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Month July 2013

Tracking down localized modes in PT-symmetric Hamiltonians under the influence of a competing nonlinearity

Bijan Bagchi, Subhrajit Modak, Prasanta K. Panigrahi

The relevance of parity and time reversal (PT)-symmetric structures in optical systems is known for sometime with the correspondence existing between the Schrodinger equation and the paraxial equation of diffraction where the time parameter represents the propagating distance and the refractive index acts as the complex potential. In this paper, we systematically analyze a normalized form of the nonlinear Schrodinger system with two new families of PT-symmetric potentials in the presence of competing nonlinearities. We generate a class of localized eigenmodes and carry out a linear stability analysis on the solutions. In particular, we find an interesting feature of bifurcation charaterized by the parameter of perturbative growth rate passing through zero where a transition to imaginary eigenvalues occurs.

http://arxiv.org/abs/1307.7246
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)

PT-Symmetric Dimer of Coupled Nonlinear Oscillators

J. Cuevas, P.G. Kevrekidis, A. Saxena, A. Khare

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend considerations to the nonlinear case for both soft and hard cubic nonlinearities identifying symmetric and anti-symmetric breather solutions, as well as symmetry breaking variants thereof. We propose a reduction of the system to a Schr\”odinger type PT-symmetric dimer, whose detailed earlier understanding can explain many of the phenomena observed herein, including the PT phase transition. Nevertheless, there are also significant parametric as well as phenomenological potential differences between the two models and we discuss where these arise and where they are most pronounced. Finally, we also provide examples of the evolution dynamics of the different states in their regimes of instability.

http://arxiv.org/abs/1307.6047

Pattern Formation and Solitons (nlin.PS)

Some remarks on quasi-Hermitian operators

Jean-Pierre Antoine, Camillo Trapani

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally we discuss their application in the so-called pseudo-Hermitian quantum mechanics.

http://arxiv.org/abs/1307.5644
Mathematical Physics (math-ph)

Double-Scaling Limit of the O(N)-Symmetric Anharmonic Oscillator

Carl M. Bender, Sarben Sarkar

In an earlier paper it was argued that the conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative and thus the integral representing the partition function of the critical theory does not exist. In this earlier paper it was shown that for an O(N)-symmetric quantum field theory in zero-dimensional spacetime one can avoid this difficulty if one replaces the original quartic theory by its PT-symmetric analog. In the current paper an O(N)-symmetric quartic quantum field theory in one-dimensional spacetime [that is, O(N)-symmetric quantum mechanics] is studied using the Schroedinger equation. It is shown that the global PT-symmetric formulation of this differential equation provides a consistent way to perform the double-scaling limit of the O(N)-symmetric anharmonic oscillator. The physical nature of the critical behavior is explained by studying the PT-symmetric quantum theory and the corresponding and equivalent Hermitian quantum theory.

http://arxiv.org/abs/1307.4348

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

On the equivalence of lossy evolution and POVM generalized quantum measurements

Raam Uzdin

Loss induced generalized measurements have been introduced years ago as a mean to implement generalized quantum measurements (POVM). Here the original idea is extended to a complete equivalence of lossy evolution and a certain widely used class of POVM. This class includes POVM used for unambiguous state discrimination and entanglement concentration. One implication of this equivalence is that unambiguous state discrimination schemes based on PT-symmetric and non-Hermitian Hamiltonians have the same performance as those of standard POVM. After discussing several key points of this equivalence we illustrate our findings in two elementary physical realizations. Finally, we discuss several implications of this equivalence.

http://arxiv.org/abs/1307.3927
Quantum Physics (quant-ph)

Information Geometry of Complex Hamiltonians and Exceptional Points

Dorje C. Brody, Eva-Maria Graefe

Information geometry provides a tool to systematically investigate parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric.

http://arxiv.org/abs/1307.4017
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

Nonlinear dynamics in PT-symmetric lattices

Panayotis G. Kevrekidis, Dmitry E. Pelinovsky, Dmitry Y.Tyugin

We consider nonlinear dynamics in a finite parity-time-symmetric chain of the discrete nonlinear Schrodinger (dNLS) type. We work in the range of the gain and loss coefficient when the zero equilibrium state is neutrally stable. We prove that the solutions of the dNLS equation do not blow up in a finite time and the trajectories starting with small initial data remain bounded for all times. Nevertheless, for arbitrary values of the gain and loss parameter, there exist trajectories starting with large initial data that grow exponentially fast for larger times with a rate that is rigorously identified. Numerical computations illustrate these analytical results for dimers and quadrimers.

http://arxiv.org/abs/1307.2973
Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)

On the integrability of PT-symmetric dimers

J. Pickton, H. Susanto

The coupled discrete linear and Kerr nonlinear Schrodinger equations with gain and loss describing transport on dimers with parity-time (PT) symmetric potentials are considered. The model is relevant among others to experiments in optical couplers and proposals on Bose-Einstein condensates in PT symmetric double-well potentials. It is shown that the models are integrable. A pendulum equation with a linear potential and a constant force for the phase-difference between the fields is obtained, which explains the presence of unbounded solutions above a critical threshold parameter.

http://arxiv.org/abs/1307.2788
Optics (physics.optics); Quantum Gases (cond-mat.quant-gas); Exactly Solvable and Integrable Systems (nlin.SI)

PT Symmetry via Electromagnetically Induced Transparency

Huijun Li, Jianpeng Dou, Jinjin Yang, Guoxiang Huang

We propose a scheme to realize parity-time (\(\mathcal{PT}\)) symmetry via electromagnetically induced transparency (EIT). The system we consider is an ensemble of cold four-level atoms with an EIT core. We show that the cross-phase modulation contributed by an assisted field, the optical lattice potential provided by a far-detuned laser field, and the optical gain resulted from an incoherent pumping can be used to construct a \(\mathcal{PT}\)-symmetric complex optical potential for probe field propagation in a controllable way. Comparing with previous study, the present scheme uses only a single atomic species and hence is easy for the physical realization of \(\mathcal{PT}\)-symmetric Hamiltonian via atomic coherence.

http://arxiv.org/abs/1307.2695
Quantum Physics (quant-ph)

Breaking of PT-symmetry in bounded and unbounded scattering systems

Philipp Ambichl, Konstantinos G. Makris, Li Ge, Yidong Chong, A. Douglas Stone, Stefan Rotter

PT-symmetric scattering systems with balanced gain and loss can undergo a symmetry-breaking transition in which the eigenvalues of the non-unitary scattering matrix change their phase shifts from real to complex values. We relate the PT-symmetry breaking points of such an unbounded scattering system to those of underlying bounded systems. In particular, we show how the PT-thresholds in the scattering matrix of the unbounded system translate into analogous transitions in the Robin boundary conditions of the corresponding bounded systems. Based on this relation, we argue and then confirm that the PT-transitions in the scattering matrix are, under very general conditions, entirely insensitive to a variable coupling strength between the bounded region and the unbounded asymptotic region, a result which can be tested experimentally.

http://arxiv.org/abs/1307.0149

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