Zhiwei Shi, Huagang Li, Xing Zhu, Xiujuan Jiang

It is studied the bright spatial solitons in nonlocal defocusing Kerr media with parity-time (PT) symmetric potentials. We find that these solitons can exist and be stable over a different range of potential parameters. The influence of the degree of nonlocality on the solitons and the transverse energy flow within the stable solitons are also examined.

http://arxiv.org/abs/1110.5108

Optics (physics.optics)

Fabio Bagarello

We have recently proposed a strategy to produce, starting from a given hamiltonian \(h_1\) and a certain operator \(x\) for which \([h_1,xx^\dagger]=0\) and \(x^\dagger x\) is invertible, a second hamiltonian \(h_2\) with the same eigenvalues as \(h_1\) and whose eigenvectors are related to those of \(h_1\) by \(x^\dagger\). Here we extend this procedure to build up a second hamiltonian, whose eigenvalues are different from those of \(h_1\), and whose eigenvectors are still related as before. This new procedure is also extended to crypto-hermitian hamiltonians.

http://arxiv.org/abs/1110.4828

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

Zafar Ahmed

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: \(R(-k)\ne R(k)$ and $T(-k) \ne T(k)\), unless the potentials are real or PT-symmetric. For complex PT-symmetry scattering potentials, we propose that \(R_{left}(-k)=R_{right}(k)\) and \(T(-k)=T(k)\).So far the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies (\(E_*=\alpha^2,\beta^2\)) either in \(T(k)\) or in \(T(-k)\), when \(\alpha\beta>0\). Thirdly, when \(\alpha \beta <0\) it possesses one SS in \(T(k)\) and the other in \(T(-k)\). Lastly, when the potential becomes PT-symmetric \([(\alpha+\beta)=0]\), we get \(T(k)=T(-k)\), it possesses a unique SS at \(E=\alpha^2\) in both \(T(-k)\) and \(T(k)\).

http://arxiv.org/abs/1110.4485

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

Sumei Hu, Daquan Lu, Xuekai Ma, Qi Guo, Wei Hu

The existence and stability of defect solitons supported by parity-time (PT) symmetric superlattices with nonlocal nonlinearity are investigated. Unlike local PT symmetric system, the nonlocal system considered reveals unusual properties. In the semi-infinite gap, in-phase solitons can exist stably for positive defects or zero defects, but can not exist for negative defects with the strong nonlocality. In the first gap, out-phase solitons are stable for positive defects or zero defects, whereas in-phase solitons are stable for negative defects. The dependence of soliton stabilities on modulation depth of the PT potentials is studied. It is interesting that solitons can exist stably for positive and zero defects when the PT potential is above the phase transition point.

http://arxiv.org/abs/1110.4344

Optics (physics.optics)

U. D. Jentschura, B. J. Wundt

We show that it is possible to construct a tachyonic version of the Dirac equation, which contains the fifth current and reads (i gamma^\mu partial_\mu – gamma^5 m) \psi = 0. Its spectrum fulfills the dispersion relation E^2 = p^2 – m^2, where E is the energy, p is the spatial momentum, and m is the mass of the particle. The tachyonic Dirac equation is shown to be CP invariant, and T invariant. The Feynman propagator is found. In contrast to the covariant formulation, the tachyonic Hamiltonian H_5 = alpha.p + beta gamma^5 m breaks Lorentz covariance (as does the Hamiltonian formalism in general, because the time variable is singled out and treated differently from space). The tachyonic Dirac Hamiltonian H_5 breaks parity but is found to be invariant under the combined action of parity and a noncovariant time reversal operation T’. In contrast to the Lorentz-covariant T operation, T’ involves the Hermitian adjoint of the Hamiltonian. Thus, in the formalism developed by Bender et al., primarily in the context of quantum mechanics, H_5 is PT’ symmetric. The PT’ invariance (in the quantum mechanical sense) is responsible for the fact that the energy eigenvalues of the tachyonic Dirac Hamiltonian are real rather than complex. The eigenstates of the Hamiltonian are shown to approximate the helicity eigenstates of a Dirac neutrino in the massless limit.

http://arxiv.org/abs/1110.4171

High Energy Physics – Phenomenology (hep-ph)

Philip Cherian, Kumar Abhinav, P. K. Panigrahi

A family of PT -symmetric complex potentials are obtained which is isospectral to free particle in an infinite complex box in one dimension (1-D). These are generalizations to the cosec2(x) potential, isospectral to particle in a real infinite box. In the complex plane, the infinite box is extended parallel to the real axis having a real width, which is found to be an integral multiple of a constant quantum factor, arising due to boundary conditions necessary for maintaining the PT -symmetry of the superpartner. As the spectra of the particle in a box is still real, it necessarily picks out the unbroken PT -sector of its superpartner, thereby invoking a close relation between PT -symmetry and SUSY for this case.

http://arxiv.org/abs/1110.3708

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

R. Driben, B. A. Malomed

We introduce a system based on dual-core nonlinear waveguides with the balanced gain and loss acting separately in the cores. The system features a “supersymmetry” when the gain and loss are equal to the inter-core coupling. This system admits a variety of exact solutions (we focus on solitons), which are subject to a specific subexponential instability. We demonstrate that the application of a “management”, in the form of periodic simultaneous switch of the sign of the gain, loss, and inter-coupling, effectively stabilizes solitons, without destroying the supersymmetry. The management turns the solitons into attractors, for which an attraction basin is identified. The initial amplitude asymmetry and phase mismatch between the components transforms the solitons into quasi-stable breathers.

http://arxiv.org/abs/1110.2409

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

Jun-Qing Li, Yan-Gang Miao

By adding an imaginary potential proportional to \(ip_1p_2\) to the hamiltonian of an anisotropic planar oscillator, we construct a model which is described by a non-hermitian hamiltonian with PT pseudo-hermiticity. We introduce the mechanism of the spontaneous breaking of permutation symmetry of the hamiltonian for diagonalizing the hamiltonian. By applying the definition of annihilation and creation operators which are PT pseudo-hermitian adjoint to each other, we give the real spectra.

http://arxiv.org/abs/1110.2312

Quantum Physics (quant-ph)

Jun-Qing Li, Yan-Gang Miao

By adding an imaginary potential proportional to ip_1p_2 to the hamiltonian of an anisotropic planar oscillator, we construct a model which is described by a non-hermitian hamiltonian with PT pseudo-hermiticity. We introduce the mechanism of the spontaneous breaking of permutation symmetry of the hamiltonian for diagonalizing the hamiltonian. By applying the definition of annihilation and creation operators which are PT pseudo-hermitian adjoint to each other, we give the real spectra.

http://arxiv.org/abs/1110.2312

Quantum Physics (quant-ph)

Gilles Demange, Eva-Maria Graefe

Parameter dependent non-Hermitian quantum systems typically not only possess eigenvalue degeneracies, but also degeneracies of the corresponding eigenfunctions at exceptional points. While the effect of two coalescing eigenfunctions on cyclic parameter variation is well investigated, little attention has hitherto been paid to the effect of more than two coalescing eigenfunctions. Here a characterisation of behaviours of symmetric Hamiltonians with three coalescing eigenfunctions is presented, using perturbation theory for non-Hermitian operators. Two main types of parameter perturbations need to be distinguished, which lead to characteristic eigenvalue and eigenvector patterns under cyclic variation. A physical system is introduced for which both behaviours might be experimentally accessible.

http://arxiv.org/abs/1110.1489

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

Sergey V. Suchkov, Boris A. Malomed, Sergey V. Dmitriev, Yuri S. Kivshar

Dynamics of a chain of interacting parity-time invariant nonlinear dimers is investigated. A dimer is built as a pair of coupled elements with equal gain and loss. A relation between stationary soliton solutions of the model and solitons of the discrete nonlinear Schrodinger (DNLS) equation is demonstrated. Approximate solutions for solitons whose width is large in comparison to the lattice spacing are derived, using a continuum counterpart of the discrete equations. These solitons are mobile, featuring nearly elastic collisions. Stationary solutions for narrow solitons, which are immobile due to the pinning by the effective Peierls-Nabarro potential, are constructed numerically, starting from the anti-continuum limit. The solitons with the amplitude exceeding a certain critical value suffer an instability leading to blowup, which is a specific feature of the nonlinear PT-symmetric chain, making it dynamically different from DNLS lattices. A qualitative explanation of this feature is proposed. The instability threshold drops with the increase of the gain-loss coefficient, but it does not depend on the lattice coupling constant, nor on the soliton’s velocity.

http://arxiv.org/abs/1110.1501

Optics (physics.optics)