2009 Fiscal Year Final Research Report
Exact Analysis of Bose-Einstein Condensates and its Applications
| Project/Area Number |
18540368
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Mathematical physics/Fundamental condensed matter physics
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| Research Institution | Tokyo University of Science (2007-2009)
The University of Tokyo (2006) |
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Principal Investigator |
WADATI Miki Tokyo University of Science, 理学部, 教授 (60015831)
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| Project Period (FY) |
2006 – 2009
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| Keywords | ボース・アインシュタイン凝縮 / 多成分非線形シュレディンガー方程式 / 多成分可積分系 / 電磁誘導透過現象 / ベーテ仮説 / BCS-BESクロスオーバー / 量子可積分系 / 逆散乱 |
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| Research Abstract |
1. In 2004, we discovered an integrable condition of the coupling constants for the Gross-Pitaevskii (GP) equation which describes the dynamics of 3-component Bose- Einstein condensates. We analyzed the system by the inverse scattering method and clarified the collision properties of solitons. Further, for generic coupling constants, integrable structure of 3-component GP equation was investigated by the Painleve test. It was proved that there exist 2 cases for integrable conditions; one is the 3-component Manakov equation and the other is our equation. The former contains only density -density interaction, while the latter describes both density-density and spin-spin interactions. Spin dynamics of the condensates has become one of the most actively studied subjects.
2. For the delta-function interacting spin1/2 Fermi gas in one-dimension, we analyzed the Yang-Yang integral equation to obtain the ground state energy of the system. As a method of solution, we employed a series expansion
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method which was introduced for the Lieb-Liniger integral equation by M.Wadati in 2002. Among many, challenging aspects are to treat attractive interactions and to include external magnetic field. We succeeded in analying the system both in weak and strong interactions. Contrary to the common sense, the weak coupling case is known to be difficult and subtle. And, we calculated the magnetization as a function of coupling constant and the magnetic field, and classified the phases. There exist three phases ; completely paired non-polarized phase, completely polarized phase without pairs and the coexistence of the above 2 phases. Quantum transitions among the three phases are proved.
3. Self-induced transparency (SIT) in nonlinear optics was studied extensively in 1960's and gave useful information for establishing the concept of solitons. As a natural and nontrivial development, electromagnetically-induced transparency (EIT) has attracted much attention. Examining the interaction strengths among two laser lights and 3-level atoms, we found integrable condition for the system. By the Backlund transformation, soliton solutions are obtained and the collision properties are analyzed in detail. Common to the multi-component soliton systems, there occur a variety of collisions, conventional elastic collision, energy exchange, pulse compression etc. EIT can be related and combined to the multi-component Bose-Einstein condensations. Its application to quantum information processing has been suggested.
4. In quantum mechanics, it has been thought that to have real valued energies the energy operator has to be hermitian (self-conjugate). However, real energies are not necessarily due to the hermicity of the energy operator. By using a formulation of the soliton theory, we have shown explicitly that non-hermitian potentials are constructed in a systematic manner. This explains the adhoc introduction of potentials in the theory of parity-time symmetric potentials. Less
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