Stefano Longhi
Bragg scattering in sinusoidal PT-symmetric complex crystals of finite thickness is theoretically investigated by the derivation of exact analytical expressions for reflection and transmission coefficients in terms of modified Bessel functions of first kind. The analytical results indicate that unidirectional invisibility, recently predicted for such crystals by coupled-mode theory [Z. Lin et al., Phys. Rev. Lett. 106, 213901 (2011)], breaks down for crystals containing a large number of unit cells. In particular, for a given modulation depth in a shallow sinusoidal potential, three regimes are encountered as the crystal thickness is increased. At short lengths the crystal is reflectionless and invisible when probed from one side (unidirectional invisibility), whereas at intermediate lengths the crystal remains reflectionless but not invisible; for longer crystals both unidirectional reflectionless and invisibility properties are broken.
http://arxiv.org/abs/1111.3448
Quantum Physics (quant-ph)
Miloslav Znojil
Two discrete N-level alternatives to the popular imaginary cubic oscillator are proposed and studied. In a certain domain \({\cal D}\) of parameters \(a\) and \(z\) of the model, the spectrum of energies is shown real (i.e., potentially, observable) and the unitarity of the evolution is shown mediated by the construction of a (non-unique) physical, ad hoc Hilbert space endowed with a nontrivial, Hamiltonian-dependent inner-product metric \(\Theta\). Beyond \({\cal D}\) the complex-energy curves are shown to form a “Fibonacci-numbered” geometric pattern and/or a “topologically complete” set of spectral loci. The dynamics-determining construction of the set of the eligible metrics is shown tractable by a combination of the computer-assisted algebra with the perturbation and extrapolation techniques. Confirming the expectation that for the local potentials the effect of the metric cannot be short-ranged.
http://arxiv.org/abs/1111.0484
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Stefano Longhi
Spatial and/or temporal propagation of light waves in periodic optical structures offers a rather unique possibility to realize in a purely classical setting the optical analogues of a wide variety of quantum phenomena rooted in relativistic wave equations. In this work a brief overview of a few optical analogues of relativistic quantum phenomena, based on either spatial light transport in engineered photonic lattices or on temporal pulse propagation in Bragg grating structures, is presented. Examples include spatial and temporal photonic analogues of the Zitterbewegung of a relativistic electron, Klein tunneling, vacuum decay and pair-production, the Dirac oscillator, the relativistic Kronig-Penney model, and optical realizations of non-Hermitian extensions of relativistic wave equations.
http://arxiv.org/abs/1111.3461
Optics (physics.optics)
Eva-Maria Graefe, Roman Schubert
The complex geometry underlying the Schr\”odinger dynamics of coherent states for non-Hermitian Hamiltonians is investigated. In particular two seemingly contradictory approaches are compared: (i) a complex WKB formalism, for which the centres of coherent states naturally evolve along complex trajectories, which leads to a class of complexified coherent states; (ii) the investigation of the dynamical equations for the real expectation values of position and momentum, for which an Ehrenfest theorem has been derived in a previous paper, yielding real but non-Hamiltonian classical dynamics on phase space for the real centres of coherent states. Both approaches become exact for quadratic Hamiltonians. The apparent contradiction is resolved building on an observation by Huber, Heller and Littlejohn, that complexified coherent states are equivalent if their centres lie on a specific complex Lagrangian manifold. A rich underlying complex symplectic geometry is unravelled. In particular a natural complex structure is identified that defines a projection from complex to real phase space, mapping complexified coherent states to their real equivalents.
http://arxiv.org/abs/1111.1877
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
H. F. Jones
Propagation of light through media with a complex refractive index in which gain and loss are engineered to be PT symmetric has many remarkable features. In particular the usual unitarity relations are not satisfied, so that the reflection coefficients can be greater than one, and in general are not the same for left or right incidence. Within the class of optical potentials of the form \(v(x)=v_1\cos(2\beta x)+iv_2\sin(2\beta x)\) the case \(v_2=v_1\) is of particular interest, as it lies on the boundary of PT-symmetry breaking. It has been shown in a recent paper by Lin et al. that in this case one has the property of “unidirectional invisibility”, while for propagation in the other direction there is a greatly enhanced reflection coefficient proportional to \(L^2\), where \(L\) is the length of the medium in the direction of propagation.
For this potential we show how analytic expressions can be obtained for the various transmission and reflection coefficients, which are expressed in a very succinct form in terms of modified Bessel functions. While our numerical results agree very well with those of Lin et al. we find that the invisibility is not quite exact, in amplitude or phase. As a test of our formulas we show that they identically satisfy a modified version of unitarity appropriate for PT-symmetric potentials. We also examine how the enhanced transmission comes about for a wave-packet, as opposed to a plane wave, finding that the enhancement now arises through an increase, of \(O(L)\), in the pulse length, rather than the amplitude.
http://arxiv.org/abs/1111.2041
Optics (physics.optics); Quantum Physics (quant-ph)
D. A. Zezyulin, Y. V. Kartashov, V. V. Konotop
We report on detailed investigation of the stability of localized modes in the nonlinear Schrodinger equations with a nonlinear parity-time (alias PT) symmetric potential. We are particularly focusing on the case where the spatially-dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and \(\tanh\)-shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, what suggests that the relation between width of the modes and spatial size of the complex potential define the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples.
http://arxiv.org/abs/1111.0898
Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph); Optics (physics.optics)
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)