August 2012
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Nonlinear dynamics of wave packets in PT-symmetric optical lattices near the phase transition point

Sean Nixon, Yi Zhu, Jianke Yang

Nonlinear dynamics of wave packets in PT-symmetric optical lattices near the phase-transition point are analytically studied. A nonlinear Klein-Gordon equation is derived for the envelope of these wave packets. A variety of novel phenomena known to exist in this envelope equation are shown to also exist in the full equation including wave blowup, periodic bound states and solitary wave solutions.

http://arxiv.org/abs/1208.5995
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

PT Symmetry in Classical and Quantum Statistical Mechanics

Peter N. Meisinger, Michael C. Ogilvie

PT-symmetric Hamiltonians and transfer matrices arise naturally in statistical mechanics. These classical and quantum models often require the use of complex or negative weights and thus fall outside of the conventional equilibrium statistical mechanics of Hermitian systems. PT-symmetric models form a natural class where the partition function is necessarily real, but not necessarily positive. The correlation functions of these models display a much richer set of behaviors than Hermitian systems, displaying sinusoidally-modulated exponential decay, as in a dense fluid, or even sinusoidal modulation without decay. Classical spin models with PT symmetry include Z(N) models with a complex magnetic field, the chiral Potts model and the anisotropic next-nearest-neighbor Ising (ANNNI) model. Quantum many-body problems with a non-zero chemical potential have a natural PT-symmetric representation related to the sign problem. Two-dimensional QCD with heavy quarks at non-zero chemical potential can be solved by diagonalizing an appropriate PT-symmetric Hamiltonian.

http://arxiv.org/abs/1208.5077
Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics – Lattice (hep-lat)

Mixed-state evolution in the presence of gain and loss

Dorje C. Brody, Eva-Maria Graefe

A model is proposed that describes the evolution of a mixed state of a quantum system for which gain and loss of energy or amplitude are present. Properties of the model are worked out in detail. In particular, invariant subspaces of the space of density matrices corresponding to the fixed points of the dynamics are identified, and the existence of a transition between the phase in which gain and loss are balanced and the phase in which this balance is lost is illustrated in terms of the time average of observables. The model is extended to include a noise term that results from a uniform random perturbation generated by white noise. Numerical studies of example systems show the emergence of equilibrium states that suppress the phase transition.

http://arxiv.org/abs/1208.5297

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

Approximate Dirac solutions of complex -symmetric Pöschl-Teller potential in view of spin and pseudospin symmetries

Sameer M. Ikhdair, Majid Hamzavi

By employing an exponential-type approximation scheme to replace the centrifugal term, we have approximately solved the Dirac equation for spin-particle subject to the complex-symmetric scalar and vector Poschl-Teller (PT) potentials with arbitrary spin-orbit -wave states in view of spin and pseudospin (p-spin) symmetries. The real bound-state energy eigenvalue equation and the corresponding two-spinor components wave function expressible in terms of the hypergeometric functions are obtained by means of the wave function analysis. The spin-Dirac equation and the spin-Klein-Gordon (KG) equation with the complex Poschl-Teller potentials share the same energy spectrum under the choice of (i.e., exact spin and p-spin symmetries).

http://arxiv.org/abs/1208.4960
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

Spectral singularities in PT-symmetric periodic finite-gap systems

Francisco Correa, Mikhail S. Plyushchay

The origin of spectral singularities in finite-gap singly periodic PT-symmetric quantum systems is investigated. We show that they emerge from a limit of band-edge states in a doubly periodic finite gap system when the imaginary period tends to infinity. In this limit, the energy gaps are contracted and disappear, every pair of band states of the same periodicity at the edges of a gap coalesces and transforms into a singlet state in the continuum. As a result, these spectral singularities turn out to be analogous to those in the non-periodic systems, where they appear as zero-width resonances. The specific degeneration related to the presence of finite number of spectral singularities in periodic quantum systems with compact and non-compact topologies is shown to be coherently reflected by a hidden, bosonized nonlinear supersymmetry.

http://arxiv.org/abs/1208.4448
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Antisymmetric PT-photonic structures with balanced positive and negative index materials

Li Ge, H. E. Tureci

In this Letter we study a new class of synthetic materials in which the refractive index satisfies a special symmetry, $$n(-x)=-n^*(x)$$, which we term antisymmetric parity-time (APT) systems. Unlike PT-symmetric systems which require balanced gain and loss, i.e. $$n(-x)=n^*(x)$$, APT systems consist of balanced positive and negative index materials (NIMs). Despite the seemingly PT-symmetric optical potential $$V(x)\equiv n(x)^2\omega^2/c^2$$, such systems are not invariant under combined PT operation due to the discontinuity of the spatial derivative of the wavefunction. We show that APT systems display intriguing properties such as spontaneous phase transition of the scattering matrix, bidirectional invisibility, and a continuous lasing spectrum.

http://arxiv.org/abs/1208.4644
Optics (physics.optics)

Quantum search using non-Hermitian adiabatic evolution

Alexander I. Nesterov, Gennady P. Berman

We propose a non-Hermitian quantum annealing algorithm which can be useful for solving complex optimization problems. We demonstrate our approach on Grover’s problem of finding a marked item inside of unsorted database. We show that the energy gap between the ground and excited states depends on the relaxation parameters, and is not exponentially small. This allows a significant reduction of the searching time. We discuss the relations between the probabilities of finding the ground state and the survival of a quantum computer in a dissipative environment.

http://arxiv.org/abs/1208.4642
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph)

Solitons and their ghosts in PT-symmetric systems with defocusing nonlinearities

V. Achilleos, P. G.Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez

We examine a prototypical nonlinear Schrodinger model bearing a defocusing nonlinearity and Parity-Time (PT) symmetry. For such a model, the solutions can be identified numerically and characterized in the perturbative limit of small gain/loss. There we find two fundamental phenomena. First, the dark solitons that persist in the presence of the PT-symmetric potential are destabilized via a symmetry breaking (pitchfork) bifurcation. Second, the ground state and the dark soliton die hand-in-hand in a saddle-center bifurcation (a nonlinear analogue of the PT-phase transition) at a second critical value of the gain/loss parameter. The daughter states arising from the pitchfork are identified as “ghost states”, which are not exact solutions of the original system, yet they play a critical role in the system’s dynamics. A similar phenomenology is also pairwise identified for higher excited states, with e.g. the two-soliton structure bearing similar characteristics to the zero-soliton one, and the three-soliton state having the same pitchfork destabilization mechanism and saddle-center collision (in this case with the two-soliton) as the one-dark soliton. All of the above notions are generalized in two-dimensional settings for vortices, where the topological charge enforces the destabilization of a two-vortex state and the collision of a no-vortex state with a two-vortex one, of a one-vortex state with a three-vortex one, and so on. The dynamical manifestation of the instabilities mentioned above is examined through direct numerical simulations.

http://arxiv.org/abs/1208.2445
Pattern Formation and Solitons (nlin.PS); Soft Condensed Matter (cond-mat.soft)

Parity anomaly and Landau-level lasing in strained photonic honeycomb lattices

Henning Schomerus, Nicole Yunger Halpern

We describe the formation of highly degenerate, Landau-level-like amplified states in a strained photonic honeycomb lattice in which amplification breaks the sublattice symmetry. As a consequence of the parity anomaly, the zeroth Landau level is localized on a single sublattice and possesses an enhanced or reduced amplification rate. The spectral properties of the higher Landau levels are constrained by a generalized time-reversal symmetry. In the setting of two-dimensional photonic crystal lasers, the anomaly directly affects the mode selection and lasing threshold while in three-dimensional photonic lattices it can be probed via beam dynamics.

http://arxiv.org/abs/1208.2901

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

Parity–time synthetic photonic lattices

Alois Regensburger, Christoph Bersch, Mohammad-Ali Miri, Georgy Onishchukov, Demetrios N. Christodoulides, Ulf Peschel

The development of new artificial structures and materials is today one of the major research challenges in optics. In most studies so far, the design of such structures has been based on the judicious manipulation of their refractive index properties. Recently, the prospect of simultaneously using gain and loss was suggested as a new way of achieving optical behaviour that is at present unattainable with standard arrangements. What facilitated these quests is the recently developed notion of ‘parity–time symmetry’ in optical systems, which allows a controlled interplay between gain and loss. Here we report the experimental observation of light transport in large-scale temporal lattices that are parity–time symmetric. In addition, we demonstrate that periodic structures respecting this symmetry can act as unidirectional invisible media when operated near their exceptional points. Our experimental results represent a step in the application of concepts from parity–time symmetry to a new generation of multifunctional optical devices and networks.

http://dx.doi.org/10.1038/nature11298