July 2012
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Day July 24, 2012

PT-Symmetric Pseudo-Hermitian Relativistic Quantum Mechanics With a Maximal Mass

V. N. Rodionov

The quantum-field model described by non-Hermitian, but a \({\cal PT}\)-symmetric Hamiltonian is considered. It is shown by the algebraic way that the limiting of the physical mass value \(m \leq m_{max}= {m_1}^2/2m_2\) takes place for the case of a fermion field with a \(\gamma_5\)-dependent mass term (\(m\rightarrow m_1 +\gamma_5 m_2 \)). In the regions of unbroken \(\cal PT\) symmetry the Hamiltonian \(H\) has another symmetry represented by a linear operator \( \cal C\). We exactly construct this operator by using a non-perturbative method. In terms of \( \cal C\) operator we calculate a time-independent inner product with a positive-defined norm. As a consequence of finiteness mass spectrum we have the \(\cal PT\)-symmetric Hamiltonian in the areas \((m\leq m_{max})\), but beyond this limits \(\cal PT\)-symmetry is broken. Thus, we obtain that the basic results of the fermion field model with a \(\gamma_5\)-dependent mass term is equivalent to the Model with a Maximal Mass which for decades has been developed by V.Kadyshevsky and his colleagues. In their numerous papers the condition of finiteness of elementary particle mass spectrum was introduced in a purely geometric way, just as the velocity of light is a maximal velocity in the special relativity. The adequate geometrical realization of the limiting mass hypothesis is added up to the choice of (anti) de Sitter momentum space of the constant curvature.

Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

Time-dependent Hamiltonians with 100% evolution speed efficiency

Raam Uzdin, Uwe Guenther, Saar Rahav, Nimrod Moiseyev

The evolution speed in projective Hilbert space is considered for Hermitian Hamiltonians and for non-Hermitian (NH) ones. Based on the Hilbert-Schmidt norm and the spectral norm of a Hamiltonian, resource-related upper bounds on the evolution speed are constructed. These bounds are valid also for NH Hamiltonians and they are illustrated for an optical NH Hamiltonian and for a non-Hermitian \(\mathcal{PT}\)-symmetric matrix Hamiltonian. Furthermore, the concept of quantum speed efficiency is introduced as measure of the system resources directly spent on the motion in the projective Hilbert space. A recipe for the construction of time-dependent Hamiltonians which ensure 100% speed efficiency is given. Generally these efficient Hamiltonians are NH but there is a Hermitian efficient Hamiltonian as well. Finally, the extremal case of a non-Hermitian non-diagonalizable Hamiltonian with vanishing energy difference is shown to produce a 100% efficient evolution with minimal resources consumption.


Quantum Physics (quant-ph)

Breakdown of adiabatic transfer schemes in the presence of decay

Eva-Maria Graefe, Alexei A. Mailybaev, Nimrod Moiseyev

In atomic physics, adiabatic evolution is often used to achieve a robust and efficient population transfer. Many adiabatic schemes have also been implemented in optical waveguide structures. Recently there has been increasing interests in the influence of decay and absorption, and their engineering applications. Here it is shown that contrary to what is often assumed, even a small decay can significantly influence the dynamical behaviour of a system, above and beyond a mere change of the overall norm. In particular, a small decay can lead to a breakdown of adiabatic transfer schemes, even when both the spectrum and the eigenfunctions are only sightly modified. This is demonstrated for the decaying version of a STIRAP scheme that has recently been implemented in optical waveguide structures. It is found that the transfer property of the scheme breaks down at a sharp threshold, which can be estimated by simple analytical arguments.

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)