{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T14:54:09Z","timestamp":1766415249553,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,23]],"date-time":"2023-02-23T00:00:00Z","timestamp":1677110400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2019YFA0307700","2022YFE0134200","12074145","12204214","11627807","20220101003JC","LJKMZ20221435"],"award-info":[{"award-number":["2019YFA0307700","2022YFE0134200","12074145","12204214","11627807","20220101003JC","LJKMZ20221435"]}]},{"name":"National Natural Science Foundation of China (NSFC)","award":["2019YFA0307700","2022YFE0134200","12074145","12204214","11627807","20220101003JC","LJKMZ20221435"],"award-info":[{"award-number":["2019YFA0307700","2022YFE0134200","12074145","12204214","11627807","20220101003JC","LJKMZ20221435"]}]},{"name":"Jilin Provincial Research Foundation for Basic Research, China","award":["2019YFA0307700","2022YFE0134200","12074145","12204214","11627807","20220101003JC","LJKMZ20221435"],"award-info":[{"award-number":["2019YFA0307700","2022YFE0134200","12074145","12204214","11627807","20220101003JC","LJKMZ20221435"]}]},{"name":"Foundation of Education Department of Liaoning Province, China","award":["2019YFA0307700","2022YFE0134200","12074145","12204214","11627807","20220101003JC","LJKMZ20221435"],"award-info":[{"award-number":["2019YFA0307700","2022YFE0134200","12074145","12204214","11627807","20220101003JC","LJKMZ20221435"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>High-order harmonic generation of atoms irradiated by an ultrashort laser pulse was calculated by numerically solving the time-dependent Schr\u00f6dinger equation and the Bohmian trajectory scheme. The harmonic spectra with the two schemes are quantitatively consistent. Using the wavelet behavior of the Bohmian trajectory, the spatio-temporal features of harmonic emission from different energy regions are analyzed. It is found that the spatio-temporal distribution of the harmonic well revealed the physical mechanism of harmonic generation. This method will contribute to the understanding of harmonic emission mechanisms in complex systems, which include many atoms.<\/jats:p>","DOI":"10.3390\/sym15030581","type":"journal-article","created":{"date-parts":[[2023,2,23]],"date-time":"2023-02-23T03:25:56Z","timestamp":1677122756000},"page":"581","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Investigation of the Spatio-Temporal Characteristics of High-Order Harmonic Generation Using a Bohmian Trajectory Scheme"],"prefix":"10.3390","volume":"15","author":[{"given":"Simeng","family":"Song","sequence":"first","affiliation":[{"name":"Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China"},{"name":"Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun 130012, China"}]},{"given":"Linyan","family":"Wu","sequence":"additional","affiliation":[{"name":"Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China"},{"name":"Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun 130012, China"}]},{"given":"Yue","family":"Qiao","sequence":"additional","affiliation":[{"name":"Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China"},{"name":"Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun 130012, China"}]},{"given":"Shushan","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Physics and Electronic Technology, Liaoning Normal University, Dalian 116029, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3039-6371","authenticated-orcid":false,"given":"Jun","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China"},{"name":"Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun 130012, China"}]},{"given":"Fuming","family":"Guo","sequence":"additional","affiliation":[{"name":"Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China"},{"name":"Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun 130012, China"}]},{"given":"Yujun","family":"Yang","sequence":"additional","affiliation":[{"name":"Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China"},{"name":"Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun 130012, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"35444","DOI":"10.1364\/OE.470495","article-title":"Calculation of high-order harmonic generation of atoms and molecules by combining time series prediction and neural networks","volume":"30","author":"Yan","year":"2022","journal-title":"Opt. Express"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1537","DOI":"10.1088\/0256-307X\/24\/6\/029","article-title":"Ultrahigh harmonic generation from an atom with superposition of ground state and highly excited states","volume":"24","author":"Yang","year":"2007","journal-title":"Chin. Phys. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3535","DOI":"10.1103\/PhysRevLett.68.3535","article-title":"High-order harmonic generation from atoms and ions in the high intensity regime","volume":"68","author":"Krause","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1088\/0953-4075\/26\/4\/003","article-title":"Atoms in ultra-intense laser fields","volume":"26","author":"Burnett","year":"1993","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1103\/RevModPhys.81.163","article-title":"Attosecond physics","volume":"81","author":"Krausz","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"023510","DOI":"10.1103\/PhysRevA.106.023510","article-title":"High-order harmonic generation of 1-nonene under linearly polarized laser pulses","volume":"106","author":"Zhou","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1038\/nature03183","article-title":"Tomographic imaging of molecular orbitals","volume":"432","author":"Itatani","year":"2004","journal-title":"Nature"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"822","DOI":"10.1038\/nphys2029","article-title":"Generalized molecular orbital tomography","volume":"7","author":"Vozzi","year":"2011","journal-title":"Nat. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"043004","DOI":"10.1103\/PhysRevLett.108.043004","article-title":"Molecular imaging using high-order harmonic generation and above-threshold ionization","volume":"108","author":"Lein","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","first-page":"231","article-title":"High-order harmonic generation from organic molecules in ultra-short pulses","volume":"14","author":"Hay","year":"2001","journal-title":"Eur. Phys. J. D At. Mol. Opt. Plasma Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"214107","DOI":"10.1063\/1.3031214","article-title":"Molecular dynamics with time dependent quantum Monte Carlo","volume":"129","author":"Christov","year":"2008","journal-title":"J. Chem. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"9971","DOI":"10.1364\/OE.446432","article-title":"All-optical reconstruction of three-band transition dipole moments by the crystal harmonic spectrum from a two-color laser pulse","volume":"30","author":"Qiao","year":"2022","journal-title":"Opt. Express"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"155201","DOI":"10.1103\/PhysRevB.102.155201","article-title":"Smooth periodic gauge satisfying crystal symmetry and periodicity to study high-harmonic generation in solids","volume":"102","author":"Jiang","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Qiao, Y., Chen, J., and Chen, J. (2022). Review on the Reconstruction of Transition Dipole Moments by Solid Harmonic Spectrum. Symmetry, 14.","DOI":"10.3390\/sym14122646"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"075201","DOI":"10.1103\/PhysRevB.107.075201","article-title":"Robust retrieval method of crystal transition dipole moments by high-order harmonic spectrum","volume":"107","author":"Qiao","year":"2023","journal-title":"Phys. Rev. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"093902","DOI":"10.1103\/PhysRevLett.105.093902","article-title":"Isolated attosecond pulse generation without the need to stabilize the carrier-envelope phase of driving lasers","volume":"105","author":"Gilbertson","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1126\/science.1088654","article-title":"Coherent soft X-ray generation in the water window with quasi-phase matching","volume":"302","author":"Gibson","year":"2003","journal-title":"Science"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1038\/nature10711","article-title":"Direct frequency comb spectroscopy in the extreme ultraviolet","volume":"482","author":"Yost","year":"2012","journal-title":"Nature"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"063428","DOI":"10.1103\/PhysRevA.100.063428","article-title":"High-order harmonic generation from H 2+ irradiated by a co-rotating two-color circularly polarized laser field","volume":"100","author":"Qiao","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1126\/science.1163439","article-title":"Attosecond ionization and tunneling delay time measurements in helium","volume":"322","author":"Eckle","year":"2008","journal-title":"Science"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1126\/science.1260311","article-title":"Attosecond band-gap dynamics in silicon","volume":"346","author":"Schultze","year":"2014","journal-title":"Science"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1126\/science.aab2160","article-title":"Measurement and laser control of attosecond charge migration in ionized iodoacetylene","volume":"350","author":"Kraus","year":"2015","journal-title":"Science"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1038\/nature16528","article-title":"Optical attosecond pulses and tracking the nonlinear response of bound electrons","volume":"530","author":"Hassan","year":"2016","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"142001","DOI":"10.1088\/0953-4075\/49\/14\/142001","article-title":"Charge migration induced by attosecond pulses in bio-relevant molecules","volume":"49","author":"Calegari","year":"2016","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"18106","DOI":"10.1364\/OE.23.018106","article-title":"Polarization properties of below-threshold harmonics from aligned molecules H2+ in linearly polarized laser fields","volume":"23","author":"Dong","year":"2015","journal-title":"Opt. Express"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1614","DOI":"10.1126\/science.1157846","article-title":"Single-cycle nonlinear optics","volume":"320","author":"Goulielmakis","year":"2008","journal-title":"Science"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5208","DOI":"10.1364\/OL.37.005208","article-title":"Tomographic imaging of asymmetric molecular orbitals with a two-color multicycle laser field","volume":"37","author":"Qin","year":"2012","journal-title":"Opt. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"9776","DOI":"10.1073\/pnas.1919360117","article-title":"Detecting electronic coherences by time-domain high-harmonic spectroscopy","volume":"117","author":"Jiang","year":"2020","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1994","DOI":"10.1103\/PhysRevLett.71.1994","article-title":"Plasma perspective on strong field multiphoton ionization","volume":"71","author":"Corkum","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"073906","DOI":"10.1103\/PhysRevLett.96.073906","article-title":"Three-step model for high-harmonic generation in many-electron systems","volume":"96","author":"Santra","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1130002","DOI":"10.1142\/S021798491130002X","article-title":"High-order harmonic generation in diatomic molecules: Quantum interference, nodal structures and multiple orbitals","volume":"26","author":"Augstein","year":"2012","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"023411","DOI":"10.1103\/PhysRevA.83.023411","article-title":"Medium propagation effects in high-order harmonic generation of Ar and N2","volume":"83","author":"Jin","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"132001","DOI":"10.1088\/0953-4075\/41\/13\/132001","article-title":"Macroscopic aspects of attosecond pulse generation","volume":"41","author":"Gaarde","year":"2008","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"123901","DOI":"10.1103\/PhysRevLett.116.123901","article-title":"Strong-field photoionization as excited-state tunneling","volume":"116","author":"Serebryannikov","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"032506","DOI":"10.1103\/PhysRevA.86.032506","article-title":"Ab initio study of high-lying doubly excited states of helium in static electric fields: Complex-scaling generalized pseudospectral method in hyperspherical coordinates","volume":"86","author":"Heslar","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1038\/s42254-019-0126-3","article-title":"Numerical methods every atomic and molecular theorist should know","volume":"2","author":"Schneider","year":"2020","journal-title":"Nat. Rev. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"044203","DOI":"10.1088\/0256-307X\/39\/4\/044203","article-title":"Laser-Induced Electron Fresnel Diffraction in Tunneling and Over-Barrier Ionization","volume":"39","author":"Geng","year":"2022","journal-title":"Chin. Phys. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"053301","DOI":"10.1088\/0256-307X\/39\/5\/053301","article-title":"Decoherence Effects of Terahertz Generation in Solids under Two-Color Femtosecond Laser Fields","volume":"39","author":"Peng","year":"2022","journal-title":"Chin. Phys. Lett."},{"key":"ref_39","first-page":"FW4D\u20134","article-title":"A Robust Scaling Up Method of Output Energy and Photon Energy on High-Order Harmonic Generation: Towards Sub-pJ Water Window Soft X-rays. CLEO: QELS_Fundamental Science","volume":"2020","author":"Nishimura","year":"2020","journal-title":"Opt. Soc. Am."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"113035","DOI":"10.1088\/1367-2630\/11\/11\/113035","article-title":"From a quantum to a classical description of intense laser\u2013atom physics with Bohmian trajectories","volume":"11","author":"Lai","year":"2009","journal-title":"New J. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"074309","DOI":"10.1063\/1.3553178","article-title":"Visualization and interpretation of attosecond electron dynamics in laser-driven hydrogen molecular ion using Bohmian trajectories","volume":"134","author":"Takemoto","year":"2011","journal-title":"J. Chem. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"033424","DOI":"10.1103\/PhysRevA.86.033424","article-title":"Investigation of the generation of high-order harmonics through Bohmian trajectories","volume":"86","author":"Song","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"023415","DOI":"10.1103\/PhysRevA.88.023415","article-title":"Local dynamics in high-order-harmonic generation using Bohmian trajectories","volume":"88","author":"Wu","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"263002","DOI":"10.1103\/PhysRevLett.99.263002","article-title":"Binary and recoil collisions in strong field double ionization of helium","volume":"99","author":"Staudte","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1088\/0034-4885\/60\/4\/001","article-title":"Atomic physics with super-high intensity lasers","volume":"60","author":"Protopapas","year":"1997","journal-title":"Rep. Prog. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"053419","DOI":"10.1103\/PhysRevA.88.053419","article-title":"Investigation of atomic radiative recombination processes by the Bohmian- mechanics method","volume":"88","author":"Song","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1016\/S1386-1425(97)00022-X","article-title":"The calculation of accurate harmonic frequencies of large molecules: The polycyclic aromatic hydrocarbons, a case study","volume":"53","author":"Bauschlicher","year":"1997","journal-title":"Spectrochim. Acta Part A Mol. Biomol. Spectrosc."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"013201","DOI":"10.1088\/1674-1056\/aca3a1","article-title":"High-order harmonic generation of the cyclo[18]carbon molecule irradiated by circularly polarized laser pulse","volume":"32","author":"Zhou","year":"2022","journal-title":"Chin. Phys. B"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"033203","DOI":"10.1088\/1674-1056\/22\/3\/033203","article-title":"High-intensity molecular harmonic generation without ionization","volume":"22","author":"Wang","year":"2013","journal-title":"Chin. Phys. B"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"053202","DOI":"10.1088\/1674-1056\/23\/5\/053202","article-title":"Investigation on the influence of atomic potentials on the above threshold ionization","volume":"23","author":"Tian","year":"2014","journal-title":"Chin. Phys. B"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"063418","DOI":"10.1103\/PhysRevA.87.063418","article-title":"Dynamic stabilization of ionization for an atom irradiated by high-frequency laser pulses studied with the Bohmian-trajectory scheme","volume":"87","author":"Wei","year":"2013","journal-title":"Phys. Rev. A"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/581\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:40:04Z","timestamp":1760121604000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/581"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,23]]},"references-count":51,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["sym15030581"],"URL":"https:\/\/doi.org\/10.3390\/sym15030581","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,2,23]]}}}