processes in solids, opening new paths for ultrafast spectroscopy and valley manipulation.

Orbital perspective on high-harmonic generation from solids

High-harmonic generation in solids allows probing and controlling electron dynamics in crystals on few femtosecond timescales, paving the way to lightwave electronics. In the spatial domain, recent advances in the real-space interpretation of high-harmonic emission in solids allows imaging the field-free, static, potential of the valence electrons with picometer resolution. The combination of such extreme spatial and temporal resolutions to measure and control strong-field dynamics in solids at the atomic scale is poised to unlock a new frontier of lightwave electronics. Here, we report a strong intensity-dependent anisotropy in the high-harmonic generation from ReS2 that we attribute to angle-dependent interference of currents from the different atoms in the unit cell. Furthermore, we demonstrate how the laser parameters control the relative contribution of these atoms to the high-harmonic emission. Our findings provide an unprecedented atomic perspective on strong-field dynamics in crystals, revealing key factors to consider in the route towards developing efficient harmonic emitters.

Revealing the Interplay between Strong Field Selection Rules and Crystal Symmetries

Symmetries are ubiquitous in condensed matter physics, playing an important role in the appearance of different phases of matter. Nonlinear light matter interactions serve as a coherent probe for resolving symmetries and symmetry breaking via their link to selection rules of the interaction. In the extreme nonlinear regime, high harmonic generation (HHG) spectroscopy offers a unique spectroscopic approach to study this link, probing the crystal spatial properties with high sensitivity while opening new paths for selection rules in the XUV regime. In this Letter we establish an advanced HHG polarimetry scheme, driven by a multicolor strong laser field, to observe the structural symmetries of solids and their interplay with the HHG selection rules. By controlling the crystal symmetries, we resolve nontrivial polarization states associated with new spectral features in the HHG spectrum. Our scheme opens new opportunities in resolving the symmetries of quantum materials, as well as ultrafast light driven symmetries in condensed matter systems.

Band Nonlinearity-Enabled Manipulation of Dirac Nodes, Weyl Cones, and Valleytronics with Intense Linearly Polarized Light

We study low-frequency linearly polarized laser-dressing in materials with valley (graphene and hexagonal-Boron-Nitride) and topological (Dirac- and Weyl-semimetals) properties. In Dirac-like linearly dispersing bands, the laser substantially moves the Dirac nodes away from their original position, and the movement direction can be fully controlled by rotating the laser polarization. We prove that this effect originates from band nonlinearities away from the Dirac nodes. We further demonstrate that this physical mechanism is widely applicable and can move the positions of the valley minima in hexagonal materials to tune valley selectivity, split and move Weyl cones in higher-order Weyl semimetals, and merge Dirac nodes in three-dimensional Dirac semimetals. The model results are validated with ab initio calculations. Our results directly affect efforts for exploring light-dressed electronic structure, suggesting that one can benefit from band nonlinearity for tailoring material properties, and highlight the importance of the full band structure in nonlinear optical phenomena in solids.

Control of the electron dynamics in solid-state high harmonic generation on ultrafast time scales by a polarization-skewed laser pulse

Since high-order harmonic generation (HHG) from atoms depends sensitively on the polarization of the driving laser field, the polarization gating (PG) technique was developed and applied successfully to generate isolated attosecond pulses from atomic gases. The situation is, however, different in solid-state systems as it has been demonstrated that due to collisions with neighboring atomic cores of the crystal lattice strong HHG can be generated even by elliptically- and circularly-polarized laser fields. Here we apply PG to solid-state systems and find that the conventional PG technique is inefficient for the generation of isolated ultrashort harmonic pulse bursts. In contrast, we demonstrate that a polarization-skewed laser pulse is able to confine the harmonic emission to a time window of less than one-tenth of the laser cycle. This method provides a novel way to control HHG and to generate isolated attosecond pulses in solids.

Symmetry-aware deep neural networks for high harmonic spectroscopy in solids

Neural networks are a prominent tool for identifying and modeling complex patterns, which are otherwise hard to detect and analyze. While machine learning and neural networks have been finding applications across many areas of science and technology, their use in decoding ultrafast dynamics of quantum systems driven by strong laser fields has been limited so far. Here we use standard deep neural networks to analyze simulated noisy spectra of highly nonlinear optical response of a 2-dimensional gapped graphene crystal to intense few-cycle laser pulses. We show that a computationally simple 1-dimensional system provides a useful "nursery school" for our neural network, allowing it to be retrained to treat more complex 2D systems, recovering the parametrized band structure and spectral phases of the incident few-cycle pulse with high accuracy, in spite of significant amplitude noise and phase jitter. Our results offer a route for attosecond high harmonic spectroscopy of quantum dynamics in solids with a simultaneous, all-optical, solid-state based complete characterization of few-cycle pulses, including their nonlinear spectral phase and the carrier envelope phase.

Characterizing Anomalous High-Harmonic Generation in Solids

Anomalous high-harmonic generation (HHG) arises in certain solids when irradiated by an intense laser field, originating from a Berry-curvature-induced perpendicular anomalous current. The observation of pure anomalous harmonics is, however, often prohibited by contamination from harmonics stemming from interband coherences. Here, we fully characterize the anomalous HHG mechanism, via development of an ab initio methodology for strong-field laser-solid interaction that allows a rigorous decomposition of the total current. We identify two unique properties of the anomalous harmonic yields: an overall yield increase with laser wavelength; and pronounced minima at certain laser wavelengths and laser intensities around which the spectral phases drastically change. Such signatures can be exploited to disentangle the anomalous harmonics from competing HHG mechanisms, and thus pave the way for the experimental identification and time-domain control of pure anomalous harmonics, as well as reconstruction of Berry curvatures.

Trajectory-controlled high-order harmonic generation in ZnO crystals

We experimentally and theoretically study high-order harmonic generation in zinc oxide crystals irradiated by mid-infrared lasers. The trajectories are mapped to the far field spatial distribution of harmonics. The divergence angles of on-axis and off-axis parts exhibit different dependences on the order of the harmonics. This observation can be theoretically reproduced by the coherent interference between the short and long trajectories with dephasing time longer than 0.5 optical cycle. Further, the relative contribution of the short and long trajectories is demonstrated to be accurately controlled by a one-color or two-color laser on the attosecond time scale. This work provides a reliable method to determine the electron dephasing time and demonstrates a versatile control of trajectory interference in the solid high-order harmonic generation.

All-optical sampling of ultrashort laser pulses based on perturbed transient grating

We propose and demonstrate an all-optical pulse sampling technique based on the transient grating (TG) procedure with perturbation, which provides a simple and robust manner to characterize an ultrashort laser pulse without employing a retrieval algorithm. In our approach, a two-orders weaker perturbation pulse perturbs the diffracted pulse from the TG, which is generated by another strong fundamental pulse. The modulation of the diffracted pulse energy directly represents the temporal profile of the perturbation pulse. We have successfully characterized few-cycle and multi-cycle pulses, which is consistent with the results verified by the widely employed frequency-resolved optical gating (FROG) method. Our method provides a potential way to characterize ultrashort laser waveform from the deep-UV to far-infrared region.

Probing phonon dynamics with multidimensional high harmonic carrier-envelope-phase spectroscopy

We explore pump-probe high harmonic generation (HHG) from monolayer hexagonal-boron-nitride, where a terahertz pump excites coherent optical phonons that are subsequently probed by an intense infrared pulse that drives HHG. We find, through state-of-the-art ab initio calculations, that the structure of the emission spectrum is attenuated by the presence of coherent phonons and no longer comprises discrete harmonic orders, but rather a continuous emission in the plateau region. The HHG yield strongly oscillates as a function of the pump-probe delay, corresponding to ultrafast changes in the lattice such as specific bond compression or stretching dynamics. We further show that in the regime where the excited phonon period and the pulse duration are of the same order of magnitude, the HHG process becomes sensitive to the carrier-envelope phase (CEP) of the driving field, even though the pulse duration is so long that no such sensitivity is observed in the absence of coherent phonons. The degree of CEP sensitivity versus pump-probe delay is shown to be a highly selective measure for instantaneous structural changes in the lattice, providing an approach for ultrafast multidimensional HHG spectroscopy. Remarkably, the obtained temporal resolution for phonon dynamics is ∼1 femtosecond, which is much shorter than the probe pulse duration because of the inherent subcycle contrast mechanism. Our work paves the way toward routes of probing phonons and ultrafast material structural changes with subcycle temporal resolution and provides a mechanism for controlling the HHG spectrum.

Role of Majorana fermions in high-harmonic generation from Kitaev chain

The observation of Majorana fermions as collective excitations in condensed-matter systems is an ongoing quest, and several state-of-the-art experiments have been performed in the last decade. As a potential avenue in this direction, we simulate the high-harmonic spectrum of Kitaev's superconducting chain model that hosts Majorana edge modes in its topological phase. It is well-known that this system exhibits a topological-trivial superconducting phase transition. We demonstrate that high-harmonic spectroscopy is sensitive to the phase transition in presence of open boundary conditions due to the presence or absence of these edge modes. The population dynamics of the Majorana edge modes are different from the bulk modes, which is the underlying reason for the distinct harmonic profile of both the phases. On the contrary, in presence of periodic boundary conditions with only bulk modes, high-harmonic spectroscopy becomes insensitive to the phase transition with similar harmonic profiles in both phases.

High-Order Harmonic Generation and Its Unconventional Scaling Law in the Mott-Insulating Ca_{2}RuO_{4}

Competition and cooperation among orders is at the heart of many-body physics in strongly correlated materials and leads to their rich physical properties. It is crucial to investigate what impact many-body physics has on extreme nonlinear optical phenomena, with the possibility of controlling material properties by light. However, the effect of competing orders and electron-electron correlations on highly nonlinear optical phenomena has not yet been experimentally clarified. Here, we investigated high-order harmonic generation from the Mott-insulating phase of Ca_{2}RuO_{4}. Changing the gap energy in Ca_{2}RuO_{4} as a function of temperature, we observed a strong enhancement of high order harmonic generation at 50 K, increasing up to several hundred times compared to room temperature. We discovered that this enhancement can be well reproduced by an empirical scaling law that depends only on the material gap energy and photon emission energy. Such a scaling law can hardly be explained by the electronic structure change in the single particle model and has not been predicted by previous theoretical studies on HHG in the simple Mott-Hubbard model. Our results suggest that the highly nonlinear optical response of strongly correlated materials is influenced by competition among the multiple degrees of freedom and electron-electron correlations.

All-optical reconstruction of three-band transition dipole moments by the crystal harmonic spectrum from a two-color laser pulse

When a bulk solid is irradiated by an intense laser pulse, transition dipole moments (TDMs) between different energy bands have an important influence on the ultra-fast dynamic process. In this paper, we propose a new all-optical method to reconstruct the k-dependent TDMs between multi-bands using a crystal high-order harmonic generation (HHG). Taking advantage of an obvious separation of bandgaps between three energy bands of an MgO crystal along the <001 > direction, a continuous harmonic spectrum with two plateaus can be generated by a two-color laser pulse. Furthermore, the first harmonic platform is mainly dominated by the polarization between the first conduction band and the valence band, and the second one is largely attributed to the interband HHG from the second conduction band and the valence band. Therefore, the harmonic spectrum from a single quantum trajectory can be adopted to map TDMs between the first, second conduction bands, and the valence one. Our work is of great significance for understanding the instantaneous properties of solid materials in the strong laser field, and will strongly promote the development of the HHG detection technology.

RhoDyn: A ρ-TD-RASCI Framework to Study Ultrafast Electron Dynamics in Molecules

This article presents the program module RhoDyn as part of the OpenMOLCAS project intended to study ultrafast electron dynamics within the density-matrix-based time-dependent restricted active space configuration interaction framework (ρ-TD-RASCI). The formalism allows for the treatment of spin-orbit coupling effects, accounts for nuclear vibrations in the form of a vibrational heat bath, and naturally incorporates (auto)ionization effects. Apart from describing the theory behind and the program workflow, the paper also contains examples of its application to the simulations of the linear L_2,3 absorption spectra of a titanium complex, high harmonic generation in the hydrogen molecule, ultrafast charge migration in benzene and iodoacetylene, and spin-flip dynamics in the core excited states of iron complexes.

The 2021 ultrafast spectroscopic probes of condensed matter roadmap

In the 60 years since the invention of the laser, the scientific community has developed numerous fields of research based on these bright, coherent light sources, including the areas of imaging, spectroscopy, materials processing and communications. Ultrafast spectroscopy and imaging techniques are at the forefront of research into the light-matter interaction at the shortest times accessible to experiments, ranging from a few attoseconds to nanoseconds. Light pulses provide a crucial probe of the dynamical motion of charges, spins, and atoms on picosecond, femtosecond, and down to attosecond timescales, none of which are accessible even with the fastest electronic devices. Furthermore, strong light pulses can drive materials into unusual phases, with exotic properties. In this roadmap we describe the current state-of-the-art in experimental and theoretical studies of condensed matter using ultrafast probes. In each contribution, the authors also use their extensive knowledge to highlight challenges and predict future trends.

Tuning the electronic band structure of metal surfaces for enhancing high-order harmonic generation

High-harmonic generation (HHG) from the condensed matter phase holds promise to promote future cutting-edge research in the emerging field of attosecond nanoscopy. The key for the progress of the field relies on the capability of the existing schemes to enhance the harmonic yield and to push the photon energy cutoff to the extreme-ultraviolet (XUV, 10-100 eV) regime and beyond toward the spectral "water window" region (282-533 eV). Here, we demonstrate a coherent control scheme of HHG, which we show to give rise to quantum modulations in the XUV region. These modulations are shown to be caused by quantum-path interferences and are found to exhibit a strong sensitivity to the delocalized character of bulk states of the material. The control scheme is based on exploring surface states in transition-metal surfaces and, specifically, tuning the electronic structure of the metal surface itself together with the use of optimal chirped pulses. Moreover, we show that the use of such pulses having moderate intensities permits us to push the harmonic cutoff further to the spectral water window region and that the extension is found to be robust against the change in the intrinsic properties of the material. The scenario is numerically implemented using a minimal model by solving the time-dependent Schrödinger equation for the metal surface Cu(111) initially prepared in the surface state. Our findings elucidate the importance of metal surfaces for generating coherent isolated attosecond XUV and soft-x-ray pulses and for designing compact solid-state HHG devices.

Chiral high-harmonic generation and spectroscopy on solid surfaces using polarization-tailored strong fields

Strong-field methods in solids enable new strategies for ultrafast nonlinear spectroscopy and provide all-optical insights into the electronic properties of condensed matter in reciprocal and real space. Additionally, solid-state media offers unprecedented possibilities to control high-harmonic generation using modified targets or tailored excitation fields. Here we merge these important points and demonstrate circularly-polarized high-harmonic generation with polarization-matched excitation fields for spectroscopy of chiral electronic properties at surfaces. The sensitivity of our approach is demonstrated for structural helicity and termination-mediated ferromagnetic order at the surface of silicon-dioxide and magnesium oxide, respectively. Circularly polarized radiation emanating from a solid sample now allows to add basic symmetry properties as chirality to the arsenal of strong-field spectroscopy in solids. Together with its inherent temporal (femtosecond) resolution and non-resonant broadband spectrum, the polarization control of high harmonics from condensed matter can illuminate ultrafast and strong field dynamics of surfaces, buried layers or thin films.

Strong nonlinearities in solid state materials can lead to interesting applications in photonics. Here the authors study chiral high-harmonic generation at SiO₂ and MgO surfaces using bi-circular two-color driving fields and extract information on crystal properties.

Multilevel quantum interference in the formation of high-order fractional molecular alignment echoes

We theoretically investigate the formation of the high-order fractional alignment echo in OCS molecule and systematically study the dependence of echo intensity on the intensities and time delay of the two excitation pulses. Our simulations reveal an intricate dependence of the intensity of high-order fractional alignment echo on the laser conditions. Based on the analysis with rotational density matrix, this intricate dependence is further demonstrated to arise from the interference of multiple quantum pathways that involve multilevel rotational transitions. Our result provides a comprehensive multilevel picture of the quantum dynamics of high-order fractional alignment echo in molecular ensembles, which will facilitate the development of "rotational echo spectroscopy."