Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons
We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance...
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MDPI AG
2021-12-01
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| author | Kayo Kinjo Eriko Kaminishi Takashi Mori Jun Sato Rina Kanamoto Tetsuo Deguchi |
| author_facet | Kayo Kinjo Eriko Kaminishi Takashi Mori Jun Sato Rina Kanamoto Tetsuo Deguchi |
| author_sort | Kayo Kinjo |
| collection | DOAJ |
| description | We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the <i>N</i>-particle and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons. |
| first_indexed | 2024-03-10T00:24:38Z |
| format | Article |
| id | doaj.art-46e39eacff594ef782aa7e819123cd6a |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-10T00:24:38Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-46e39eacff594ef782aa7e819123cd6a2023-11-23T15:36:45ZengMDPI AGUniverse2218-19972021-12-0181210.3390/universe8010002Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-SolitonsKayo Kinjo0Eriko Kaminishi1Takashi Mori2Jun Sato3Rina Kanamoto4Tetsuo Deguchi5Department of Physics, Graduate School of Humanities and Sciences, Ochanomizu University, Bunkyo-ku, Tokyo 112-8610, JapanQuantum Computing Center, Keio University, Yokohama 223-8522, JapanRIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, JapanFaculty of Engineering, Tokyo Polytechnic University, Atsugi 243-0297, JapanDepartment of Physics, Meiji University, Kawasaki 214-8571, JapanDepartment of Physics, Faculty of Core Research, Ochanomizu University, Bunkyo-ku, Tokyo 112-8610, JapanWe study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the <i>N</i>-particle and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.https://www.mdpi.com/2218-1997/8/1/2dark solitonquantum soliton1D Bose gas |
| spellingShingle | Kayo Kinjo Eriko Kaminishi Takashi Mori Jun Sato Rina Kanamoto Tetsuo Deguchi Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons Universe dark soliton quantum soliton 1D Bose gas |
| title | Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons |
| title_full | Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons |
| title_fullStr | Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons |
| title_full_unstemmed | Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons |
| title_short | Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons |
| title_sort | quantum dark solitons in the 1d bose gas from single to double dark solitons |
| topic | dark soliton quantum soliton 1D Bose gas |
| url | https://www.mdpi.com/2218-1997/8/1/2 |
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