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<!DOCTYPE html> <html lang="en" class="grade-c"> <head> <title>Monitoring ultrafast vibrational dynamics of isotopic molecules with frequency modulation of high-order harmonics | Nature Communications</title> <link rel="alternate" type="application/rss+xml" href="https://www.nature.com/ncomms.rss"/> <script id="save-data-connection-testing"> function hasConnection() { return navigator.connection || navigator.mozConnection || navigator.webkitConnection || navigator.msConnection; } function createLink(src) { var preloadLink = document.createElement("link"); preloadLink.rel = "preload"; preloadLink.href = src; preloadLink.as = "font"; preloadLink.type = "font/woff2"; preloadLink.crossOrigin = ""; document.head.insertBefore(preloadLink, document.head.firstChild); } var connectionDetail = { saveDataEnabled: false, slowConnection: false }; var connection = hasConnection(); if (connection) { connectionDetail.saveDataEnabled = connection.saveData; if (/\slow-2g|2g/.test(connection.effectiveType)) { connectionDetail.slowConnection = true; } } if (!(connectionDetail.saveDataEnabled || connectionDetail.slowConnection)) { createLink("/static/fonts/HardingText-Regular-Web-cecd90984f.woff2"); } else { document.documentElement.classList.add('save-data'); } </script> <link rel="preconnect" href="https://cmp.nature.com" crossorigin> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width,initial-scale=1.0,maximum-scale=5,user-scalable=yes"> <meta name="360-site-verification" content="5a2dc4ab3fcb9b0393241ffbbb490480" /> <script data-test="dataLayer"> window.dataLayer = [{"content":{"category":{"contentType":"article","legacy":{"webtrendsPrimaryArticleType":"research","webtrendsSubjectTerms":"atomic-and-molecular-collision-processes;atomic-and-molecular-interactions-with-photons;atomic-and-molecular-physics","webtrendsContentCategory":null,"webtrendsContentCollection":null,"webtrendsContentGroup":"Nature Communications","webtrendsContentGroupType":null,"webtrendsContentSubGroup":"Article","status":null}},"article":{"doi":"10.1038/s41467-018-03568-3"},"attributes":{"cms":null,"deliveryPlatform":"oscar","copyright":{"open":true,"legacy":{"webtrendsLicenceType":"http://creativecommons.org/licenses/by/4.0/"}}},"contentInfo":{"authors":["Lixin He","Qingbin Zhang","Pengfei Lan","Wei Cao","Xiaosong Zhu","Chunyang Zhai","Feng Wang","Wenjing Shi","Muzi Li","Xue-Bin Bian","Peixiang Lu","André D. 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However, the real-time observation of the vibrational motion of isotopic nuclei in molecules is still challenging due to its ultrashort time scale. Here we demonstrate a method to monitor the nuclear vibration of isotopic molecules with the frequency modulation of high-order harmonic generation (HHG) during the laser-molecule interaction. In the proof-of-principle experiment, we report a red shift in HHG from H2 and D2. The red shift is ascribed to dominant HHG from the stretched isotopic molecules at the trailing edge of the laser pulse. By utilizing the observed frequency shift, the laser-driven nuclear vibrations of H2 and D2 are retrieved. These findings pave an accessible route toward monitoring the ultrafast nuclear dynamics and even tracing a chemical reaction in real time. Previous studies on high harmonic generation from molecules have been used to identify the spectral properties and orbital contributions. Here the authors measure the isotopic effects in the energy shift of the HHG spectra caused by the nuclear motion of the molecules.","datePublished":"2018-03-16T00:00:00Z","dateModified":"2018-03-16T00:00:00Z","pageStart":"1","pageEnd":"7","license":"http://creativecommons.org/licenses/by/4.0/","sameAs":"https://doi.org/10.1038/s41467-018-03568-3","keywords":["Atomic and molecular collision processes","Atomic and molecular interactions with photons","Atomic and molecular physics","Science","Humanities and Social 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Bandrauk","affiliation":[{"name":"Université de Sherbrooke","address":{"name":"Laboratoire de chimie théorique, Département de Chimie, Université de Sherbrooke, Sherbrooke, Canada","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> <link rel="canonical" href="https://www.nature.com/articles/s41467-018-03568-3"> <meta name="journal_id" content="41467"/> <meta name="dc.title" content="Monitoring ultrafast vibrational dynamics of isotopic molecules with frequency modulation of high-order harmonics"/> <meta name="dc.source" content="Nature Communications 2018 9:1"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Nature Publishing Group"/> <meta name="dc.date" content="2018-03-16"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2018 The Author(s)"/> <meta name="dc.rights" content="2018 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="Molecules constituted by different isotopes are different in vibrational modes, making it possible to elucidate the mechanism of a chemical reaction via the kinetic isotope effect. 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Here the authors measure the..."/> <meta name="twitter:image" content="https://media.springernature.com/full/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig1_HTML.jpg"/> <meta property="og:url" content="https://www.nature.com/articles/s41467-018-03568-3"/> <meta property="og:type" content="article"/> <meta property="og:site_name" content="Nature"/> <meta property="og:title" content="Monitoring ultrafast vibrational dynamics of isotopic molecules with frequency modulation of high-order harmonics - Nature Communications"/> <meta property="og:description" content="Previous studies on high harmonic generation from molecules have been used to identify the spectral properties and orbital contributions. 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data-test="article-title" data-article-title="">Monitoring ultrafast vibrational dynamics of isotopic molecules with frequency modulation of high-order harmonics</h1> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Lixin-He-Aff1" data-author-popup="auth-Lixin-He-Aff1" data-author-search="He, Lixin">Lixin He</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup><sup class="u-js-hide"> <a href="#na1">na1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Qingbin-Zhang-Aff1" data-author-popup="auth-Qingbin-Zhang-Aff1" data-author-search="Zhang, Qingbin">Qingbin Zhang</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup><sup class="u-js-hide"> <a href="#na1">na1</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Pengfei-Lan-Aff1" data-author-popup="auth-Pengfei-Lan-Aff1" data-author-search="Lan, Pengfei" data-corresp-id="c1">Pengfei Lan<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0003-1111-5308"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0003-1111-5308</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Wei-Cao-Aff1" data-author-popup="auth-Wei-Cao-Aff1" data-author-search="Cao, Wei">Wei Cao</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Xiaosong-Zhu-Aff1" data-author-popup="auth-Xiaosong-Zhu-Aff1" data-author-search="Zhu, Xiaosong">Xiaosong Zhu</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Chunyang-Zhai-Aff1" data-author-popup="auth-Chunyang-Zhai-Aff1" data-author-search="Zhai, Chunyang">Chunyang Zhai</a><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a 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href="#Aff3">3</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Xue_Bin-Bian-Aff2" data-author-popup="auth-Xue_Bin-Bian-Aff2" data-author-search="Bian, Xue-Bin" data-corresp-id="c2">Xue-Bin Bian<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff2">2</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Peixiang-Lu-Aff1-Aff4" data-author-popup="auth-Peixiang-Lu-Aff1-Aff4" data-author-search="Lu, Peixiang" data-corresp-id="c3">Peixiang Lu<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><sup class="u-js-hide"><a href="#Aff1">1</a>,<a href="#Aff4">4</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 12 authors for this article" title="Show all 12 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Andr__D_-Bandrauk-Aff5" data-author-popup="auth-Andr__D_-Bandrauk-Aff5" data-author-search="Bandrauk, André D.">André D. Bandrauk</a><sup class="u-js-hide"><a href="#Aff5">5</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-down-medium"></use></svg><span>Show authors</span></button> <p class="c-article-info-details" data-container-section="info"> <a data-test="journal-link" href="/ncomms" data-track="click" data-track-action="journal homepage" data-track-category="article body" data-track-label="link"><i data-test="journal-title">Nature Communications</i></a> <b data-test="journal-volume"><span class="u-visually-hidden">volume</span> 9</b>, Article number: <span data-test="article-number">1108</span> (<span data-test="article-publication-year">2018</span>) <a href="#citeas" class="c-article-info-details__cite-as u-hide-print" data-track="click" data-track-action="cite this article" data-track-label="link">Cite this article</a> </p> <div class="c-article-metrics-bar__wrapper u-clear-both"> <ul class="c-article-metrics-bar u-list-reset"> <li class=" c-article-metrics-bar__item" data-test="access-count"> <p class="c-article-metrics-bar__count">5591 <span class="c-article-metrics-bar__label">Accesses</span></p> </li> <li class="c-article-metrics-bar__item" data-test="citation-count"> <p class="c-article-metrics-bar__count">104 <span class="c-article-metrics-bar__label">Citations</span></p> </li> <li class="c-article-metrics-bar__item" data-test="altmetric-score"> <p class="c-article-metrics-bar__count">1 <span class="c-article-metrics-bar__label">Altmetric</span></p> </li> <li class="c-article-metrics-bar__item"> <p class="c-article-metrics-bar__details"><a href="/articles/s41467-018-03568-3/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Metrics <span class="u-visually-hidden">details</span></a></p> </li> </ul> </div> </header> <div class="u-js-hide" data-component="article-subject-links"> <h3 class="c-article__sub-heading">Subjects</h3> <ul class="c-article-subject-list"> <li class="c-article-subject-list__subject"><a href="/subjects/atomic-and-molecular-collision-processes" data-track="click" data-track-action="view subject" data-track-label="link">Atomic and molecular collision processes</a></li><li class="c-article-subject-list__subject"><a href="/subjects/atomic-and-molecular-interactions-with-photons" data-track="click" data-track-action="view subject" data-track-label="link">Atomic and molecular interactions with photons</a></li><li class="c-article-subject-list__subject"><a href="/subjects/atomic-and-molecular-physics" data-track="click" data-track-action="view subject" data-track-label="link">Atomic and molecular physics</a></li> </ul> </div> </div> <div class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>Molecules constituted by different isotopes are different in vibrational modes, making it possible to elucidate the mechanism of a chemical reaction via the kinetic isotope effect. However, the real-time observation of the vibrational motion of isotopic nuclei in molecules is still challenging due to its ultrashort time scale. Here we demonstrate a method to monitor the nuclear vibration of isotopic molecules with the frequency modulation of high-order harmonic generation (HHG) during the laser-molecule interaction. In the proof-of-principle experiment, we report a red shift in HHG from H<sub>2</sub> and D<sub>2</sub>. The red shift is ascribed to dominant HHG from the stretched isotopic molecules at the trailing edge of the laser pulse. By utilizing the observed frequency shift, the laser-driven nuclear vibrations of H<sub>2</sub> and D<sub>2</sub> are retrieved. These findings pave an accessible route toward monitoring the ultrafast nuclear dynamics and even tracing a chemical reaction in real time.</p></div></div></section> <noscript> </noscript> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41567-024-02640-8/MediaObjects/41567_2024_2640_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41567-024-02640-8?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1038/s41567-024-02640-8">Ultrafast high-harmonic spectroscopy of solids </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">10 October 2024</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41566-024-01457-4/MediaObjects/41566_2024_1457_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41566-024-01457-4?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1038/s41566-024-01457-4">High-harmonic spectroscopy probes lattice dynamics </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">10 June 2024</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3A10.1038%2Fs41467-022-33477-5/MediaObjects/41467_2022_33477_Fig1_HTML.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://www.nature.com/articles/s41467-022-33477-5?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1038/s41467-022-33477-5">Sub-optical-cycle light-matter energy transfer in molecular vibrational spectroscopy </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">06 October 2022</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1732372882, embedded_user: 'null' } }); </script> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1">Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>Since Soddy first suggested the existence of isotopes in 1913<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Soddy, F. Intra-atomic charge. Nature 92, 399–400 (1913)." href="/articles/s41467-018-03568-3#ref-CR1" id="ref-link-section-d106621839e505">1</a></sup>, isotopes have drawn a great deal of attention due to its application in the fields of physics, chemistry, biomedicine, and geology. Generally, isotopes with different nuclear masses could change the energy levels within isotopic atoms and molecules, thus lead to a frequency shift in the atomic or molecular spectrum<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="King, W. H. Isotope Shifts in Atomic Spectra (Plenum, New York, 1984)." href="/articles/s41467-018-03568-3#ref-CR2" id="ref-link-section-d106621839e509">2</a></sup>, which has been widely used to identify the species of the isotopes and to investigate the static structure of the isotopologues. Moreover, for isotopic molecules, the vibrational modes depend sensitively on the masses of its constituent isotopic atoms, which provides an important method to determine the mechanism of a chemical reaction via the kinetic isotope effect<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Westheimer, F. H. The magnitude of the primary kinetic isotope effect for compounds of hydrogen and deuterium. Chem. Rev. 61, 265–273 (1961)." href="/articles/s41467-018-03568-3#ref-CR3" id="ref-link-section-d106621839e513">3</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Liu, Y. P. et al. Molecular modeling of the kinetic isotope effect for the [1, 5]-sigmatropic rearrangement of cis-1, 3-pentadiene. J. Am. Chem. Soc. 115, 2408–2415 (1993)." href="/articles/s41467-018-03568-3#ref-CR4" id="ref-link-section-d106621839e516">4</a></sup>, namely, the fact that heavier isotopes tend to react more slowly than lighter ones. However, a real-time measurement of the motions of the isotopic atoms in molecule (molecular vibration) is a long-standing challenge over the last century, due to the awesome rapidity of the molecular vibration.</p><p>Recent advances in strong-field physics have provided efficient approaches to probe both the molecular structure and dynamics using the table-top laser. These new methods rely on the recollision of an electron, removed from the molecule by a strong laser field, with its parent ion. The molecular structure and dynamics are encoded in the amplitude and phase of the emitted high-order harmonics. It stimulates the development of high-order harmonic spectroscopy (HHS)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Itatani, J. et al. Tomographic imaging of molecular orbitals. Nature 432, 867–871 (2004)." href="#ref-CR5" id="ref-link-section-d106621839e523">5</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Haessler, S. et al. Attosecond imaging of molecular electronic wavepackets. Nat. Phys. 6, 200–206 (2010)." href="#ref-CR6" id="ref-link-section-d106621839e523_1">6</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Kraus, P. M., Rupenyan, A. & Wörner, H. J. High-harmonic spectroscopy of oriented OCS molecules: emission of even and odd harmonics. Phys. Rev. Lett. 109, 233903 (2012)." href="#ref-CR7" id="ref-link-section-d106621839e523_2">7</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Frumker, E. et al. Probing polar molecules with high harmonic spectroscopy. Phys. Rev. Lett. 109, 249902 (2012)." href="#ref-CR8" id="ref-link-section-d106621839e523_3">8</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Zhai, C. et al. Diffractive molecular-orbital tomography. Phys. Rev. A 95, 033420 (2017)." href="/articles/s41467-018-03568-3#ref-CR9" id="ref-link-section-d106621839e526">9</a></sup> as an emerging tool for ultrafast detection with femtosecond to attosecond time resolutions. Apart from HHS, some other techniques based on strong-field ionization, such as photoelectron holography and photoelectron diffraction<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Zuo, T., Bandrauk, A. D. & Corkum, P. B. Laser-induced electron diffraction: a new tool for probing ultrafast molecular dynamics. Chem. Phys. Lett. 259, 313–320 (1996)." href="#ref-CR10" id="ref-link-section-d106621839e530">10</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Meckel, M. et al. Laser-induced electron tunneling and diffraction. Science 320, 1478–1482 (2008)." href="#ref-CR11" id="ref-link-section-d106621839e530_1">11</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Peters, M. et al. Laser-induced electron diffraction: a tool for molecular orbital imaging. Phys. Rev. A 85, 053417 (2012)." href="#ref-CR12" id="ref-link-section-d106621839e530_2">12</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Blaga, C. I. et al. Imaging ultrafast molecular dynamics with laser-induced electron diffraction. Nature 483, 194–197 (2012)." href="#ref-CR13" id="ref-link-section-d106621839e530_3">13</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Pullen, MichaelG. et al. Imaging an aligned polyatomic molecule with laser-induced electron diffraction. Nat. Commun. 6, 7262 (2015)." href="#ref-CR14" id="ref-link-section-d106621839e530_4">14</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Puthumpally-Joseph, R. et al. Inversion of strong-field photoelectron spectra for molecular orbital imaging. Phys. Rev. A 94, 023421 (2016)." href="#ref-CR15" id="ref-link-section-d106621839e530_5">15</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Xu, J. et al. Time-resolved molecular imaging. J. Phys. B 49, 112001 (2016)." href="#ref-CR16" id="ref-link-section-d106621839e530_6">16</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Krasniqi, F. et al. Imaging molecules from within: ultrafast angström-scale structure determination of molecules via photoelectron holography using free-electron lasers. Phys. Rev. A 81, 033411 (2010)." href="/articles/s41467-018-03568-3#ref-CR17" id="ref-link-section-d106621839e533">17</a></sup>, and so on, have also been demonstrated to image the molecular structure and dynamics. Up to now, many works have been carried out to investigate the effects of nuclear motion in strong-field ionization<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Zuo, T. & Bandrauk, A. D. Charge-resonance-enhanced ionization of diatomic molecular ions by intense lasers. Phys. Rev. A 52, R2511–R2514 (1995)." href="#ref-CR18" id="ref-link-section-d106621839e537">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Seideman, T., Ivanov, M., Yu. & Corkum, P. B. Role of electron localization in intense-field molecular ionization. Phys. Rev. Lett. 75, 2819–2822 (1995)." href="#ref-CR19" id="ref-link-section-d106621839e537_1">19</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Gibson, G. N., Li, M., Guo, C. & Neira, J. Strong-field dissociation and ionization of H2 using ultrashort laser pulses. Phys. Rev. Lett. 79, 2022–2025 (1997)." href="#ref-CR20" id="ref-link-section-d106621839e537_2">20</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Tolstikhin, O., Wörner, H. & Morishita, T. Effect of nuclear motion on tunneling ionization rates of molecules. Phys. Rev. A 87, 041401(R) (2013)." href="/articles/s41467-018-03568-3#ref-CR21" id="ref-link-section-d106621839e540">21</a></sup> and molecular high-order harmonic generation (MHOHG)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Bandrauk, A. D., Chelkowski, S., Kawai, S. & Lu, H. Effect of nuclear motion on molecular high-order harmonics and on generation of attosecond pulses in intense laser pulses. Phys. Rev. Lett. 101, 153901 (2008)." href="#ref-CR22" id="ref-link-section-d106621839e544">22</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Bandrauk, A. D., Chelkowski, S. & Lu, H. Signatures of nuclear motion in molecular high-order harmonics and in the generation of attosecond pulse trains by ultrashort intense laser pulses. J. Phys. B 42, 075602 (2009)." href="#ref-CR23" id="ref-link-section-d106621839e544_1">23</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Feng, L. & Chu, T. Nuclear signatures on the molecular harmonic emission and the attosecond pulse generation. J. Chem. Phys. 136, 054102 (2012)." href="#ref-CR24" id="ref-link-section-d106621839e544_2">24</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Ge, X. L., Wang, T., Guo, J. & Liu, X. S. Quantum-path control and isolated-attosecond-pulse generation using H2 molecules with moving nuclei in few-cycle laser pulses. Phys. Rev. A 89, 023424 (2014)." href="#ref-CR25" id="ref-link-section-d106621839e544_3">25</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Ahmadi, H. et al. Effect of nuclear motion on high-order-harmonic generation of H2 in intense ultrashort laser pulses. Phys. Rev. A 90, 043411 (2014)." href="/articles/s41467-018-03568-3#ref-CR26" id="ref-link-section-d106621839e547">26</a></sup>. In 2005, Lein showed theoretically that the laser-driven nuclear motion will introduce an amplitude modulation (AM) (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig1">1</a>) in harmonic signals via the nuclear autocorrelation function<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Lein, M. Attosecond probing of vibrational dynamics with high-harmonic generation. Phys. Rev. Lett. 94, 053004 (2005)." href="/articles/s41467-018-03568-3#ref-CR27" id="ref-link-section-d106621839e555">27</a></sup>, which denotes the overlap between the initial and time-dependent nuclear wave function that evolves from the moment of ionization until the recollision. By analyzing the AMs in high-order harmonic generation (HHG) from isotopic molecules (H<sub>2</sub> and D<sub>2</sub>), the intracycle nuclear dynamics has been theoretically predicted<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Lein, M. Attosecond probing of vibrational dynamics with high-harmonic generation. Phys. Rev. Lett. 94, 053004 (2005)." href="/articles/s41467-018-03568-3#ref-CR27" id="ref-link-section-d106621839e563">27</a></sup> and experimentally detected<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Baker, S. et al. Probing proton dynamics in molecules on an attosecond time scale. Science 312, 424–427 (2006)." href="/articles/s41467-018-03568-3#ref-CR28" id="ref-link-section-d106621839e567">28</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Lan, P. et al. Attosecond probing of nuclear dynamics with trajectory-resolved high-harmonic spectroscopy. Phys. Rev. Lett. 199, 033201 (2017)." href="/articles/s41467-018-03568-3#ref-CR29" id="ref-link-section-d106621839e570">29</a></sup>. Nevertheless, this method is restricted because the propagation and other inherent physical factors, such as two-center interference<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Lein, M. Attosecond probing of vibrational dynamics with high-harmonic generation. Phys. Rev. Lett. 94, 053004 (2005)." href="/articles/s41467-018-03568-3#ref-CR27" id="ref-link-section-d106621839e574">27</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Baker, S. et al. Dynamic two-center interference in high-order harmonic generation from molecules with attosecond nuclear motion. Phys. Rev. Lett. 101, 053901 (2008)." href="/articles/s41467-018-03568-3#ref-CR30" id="ref-link-section-d106621839e577">30</a></sup> and energy-dependent rescattering cross sections<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Li, M. Z., Jia, G. R. & Bian, X. B. Alignment dependent ultrafast electron-nuclear dynamics in molecular high-order harmonic generation. J. Chem. Phys. 146, 084305 (2017)." href="/articles/s41467-018-03568-3#ref-CR31" id="ref-link-section-d106621839e582">31</a></sup>, may affect the harmonic intensity. Moreover, in the presence of intense lasers, the nuclear motion will lead to larger internuclear distances <i>R</i> and a decrease in the ionization potential <i>I</i><sub>p</sub><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Zuo, T. & Bandrauk, A. D. Charge-resonance-enhanced ionization of diatomic molecular ions by intense lasers. Phys. Rev. A 52, R2511–R2514 (1995)." href="#ref-CR18" id="ref-link-section-d106621839e592">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Seideman, T., Ivanov, M., Yu. & Corkum, P. B. Role of electron localization in intense-field molecular ionization. Phys. Rev. Lett. 75, 2819–2822 (1995)." href="#ref-CR19" id="ref-link-section-d106621839e592_1">19</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Gibson, G. N., Li, M., Guo, C. & Neira, J. Strong-field dissociation and ionization of H2 using ultrashort laser pulses. Phys. Rev. Lett. 79, 2022–2025 (1997)." href="/articles/s41467-018-03568-3#ref-CR20" id="ref-link-section-d106621839e595">20</a></sup>, which can result in an increase in the ionization rate and thus a strong AM in MHOHG. These factors complicate the retrieval of nuclear dynamics by AM. Apart from AM, frequency modulation (FM) is an alternative way commonly used in various applications, e.g., signal processing and telecommunications. By considering the frequency shift in the atomic spectrum of isotopes, it stimulates us to ask whether the nuclear motion in intense laser fields can induce a frequency shift in the MHOHG spectrum (see Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig1">1</a>). Compared to AM, FM is more stable and insensitive to the laser parameters provided that the ionization saturation is avoided and the pulse length is properly adopted. It thus can provide an alternative powerful way to identify the nuclear dynamics. After the prediction by Bian and Bandrauk in ref. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Bian, X. B. & Bandrauk, A. D. Probing nuclear motion by frequency modulation of molecular high-order harmonic generation. Phys. Rev. Lett. 113, 193901 (2014)." href="/articles/s41467-018-03568-3#ref-CR32" id="ref-link-section-d106621839e602">32</a></sup>, FM has received a lot of attention in theoretical studies<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Li, M. Z., Jia, G. R. & Bian, X. B. Alignment dependent ultrafast electron-nuclear dynamics in molecular high-order harmonic generation. J. Chem. Phys. 146, 084305 (2017)." href="/articles/s41467-018-03568-3#ref-CR31" id="ref-link-section-d106621839e607">31</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lara-Astiaso, M. et al. Enhancing high-order harmonic generation in light molecules by using chirped pulses. Phys. Rev. Lett. 117, 093003 (2016)." href="#ref-CR33" id="ref-link-section-d106621839e610">33</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Miller, M. R., Xia, Y., Becker, A. & Jaroń-Becker, A. Laser-driven nonadiabatic electron dynamics in molecules. Optica 3, 259–269 (2016)." href="#ref-CR34" id="ref-link-section-d106621839e610_1">34</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Silva, R. E. F. et al. Even harmonic generation in isotropic media of dissociating homonuclear molecules. Sci. Rep. 6, 32653 (2016)." href="/articles/s41467-018-03568-3#ref-CR35" id="ref-link-section-d106621839e613">35</a></sup>. However, the FM in isotopic MHOHG has never been observed in experiment and the measurement of nuclear motion based on the FM is not addressed.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Fig. 1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-018-03568-3/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig1_HTML.jpg?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig1_HTML.jpg" alt="figure 1" loading="lazy" width="685" height="211"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>Sketch of AM and FM in MHOHG. <b>a</b> Schematic diagram of HHG from atoms. The harmonic spectrum generated with atom is composed by series regular odd-order harmonics. For different harmonics (<i>q</i><sub>1</sub> and <i>q</i><sub>2</sub>) in the plateau region, the harmonic intensities are comparable (<i>A</i><sub>1</sub> ≈ <i>A</i><sub>2</sub>). <b>b</b> Schematic diagram of HHG from molecules. For molecules, the laser-driven nuclear motion introduces an additional degree of freedom and will modulate the MHOHG. As demonstrated in refs. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lein, M. Attosecond probing of vibrational dynamics with high-harmonic generation. Phys. Rev. Lett. 94, 053004 (2005)." href="#ref-CR27" id="ref-link-section-d106621839e650">27</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Baker, S. et al. Probing proton dynamics in molecules on an attosecond time scale. Science 312, 424–427 (2006)." href="#ref-CR28" id="ref-link-section-d106621839e650_1">28</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Lan, P. et al. Attosecond probing of nuclear dynamics with trajectory-resolved high-harmonic spectroscopy. Phys. Rev. Lett. 199, 033201 (2017)." href="/articles/s41467-018-03568-3#ref-CR29" id="ref-link-section-d106621839e653">29</a></sup>, the harmonic intensity in the plateau is approximately proportional to the square of modulus of the nuclear correlation function, which depends sensitively on the traveling time of the electron in the continuum. For different harmonics <i>q</i><sub>1</sub> and <i>q</i><sub>2</sub>, the traveling times of the electrons are different, thus leading to an AM in the spectrum (<i>A</i>′<sub>1</sub> ≠ <i>A</i>′<sub>2</sub>). On the other hand, the laser-driven nuclear motion will enhance the ionization rate and thus strengthen the harmonic emission at the trailing edge of the laser pulse. Due to the laser-driven nonadiabatic effect, harmonics dominated at the trailing edge will emerge a FM (red shift) in the spectrum (<i>q</i><sub>1</sub><i>ω</i><sub>0</sub> → <i>q</i><sub>1</sub><i>ω</i><sub>0</sub> − Δ<i>ω</i><sub>1</sub>, <i>q</i><sub>2</sub><i>ω</i><sub>0</sub> → <i>q</i><sub>2</sub><i>ω</i><sub>0</sub> − Δ<i>ω</i><sub>2</sub>)</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-018-03568-3/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>In the present work, we report the experimental observation of FM in MHOHG from isotopic molecules H<sub>2</sub> and D<sub>2</sub>. High-order harmonics generated from isotopic molecules show obvious red shift with respect to those from Ar atom. The red shift is demonstrated to originate from the laser-induced nuclear motion of isotopic molecules, which strengthens harmonic emission at the trailing edge of the laser pulse. From the observed frequency shift, the nuclear motions of H<sub>2</sub> and D<sub>2</sub> are successfully retrieved, which agree well with the calculations from non-Born–Oppenheimer time-dependent Schrödinger equation (NBO–TDSE).</p></div></div></section><section data-title="Results"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2">Results</h2><div class="c-article-section__content" id="Sec2-content"><h3 class="c-article__sub-heading" id="Sec3">Experimental observation of FM</h3><p>The experiment is carried out by adopting a Ti:sapphire laser, and H<sub>2</sub> and D<sub>2</sub> molecules (see the methods). These isotopes have attracted extensive interest as a prototype. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig2">2a–c</a> displays the spatially resolved harmonic spectra generated from atomic gas Ar and the hydrogen isotopes H<sub>2</sub> and D<sub>2</sub>, respectively. Their ionization potentials are very close. The spatially integrated HHG signals are presented by the dash-dotted (Ar), solid (H<sub>2</sub>), and dashed (D<sub>2</sub>) lines in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig2">2d</a>. One can see that the harmonic intensities from D<sub>2</sub> are higher than those from H<sub>2</sub>, which is in consistent with previous studies<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lein, M. Attosecond probing of vibrational dynamics with high-harmonic generation. Phys. Rev. Lett. 94, 053004 (2005)." href="#ref-CR27" id="ref-link-section-d106621839e769">27</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Baker, S. et al. Probing proton dynamics in molecules on an attosecond time scale. Science 312, 424–427 (2006)." href="#ref-CR28" id="ref-link-section-d106621839e769_1">28</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Lan, P. et al. Attosecond probing of nuclear dynamics with trajectory-resolved high-harmonic spectroscopy. Phys. Rev. Lett. 199, 033201 (2017)." href="/articles/s41467-018-03568-3#ref-CR29" id="ref-link-section-d106621839e772">29</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Mizutani, H., Minemoto, S., Oguchi, Y. & Sakai, H. Effect of nuclear motion observed in high-order harmonic generation from D2/H2 molecules with intense multi-cycle 1300 nm and 800 nm pulses. J. Phys. B 44, 081002 (2011)." href="/articles/s41467-018-03568-3#ref-CR36" id="ref-link-section-d106621839e775">36</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 37" title="Kanai, T., Takahashi, E. J., Nabekawa, Y. & Midorikawa, K. Observing the attosecond dynamics of nuclear wavepackets in molecules by using high harmonic generation in mixed gases. New J. Phys. 10, 025036 (2008)." href="/articles/s41467-018-03568-3#ref-CR37" id="ref-link-section-d106621839e778">37</a></sup>. More importantly, the measured harmonics from H<sub>2</sub> and D<sub>2</sub> present obvious frequency shift with respect to those from Ar. As shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig2">2e–h</a>, each harmonic from both H<sub>2</sub> and D<sub>2</sub> shows a red shift relative to that from Ar. While for D<sub>2</sub>, the frequency shift is larger than that of H<sub>2</sub>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Fig. 2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-018-03568-3/figures/2" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig2_HTML.jpg?as=webp"><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig2_HTML.jpg" alt="figure 2" loading="lazy" width="685" height="351"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Experimentally measured harmonic signals. <b>a</b>–<b>c</b> are the spatially resolved harmonic spectra of Ar, H<sub>2</sub>, and D<sub>2</sub>. <b>d</b> shows the spatially integrated HHG signals for the spectra in <b>a</b> (dash-dotted line), <b>b</b> (solid line), and <b>c</b> (dashed line), respectively. For clarity, the dash-dotted line is multiplied by a factor of 0.2. <b>e</b>–<b>h</b> are the normalized harmonic signals of H15-H21 for Ar (dash-dotted line), H<sub>2</sub> (solid line), and D<sub>2</sub> (dashed line), respectively. Here, the laser intensity is 1.5 × 10<sup>14</sup> W cm<sup>−2</sup> and the pulse duration is 30 fs</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-018-03568-3/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="Sec4">Gas pressure dependence of FM</h3><p>HHG in gas medium includes the individual response, as well as the copropagation of laser and harmonic fields. The propagation effect can possibly induce a frequency shift in HHG<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Wood, W. M., Siders, C. W. & Downer, M. C. Measurement of femtosecond ionization dynamics of atmospheric density gases by spectral blueshifting. Phys. Rev. Lett. 67, 3523–3526 (1991)." href="#ref-CR38" id="ref-link-section-d106621839e863">38</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Brandi, F., Giammanco, F. & Ubachs, W. Spectral redshift in harmonic generation from plasma dynamics in the laser focus. Phys. Rev. Lett. 96, 123904 (2006)." href="#ref-CR39" id="ref-link-section-d106621839e863_1">39</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title="Shin, H. J., Lee, D. G., Cha, Y. H., Hong, K. H. & Nam, C. H. Generation of nonadiabatic blueshift of high harmonics in an intense femtosecond laser field. Phys. Rev. Lett. 83, 2544–2547 (1999)." href="/articles/s41467-018-03568-3#ref-CR40" id="ref-link-section-d106621839e866">40</a></sup>, which depends sensitively on the gas pressure. However, in our experiment the ionizations of the three gases are weak (below 4%), and also the gas pressure is low. Then the frequency shift induced by the propagation effect will be inappreciable. To check this effect, we measured the harmonic spectra generated from Ar, H<sub>2</sub>, and D<sub>2</sub> at different gas pressures. With the gas pressure changing from 15 to 35 torr, the intensity of each harmonic from these three gases exhibits a quadratic increase, which indicates a good phase matching in our experiment. More than that, the central wavelengths of each harmonic from the three gases are nearly unchanged as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig3">3a–c</a>. For a clear insight, in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig3">3d</a>, we present the central wavelength of H17 for Ar (diamonds), H<sub>2</sub> (squares), and D<sub>2</sub> (circles) as a function of the gas pressure. The frequency shift of H<sub>2</sub> and D<sub>2</sub> relative to that of Ar keeps almost constant as the gas pressure varies. These results indicate that the influence of propagation effect on the harmonic frequency shift is negligible in our experiment. Besides, the experimental conditions used for HHG from H<sub>2</sub> and D<sub>2</sub> are exactly the same, the differences in the harmonic spectra can be mainly attributed to the individual response of isotope molecules in the driving laser field.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Fig. 3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-018-03568-3/figures/3" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig3_HTML.jpg?as=webp"><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig3_HTML.jpg" alt="figure 3" loading="lazy" width="685" height="553"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>Gas pressure-dependent HHG. <b>a</b>–<b>c</b> Measured harmonic spectra from Ar (<b>a</b>), H<sub>2</sub> (<b>b</b>), and D<sub>2</sub> (<b>c</b>) at different gas pressures. <b>d</b> The central wavelength of H17 from Ar (diamonds), H<sub>2</sub> (squares), and D<sub>2</sub> (circles) as a function of the gas pressure. Shaded areas in <b>d</b> represent the standard deviation of nine independent measurements</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-018-03568-3/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="Sec5">Theoretical simulation of FM</h3><p>It has been reported that the nonadiabatic effect of the time-dependent laser intensity can induce a blue or red shift when HHG is dominant at the leading or trailing edge of the laser pulse<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Schafer, K. J. & Kulander, K. C. High harmonic generation from ultrafast pump lasers. Phys. Rev. Lett. 78, 638–641 (1997)." href="#ref-CR41" id="ref-link-section-d106621839e950">41</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Geissler, M., Tempea, G. & Brabec, T. Phase-matched high-order harmonic generation in the nonadiabatic limit. Phys. Rev. A 62, 033817 (2000)." href="#ref-CR42" id="ref-link-section-d106621839e950_1">42</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 43" title="Bian, X. B. & Bandrauk, A. D. Nonadiabatic molecular high-order harmonic generation from polar molecules: spectral redshift. Phys. Rev. A 83, 041403 (2011)." href="/articles/s41467-018-03568-3#ref-CR43" id="ref-link-section-d106621839e953">43</a></sup>. For H<sub>2</sub> (or D<sub>2</sub>), the ionization rate depends sensitively on the internuclear distance <i>R</i><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Zuo, T. & Bandrauk, A. D. Charge-resonance-enhanced ionization of diatomic molecular ions by intense lasers. Phys. Rev. A 52, R2511–R2514 (1995)." href="/articles/s41467-018-03568-3#ref-CR18" id="ref-link-section-d106621839e963">18</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Seideman, T., Ivanov, M., Yu. & Corkum, P. B. Role of electron localization in intense-field molecular ionization. Phys. Rev. Lett. 75, 2819–2822 (1995)." href="/articles/s41467-018-03568-3#ref-CR19" id="ref-link-section-d106621839e966">19</a></sup>. Due to the laser-driven nuclear motion, the average internuclear distance at the trailing edge can be larger than that at the leading edge of the laser pulse, which makes the ionization, as well as the HHG signals stronger at the trailing edge, and therefore induces a red shift in the harmonic spectrum. In contrast, since the laser intensity used in our experiment is far smaller than the ionization saturation threshold of Ar, the ionization and HHG of Ar atom mainly occurs at central part of the laser pulse and is symmetric with respect to the pulse center (<i>t</i> = 0). Then no obvious net shift exists in the harmonics from Ar. Since the nuclear dynamics is avoided for Ar, it can serve as a benchmark to evaluate the frequency shift of harmonics from the two isotopic molecules. For a given harmonic order, the frequency shift caused by the nonadiabatic effect can be obtained via the time derivative of the laser pulse, namely, <span class="mathjax-tex">$\Delta \omega = \alpha _q\frac{{\partial I(t)}}{{\partial t}}|_{t = t_{\mathrm{i}}}$</span><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Gaarde, M. B. et al. Spatiotemporal separation of high harmonic radiation into two quantum path components. Phys. Rev. A 59, 1367–1373 (1999)." href="#ref-CR44" id="ref-link-section-d106621839e1068">44</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Salières, P. et al. Study of the spatial and temporal coherence of high order harmonics. Adv. At. Mol. Opt. Phys. 41, 83–142 (1999)." href="#ref-CR45" id="ref-link-section-d106621839e1068_1">45</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="He, L. et al. Spectrally resolved spatiotemporal features of quantum paths in high-order-harmonic generation. Phys. Rev. A 92, 043403 (2015)." href="/articles/s41467-018-03568-3#ref-CR46" id="ref-link-section-d106621839e1071">46</a></sup>. Here, <span class="mathjax-tex">$I(t) = I_0{\mathrm{exp}}\left( {\frac{{ - 4{\mathrm{ln}}(2)t^2}}{{\tau ^2}}} \right)$</span> with <i>I</i><sub>0</sub> = 1.5 × 10<sup>14</sup> W cm<sup>−2</sup>, <i>τ</i> = 30 fs is the envelope of the laser pulse, <i>t</i><sub>i</sub> is the ionization moment of the given harmonic, and <i>α</i><sub> <i>q</i> </sub> is its phase coefficient, which can be evaluated according to the strong-field approximation (SFA) model<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Lewenstein, M. et al. Theory of high-harmonic generation by low-frequency laser fields. Phys. Rev. A 49, 2117–2132 (1994)." href="/articles/s41467-018-03568-3#ref-CR47" id="ref-link-section-d106621839e1209">47</a></sup>. The time derivative with a positive (negative) sign means a blue (red) shift of this harmonic. Owing to the slower nuclear motion of heavier nuclei, the dominant harmonic emission of D<sub>2</sub> occurs later than that of H<sub>2</sub> (see Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">1</a> and Supplementary Figure <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">1</a>). As a result, the HHG from D<sub>2</sub> experiences a more rapid change of the effective laser intensity (namely, a larger value of <span class="mathjax-tex">$\left| {\frac{{\partial I(t)}}{{\partial t}}} \right|$</span>), which therefore gives rise to a larger red shift in the harmonic spectrum as observed in our experiment. Besides the nonadiabatic effect, the nuclear motion can lead to the variation of the ionization potential and the complex recombination dipole, which may affect the harmonic phase accumulated during the electron excursion and influence the MHOHG<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Le, Anh-Thu, Morishita, T., Lucchese, R. R. & Lin, C. D. Theory of high harmonic generation for probing time-resolved large-amplitude molecular vibrations with ultrashort intense lasers. Phys. Rev. Lett. 109, 203004 (2012)." href="/articles/s41467-018-03568-3#ref-CR48" id="ref-link-section-d106621839e1278">48</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 49" title="Ferré, A. et al. Two-dimensional frequency resolved optomolecular gating of high-order harmonic generation. Phys. Rev. Lett. 116, 053002 (2016)." href="/articles/s41467-018-03568-3#ref-CR49" id="ref-link-section-d106621839e1281">49</a></sup>. To evaluate these influences, we have performed simulations with the modified SFA model<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Le, Anh-Thu, Morishita, T., Lucchese, R. R. & Lin, C. D. Theory of high harmonic generation for probing time-resolved large-amplitude molecular vibrations with ultrashort intense lasers. Phys. Rev. Lett. 109, 203004 (2012)." href="/articles/s41467-018-03568-3#ref-CR48" id="ref-link-section-d106621839e1286">48</a></sup>, which indicates that the frequency shift induced by these two effects is far smaller than our experimental observations (see Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">2</a> and Supplementary Figures <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">2</a> and <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig3">3</a>). Moreover, the laser-driven nonadiabatic alignment may also lead to a red shift in MHOHG. We have evaluated this influence by considering the time-dependence of the laser-driven alignment under our experiment condition (see Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">3</a> and Supplementary Figure <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">4</a>). Our calculations show that the red shift induced by the molecular alignment is about one order of magnitude smaller than our experimental observations. Note also that the fluctuation of the laser carrier-envelope phase, which is not fixed in our experiment, will not affect the measured frequency shift of MHOHG because a multi-cycle laser pulse is used in our experiment. Therefore, the main contribution to the frequency shift shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig2">2</a> is attributed to the nonadiabatic effect induced by the nuclear motion<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Bian, X. B. & Bandrauk, A. D. Probing nuclear motion by frequency modulation of molecular high-order harmonic generation. Phys. Rev. Lett. 113, 193901 (2014)." href="/articles/s41467-018-03568-3#ref-CR32" id="ref-link-section-d106621839e1309">32</a></sup>.</p><p>In Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig4">4a</a>, we present the relative frequency shift of H15-H23 for H<sub>2</sub> (squares) and D<sub>2</sub> (circles). The relative frequency shifts gradually decrease as the harmonic order increases. The experiment is also simulated by solving the NBO–TDSE (see the methods). The calculated frequency shifts of H<sub>2</sub> and D<sub>2</sub> are presented by the dashed lines in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig4">4a</a>, which are in agreement with the experimental observations. Some difference in quantity may arise from the uncertainties of experimental parameters. In Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig4">4b</a>, we have calculated the asymmetry coefficients of HHG signals for H<sub>2</sub> (squares) and D<sub>2</sub> (circles). Here, the HHG asymmetry is defined as <i>η</i>(<i>ω</i>) = (<i>P</i><sub>+</sub>(<i>ω</i>) − <i>P</i><sub>−</sub>(<i>ω</i>))/(<i>P</i><sub>+</sub>(<i>ω</i>) + <i>P</i><sub>−</sub>(<i>ω</i>))<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Li, M. Z., Jia, G. R. & Bian, X. B. Alignment dependent ultrafast electron-nuclear dynamics in molecular high-order harmonic generation. J. Chem. Phys. 146, 084305 (2017)." href="/articles/s41467-018-03568-3#ref-CR31" id="ref-link-section-d106621839e1374">31</a></sup>, where <span class="mathjax-tex">$P_ + (w) = {\int}_0^{ + \infty } g(\omega ,t)\mathrm{d}t$</span> and <span class="mathjax-tex">$P_ - (w) = {\int}_{ - \infty }^0 g(\omega ,t)\mathrm{d}t$</span> are the amount of harmonic <i>ω</i> generated at the trailing and leading edges of the laser pulse, respectively. <i>g</i>(<i>ω</i>, <i>t</i>) is the time-frequency spectrogram calculated with the Gabor transform. In our calculation, the width of the time window used in the Gabor transform is 0.1 fs, which corresponds to a filter with the width of 10<i>ω</i><sub>0</sub> (<i>ω</i><sub>0</sub> is laser frequency) in the frequency domain. One can see that for each harmonic order, the harmonic emission is more pronounced at the trailing edge of the laser pulse (namely, <i>η</i>(<i>ω</i>) > 0). Therefore, all the harmonics exhibit a red shift in the spectrum as shown in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig4">4a</a>. Moreover, the trend of the red shift agrees qualitatively with the asymmetry coefficients. This agreement suggests that the observed red shift indeed results from the delayed emission of HHG with respect to the center of the laser pulse.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Fig. 4"><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 4</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/articles/s41467-018-03568-3/figures/4" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig4_HTML.jpg?as=webp"><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_Fig4_HTML.jpg" alt="figure 4" loading="lazy" width="685" height="396"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>Measured and calculated frequency shifts and nuclear motions of H<sub>2</sub> and D<sub>2</sub>. <b>a</b> Measured and calculated red shift Δ<i>ω</i> in MHOHG with respect to harmonics of Ar as a function of the harmonic order. <b>b</b> The HHG asymmetry coefficients <i>η</i> calculated for H<sub>2</sub> and D<sub>2</sub>. <i>η</i>(<i>ω</i>) > 0(<0) means the harmonic emission is more pronounced at the trailing (leading) edge of the laser pulse. Squares and circles are for H<sub>2</sub> and D<sub>2</sub>, respectively. <b>c</b> Calculated (solid line) and experimentally retrieved (dash-dotted line) nuclear vibration of H<sub>2</sub>. <b>d</b> Same as <b>c</b>, but for D<sub>2</sub>. Shaded areas in <b>a</b> represent the standard deviation of nine independent measurements</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/articles/s41467-018-03568-3/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><h3 class="c-article__sub-heading" id="Sec6">Monitoring the nuclear dynamics by FM</h3><p>As mentioned above, the frequency shift mainly arises from the asymmetry of the ionization (and so HHG) with respect to the center of laser pulse (<i>t</i> = 0 fs)<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Bian, X. B. & Bandrauk, A. D. Probing nuclear motion by frequency modulation of molecular high-order harmonic generation. Phys. Rev. Lett. 113, 193901 (2014)." href="/articles/s41467-018-03568-3#ref-CR32" id="ref-link-section-d106621839e1664">32</a></sup>. Previous studies have shown that the ionization rate of H<sub>2</sub> (D<sub>2</sub>) is approximately linearly dependent on internuclear distance <i>R</i> before it reaches 2 a.u.<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Saenz, A. Behavior of molecular hydrogen exposed to strong DC, AC, or low-frequency laser fields. II. Comparison of ab initio and Ammosov-Delone-Krainov rates. Phys. Rev. A 66, 063408 (2002)." href="/articles/s41467-018-03568-3#ref-CR50" id="ref-link-section-d106621839e1676">50</a></sup>. Thus the relative frequency shift of the <i>q</i>-th harmonic with respect to <i>qω</i><sub>0</sub> can be estimated as:</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\frac{{\Delta \omega }}{{\omega _0}} = \frac{{\mathop {\sum}\limits_{t_{\mathrm{i}} < 0} R(t_{\mathrm{i}}) - \mathop {\sum}\limits_{t_{\mathrm{i}} > 0} R(t_{\mathrm{i}})}}{{\mathop {\sum}\limits_{t_{\mathrm{i}}} R(t_{\mathrm{i}})}},$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>where <i>ω</i><sub>0</sub> is the frequency of the driving laser, <i>t</i><sub>i</sub> is the ionization moment of the electron (contributing to the <i>q</i>-th harmonic generation) in each half optical cycle. <i>t</i><sub>i</sub> < 0 and <i>t</i><sub>i</sub> > 0 mean the ionization occurs at the leading and trailing edges of laser pulse, respectively. For a given harmonic, <i>t</i><sub>i</sub> can be calculated according to the three-step model<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Lewenstein, M. et al. Theory of high-harmonic generation by low-frequency laser fields. Phys. Rev. A 49, 2117–2132 (1994)." href="/articles/s41467-018-03568-3#ref-CR47" id="ref-link-section-d106621839e1934">47</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Corkum, P. B. Plasma perspective on strong field multiphoton ionization. Phys. Rev. Lett. 71, 1994–1997 (1993)." href="/articles/s41467-018-03568-3#ref-CR51" id="ref-link-section-d106621839e1937">51</a></sup>. To retrieve the nuclear motion, we consider to employ the commonly used linear harmonic oscillator model<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 52" title="Landau, L. D. & Lifshitz, E. M. Quantum Mechanics (Pergamon, New York, 1958)." href="/articles/s41467-018-03568-3#ref-CR52" id="ref-link-section-d106621839e1941">52</a></sup> to describe the two-body vibrations of H<sub>2</sub> and D<sub>2</sub>. The simulations with the NBO–TDSE show that the harmonic oscillator model works well in a low-ionization case. In our experiment, the ionization is below 4%, therefore the harmonic oscillator model is applicable. In the harmonic oscillator model, the potential <i>V</i>(<i>r</i>) of H<sub>2</sub> (or D<sub>2</sub>) can be approximatively expressed as <span class="mathjax-tex">$V(r) = V_0 + \frac{k}{2}(r - R_e)^2$</span>, where <i>V</i><sub>0</sub> and <i>k</i> are constants and <i>R</i><sub>e</sub> is the equilibrium internuclear distance of H<sub>2</sub> and D<sub>2</sub>. Then the laser-driven nuclear motion can be derived in the form of (for details, see Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">4</a>)</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$R(t) = A{\mathrm{sin}}({\mathrm{\Omega }}t + \phi ) + BI(t) + R_{\mathrm{e}}.$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>On the right side of Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-018-03568-3#Equ2">2</a>), the first term denotes the inherent harmonic vibration, <i>A</i>, Ω, and <i>ϕ</i> are the corresponding amplitude, frequency, and phase of the vibration. The second term represents the laser-nucleus interaction. Inserting Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-018-03568-3#Equ2">2</a>) into Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-018-03568-3#Equ1">1</a>), the frequency shift of a specific harmonic can be expressed as a function of <i>A</i>,<i> B</i>, Ω, and <i>ϕ</i>. By fitting the observed frequency shifts of H15-H23 to Eq. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/articles/s41467-018-03568-3#Equ1">1</a>) with the least square method, the four parameters can be determined. Then the nuclear motion <i>R</i>(<i>t</i>) can be retrieved. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig4">4c–d</a> shows the retrieved nuclear vibrations (dash-dotted line) of H<sub>2</sub> and D<sub>2</sub>, respectively. As shown in this figure, the maximum of the retrieved <i>R</i>(<i>t</i>) of H<sub>2</sub> is about 1.505 a.u., which is slightly larger than that of D<sub>2</sub> (1.485 a.u.). Moreover, the retrieved <i>R</i>(<i>t</i>) of H<sub>2</sub> oscillates with a period of 8.2 fs. In contrast, it is 11.4 fs for D<sub>2</sub>. The retrieved oscillation periods of H<sub>2</sub> and D<sub>2</sub> are very close to the vibrational periods of H<sub>2</sub> and D<sub>2</sub> in their ground electronic state (7.5 and 10.6 fs). The ratio of these two retrieved periods is also very close to the expected mass ratio of <span class="mathjax-tex">$\sqrt 2$</span>. The results calculated from the NBO–TDSE are also presented as the solid lines. From Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig4">4c, d</a>, one can see that due to the inherent harmonic vibration of the harmonic oscillator model, the retrieved nuclear motion shows much deeper modulation at the beginning when compared to the simulated one. While with the increase of the laser intensity, the simulated nuclear motion also turns to oscillate after <i>t</i> = − 20 fs due to vibrational excitation. Despite the initial oscillation, the main structures of the retrieved nuclear motion <i>R</i>(<i>t</i>), such as the dynamic range and the overall trend, can agree well with the theoretical predictions in the range of [−20, 20] fs where most of the HHG signals are generated. It should be explained that to compare with the NBO–TDSE calculations, the initial internuclear distance used in the experimental fitting is obtained from the NBO–TDSE simulation [namely, the initial values of the solid lines in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig4">4c, d</a>]. It is given by the expectation value of <i>R</i> with the ground state wavefunction Ψ<sub>0</sub>, namely, 〈Ψ<sub>0</sub>|<i>R</i>|Ψ<sub>0</sub>〉. Note that the expectation values 〈Ψ<sub>0</sub>|<i>R</i>|Ψ<sub>0</sub>〉 are different for H<sub>2</sub> and D<sub>2</sub> due to their different field-free Hamiltonians (depending on the nuclear mass). Moreover, the so-called equilibrium internuclear distance <i>R</i><sub>e</sub> is defined as the minimum of the BO potential <i>V</i><sub>BO</sub>(<i>R</i>), namely, where d<i>V</i><sub>BO</sub>(<i>R</i>)/d<i>R</i> = 0<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 53" title="Shaik, S. S. & Hiberty, P. C. A Chemist’s Guide to Valence Bond Theory (Wiley, New York, 2007)." href="/articles/s41467-018-03568-3#ref-CR53" id="ref-link-section-d106621839e2336">53</a></sup>. Since the BO potential <i>V</i><sub>BO</sub>(<i>R</i>) is slightly asymmetric with respect to <i>R</i><sub>e</sub>, the expectation value of <i>R</i> obtained from the NBO–TDSE simulation (1.44 a.u. for H<sub>2</sub> and 1.43 a.u. for D<sub>2</sub>) is slightly different from the so-called equilibrium internuclear distance <i>R</i><sub>e</sub> (1.4 a.u. for both H<sub>2</sub> and D<sub>2</sub><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 54" title="Kol/os, W. O., Szalewicz, K. & Monkhorst, H. J. New Born-Oppenheimer potential energy curve and vibrational energies for the electronic ground state of the hydrogen molecule. J. Chem. Phys. 84, 3278–3283 (1986)." href="/articles/s41467-018-03568-3#ref-CR54" id="ref-link-section-d106621839e2367">54</a></sup>). To study the stability of the retrievals, we have also performed the fitting with different harmonic orders or using the known values of the oscillation frequency Ω. The obtained results are all in good agreement with the NBO–TSDE simulations (see Supplementary Note <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">4</a> and Supplementary Figures <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">5</a> and <a data-track="click" data-track-label="link" data-track-action="supplementary material anchor" href="/articles/s41467-018-03568-3#MOESM1">6</a>). Considering the simplicity of the harmonic oscillator model and the uncertainty of experimental parameters, the agreement of the retrieved nuclear motions with the TDSE predictions is very satisfying.</p></div></div></section><section data-title="Discussion"><div class="c-article-section" id="Sec7-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec7">Discussion</h2><div class="c-article-section__content" id="Sec7-content"><p>In summary, we experimentally observed the red shift in HHG from isotopic molecules H<sub>2</sub> and D<sub>2</sub>. The red shift is primarily attributed to the laser-driven nuclear motion in H<sub>2</sub> and D<sub>2</sub>, which strengthens the ionization rate and harmonic emission due to larger internuclear distance <i>R</i> and lower <i>I</i><sub>p</sub> at the trailing edge of the laser pulse. By using a linear harmonic oscillator model, the nuclear vibrations of H<sub>2</sub> and D<sub>2</sub> are successfully retrieved from the observed frequency shift. The FM effect in MHOHG is universal, which can be directly applied to other light molecules if the ionization rate is sensitive to nuclear motion. Moreover, in our experiment the molecules are not pre-aligned, the alignment effect is negligible. In principle, the FM technology can be extended to aligned molecules with any alignment angles with respect to the laser polarization. The alignment-angle-dependent FM can not only be used to extract the ultrafast electron-nuclear dynamics, but also be possible to image molecular structure.</p><p>In previous studies of AM, the intensity ratios of HHG from isotopic molecules reveal the nuclear dynamics of <span class="mathjax-tex">${\mathrm{H}}_2^ +$</span> and <span class="mathjax-tex">${\mathrm{D}}_2^ +$</span> within the time window from ionization to recombination in one laser cycle, namely, intracycle dynamics. In contrast, in the present work, the observed frequency shift provides a monitoring of the nuclear vibrations of H<sub>2</sub> and D<sub>2</sub> at each ionization moment in the laser pulse, namely, intercycle dynamics. Therefore, FM in MHOHG reveals a different physical process and is complementary with the method of AM<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Lein, M. Attosecond probing of vibrational dynamics with high-harmonic generation. Phys. Rev. Lett. 94, 053004 (2005)." href="#ref-CR27" id="ref-link-section-d106621839e2489">27</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Baker, S. et al. Probing proton dynamics in molecules on an attosecond time scale. Science 312, 424–427 (2006)." href="#ref-CR28" id="ref-link-section-d106621839e2489_1">28</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Lan, P. et al. Attosecond probing of nuclear dynamics with trajectory-resolved high-harmonic spectroscopy. Phys. Rev. Lett. 199, 033201 (2017)." href="/articles/s41467-018-03568-3#ref-CR29" id="ref-link-section-d106621839e2492">29</a></sup> for probing the nuclear dynamics. These findings may provide a deep insight into some of the most fundamental events in chemistry and facilitate the development of HHS.</p></div></div></section><section data-title="Methods"><div class="c-article-section" id="Sec8-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec8">Methods</h2><div class="c-article-section__content" id="Sec8-content"><h3 class="c-article__sub-heading" id="Sec9">Experimental methods</h3><p>The experiment is performed by using a commercial Ti:sapphire laser system (Legend Elite-Duo, Coherent, Inc.), which delivers the 30 fs, 800 nm pulses at a repetition rate of 1 kHz. The output laser pulse is focused to a 2-mm-long gas cell by a 600-mm focal-length lens. In Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/articles/s41467-018-03568-3#Fig2">2</a>, the stagnation pressure of the gases is 30 torr and the gas cell is placed 2 mm after the laser focus to ensure the phase matching of the short quantum path. The laser energy used in our experiment is maintained at 1.5 mJ and the corresponding intensity is estimated to be 1.5 × 10<sup>14</sup> W cm<sup>−2</sup>. The generated harmonic spectrum is detected by a homemade flat-field soft x-ray spectrometer consisting of a flat-field grating (1200 grooves mm<sup>−1</sup>) and a slit with a width of about 0.1 mm and height of 15 mm. High-order harmonics are dispersed by the grating and imaged onto the microchannel plate (MCP) fitted with a phosphor screen. The image on the screen is read out by a CCD camera.</p><p>To accurately evaluate the frequency shift in MHOHG, we have calibrated the spectrometer by using the atomic lines of carbon in terms of a procedure similar to that in refs. <sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 55" title="Shiner, A. D. et al. Probing collective multi-electron dynamics in xenon with high-harmonic spectroscopy. Nat. Phys. 7, 464–467 (2011)." href="/articles/s41467-018-03568-3#ref-CR55" id="ref-link-section-d106621839e2520">55</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 56" title="Farrell, J. P., McFarland, B. K., Bucksbaum, P. H. & Gühr, M. Calibration of a high harmonic spectrometer by laser induced plasma emission. Opt. Express 17, 15134–15144 (2009)." href="/articles/s41467-018-03568-3#ref-CR56" id="ref-link-section-d106621839e2523">56</a></sup>. The atomic lines are produced by focusing several millijoules of the driving laser pulse to interact with a 0.5-mm-thick graphite sheet placed at the position where HHG occurs. We record the generated atomic lines and read their coordinates on the phosphor screen. By assigning the observed atomic lines to the known literature data of carbon, we can then achieve the calibration of the spectrometer. Details of the calibration are provided in Supplementary Methods.</p><h3 class="c-article__sub-heading c-article__sub-heading--divider" id="Sec10">Theoretical methods</h3><p>To simulate the HHG process and nuclear dynamics of H<sub>2</sub> and D<sub>2</sub>, we numerically solve the NBO–TDSE with one active electron<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Lein, M. Attosecond probing of vibrational dynamics with high-harmonic generation. Phys. Rev. Lett. 94, 053004 (2005)." href="/articles/s41467-018-03568-3#ref-CR27" id="ref-link-section-d106621839e2539">27</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 57" title="Chiril, C. C. & Lein, M. Influence of nuclear vibration on harmonic generation in molecules. J. Phys. B 39, S437–S444 (2006)." href="/articles/s41467-018-03568-3#ref-CR57" id="ref-link-section-d106621839e2542">57</a></sup>. Since the electron and nuclear motions follow the linearly polarized laser field, we adopt the one-dimensional model,</p><div id="Equa" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$i\frac{{\partial \Psi (z,R,t)}}{{\partial t}} = [H_{\mathrm{e}}(t) + H_{\mathrm{n}}(t) - E(t)z]\Psi (z,R,t),$$</span></div></div><div id="Equb" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$H_{\mathrm{n}} = - \frac{1}{{2\mu }}\frac{{\partial ^2}}{{\partial R^2}} + \frac{1}{R},$$</span></div></div><div id="Equc" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$H_{\mathrm{e}} = - \frac{1}{2}\frac{{\partial ^2}}{{\partial z^2}} + V_{{\mathrm{en}}}(R,z),$$</span></div></div><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$V_{{\mathrm{en}}}(R,z) = - \frac{{Z(R,|z + R/2|)}}{{\sqrt {(z + R/2)^2 + 0.5} }} - \frac{{Z(R,|z - R/2|)}}{{\sqrt {(z - R/2)^2 + 0.5} }} + V_{{\mathrm{BO}}}^ + (R).$$</span></div><div class="c-article-equation__number"> (3) </div></div><p>Here, <i>z</i> is the electron coordinate, <i>R</i> is the internuclear distance, and <i>E</i>(<i>t</i>) is the driving laser field. <i>H</i><sub>n</sub> and <i>H</i><sub>e</sub> are the Hamiltonians for the nuclei and electron, respectively. <i>V</i><sub>en</sub>(<i>R</i>, <i>z</i>) is the Coulomb potential of the electron–nucleus interaction. <i>μ</i> is the reduced mass of two nuclei and <span class="mathjax-tex">$V_{{\mathrm{BO}}}^ + (R)$</span> is the lowest BO potential of H<span class="mathjax-tex">$_2^ +$</span>. In order to faithfully mimic the nuclear dynamics of H<sub>2</sub> and D<sub>2</sub>, we have adopted an effective nuclear charge <span class="mathjax-tex">$Z(R,\xi ) = \left[ {1 + e^{ - \xi ^2/\sigma ^2(R)}} \right]/2$</span>, where <i>σ</i>(<i>R</i>) is an <i>R</i>-dependent screening parameter. By adjusting the parameter <i>σ</i>(<i>R</i>) at each internuclear distance, the energy of the ground electronic state of hydrogen and hydrogen ion can be well reproduced.</p><p>The TDSE is solved using the <i>B</i>-spline method with Crank–Nicolson scheme<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 58" title="Bian, X. B. Photoionization of atoms and molecules studied by the Crank-Nicolson method. Phys. Rev. A 90, 033403 (2014)." href="/articles/s41467-018-03568-3#ref-CR58" id="ref-link-section-d106621839e3429">58</a></sup>. The initial wave function is taken as the ground state of the hydrogen molecule calculated using the imaginary-time propagation method. The wave function is numerically discretized in the grid with a size of −50 ≤ <i>z</i> ≤ 50 a.u. and <i>R</i> ≤ 18 a.u. The discrete steps for the electron coordinate and nuclear coordinate are <i>δz</i> = 0.1 and <i>δR</i> = 0.05 a.u., respectively. The temporal step is <i>δt</i> = 0.05 a.u. Then the harmonic spectrum can be obtained by performing the Fourier transform of the laser-induced electron dipole moment <i>d</i>(<i>t</i>) = 〈Ψ(<i>z</i>,<i>R</i>,<i>t</i>)|<i>z</i>|Ψ(<i>z</i>,<i>R</i>,<i>t</i>)〉. The expectation of the internuclear distance is calculated by <span class="mathjax-tex">$R(t) = \frac{{\left\langle {\Psi \left( {z,R,t} \right)\left| R \right|\Psi \left( {z,R,t} \right)} \right\rangle }}{{\left\langle {\Psi \left( {z,R,t} \right)\Psi \left( {z,R,t} \right)} \right\rangle }}$</span><sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 59" title="Kulander, K. C., Mies, F. H. & Schafer, K. J. Model for studies of laser-induced nonlinear processes in molecules. Phys. Rev. A 53, 2562–2570 (1996)." href="/articles/s41467-018-03568-3#ref-CR59" id="ref-link-section-d106621839e3611">59</a></sup>.</p><p>To confirm the results of the one-electron model, we have also solved the NBO–TDSE with two electrons<sup><a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Bandrauk, A. D., Chelkowski, S., Kawai, S. & Lu, H. Effect of nuclear motion on molecular high-order harmonics and on generation of attosecond pulses in intense laser pulses. Phys. Rev. Lett. 101, 153901 (2008)." href="/articles/s41467-018-03568-3#ref-CR22" id="ref-link-section-d106621839e3618">22</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Bandrauk, A. D., Chelkowski, S. & Lu, H. Signatures of nuclear motion in molecular high-order harmonics and in the generation of attosecond pulse trains by ultrashort intense laser pulses. J. Phys. B 42, 075602 (2009)." href="/articles/s41467-018-03568-3#ref-CR23" id="ref-link-section-d106621839e3621">23</a></sup>, which reads</p><div id="Equd" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$i\frac{{\partial \psi (z_1,z_2,R,t)}}{{\partial t}} = \left[ {H_{\mathrm{e}} + H_{\mathrm{n}} + E(t)(z_1 + z_2)} \right]\psi (z_1,z_2,R,t),$$</span></div></div><div id="Eque" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$H_{\mathrm{n}} = - \frac{1}{{2\mu }}\frac{{\partial ^2}}{{\partial R^2}} + \frac{1}{R},$$</span></div></div><div id="Equf" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$H_{\mathrm{e}} = - \frac{1}{2}\frac{{\partial ^2}}{{\partial z_1^2}} - V_{{\mathrm{en}}}(z_1) - \frac{1}{2}\frac{{\partial ^2}}{{\partial z_2^2}} - V_{{\mathrm{en}}}(z_2) + V_{{\mathrm{ee}}}(z_1 - z_2),$$</span></div></div><div id="Equg" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$V_{{\mathrm{en}}}(z_i) = \frac{1}{{\sqrt {(z_i + R/2)^2 + \beta (R)^2} }} + \frac{1}{{\sqrt {(z_i - R/2)^2 + \beta (R)^2} }},$$</span></div></div><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$V_{{\mathrm{ee}}} = \frac{1}{{\sqrt {(z_1 - z_2)^2 + \gamma (R)^2} }}.$$</span></div><div class="c-article-equation__number"> (4) </div></div><p><i>H</i><sub>n</sub> and <i>H</i><sub>e</sub> are the Hamiltonians for the nuclei and electrons. <i>V</i><sub>en</sub>(<i>z</i><sub> <i>i</i> </sub>) and <i>V</i><sub>ee</sub> are the Coulomb potential for the electron–nucleus and electron–electron interactions. <i>β</i>(<i>R</i>), <i>γ</i>(<i>R</i>) are the <i>R</i>-dependent softening parameters. Due the huge computation of the two-electron model, we calculate only the harmonic spectrum for H<sub>2</sub>. The obtained red shift is consistent with the one-electron model, as well as the experimental observations.</p><h3 class="c-article__sub-heading c-article__sub-heading--divider" id="Sec11">Data availability</h3><p>All the data that support the findings of this study are available from the corresponding author upon reasonable request.</p></div></div></section> </div> <div> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Soddy, F. 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Lein and T. Ozaki for valuable discussions. This work was supported by the National Natural Science Foundation of China under Grants No. 11234004, 61475055, 11422435, 11404123, 11404376, and 11674363. Numerical simulations presented in this paper were carried out using the high performance computing experimental testbed in SCTS/CGCL (see <a href="http://grid.hust.edu.cn/hpcc">http://grid.hust.edu.cn/hpcc</a>) and massively parallel computer clusters of RQCHP and Compute Canada.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><span class="c-article-author-information__subtitle u-visually-hidden" id="author-notes">Author notes</span><ol class="c-article-author-information__list"><li class="c-article-author-information__item" id="na1"><p>These authors contributed equally: Lixin He, Qingbin Zhang.</p></li></ol><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, 430074, Wuhan, China</p><p class="c-article-author-affiliation__authors-list">Lixin He, Qingbin Zhang, Pengfei Lan, Wei Cao, Xiaosong Zhu, Chunyang Zhai, Feng Wang, Wenjing Shi & Peixiang Lu</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, 430071, Wuhan, China</p><p class="c-article-author-affiliation__authors-list">Muzi Li & Xue-Bin Bian</p></li><li id="Aff3"><p class="c-article-author-affiliation__address">University of Chinese Academy of Sciences, 100049, Beijing, China</p><p class="c-article-author-affiliation__authors-list">Muzi Li</p></li><li id="Aff4"><p class="c-article-author-affiliation__address">Laboratory of Optical Information Technology, Wuhan Institute of Technology, 430205, Wuhan, China</p><p class="c-article-author-affiliation__authors-list">Peixiang Lu</p></li><li id="Aff5"><p class="c-article-author-affiliation__address">Laboratoire de chimie théorique, Département de Chimie, Université de Sherbrooke, Sherbrooke, J1K 2R1, Quebéc, Canada</p><p class="c-article-author-affiliation__authors-list">André D. Bandrauk</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Lixin-He-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Lixin He</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=Lixin%20He" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Lixin%20He" data-track="click" data-track-action="author link - 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Bandrauk</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?author=Andr%C3%A9%20D.%20Bandrauk" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Andr%C3%A9%20D.%20Bandrauk" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Andr%C3%A9%20D.%20Bandrauk%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="contributions">Contributions</h3><p>P.F.L. and P.X.L. conceived and designed the experiment. L.X.H., Q.B.Z., P.F.L., C.Y.Z., F.W., and W.J.S. performed the experiments. X.B.B, M.Z.L., L.X.H., and P.F.L. performed the simulations. W.C., X.S.Z., and A.D.B. participated in the discussions and revised the manuscript.</p><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding authors</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:pengfeilan@hust.edu.cn">Pengfei Lan</a>, <a id="corresp-c2" href="mailto:xuebin.bian@wipm.ac.cn">Xue-Bin Bian</a> or <a id="corresp-c3" href="mailto:lupeixiang@hust.edu.cn">Peixiang Lu</a>.</p></div></div></section><section data-title="Ethics declarations"><div class="c-article-section" id="ethics-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="ethics">Ethics declarations</h2><div class="c-article-section__content" id="ethics-content"> <h3 class="c-article__sub-heading" id="FPar1">Competing interests</h3> <p>The authors declare no competing interests.</p> </div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><p><b>Publisher's note:</b> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></div></div></section><section data-title="Electronic supplementary material"><div class="c-article-section" id="Sec12-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec12">Electronic supplementary material</h2><div class="c-article-section__content" id="Sec12-content"><div data-test="supplementary-info"><div id="figshareContainer" class="c-article-figshare-container" data-test="figshare-container"></div><div class="c-article-supplementary__item" data-test="supp-item" id="MOESM1"><h3 class="c-article-supplementary__title u-h3"><a class="print-link" data-track="click" data-track-action="view supplementary info" data-test="supp-info-link" data-track-label="supplementary information(pdf 1032 kb)" href="https://static-content.springer.com/esm/art%3A10.1038%2Fs41467-018-03568-3/MediaObjects/41467_2018_3568_MOESM1_ESM.pdf" data-supp-info-image="">Supplementary Information(PDF 1032 kb)</a></h3></div></div></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p><b>Open Access</b> This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 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id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">He, L., Zhang, Q., Lan, P. <i>et al.</i> Monitoring ultrafast vibrational dynamics of isotopic molecules with frequency modulation of high-order harmonics. <i>Nat Commun</i> <b>9</b>, 1108 (2018). https://doi.org/10.1038/s41467-018-03568-3</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1038/s41467-018-03568-3?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2017-09-04">04 September 2017</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2018-02-23">23 February 2018</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2018-03-16">16 March 2018</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1038/s41467-018-03568-3</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box u-display-none" 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