Resumen
Light and matter interaction models, describing the interaction between photons and two-level atoms, have a rich mathematical structure combining areas such as number theory, geometry, and representation theory. In this talk, we focus on the quantum Rabi mode (QRM), its asymmetric version (AQRM), and the non-commutative harmonic oscillator (NCHO). The latter may be considered a covering model of the former ones via the corresponding Fuchsian ODE pictures. Quite recently, the eigenvalue problem of NCHO was actually shown to be equivalent to the two-photon QRM (2pQRM). For the AQRM, we discuss the problem of spectral degeneracy and hidden symmetry via (hyper) elliptic curves, and its heat kernel (propagator)/partition function. Here, the analytical formula for the heat kernel, given by an infinite series expression (“discrete path integral”), can be interpreted as the irreducible decomposition of the infinite symmetry group naturally acting on Z_2♾️, Z_2 being the binary field. In addition, the special values of the spectral zeta function of the NCHO (obtained from the Mellin transform of the partition function) give natural analogues of the Apéry numbers, and these are intimately connected to (non-holomorphic) modular forms for a specific congruence subgroup.
The talk is based mostly on recent research with Kazufumi Kimoto and Cid Reyes Bustos.
Imparte
Prof. Dr. Masato Wakayama
NTT IFM, Kyushu University, Japan
Organiza: Dr. Ricardo A. Weder