Sanjib Dey, Andreas Fring, Thilagarajah Mathanaranjan

We propose a noncommutative version of the Euclidean Lie algebra \(E_2\). Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra.

http://arxiv.org/abs/1407.8097

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

Richard J. Potton

In crystal optics the special status of the rest frame of the crystal means that space-time symmetry is less restrictive of electrodynamic phenomena than it is of static electromagnetic effects. A relativistic justification for this claim is provided and its consequences for the analysis of optical activity are explored. The discrete space-time symmetries P and T that lead to classification of static property tensors as polar or axial, time-invariant (-i) or time-change (-c) are shown to be connected by orientation considerations. The connection finds expression in the dynamic phenomenon of gyrotropy in certain, symmetry determined, crystal classes. In particular, the degeneracies of forward and backward waves in optically active crystals arise from the covariance of the wave equation under space-time (PT) reversal.

http://arxiv.org/abs/1407.6797

Optics (physics.optics)

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 \(H\).

http://arxiv.org/abs/1407.4633

Mathematical Physics (math-ph)

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.

http://arxiv.org/abs/1407.4535

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

C. Poli, M. Bellec, U.Kuhl, F. Mortessagne, H. Schomerus

An attractive mechanism to induce robust spatially confined states utilizes interfaces between regions with topologically distinct gapped band structures. For electromagnetic waves, this mechanism can be realized in two dimensions by breaking symmetries in analogy to the quantum Hall effect or by employing analogies to the quantum spin Hall effect, while in one dimension it can be obtained by geometric lattice modulation. Induced by the presence of the interface, a topologically protected, exponentially confined state appears in the middle of the band gap. The intrinsic robustness of such states raises the question whether their properties can be controlled and modified independently of the other states in the system. Here, we draw on concepts from passive non-hermitian parity-time (PT)-symmetry to demonstrate the selective control and enhancement of a topologically induced state in a one-dimensional microwave set-up. In particular, we show that the state can be isolated from losses that affect all other modes in the system, which enhances its visibility in the temporal evolution of a pulse. The intrinsic robustness of the state to structural disorder persists in the presence of the losses. The combination of concepts from topology and non-hermitian symmetry is a promising addition to the set of design tools for optical structures with novel functionality.

http://arxiv.org/abs/1407.3703

Optics (physics.optics); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

P.A. Kalozoumis, G. Pappas, F.K. Diakonos, P. Schmelcher

Recently [Phys. Rev. Lett. **106**, 093902 (2011)] it has been shown that PT-symmetric scattering systems with balanced gain and loss, undergo a transition from PT-symmetric scattering eigenstates, which are norm preserving, to symmetry broken pairs of eigenstates exhibiting net amplification and loss. In the present work we derive the existence of an invariant non-local current which can be directly associated with the observed transition playing the role of an “order parameter”. The use of this current for the description of the PT-symmetry breaking allows the extension of the known phase diagram to higher dimensions incorporating scattering states which are not eigenstates of the scattering matrix.

http://arxiv.org/abs/1407.2655

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

João-Paulo Dias, Mário Figueira, Vladimir V. Konotop, Dmitry A. Zezyulin

We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrodinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonlinearity and the nonlinear coupling is repulsive, as well. The proofs are based on the virial technique arguments. Several particular cases are also illustrated numerically.

http://arxiv.org/abs/1407.2438

Analysis of PDEs (math.AP); Optics (physics.optics)

Philip D. Mannheim

We present some interesting connections between \(PT\) symmetry and conformal symmetry. We use them to develop a metricated theory of electromagnetism in which the electromagnetic field is present in the geometric connection. However, unlike Weyl who first advanced this possibility, we do not take the connection to be real but to instead be \(PT\) symmetric, with it being \(iA_{\mu}\) rather than \(A_{\mu}\) itself that then appears in the connection. With this modification the standard minimal coupling of electromagnetism to fermions is obtained. Through the use of torsion we obtain a fully metricated theory of electromagnetism that treats its electric and magnetic sectors completely symmetrically, with a conformal invariant theory of gravity being found to emerge.

http://arxiv.org/abs/1407.1820

High Energy Physics – Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)

Jiun-Yi Lien, Yueh-Nan Chen, Natsuko Ishida, Hong-Bin Chen, Chi-Chuan Hwang, Franco Nori

Exciton-polaritons can condense to a macroscopic quantum state through a non-equilibrium process of pumping and decay. In recent experiments, polariton condensates are used to observe, for a short time, nonlinear Josephson phenomena by coupling two condensates. However, it is still not clear how these phenomena are affected by the pumping and decay at long times and how the coupling alters the polariton condensation. Here, we consider a polariton Josephson junction pumped on one side and study its dynamics within a mean-field theory. The Josephson current is found to give rise to multi-stability of the stationary states, which are sensitive to the initial conditions and incoherent noises. These states can be attributed to either the self-trapping effect or the parity-time (PT) symmetry of the system. These results can be used to explain the emission spectra and the \(\pi\)-phase locking observed in recent experiments. We further predict that the multi-stability can reduce to the self-trapped state if the PT symmetry is broken. Moreover, the polaritons can condense even below the threshold, exhibiting hysteresis.

http://arxiv.org/abs/1407.1271

Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

Ali Mostafazadeh

We give a complete solution of the problem of constructing a scattering potential v(x) that possesses scattering properties of one’s choice at an arbitrary prescribed wavenumber. Our solution involves expressing v(x) as the sum of at most six unidirectionally invisible finite-range potentials for which we give explicit formulas. Our results can be employed for designing optical potentials. We discuss its application in modeling threshold lasers, coherent perfect absorbers, and bidirectionally and unidirectionally reflectionless absorbers, amplifiers, and phase shifters.

http://arxiv.org/abs/1407.1760

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