Phase Space Approach to Signal Processing and Quantum Mechanical Calculations: Thinking Inside the Box

Seminar
Quantum Optics Resnick seminar
Speaker
Prof. David Tannor, Department of Chemical Physics, Weizmann Institute of Science
Date
03/04/2013 - 16:00 - 15:00Add to Calendar 2013-04-03 15:00:00 2013-04-03 16:00:00 Phase Space Approach to Signal Processing and Quantum Mechanical Calculations: Thinking Inside the Box   In 1946, Gabor proposed using a lattice of  Gaussians in the time-frequency phase space to provide an intuitive and compact representation of arbitrary signals. The same idea had actually been discovered fifteen years earlier by his countryman von Neumann, in searching for a way to represent arbitrary quantum mechanical wavefunctions in the position-momentum phase space.  Despite the great interest in this approach in both the signal processing and quantum mechanical communities, the method has never succeeded as hoped due to severe convergence problems.  We have recently discovered a simple and surprising solution to the convergence problem, based on introducing periodic boundary conditions into the Gabor/von Neumann lattice. The resulting method provides a simple and compact representation of arbitrary signals and images, and opens the door to unprecedentedly large quantum calculations based on exploiting the underlying classical phase space structure. In the classical limit the method reaches the remarkable efficiency of 1 basis function per 1 eigenstate. We illustrate the method by calculating the vibrational eigenstates of a polyatomic with 104 bound states and by simulating attosecond electron dynamics in the presence of combined strong XUV and NIR laser fields. We also present examples of audio and image compression where we show that the method is competitive with or superior to state-of-the-art wavelet methods.   1. A. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-independent Schrödinger Equation, Phys. Rev. Lett. 109, 070402 (2012). 2. N. Takemoto, A. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-dependent Schrödinger Equation: Application to the Simulation and Control of Attosecond Electron Dynamics in the Presence of a Strong Laser Field, J. Chem. Phys. 137, 011102 (2012) (Communication). 3. A. Shimshovitz and D. J. Tannor, Phase Space Wavelets for Solving Coulomb Problems, J. Chem. Phys. 137, 101103 (2012) (Communication). 4. A. Shimshovitz and D.J. Tannor, Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression, IEEE Signal Processing (submitted);  arXiv:1207.0632 [math:FA] Nano center, 9th floor seminar room Department of Physics physics.dept@mail.biu.ac.il Asia/Jerusalem public
Place
Nano center, 9th floor seminar room
Abstract

 

In 1946, Gabor proposed using a lattice of  Gaussians in the time-frequency phase space to provide an intuitive and compact representation of arbitrary signals. The same idea had actually been discovered fifteen years earlier by his countryman von Neumann, in searching for a way to represent arbitrary quantum mechanical wavefunctions in the position-momentum phase space.  Despite the great interest in this approach in both the signal processing and quantum mechanical communities, the method has never succeeded as hoped due to severe convergence problems.  We have recently discovered a simple and surprising solution to the convergence problem, based on introducing periodic boundary conditions into the Gabor/von Neumann lattice. The resulting method provides a simple and compact representation of arbitrary signals and images, and opens the door to unprecedentedly large quantum calculations based on exploiting the underlying classical phase space structure. In the classical limit the method reaches the remarkable efficiency of 1 basis function per 1 eigenstate. We illustrate the method by calculating the vibrational eigenstates of a polyatomic with 104 bound states and by simulating attosecond electron dynamics in the presence of combined strong XUV and NIR laser fields. We also present examples of audio and image compression where we show that the method is competitive with or superior to state-of-the-art wavelet methods.

 

1. A. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-independent Schrödinger Equation, Phys. Rev. Lett. 109, 070402 (2012).

2. N. Takemoto, A. Shimshovitz and D. J. Tannor, Phase Space Approach to Solving the Time-dependent Schrödinger Equation: Application to the Simulation and Control of Attosecond Electron Dynamics in the Presence of a Strong Laser Field, J. Chem. Phys. 137, 011102 (2012) (Communication).

3. A. Shimshovitz and D. J. Tannor, Phase Space Wavelets for Solving Coulomb Problems, J. Chem. Phys. 137, 101103 (2012) (Communication).

4. A. Shimshovitz and D.J. Tannor, Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression, IEEE Signal Processing (submitted);  arXiv:1207.0632 [math:FA]

Last Updated Date : 31/03/2013