Mathematical Horizons for Quantum Physics

(28 Jul - 21 Sep 2008)

... Jointly organized with Centre for Quantum Technologies, NUS
Partially supported by Lee Foundation and Faculty of Science, NUS

~ Abstracts ~

Control of the molecular orientation with laser pulses
Arne Keller, Universite Paris-Sud, France

• Motivations
• Control at zero temperature - pure state unitary control
• Control at non zero temperature - density matrix unitary control
• Towards dissipative control



Introduction to classical and quantum chaos
Stephan de Bievre, UFR de Mathématiques et Laboratoire CNRS Paul Painlevé, France

• Introduction
• Ergodicity and mixing in classical hamiltonian systems
• The example of maps (discrete time dynamics)
• Quantum maps
• Long time dynamics
• Semiclassical eigenfunction behaviour
• Outlook


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Introduction to quantum computing and devices
Goong Chen, Texas A&M University, USA

• Elementary logic gates
• Heat generation; irreversible and reversible computing
• Quantum phenomena: the Stern–Gerlach experiment
• Two-level atoms; the Schrödinger equation
• The coupling of the Schrödinger equation and the Maxwell equations
• The simple harmonic oscillator
• Quantum devices, cavity QED, ion and atom traps,quantum dots, linear optics and SQUIDS


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The dynamics of the interaction between atoms, molecules and electromagnetic fields
Hans-Rudolf Jauslin, Université de Bourgogne, France

• Photons and classical electromagnetic fields
• The Floquet representation
• Control by adiabatic processes
• Robust processes : geometrical and topological characterization
• Resonances
• KAM techniques in quantum mechanics

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Spectral relationships between kicked Harper and on-resonance double kicked rotor operators
Wayne Lawton, National University of Singapore
Anders Mouritzen, National University of Singapore

Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectrums of these operators. In this paper we apply C*-algebra methods to explain this resemblance. We show that each pair of corresponding operators belong to a common rotation C*-algebra Bα, prove that their spectrums are equal if α is irrational, and prove that the Hausdorff distance between their spectrums converges to zero as q increases if α = p/q with p and q coprime integers. Moreover, we show that corresponding operators in Bα are homomorphic images of mother operators in the universal rotation C*-algebra Aα that are unitarily equivalent and hence have identical spectrums. These results extend analogous results for almost Mathieu operators. We also utilize the C*-algebraic framework to develop efficient algorithms to compute the spectrums of these mother operators for rational α and present preliminary numerical results that support the conjecture that their spectrums are Cantor sets if α is irrational. This conjecture for almost Mathieu operators, called the Ten Martini Problem, was recently proved after intensive efforts over several decades. The proof of this conjecture for almost Mathieu operators utilized transfer matrix methods. These methods do not exist for the kicked operators. We outline a strategy, based on a special property of loop groups of semisimple Lie groups, to prove this conjecture for the kicked operators.

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Toy renormalization
Jan Derezinski, University of Warsaw, Poland

I will discuss a number of examples of self-adjoint operators whose definition requires an infinite renormalization. Some of them are often used in quantum physics, even though they are tricky.

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