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Spectral analysis of non-commutative harmonic oscillators and quantum devices

2012.12.20

2012 Workshop "Cryptographic Technologies suitable for Cloud Computing"
Spectral analysis of non-commutative harmonic oscillators and quantum devices

Supported by
Ministry of Education, Culture, Sports, Science & Technology (MEXT)
Faculty of Mathematics, Kyushu University
Institute of Mathematics for Industry, Kyushu University
Global COE Program Education-and-Research Hub for Mathematics-for-Industry
Kyushu University

Abstracts：
The non-commutative harmonic oscillator (NCHO) is a self-adjoint operator indexed by two real parameters \alpha and \beta, with purely discrete spectrum. The spectrum has been studied from a number theoretic point of view. The spectral zeta function \zeta_Q is defined by the infinite sum of power of eigenvalues of NCHO, which can be regarded as a q-deformation of the Riemann zeta function. \zeta_Q coincides with the Riemann zeta function when special values of \alpha and \beta are chosen. The importance of \zeta_Q has been recognised only recently.

On the other hand, NCHO can be regarded as a Hamiltonian describing an interaction between a one-mode photon and a two-level atom. The 2012 Nobel Prize in physics was awarded jointly to S. Haroche and D. Wineland for their experimental success in studying this quantum system.

The eigenvalues of NCHO build a continuous curve with arguments \alpha and \beta. It comes as an important problem of mathematics to analyse the behaviour of eigenvalue curves, in particular, a main issue of present day research addresses the characterisation of crossing/avoided crossing of eigenvalue curves. In recent years Japanese industry progressed to allow creating two-level atoms artificially, which enables researchers to observe the interaction of two-level atoms with one-mode photon controlled with high precision. This effect is called strong interaction. Surprisingly, it has been realised that there is a deep relationship between strong interaction and the crossing/avoided crossing of eigenvalue curves.

The purpose of this meeting is to clarify the relationship between crossing/avoided crossing and existence/absence of strong interaction, and to promote feedback from mathematics to experimental physics.