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Day July 18, 2014

Isospectral Hermitian counterpart of complex non Hermitian Hamiltonian \(p^{2}-gx^{4}+a/x^{2}\)

Asiri Nanayakkara, Thilagarajah Mathanaranjan

In this paper we show that the non-Hermitian Hamiltonians \(H=p^{2}-gx^{4}+a/x^2\) and the conventional Hermitian Hamiltonians \(h=p^2+4gx^{4}+bx\) (\(a,b\in \mathbb{R}\)) are isospectral if \(a=(b^2-4g\hbar^2)/16g\) and \(a\geq -\hbar^2/4\). This new class includes the equivalent non-Hermitian – Hermitian Hamiltonian pair, \(p^{2}-gx^{4}\) and \(p^{2}+4gx^{4}-2\hbar \sqrt{g}x\), found by Jones and Mateo six years ago as a special case. When \(a=\left(b^{2}-4g\hbar ^{2}\right) /16g\) and \(a<-\hbar^2/4\), although \(h\) and \(H\) are still isospectral, \(b\) is complex and \(h\) is no longer the Hermitian counterpart of \(H\).

Mathematical Physics (math-ph)

PT spectroscopy of the Rabi problem

Yogesh N. Joglekar, Rahul Marathe, P. Durganandini, Rajeev K. Pathak

We investigate the effects of a time-periodic, non-hermitian, PT-symmetric perturbation on a system with two (or few) levels, and obtain its phase diagram as a function of the perturbation strength and frequency. We demonstrate that when the perturbation frequency is close to one of the system resonances, even a vanishingly small perturbation leads to PT symmetry breaking. We also find a restored PT-symmetric phase at high frequencies, and at moderate perturbation strengths, we find multiple frequency windows where PT-symmetry is broken and restored. Our results imply that the PT-symmetric Rabi problem shows surprisingly rich phenomena absent in its hermitian or static counterparts.


Optics (physics.optics); Other Condensed Matter (cond-mat.other); Quantum Physics (quant-ph)