## Analytic Solution for PT-Symmetric Volume Gratings

Mykola Kulishov, H. F. Jones, Bernard Kress

We study the diffraction produced by a PT-symmetric volume Bragg grating that combines modulation of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for reflections, be properly implemented. Using our solution we analyze the properties of such a grating in a wide variety of configurations.

http://arxiv.org/abs/1412.0506
Optics (physics.optics)

## Singular Mapping for a PT-Symmetric Sinusoidal Optical Lattice at the Symmetry-Breaking Threshold

H. F. Jones

A popular PT-symmetric optical potential (variation of the refractive index) that supports a variety of interesting and unusual phenomena is the imaginary exponential, the limiting case of the potential $$V_0[\cos(2\pi x/a)+i\lambda\sin(2\pi x/a)]$$ as $$\lambda\to1$$, the symmetry-breaking point. For $$\lambda<1$$, when the spectrum is entirely real, there is a well-known mapping by a similarity transformation to an equivalent Hermitian potential. However, as $$\lambda\to1$$, the spectrum, while remaining real, contains Jordan blocks in which eigenvalues and the corresponding eigenfunctions coincide. In this limit the similarity transformation becomes singular. Nonetheless, we show that the mapping from the original potential to its Hermitian counterpart can still be implemented; however, the inverse mapping breaks down. We also illuminate the role of Jordan associated functions in the original problem, showing that they map onto eigenfunctions in the associated Hermitian problem.

http://arxiv.org/abs/1411.6451
Optics (physics.optics); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

## Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

Eva-Maria Graefe, Hans Jürgen Korsch, Alexander Rush, Roman Schubert

The non-Hermitian quadratic oscillator studied by Swanson is one of the popular PT-symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine the classical symplectic flow for Hermitian systems with a dissipative metric flow for the anti-Hermitian part. Closed form expressions for the metric and phase-space trajectories are presented which are found to be periodic in time. Since the Hamiltonian is only quadratic the classical dynamics exactly describes the quantum dynamics of Gaussian wave packets. It is shown that the classical metric and trajectories as well as the quantum wave functions can diverge in finite time even though the PT-symmetry is unbroken, i.e., the eigenvalues are purely real.

http://arxiv.org/abs/1409.6456
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

## Infinitely many inequivalent field theories from one Lagrangian

Carl M. Bender, Daniel W. Hook, Nick E. Mavromatos, Sarben Sarkar

Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field $$\phi$$. In Euclidean space the Lagrangian of such a theory, $$L=\frac{1}{2}(\nabla\phi)^2−ig\phi \exp(ia\phi)$$, is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the $$m$$th energy level in the $$n$$th sector is given by $$E_{m,n}∼(m+1/2)^2a^2/(16n^2)$$.

http://arxiv.org/abs/1408.2432
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

## Unique optical characteristics of a Fabry-Perot resonator with embedded PT-symmetrical grating

Mykola Kulishov, Bernand Kress, H. F. Jones

We explore the optical properties of a Fabry-Perot resonator with an embedded Parity-Time (PT) symmetrical grating. This PT-symmetrical grating is non diffractive (transparent) when illuminated from one side and diffracting (Bragg reflection) when illuminated from the other side, thus providing a unidirectional reflective functionality. The incorporated PT-symmetrical grating forms a resonator with two embedded cavities. We analyze the transmission and reflection properties of these new structures through a transfer matrix approach. Depending on the resonator geometry these cavities can interact with different degrees of coherency: fully constructive interaction, partially constructive interaction, partially destructive interaction, and finally their interaction can be completely destructive. A number of very unusual (exotic) nonsymmetrical absorption and amplification behaviors are observed. The proposed structure also exhibits unusual lasing performance. Due to the PT-symmetrical grating, there is no chance of mode hopping; it can lase with only a single longitudinal mode for any distance between the distributed reflectors.

http://arxiv.org/abs/1405.6024
Optics (physics.optics)

## Complex classical motion in potentials with poles and turning points

Carl M. Bender, Daniel W. Hook

Complex trajectories for Hamiltonians of the form H=p^n+V(x) are studied. For n=2 time-reversal symmetry prevents trajectories from crossing. However, for n>2 trajectories may indeed cross, and as a result, the complex trajectories for such Hamiltonians have a rich and elaborate structure. In past work on complex classical trajectories it has been observed that turning points act as attractors; they pull on complex trajectories and make them veer towards the turning point. In this paper it is shown that the poles of V(x) have the opposite effect — they deflect and repel trajectories. Moreover, poles shield and screen the effect of turning points.

http://arxiv.org/abs/1402.3852
Mathematical Physics (math-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)

## Mean-field approximation for a Bose-Hubbard dimer with complex interaction strength

Eva-Maria Graefe, Chiara Liverani

A generalised mean-field approximation for non-Hermitian many-particle systems has been introduced recently for a Bose-Hubbard dimer with complex on-site energies. Here we apply this approximation to a Bose-Hubbard dimer with a complex particle interaction term, modelling losses due to interactions in a two mode Bose-Einstein condensate. We derive the mean-field equations of motion leading to nonlinear dissipative Bloch dynamics, related to a nontrivial complex generalisation of the nonlinear Schrodinger equation. It is shown that depending on the parameter values there can be up to six stationary states. Further, for small values of the interaction strength the dynamics shows limit cycles.

http://arxiv.org/abs/1305.7160
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

## Vector Models in PT Quantum Mechanics

Katherine Jones-Smith, Rudolph Kalveks

We present two examples of non-Hermitian Hamiltonians which consist of an unperturbed part plus a perturbation that behaves like a vector, in the framework of PT quantum mechanics. The first example is a generalization of the recent work by Bender and Kalveks, wherein the E2 algebra was examined; here we consider the E3 algebra representing a particle on a sphere, and identify the critical value of coupling constant which marks the transition from real to imaginary eigenvalues. Next we analyze a model with SO(3) symmetry, and in the process extend the application of the Wigner-Eckart theorem to a non-Hermitian setting.

http://arxiv.org/abs/1304.5692
Quantum Physics (quant-ph)

## The Floquet Method for PT-symmetric Periodic Potentials

H. F. Jones

By the general theory of PT-symmetric quantum systems, their energy levels are either real or occur in complex-conjugate pairs, which implies that the secular equation must be real. However, for periodic potentials it is by no means clear that the secular equation arising in the Floquet method is indeed real, since it involves two linearly independent solutions of the Schrodinger equation. In this brief note we elucidate how that reality can be established.

http://arxiv.org/abs/1211.1560
Mathematical Physics (math-ph); Quantum Physics (quant-ph)