What is it about?
Dimer is the discrete nonlinear Schroedinger equation defined on a lattice consisting just of 2 sites. The PT symmetric dimer has one site gaining and the other losing energy at the same rate. This paper is about two things, one physical and one mathematical. The principal mathematical conclusion is that the PT-symmetric extensions of all conservative nonlinear Schrödinger dimers remain completely integrable Hamiltonian systems. The main physical upshot is that there are broad classes of PT-symmetric dimers that confine all their trajectories regardless of the value of the gain–loss parameter γ. The PT-symmetry, which is broken at the level of the underlying linear equation, becomes spontaneously restored thanks to the nonlinear coupling.
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Why is it important?
The fact that an open system (a system with gain and loss of energy) can be Hamiltonian, is highly nontrivial. In fact, we show more: all PT symmetric Schroedinger dimers are completely integrable (that is, exhibit high regularity of motion and admit exhaustive solution). As for the PT symmetry restoration, it is a new phenomenon with a range of potential applications. No input can trigger an uncontrollable growth of optical modes in a dimer with the nonlinearly-restored PT symmetry.
Perspectives
Personally, I am intrigued by the following two aspects. First, by a new set of coordinates for the dimer - the canonical coordinates - which allow to represent the system in the hamiltonian form. Second, by an amazing relation between a particular PT symmetric dimer and a pair of coupled pendula with the periodically varied coupling. Who could have thought that such an elementary mechanical structure would give rise to a mathematical sibling of an optical coupler with gain and loss, or boson condensate trapped in a pair of potential wells!
Professor Igor Barashenkov
University of Cape Town
Read the Original
This page is a summary of: Dimer with gain and loss: Integrability and ${\mathcal{P}}{\mathcal{T}}$-symmetry restoration, Journal of Physics A Mathematical and Theoretical, July 2015, Institute of Physics Publishing,
DOI: 10.1088/1751-8113/48/32/325201.
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