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Center for Nonlinear Studies |
Given an initial quantum state |I> and a final quantum state |F> in a Hilbert space, there exist Hamiltonians H under which |I> evolves into |F>. Consider the following quantum brachistochrone problem: Subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time T? For Hermitian Hamiltonians, T has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, T can be made arbitrarily small without violating the time-energy uncertainty principle. This is because the complex path from |I> to |F> can be made arbitrarily small. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made arbitrarily small if they are connected by a wormhole. This result may have applications in quantum computing. If time permits, the nature of the classical theory that underlies the PT-Symmetric quantum theory will be discussed. The solutions to the classical equations of motion have remarkable properties and the particle trajectories can visit multiple sheets of a Riemann surface. In keeping with the mission of the CNLS, nonlinear wave equations that exhibit PT symmetry will also be discussed, the most notable being the Korteweg-de Vries equation.
Host: Robert Ecke, T-CNLS