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Spectral analysis of non-commutative harmonic oscillators and quantum devices

Hold Date
2012-11-26 10:00〜2012-11-27 17:30
Faculty of Mathematics・IMI Building

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
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.
 Date November 26 (Monday) - 27 (Tuesday), 2012
 Place 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
 Faculty of Mathematics・IMI Building

 Program  November 26 (Monday)

 10:00 - 11:00
 A. Parmeggiani (Università di Bologna)
 "On the spectrum of NCHOs: a survey"
 11:10 - 12:10
 Kazufumi Kimoto (University of the Ryukyus)
 "Problems in the special values of the spectral zeta functions of the non-commutative harmonic oscillator"
 13:00 - 13:20
 Fumio Hiroshima (Kyushu University)
 "Ground state of Rabi model"
 13:30 - 14:30
 Kouichi Semba (National Institute of Informatics)
 "Josephson circuit QED and hybrid quantum systems"
 15:00 - 16:00
 A. Parmeggiani (Università di Bologna)
 "On the problem of positivity estimates for systems of ΨDOs"
 16:10 - 17:10
 Daisuke Tagami (Kyushu University)
 "A consideration of numerical simulations for an eigenvalue problem of a differential operator"
 17:15 - 18:15
 Max Lein (Kyushu University)

 November 27 (Tuesday)

 10:00 - 11:00
 Taku Matsui (Kyushu University)
 "Spectral gap and entanglement entropy in 1 dim quantum systems"
 11:10 - 12:10
 Masato Wakayama (Kyushu University)
 "Eichler cohomology groups for spectrum of NcHO's"
 13:00 - 13:20
 Itaru Sasaki (Shinshu University)
 "Multiplicity of the ground state of non-commutative harmonic oscillators"
 13:30 - 14:30
 Yuji Nomura (Ehime University)
 "The spectrum of Schroedinger operators with magnetic fields invariant under the action of discrete groups"
 15:00 - 16:00
 Tsuyoshi Yamamoto (RIKEN)
 "Large dispersive shift of cavity resonance induced by a capacitively coupled superconducting flux qubit"
 16:10 - 17:10
 Masao Hirokawa (Okayama University)
 "On the coupling-strength increase of the Rabi model in the light of SUSYQM
 ---from supersymmetric ground state to spontaneous supersymmetry breaking---
 Organization  Masao Hirokawa (Okayama University)
 Fumio Hiroshima (Kyushu University)
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