Mathematical and Numerical Methods for Time-Dependent Quantum Mechanics - from Dynamics to Quantum Information (17w5010)

Arriving in Oaxaca, Mexico Sunday, August 13 and departing Friday August 18, 2017

Organizers

(University of Calgary)

(Universite de Sherbrooke)

(Carleton University)

Objectives

This workshop will focus on mathematical and numerical methods for time-dependent relativistic & nonrelativistic quantum mechanics, which is the mathematical foundation for quantum dynamics and for quantum information processing. Regimes for which highly nonlinear interactions occur at both microscopic and macroscopic levels, will be specifically addressed. As an example of the importance of applied mathematics in these areas of modern science, we mention the recent article in SIAM NEWS-vol 48(4), May 2015, “Mathematical Challenges and Opportunities in Optics and Photonics”, which emphasizes the importance of research in light-matter interaction as becoming one of the most active research fields in applied mathematics. More specifically, the topics to be developed during the workshop will involve and treat:

Time-Dependent Density Functional Theory (TDDFT) for general material sciences, Maxwell-Schrödinger/Dirac equations in nonperturbative nonlinear optics, quantum control, quantum information theory, molecular biological applications, and time-dependent N-body problems.

These topics will be studied from different mathematical perspectives and will involve several modern techniques for solving high-dimensional equations arising in time-dependent quantum physics. The following list contains some of the major mathematical methods and areas which will be addressed:

Domain decomposition methods, high-order splitting methods, spectral methods, numerical linear algebra, imaginary time methods, sparse grids, absorbing boundary conditions, high-performance computing aspects for large quantum systems (GPU-computing, ...), and quantum control. A large portion of the workshop will also be dedicated to operator algebras. Operator algebra provides valuable techniques for representing dynamics and decomposing into physically accessible operations, which is especially valuable for constructing quantum simulation algorithms for quantum computing. Various techniques such as Trotter-Suzuki expansions, truncated Taylor series expansions of Hamiltonian evolutions and complex time-integration will be addressed as new low-cost quantum simulation algorithms.

Many of the invited speakers are leaders in their respective research field. We therefore expect that the workshop will be, not only a state-of-the-art activity in mathematical and numerical quantum dynamics and information theory, but also will present new challenges and the future main research trends, in applications of highly nonlinear mathematical problems in the ever increasing use of advanced applied and numerical mathematics in modern science and high technology.

Among many questions and applications that will be discussed and addressed during this meeting, let us emphasize the time scales relevant to controlling matter at the atomic-molecular level, from femtosecond (10**-15s) for atomic motion to attosecond (10**-18s) for electron control, to zeptosecond (10**-21s) for nuclear particle control in quantum electrodynamics. Modern laser technology is today the unique source of ultrafast (few cycle) intense laser pulses, with intensities exceeding the internal electric field in atoms and molecules. The interaction of such pulses with atoms and molecules leads to a new regime of laser-matter interaction, namely a highly nonlinear nonperturbative regime where new nonlinear physical phenomena occur. As an example, High Order Harmonic Generation, HHG, from which the first shortest pulse has been created, provides the first attosecond pulse for control of the quantum wave nature of the electron.

We expect the participation of research-oriented graduate students and post-doctoral fellows from different disciplines, such as applied mathematics, quantum information theory and theoretical molecular sciences. Canada is already a world leader in Ultrafast Intense Laser Science with the existence of a national laser facility, ALLS (Advanced laser Light Source-at INRS-EMT), and in Quantum Information Theory (IQST, PI) , thus emphasizing the importance of such activities in modern mathematics. A poster session will be organized for these HPQ to present their research .

This workshop will follow and update several related workshops held in recent years:

i)“Frontiers of Intense Laser Science” (http://www.kitp.ucsb.edu) at the Kavli Institute of Theoretical Physics, Univ. California Santa Barbara, co-financed by NSF (USA) and Simons Foundation, 2014

ii) “Mathematical Methods for Filamentation”, At Centre de Recherches Mathématiques at Montréal (CRM), co-financed by the NSF (USA) and CRM, 2014.

iii) “Quantum Dynamic Imaging”, Montréal, CRM, 2013.

iv) “High Dimensional PDE's in Science and Engineering”, Montréal, CRM, 2009.