July 2012
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Month July 2012

Reciprocity and unitarity in scattering from a non-Hermitian complex PT-symmetric potential

Zafar Ahmed

In non-relativistic quantum scattering, Hermiticity is necessary for both reciprocity and unitarity. Reciprocity means that both reflectivity (R) and transmitivity (T) are insensitive to the direction of incidence of a wave (particle) at a scatterer from left/right. Unitarity means that R+T=1. In scattering from non-Hermitian PT-symmetric structures the (left/right) handedness (non-reciprocity) of reflectivity is known to be essential and unitarity remains elusive so far. Here we present a surprising occurrence of both reciprocity and unitarity in scattering from a complex PT-symmetric potential. In special cases, we show that this potential can even become invisible (R=0, T=1) from both left and right sides. We also find that this optical potential can give rise to a perfect transmission (T=1) this time without both unitarity and reciprocity (of reflectivity).

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

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)

Non-Hermitian quantum dynamics of a two-level system and models of dissipative environments

Alessandro Sergi, Konstantin G. Zloshchastiev

We consider a non-Hermitian Hamiltonian in order to effectively describe a two-level system coupled to a dissipative environment. The total Hamiltonian of the model is obtained by adding a general anti-Hermitian part, depending on four parameters, to the Hermitian Hamiltonian of a tunneling two-level system. The time evolution is formulated and derived in terms of the density matrix of the model, different types of decays are revealed and analyzed. In particular, the population difference and coherence are defined and calculated analytically. We have been able to mimic various physical situations with different properties, such as dephasing and vanishing population difference.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

PT-symmetric quantum Liouvillean dynamics

Tomaz Prosen

We discuss a combination of unitary and anti-unitary symmetry of quantum Liouvillian dynamics, in the context of open quantum systems, which implies a D2 symmetry of the complex Liovillean spectrum. For sufficiently weak system-bath coupling it implies a uniform decay rate for all coherences, i.e. off-diagonal elements of the system’s density matrix taken in the eigenbasis of the Hamiltonian. As an example we discuss symmetrically boundary driven open XXZ spin 1/2 chains.

Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

Dynamics of higher-order solitons in regular and PT-symmetric nonlinear couplers

R. Driben, B. A. Malomed

Dynamics of symmetric and antisymmetric 2-solitons and 3-solitons is studied in the model of the nonlinear dual-core coupler and its PT-symmetric version. Regions of the convergence of the injected perturbed symmetric and antisymmetric N-solitons into symmetric and asymmetric quasi-solitons are found. In the PT-symmetric system, with the balanced gain and loss acting in the two cores, borders of the stability against the blowup are identified. Notably, in all the cases the stability regions are larger for antisymmetric 2-soliton inputs than for their symmetric counterparts, on the contrary to previously known results for fundamental solitons (N=1). Dynamical regimes (switching) are also studied for the 2-soliton injected into a single core of the coupler. In particular, a region of splitting of the input into a pair of symmetric solitons is found, which is explained as a manifestation of the resonance between the vibrations of the 2-soliton and oscillations of energy between the two cores in the coupler.

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

The two dimensional harmonic oscillator on a noncommutative space with minimal uncertainties

Sanjib Dey, Andreas Fring

The two dimensional set of canonical relations giving rise to minimal uncertainties previously constructed from a q-deformed oscillator algebra is further investigated. We provide a representation for this algebra in terms of a flat noncommutative space and employ it to study the eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes complex and its PT-symmetry is spontaneously broken.

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

Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations

Sanjib Dey, Andreas Fring

We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wavepackets are assembled according to sub-Poissonian statistics. Fractional revival times, indicating the superposition of classical-like sub-wave packets are clearly identified.

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

Spectral Singularities Do Not Correspond to Bound States in the Continuum

Ali Mostafazadeh

We show that, contrary to a claim made in arXiv:1011.0645, the von Neumann-Winger bound states that lie in the continuum of the scattering states are fundamentally different from Naimark’s spectral singularities.


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