2024 Fiscal Year Research-status Report
Number theory and geometry behind quantum interaction models
| Project/Area Number |
24K16941
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| Research Institution | NTT Communication Science Laboratories |
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Principal Investigator |
REYESBUSTOS CID 日本電信電話株式会社NTTコミュニケーション科学基礎研究所, 基礎数学研究P, 研究主任 (10991565)
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| Project Period (FY) |
2024-04-01 – 2027-03-31
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| Keywords | Interaction models / quantum Rabi model / hidden symmetry / spectral degeneracy / spectral zeta function |
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| Outline of Annual Research Achievements |
The main objective of the project is to develop the theory of the geometric and algebraic structures underlying interaction models in quantum optics, with the quantum Rabi model (QRM) as the central representative.
For this purpose, in the first year we focused on investigating the algebro-geometry picture of the symmetry and degeneracy for the two-photon QRM (2pQRM), an important generalization of the QRM. In addition to obtaining us a better understanding of the symmetry-degeneracy phenomenon due to the additional mathematical properties of the 2pQRM, this research is also significant for theoretical and experimental quantum physics. We expect to finish an extensive paper on this topic in FY2025.
The second objective is to study the arithmetics of the AQRM spectral zeta function by their relation with time evolution operators. Here, the main achievement for this year is the publication of the paper "Zeta limits for the spectrum of quantum Rabi models" (Journal of Mathematical physics) on the relation of the spectral zeta function of the QRM with the Riemann zeta function using the explicit heat kernel formulas, that is, a relation between quantum interaction models and classical number theory. This result also gives a mathematical proof of the behavior of the spectrum of the QRM for large coupling, previously only known empirically.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
The project has been progressing steadily, slightly slower than the original plan due to some minor difficulties encountered due to the complexity of the research. Concretely, the difficulty of certain parts of the project (the symmetry-degeneracy picture of the two-photon quantum Rabi model) was slightly underestimated and the scope of the research has been expanded (a deeper mathematical description of the spectrum using representation theory), resulting in a longer research time than originally expected. However, the end result will be of higher scientific significance and with an expected larger impact than the initial plan. Summarizing, even with the unforeseen events and difficulties, we expect that the original research plan will not be altered significantly.
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| Strategy for Future Research Activity |
First, in FY2025 we will complete an extensive study of the algebraic-geometric picture of the two-photon quantum Rabi model (2pQRM). With the additional structure of the 2pQRM, in particular the presence of a rich Lie algebra structure, we will have a better understanding of the relation between symmetry and degeneracy and its geometric picture (posed as conjectures in the quantum Rabi model case). According to the progress on this point, we might also extend this research to the quantum Rabi-Stark model (not described in the original plan) since it has some similar features as the 2pQRM.
The next step, already described in the original plan, is to develop the theory of covering models to relate the 2pQRM effectively with the quantum Rabi model (QRM). Since the relation of the 2pQRM and the non-commutative harmonic oscillator (NCHO) has been independently studied, we plan to use the properties of the NCHO and 2pQRM to prove the conjectures on symmetry and degeneracy for the QRM. At the same time, we will use the covering relation to study the special values of the spectral zeta function of the QRM and 2pQRM using the heat kernel method, following the original plan.
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| Causes of Carryover |
The main reason for the difference between the plan and the used amount is that
an international business trip (Scheduled for Q4 FY2024) was delayed by reasons outside of my control. It has been rescheduled for FY2025 and it the funds will be used as originally expected (Travel Expenses).
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