Light propagation in optical waveguides with periodically modulated index of refraction and alternating gain and loss are investigated for linear and nonlinear systems. Based on a multiscale perturbation analysis, it is shown that for many non-parity-time (PT) symmetric waveguides, their linear spectrum is partially complex, thus light exponentially grows or decays upon propagation, and this growth or delay is not altered by nonlinearity. However, several classes of non-PT-symmetric waveguides are also identified to possess all-real linear spectrum. In the nonlinear regime longitudinally periodic and transversely quasi-localized modes are found for PT-symmetric waveguides both above and below phase transition. These nonlinear modes are stable under evolution and can develop from initially weak initial conditions.

http://arxiv.org/abs/1412.6113

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

We find that a new type of non-reciprocal modes exist at an interface between two *parity-time* (PT) symmetric magnetic domains (MDs) near the frequency of zero effective permeability. The new mode is non-propagating and purely magnetic when the two MDs are semi-infinite while it becomes propagating in the finite case. In particular, two pronounced nonreciprocal responses could be observed via the excitation of this mode: one-way optical tunneling for oblique incidence and unidirectional beam shift at normal incidence. When the two MDs system becomes finite in size, it is found that perfect-transmission mode could be achieved if PT-symmetry is maintained. The unique properties of such an unusual mode are investigated by analytical modal calculation as well as numerical simulations. The results suggest a new approach to the design of compact optical isolator.

http://arxiv.org/abs/1412.5725

Optics (physics.optics)

The time evolution of a system with a time-dependent non-Hermitian Hamiltonian is in general unstable with exponential growth or decay. A periodic driving field may stabilize the dynamics because the eigenphases of the associated Floquet operator may become all real. This possibility can emerge for a continuous range of system parameters with subtle domain boundaries. It is further shown that the issue of stability of a driven non-Hermitian Rabi model can be mapped onto the band structure problem of a class of lattice Hamiltonians. As an application, we show how to use the stability of driven non-Hermitian two-level systems (0-dimension in space) to simulate a spectrum analogous to Hofstadter’s butterfly that has played a paradigmatic role in quantum Hall physics. The simulation of the band structure of non-Hermitian superlattice potentials with parity-time reversal symmetry is also briefly discussed.

http://arxiv.org/abs/1412.3549

Quantum Physics (quant-ph)

We construct a previously unknown \(E_2\)-quasi-exactly solvable non-Hermitian model whose eigenfunctions involve weakly orthogonal polynomials obeying three-term recurrence relations that factorize beyond the quantization level. The model becomes Hermitian when one of its two parameters is fixed to a specific value. We analyze the double scaling limit of this model leading to the complex Mathieu equation. The norms, Stieltjes measures and moment functionals are evaluated for some concrete values of one of the two parameters.

http://arxiv.org/abs/1412.2800

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

Spectral singularities are certain points of the continuous spectrum of generic complex scattering potentials. We review the recent developments leading to the discovery of their physical meaning, consequences, and generalizations. In particular, we give a simple definition of spectral singularities, provide a general introduction to spectral consequences of PT-symmetry (clarifying some of the controversies surrounding this subject), outline the main ideas and constructions used in the pseudo-Hermitian representation of quantum mechanics, and discuss how spectral singularities entered in the physics literature as obstructions to these constructions. We then review the transfer matrix formulation of scattering theory and the application of complex scattering potentials in optics. These allow us to elucidate the physical content of spectral singularities and describe their optical realizations. Finally, we survey some of the most important results obtained in the subject, drawing special attention to the remarkable fact that the condition of the existence of linear and nonlinear optical spectral singularities yield simple mathematical derivations of some of the basic results of laser physics, namely the laser threshold condition and the linear dependence of the laser output intensity on the gain coefficient.

http://arxiv.org/abs/1412.0454

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

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)

A Parity-Time (PT)-symmetric system with periodically varying-in-time gain and loss modeled by two coupled Schrodinger equations (dimer) is studied. It is shown that the problem can be reduced to a perturbed pendulum-like equation. This is done by finding two constants of motion. Firstly, a generalized problem using Melnikov type analysis and topological degree arguments is studied for showing the existence of periodic (libration), shift periodic (rotation), and chaotic solutions. Then these general results are applied to the PT-symmetric dimer. It is interestingly shown that if a sufficient condition is satisfied, then rotation modes, which do not exist in the dimer with constant gain-loss, will persist. An approximate threshold for PT-broken phase corresponding to the disappearance of bounded solutions is also presented. Numerical study is presented accompanying the analytical results.

http://arxiv.org/abs/1412.0164

Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas); Classical Analysis and ODEs (math.CA); Optics (physics.optics)

Optomechanically-induced transparency (OMIT) and the associated slow-light propagation provide the basis for storing photons in nanofabricated phononic devices. Here we study OMIT in parity-time (PT)-symmetric microresonators with a tunable gain-to-loss ratio. This system features a reversed, non-amplifying transparency: inverted-OMIT. When the gain-to-loss ratio is steered, the system exhibits a transition from the PT-symmetric phase to the broken-PT-symmetric phase. We show that by tuning the pump power at fixed gain-to-loss ratio or the gain-to-loss ratio at fixed pump power, one can switch from slow to fast light and vice versa. Moreover, the presence of PT-phase transition results in the reversal of the pump and gain dependence of transmission rates. These features provide new tools for controlling light propagation using optomechanical devices.

http://arxiv.org/abs/1411.7115

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

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)

In this paper, we study the \(\phi^4\) kink scattering from a spatially localized PT-symmetric defect and the effect of the kink’s internal mode (IM) is discussed. It is demonstrated that if a kink hits the defect from the gain side, a noticeable IM is excited, while for the kink coming from the opposite direction the mode excitation is much weaker. This asymmetry is a principal finding of the present work. Similar to the case of the sine-Gordon kink studied earlier, it is found that the \(\\phi^4\) kink approaching the defect from the gain side always passes through the defect, while in the opposite case it must have sufficiently large initial velocity, otherwise it is trapped by the loss region. It is found that for the kink with IM the critical velocity is smaller, meaning that the kink bearing IM can pass more easily through the loss region. This feature, namely the “increased transparency” of the defect as regards the motion of the kink in the presence of IM is the second key finding of the present work. A two degree of freedom collective variable model offered recently by one of the co-authors is shown to be capable of reproducing both principal findings of the present work. A simpler, analytically tractable single degree of freedom collective variable method is used to calculate analytically the kink phase shift and the kink critical velocity sufficient to pass through the defect. Comparison with the numerical results suggests that the collective variable method is able to predict these parameters with a high accuracy.

http://arxiv.org/abs/1411.5857

Pattern Formation and Solitons (nlin.PS)

We propose the notion of \(E_2\)-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the complex Mathieu Hamiltonian in a double scaling limit, which enables us to compute the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions are univariate polynomials in the energy obeying a three-term recurrence relation. The latter property guarantees the existence of a linear functional such that the polynomials become orthogonal. The polynomials are shown to factorize for all levels above the quantization condition leading to vanishing norms rendering them to be weakly orthogonal. In two concrete examples we compute the explicit expressions for the Stieltjes measure.

http://arxiv.org/abs/1411.4300

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

For non-Hermitian equilateral q-pointed star-shaped quantum graphs of paper I [Can. J. Phys. 90, 1287 (2012), arXiv 1205.5211] we show that due to certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.

http://arxiv.org/abs/1411.3828

Quantum Physics (quant-ph); Spectral Theory (math.SP)

We introduce a 2D network built of PT-symmetric dimers with on-site cubic nonlinearity, the gain and loss elements of the dimers being linked by parallel square-shaped lattices. The system may be realized as a set of PT-symmetric dual-core waveguides embedded into a photonic crystal. The system supports PT-symmetric and antisymmetric fundamental solitons (FSs) and on-site-centered solitary vortices (OnVs). Stability of these discrete solitons is the central topic of the consideration. Their stability regions in the underlying parameter space are identified through the computation of stability eigenvalues, and verified by direct simulations. Symmetric FSs represent the system’s ground state, being stable at lowest values of the power, while anti-symmetric FSs and OnVs are stable at higher powers. Symmetric OnVs, which are also stable at lower powers, are remarkably robust modes: on the contrary to other PT-symmetric states, unstable OnVs do not blow up, but spontaneously rebuild themselves into stable FSs.

http://arxiv.org/abs/1411.3943

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

We study pseudo PT symmetry for a tight binding lattice with a general form of the modulation. Using high-frequency Floquet method, we show that the critical non-Hermitian degree for the reality of the spectrum can be manipulated by varying the parameters of the modulation. We study the effect of periodical and quasi-periodical nature of the modulation on the pseudo PT symmetry.

http://arxiv.org/abs/1411.3459

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

Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential, \(V(x)\), where the spectrum is again discrete and real. Showing that the Hamiltonian, \(p^2/2m+V(x)\), is pseudo-Hermitian could be challenging, if possible.

http://arxiv.org/abs/1411.3231

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

We study a system of two coupled nonlinear Schrodinger equations, where one equation includes gain and the other one includes losses. Strengths of the gain and the loss are equal, i.e., the resulting system is parity-time (PT) symmetric. The model includes both linear and nonlinear couplings, such that when all nonlinear coefficients are equal, the system represents the PT-generalization of the Manakov model. In the one-dimensional case, we prove the existence of a global solution to the Cauchy problem in energy space \(H_1\), such that the \(H_1\)-norm of the global solution may grow in time. In the Manakov case, we show analytically that the \(L_2\)-norm of the global solution is bounded for all times and numerically that the \(H_1\)-norm is also bounded. In the two-dimensional case, we obtain a constraint on the \(L_2\)-norm of the initial data that ensures the existence of a global solution in the energy space \(H_1\).

http://arxiv.org/abs/1411.2895

Analysis of PDEs (math.AP)

We make a nodal analysis of the processes of level crossings in a model of quantum mechanics with a PT-symmetric double well. We prove the existence of infinite crossings with their selection rules. At the crossing, before the PT-symmetry breaking and the localization, we have a total P-symmetry breaking of the states.

http://arxiv.org/abs/1410.8460

Mathematical Physics (math-ph)

Reciprocity is shown so far only when the scattering potential is either real or parity symmetric complex. We extend this result for parity violating complex potential by considering several explicit examples: (i) we show reciprocity for a PT symmetric (hence parity violating) complex potential which admits penetrating state solutions analytically for all possible values of incidence energy and (ii) reciprocity is shown to hold at certain discrete energies for two other parity violating complex potentials.

http://arxiv.org/abs/1410.7886

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

Controlling and reversing the effects of loss are major challenges in optical systems. For lasers losses need to be overcome by a sufficient amount of gain to reach the lasing threshold. We show how to turn losses into gain by steering the parameters of a system to the vicinity of an exceptional point (EP), which occurs when the eigenvalues and the corresponding eigenstates of a system coalesce. In our system of coupled microresonators, EPs are manifested as the loss-induced suppression and revival of lasing. Below a critical value, adding loss annihilates an existing Raman laser. Beyond this critical threshold, lasing recovers despite the increasing loss, in stark contrast to what would be expected from conventional laser theory. Our results exemplify the counterintuitive features of EPs and present an innovative method for reversing the effect of loss.

http://arxiv.org/abs/1410.7474

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

We report a study on a closed-form optical quadrimer waveguides system. We have studied the beam dynamics of the system below, at and above the PT-threshold in both the linear and nonlinear regimes. We have also explored the effects of gain/loss parameter and the strength as well as the nature of the nonlinearity (i.e. focusing or defocusing) upon the evolution of optical intensity in each of the four sites. We observe saturation behaviors in the spatial power evolution, when nonlinearity is incorporated into the system, in the gain-guides, a feature that could be exploited for various practical applications.

http://arxiv.org/abs/1410.6258

Optics (physics.optics)

It is known that when two identical waves are injected from left and right on a complex PT-symmetric scattering potential the two-port s-matrix can have uni-modular eigenvalues. If this happens for all energies, there occurs a perfect emission of waves at both ends. We call this phenomenon transparency. Using the versatile PT-Symmetric complex Scarf II potential, we demonstrate analytically that the transparency occurs when the potential has real discrete spectrum i.e., when PT-symmetry is exact(unbroken). Next, we find that exactness of PT-symmetry is only sufficient but not necessary for the transparency. Two other PT-symmetric domains of Scarf II reveal transparency without the PT-symmetry being exact. In these two cases there exist only scattering states. In one case the real part of the potential is a well devoid of real discrete spectrum and in the other real part is a barrier. Other numerically solved models also support our findings.

http://arxiv.org/abs/1410.5530

Quantum Physics (quant-ph)

We introduce a new class of \(\cal{PT}\)-symmetric complex crystals which are almost transparent and one-way reflectionless over a broad frequency range around the Bragg frequency, i.e. unidirectionally invisible, regardless of the thickness \(L\) of the crystal. The \(\cal{PT}\)-symmetric complex crystal is synthesized by a supersymmetric transformation of an Hermitian square well potential, and exact analytical expressions of transmission and reflection coefficients are given. As \(L\) is increased, the transmittance and reflectance from one side remain close to one and zero, respectively, whereas the reflectance from the other side secularly grows like ~\(L^2\) owing to unidirectional Bragg scattering. This is a distinctive feature as compared to the previously studied case of the complex sinusoidal \(\cal{PT}\)-symmetric potential \(V(x)=V_0\exp(−2ik_ox)\) at the symmetry breaking point, where transparency breaks down as \(L\to\infty\).

http://arxiv.org/abs/1410.5278

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

We show that the elementary modes of the planar harmonic oscillator can be quantised in the framework of quantum mechanics based on pseudo-hermitian hamiltonians. These quantised modes are demonstrated to act as dynamical structures behind a new Jordan – Schwinger realization of the SU(1,1) algebra. This analysis complements the conventional Jordan – Schwinger construction of the SU(2) algebra based on hermitian hamiltonians of a doublet of oscillators.

http://arxiv.org/abs/1410.4678

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

Among quantum systems with finite Hilbert space a specific role is played by systems controlled by non-Hermitian Hamiltonian matrices \(H\neq H^\dagger\) for which one has to upgrade the Hilbert-space metric by replacing the conventional form \(\Theta^{(Dirac)}=I\) of this metric by a suitable upgrade \(\Theta^{(non−Dirac)}\neq I\) such that the same Hamiltonian becomes self-adjoint in the new, upgraded Hilbert space, \(H=H\ddagger=\Theta^{−1}H^\dagger\Theta\). The problems only emerge in the context of scattering where the requirement of the unitarity was found to imply the necessity of a non-locality in the interaction, compensated by important technical benefits in the short-range-nonlocality cases. In the present paper we show that an why these technical benefits (i.e., basically, the recurrent-construction availability of closed-form Hermitizing metrics \(\Theta^{(non−Dirac)}\) can survive also in certain specific long-range-interaction models.

http://arxiv.org/abs/1410.3583

Quantum Physics (quant-ph)

We study resonant mode conversion in the PT-symmetric multimode waveguides, where symmetry breaking manifests itself in sequential destabilization (appearance of the complex eigenvalues) of the pairs of adjacent guided modes. We show that the efficient mode conversion is possible even in the presence of the resonant longitudinal modulation of the complex refractive index. The distinguishing feature of the resonant mode conversion in the PT-symmetric structure is a drastic growth of the width of the resonance curve when the gain/losses coefficient approaches a critical value, at which symmetry breaking occurs. We found that in the system with broken symmetry the resonant coupling between exponentially growing mode with stable higher-order one effectively stabilizes dynamically coupled pair of modes and remarkably diminishes the average rate of the total power growth.

http://arxiv.org/abs/1410.2422

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

This paper develops two approaches to Lax-integrbale systems with spatiotemporally varying coefficients. A technique based on extended Lax Pairs is first considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. As illustrative examples, we consider generalizations of the NLS and DNLS equations, as well as a PT-symmetric version of the NLS equation. It is demonstrated that the techniques yield Lax- or S-integrable NLPDEs with both time- AND space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must ‘guess’ a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we therefore next attempt to systematize the derivation of Lax-integrable sytems with variable coefficients. We attempt to apply the Estabrook- Wahlquist (EW) prolongation technique, a relatively self-consistent procedure requiring little prior infomation. However, this immediately requires that the technique be significantly generalized or broadened in several different ways. The new and extended EW technique which results is illustrated by algorithmically deriving generalized Lax-integrable versions of NLS, PT-symmetric NLS, and DNLS equations.

http://arxiv.org/abs/1410.0645

Analysis of PDEs (math.AP); Exactly Solvable and Integrable Systems (nlin.SI)

We propose a regular representation for a non-Hermitian operator even if the parameter space contains exceptional points (EPs), at which the operator cannot be diagonalized and the usual spectral representation ceases to exist. Our representation has a generalized Jordan block form and is written in terms of extended pseudo-eigenstates. Our method is free from the difficulty of the singularity of the spectral representation at EPs, at which multiple eigenvalues and eigenvectors coalesce and the eigenvectors cannot be normalized. Our representation improves the accuracy of numerical calculations of physical quantities near EPs. We also find that our method is applicable to various problems related to EPs in the parameter space of non-Hermitian operators. We demonstrate the usefulness of our representation by investigating Boltzmann’s collision operator in a one-dimensional quantum Lorentz gas in the weak coupling approximation.

http://arxiv.org/abs/1409.7453

Statistical Mechanics (cond-mat.stat-mech)

In the present work, we explore the case of a general PT-symmetric dimer in the context of two both linearly and nonlinearly coupled cubic oscillators. To obtain an analytical handle on the system, we first explore the rotating wave approximation converting it into a discrete nonlinear Schrodinger type dimer. In the latter context, the stationary solutions and their stability are identified numerically but also wherever possible analytically. Solutions stemming from both symmetric and anti-symmetric special limits are identified. A number of special cases are explored regarding the ratio of coefficients of nonlinearity between oscillators over the intrinsic one of each oscillator. Finally, the considerations are extended to the original oscillator model, where periodic orbits and their stability are obtained. When the solutions are found to be unstable their dynamics is monitored by means of direct numerical simulations.

http://arxiv.org/abs/1409.7218

Pattern Formation and Solitons (nlin.PS); Chaotic Dynamics (nlin.CD); Exactly Solvable and Integrable Systems (nlin.SI)

The concept of parity-time (PT) symmetry has been used to identify a novel route to nonreciprocal dynamics in optical momentum space, imposing the directionality on the flow of light. Whereas PT-symmetric potentials have been implemented under the requirement of \(V(x) = V*(-x)\), this precondition has only been interpreted within the mathematical frame for the symmetry of Hamiltonians and has not been directly linked to nonreciprocity. Here, within the context of light-matter interactions, we develop an alternative route to nonreciprocity in momentum space by employing the concept of causality. We demonstrate that potentials with real and causal momentum spectra produce unidirectional transitions of optical states inside the k-continuum, which corresponds to an exceptional point on the degree of PT-symmetry. Our analysis reveals a critical link between non-Hermitian problems and spectral theory and enables the multi-dimensional manipulation of optical states, in contrast to one-dimensional control from the use of a Schrodinger-like equation in previous PT-symmetric optics.

http://arxiv.org/abs/1409.7031

Optics (physics.optics)

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)

We present a quantum master equation describing a Bose-Einstein condensate with particle loss on one lattice site and particle gain on the other lattice site whose mean-field limit is a non-Hermitian PT-symmetric Gross-Pitaevskii equation. It is shown that the characteristic properties of PT-symmetric systems, such as the existence of stationary states and the phase shift of pulses between two lattice sites, are also found in the many-particle system. Visualizing the dynamics on a Bloch sphere allows us to compare the complete dynamics of the master equation with that of the Gross-Pitaevskii equation. We find that even for a relatively small number of particles the dynamics are in excellent agreement and the master equation with balanced gain and loss is indeed an appropriate many-particle description of a PT-symmetric Bose-Einstein condensate.

http://arxiv.org/abs/1409.6189

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)

Coupled-resonator optical waveguides (CROWs) are known to have interesting and useful dispersion properties. Here, we study the transport in these waveguides in the general case where each resonator is open and asymmetric, i.e., is leaky and possesses no mirror-reflection symmetry. Each individual resonator then exhibits asymmetric backscattering between clockwise and counterclockwise propagating waves, which in combination with the losses induces non-orthogonal eigenmodes. In a chain of such resonators, the coupling between the resonators induces an additional source of non-hermiticity, and a complex band structure arises. We show that in this situation the group velocity of wave packets differs from the velocity associated with the probability density flux, with the difference arising from a non-hermitian correction to the Hellmann-Feynman theorem. Exploring these features numerically in a realistic scenario, we find that the complex band structure comprises almost-real branches and complex branches, which are joined by exceptional points, i.e., nonhermitian degeneracies at which not only the frequencies and decay rates coalesce but also the eigenmodes themselves. The non-hermitian corrections to the group velocity are largest in the regions around the exceptional points.

http://arxiv.org/abs/1409.5037

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

We show how non-Hermitian potentials used to describe probability gain and loss in effective theories of open quantum systems can be achieved by a coupling of the system to an environment. We do this by coupling a Bose-Einstein condensate (BEC) trapped in an attractive double-delta potential to a condensate fraction outside the double well. We investigate which requirements have to be imposed on possible environments with a linear coupling to the system. This information is used to determine an environment required for stationary states of the BEC. To investigate the stability of the system we use fully numerical simulations of the dynamics. It turns out that the approach is viable and possible setups for the realization of a PT-symmetric potential for a BEC are accessible. Vulnerabilities of the whole system to small perturbations can be adhered to the singular character of the simplified delta-shaped potential used in our model.

http://arxiv.org/abs/1409.7490

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas)

In 1980 Englert examined the classic problem of the electromagnetic self-force on an oscillating charged particle. His approach, which was based on an earlier idea of Bateman, was to introduce a charge-conjugate particle and to show that the two-particle system is Hamiltonian. Unfortunately, Englert’s model did not solve the problem of runaway modes, and the corresponding quantum theory had ghost states. It is shown here that Englert’s Hamiltonian is PT symmetric, and that the problems with his model arise because the PT symmetry is broken at both the classical and quantum level. However, by allowing the charged particles to interact and by adjusting the coupling parameters to put the model into an unbroken PT-symmetric region, one eliminates the classical runaway modes and obtains a corresponding quantum system that is ghost free.

http://arxiv.org/abs/1409.3828

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

A non-Hermitian N−level quantum model with two free real parameters is proposed in which the bound-state energies are given as roots of an elementary trigonometric expression and in which they are, in a physical domain of parameters, all real. The wave function components are expressed as closed-form superpositions of two Chebyshev polynomials. In any eligible physical Hilbert space of finite dimension \(N<\infty\) our model is constructed as unitary with respect to an underlying Hilbert-space metric \(\Theta\neq I\). The simplest version of the latter metric is finally constructed, at any dimension N=2,3,…, in closed form. This version of the model may be perceived as an exactly solvable N−site lattice analogue of the \(N=\infty\) square well with complex Robin-type boundary conditions. At any \(N<\infty\) our closed-form metric becomes trivial (i.e., equal to the most common Dirac’s metric \(\Theta(Dirac)=I\)) at the special, Hermitian-Hamiltonian-limit parameters.

http://arxiv.org/abs/1409.3788

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

We examine a one-dimensional PT-symmetric binary lattice in the presence of diagonal disorder. We focus on the wave transport phenomena of localized and extended input beams for this disordered system. In the pure PT-symmetric case, we derive an exact expression for the evolution of light localization in terms of the typical parameters of the system. In this case localization is enhanced as the gain and loss parameter in increased. In the presence of disorder, we observe that the presence of gain and loss inhibits (favors) the transport for localized (extended) excitations.

http://arxiv.org/abs/1409.3412

Optics (physics.optics)

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. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP-space). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.

http://arxiv.org/abs/1409.3497

Mathematical Physics (math-ph)

We analyse some PT-symmetric oscillators with \(T_d\) symmetry that depend on a potential parameter \(g\). We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of \(g\). Pairs of eigenvalues coalesce at exceptional points \(g_c\); their magnitude roughly decreasing with the magnitude of the eigenvalues. It is difficult to estimate whether there is a phase transition at a nonzero value of g as conjectured in earlier papers. Group theory and perturbation theory enable one to predict whether a given space-time symmetry leads to real eigenvalues for sufficiently small nonzero values of \(g\).

http://arxiv.org/abs/1409.2672

Quantum Physics (quant-ph)

Space-time and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It is shown that tensor products of biquaternion algebras are associated with the each irreducible representation of the Lorentz group. Space time discrete symmetries P, T and their combination PT are generated by the fundamental automorphisms of this algebraic background (Clifford algebras). Charge conjugation C is presented by a pseudoautomorphism of the complex Clifford algebra. This description of the operation C allows one to distinguish charged and neutral particles including particle-antiparticle interchange and truly neutral particles. Quotient representations of the Lorentz group and their possible relations with P- and CP-violations are considered. Spin and charge multiplets, based on the interlocking representations of the Lorentz group, are introduced. A central point of the work is a correspondence between Wigner definition of elementary particle as an irreducible representation of the Poincar\’{e} group and SU(3)-description (quark scheme) of the particle as a vector of the supermultiplet (irreducible representation of SU(3)). This correspondence is realized on the ground of a spin-charge Hilbert space. Basic hadron supermultiplets of SU(3)-theory (baryon octet and two meson octets) are studied in this framework. It is shown that quark phenomenologies are naturally incorporated into presented scheme. The relationship between mass and spin allows one to take a new look at the problem of mass spectrum of elementary particles.

http://arxiv.org/abs/1409.1400

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

The natural environment of a localized quantum system is the continuum of scattering wavefunctions into which the system is embedded. It can be changed by external fields, however never be deleted. The control of the system’s properties by varying a certain parameter provides us information on the system. It is, in many cases, counterintuitive and points to the same phenomena in different systems in spite of the specific differences between them. In our paper, we use a schematic model in order to simulate the main features of small open quantum systems. At low level density, the system is described well by standard Hermitian quantum physics while fundamental differences appear at high level density due to the non-Hermiticity of the Hamiltonian which cannot be neglected under this condition. The influence of exceptional points, the phase rigidity of the wavefunctions and the nonlinearities in the equations are discussed by means of different numerical and (when possible) analytical results. The transition from a closed system at low level density to an open one at high level density occurs smoothly.

http://arxiv.org/abs/1409.1149

Quantum Physics (quant-ph)

We study the possibility of asymmetric transmission induced by a non-Hermitian scattering center embedded in a one-dimensional waveguide, motivated by the aim of realizing quantum diode in a non-Hermitian system. It is shown that a PT symmetric non-Hermitian scattering center always has symmetric transmission although the dynamics within the isolated center can be unidirectional, especially at its exceptional point. We propose a concrete scheme based on a flux-controlled non-Hermitian scattering center, which comprises a non-Hermitian triangular ring threaded by an Aharonov-Bohm flux. The analytical solution shows that such a complex scattering center acts as a diode at the resonant energy level of the spectral singularity, exhibiting perfect unidirectionality of the transmission. The connections between the phenomena of the asymmetric transmission and reflectionless absorption are also discussed.

http://arxiv.org/abs/1409.0420

Quantum Physics (quant-ph)

Chirality is a universal feature in nature, as observed in fermion interactions and DNA helicity. Much attention has been given to the chiral interactions of light, not only regarding its physical interpretation but also focusing on intriguing phenomena in excitation, absorption, generation, and refraction. Although recent progress in metamaterials and 3-dimensional writing technology has spurred artificial enhancements of optical chirality, most approaches are founded on the same principle of the mixing of electric and magnetic responses. However, due to the orthogonal form of electric and magnetic fields, intricate designs are commonly required for mixing. Here, we propose an alternative route to optical chirality, exploiting the nonmagnetic mixing of amplifying and decaying electric modes based on non-Hermitian theory. We show that a 1-dimensional helical eigenmode can exist singularly in a complex anisotropic material, in sharp contrast to the 2-dimensional eigenspaces employed in previous approaches. We demonstrate that exceptional interactions between propagating chiral waves result from this low-dimensionality, for example, one-way reflectionless chiral conversions and chirality reversal, each occurring for circular and linear polarization. Our proposal and experimental verification with complex polar meta-molecules not only provide a significant step for low-dimensional chirality, but also enable the dynamics of optical spin black hole.

http://arxiv.org/abs/1409.0180

Optics (physics.optics)

We investigate the approximate bound state solutions of the Schrodinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number \(\ell\).

http://arxiv.org/abs/1409.0518

Quantum Physics (quant-ph)

We have studied a three-level \(\Lambda\)-type atomic system with all the energy levels exhibiting decay. The system is described by a pseudo-Hermitian Hamiltonian and subject to certain conditions, the Hamiltonian shows parity-time (PT) symmetry. The probability amplitudes of various atomic levels both below and above the PT-theshold is worked out.

http://arxiv.org/abs/1408.6672

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

We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidly enough toward the periphery of the medium, combined with an antisymmetric gain-loss profile. Despite the absence of any symmetric modulation of the linear refractive index, which is usually required to establish a PT-symmetry in the form of a purely real spectrum of modes, we show that the PT-symmetry is never broken in the present system, and that the system always supports stable bright solitons, fundamental and multi-pole ones. Such phenomenon is connected to non-linearizability of the underlying evolution equation. The increase of the gain-losses strength results, in lieu of the PT-symmetry breaking, in merger of pairs of different soliton branches, such as fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable for all values of the gain-loss coefficient.

http://arxiv.org/abs/1408.6174

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

We provide a reduction of a set of two coupled oscillators with balanced loss and gain in their elementary modes. A possible method of quantization based on these elementary modes, in the framework of PT symmetric quantum mechanics is indicated.

http://arxiv.org/abs/1408.5038

High Energy Physics – Theory (hep-th)

We introduce the notion of PT-symmetry in magnetic nanostructures and show that they can support a new type of non-Hermitian dynamics. Using the simplest possible set-up consisting of two coupled ferromagnetic films, one with loss and another one with a balanced amount of gain, we demonstrate the existence of a spontaneous PT-symmetry breaking point where both the eigenfrequencies and eigenvectors are degenerate. Below this point the frequency spectrum is real indicating stable dynamics while above this point it is complex signaling unstable dynamics.

http://arxiv.org/abs/1408.3285

Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Mathematical Physics (math-ph); Chaotic Dynamics (nlin.CD)

The concept of the PT-symmetry, originating from the quantum field theory, has been intensively investigated in optics, stimulated by the similarity between the Schr\”odinger equation and the paraxial wave equation that governs the propagation of light in a guiding structure. We go beyond the bounds of the paraxial approximation and demonstrate, using the solution of the Maxwell’s equations for light beams propagating in deeply subwavelength waveguides and periodic lattices with “balanced” gain and loss, that the PT symmetry may stay unbroken in this setting. Moreover, the PT-symmetry in subwavelength optical structures may be restored after being initially broken upon the increase of gain and loss. Critical gain/loss levels, at which the breakup and subsequent restoration of the PT symmetry occur, strongly depend on the scale of the structure.

http://arxiv.org/abs/1408.2630

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

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)

We introduce a recursive bosonic quantization technique for generating classical PT photonic structures that possess hidden symmetries and higher order exceptional points. We study light transport in these geometries and we demonstrate that perfect state transfer is possible only for certain initial conditions. Moreover, we show that for the same propagation direction, left and right coherent transports are not symmetric with field amplitudes following two different trajectories. A general scheme for identifying the conservation laws in such PT-symmetric photonic networks is also presented.

http://arxiv.org/abs/1408.1561

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

A class of non-local non-linear Schrodinger equations(NLSE) is considered in an external potential with space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable non-local NLSE with self induced potential that is PT symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or non autonomous non-local NLSE by using similarity transformation and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the non-local NLSE without the external potential and a \(d+1\) dimensional generalization of it, admits all the symmetries of the \(d+1\) dimensional Schrodinger group. The conserved Noether charges associated with the time-translation, dilatation and special conformal transformation are shown to be real-valued in spite of being non-hermitian. Finally, dynamics of different moments are studied with an exact description of the time-evolution of the “pseudo-width” of the wave-packet for the special case when the system admits a \(O(2,1)\) conformal symmetry.

http://arxiv.org/abs/1408.0954

Exactly Solvable and Integrable Systems (nlin.SI); High Energy Physics – Theory (hep-th)

Symmetry breaking of solitons in a class of one-dimensional parity-time (PT) symmetric complex potentials with cubic nonlinearity is reported. In generic PT symmetric potentials, such symmetry breaking is forbidden. However, in a special class of PT-symmetric potentials \(V(x)=g^2(x)+αg(x)+ig′(x)\), where \(g(x)\) is a real and even function and α a real constant, symmetry breaking of solitons can occur. That is, a branch of non-PT-symmetric solitons can bifurcate out from the base branch of PT-symmetric solitons when the base branch’s power reaches a certain threshold. At the bifurcation point, the base branch changes stability, and the bifurcated branch can be stable.

http://arxiv.org/abs/1408.0687

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

It is generally difficult to study the dynamical properties of a quantum system with both inherent quantum noises and non-perturbative nonlinearity. Due to the possibly drastic intensity increase of an input coherent light in the gain-loss waveguide couplers with parity-time (PT) symmetry, the Kerr effect from a nonlinearity added into the systems can be greatly enhanced, and is expected to create the macroscopic entangled states of the output light fields with huge photon numbers. Meanwhile, the quantum noises also coexist with the amplification and dissipation of the light fields. Under the interplay between the quantum noises and nonlinearity, the quantum dynamical behaviors of the systems become rather complicated. However, the important quantum noise effects have been mostly neglected in the previous studies about nonlinear PT symmetric systems. Here we present a solution to this non-perturbative quantum nonlinear problem, showing the real-time evolution of the system observables. The enhanced Kerr nonlinearity is found to give rise to a previously unknown decoherence effect that is irrelevant to the quantum noises, and imposes a limit on the emergence of macroscopic nonclassicality. In contrast to what happen in the linear systems, the quantum noises exert significant impact on the system dynamics, and can create the nonclassical light field states in conjunction with the enhanced Kerr nonlinearity. This first study on the noise involved quantum nonlinear dynamics of the coupled gain-loss waveguides can help to better understand the quantum noise effects in the broad nonlinear systems.

http://arxiv.org/abs/1408.0565

Quantum Physics (quant-ph)

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)

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)

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)

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)

]]>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)

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)

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)

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)

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)

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)

In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd degree polynomial potentials of the form \(V(x)=(ix)^{2N+1}+\beta _{1}x^{2N}+\beta _{2}x^{2N-1}+\cdot \cdot \cdot \cdot \cdot +\beta _{2N}x\) where \(\beta _{k}^{\prime }\)s are real or complex for \(1\leq k\leq 2N\). The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters \(\beta _{1},\beta _{2}….\) and \(\beta _{2N}\) of the potential. Unlike in the even degree polynomial case, the highest order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex turning points which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.

http://arxiv.org/abs/1407.0191

Mathematical Physics (math-ph)

]]>Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as “Hofstadter’s butterfly”. We study a non-Hermitian continuation of Hofstadter’s model; as the non-Hermiticity parameter \(g\) increases past a sequence of critical values the eigenvalues successively go complex in a sequence of “double-pitchfork bifurcations” wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed.

http://arxiv.org/abs/1407.0093

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

]]>Searching for non-Hermitian (parity-time)\(\mathcal{PT}\)-symmetric Hamiltonians with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a \(\mathcal{PT}\)-symmetric non-Hermitian Hamiltonian model which is given as \(\hat{\mathcal{H}}=\omega (\hat{b}^\dagger\hat{b}+\frac{1}{2})+\alpha (\hat{b}^{2}-(\hat{b}^\dagger)^{2})\) where \(\omega\) and \(\alpha\) are real constants, \(\hat{b}\) and \(\hat{b^\dagger}\) are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of \(\mathcal{PT}\) symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian \(\mathcal{H}\) is pseudo-Hermitian, we have obtained the Hermitian equivalent of \(\mathcal{H}\), which is in Sturm- Liouville form, leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. \(\mathcal{H}\) is called pseudo-Hermitian, if there exists a Hermitian and invertible operator \(\eta\) satisfying \(\mathcal{H^\dagger}=\eta \mathcal{H} \eta^{-1}\). For the Hermitian Hamiltonian \(h\), one can write \(h=\rho \mathcal{H} \rho^{-1}\) where \(\rho=\sqrt{\eta}\) is unitary. Using this \(\rho\) we have obtained a physical Hamiltonian \(h\) for each case. Then, the Schr\”{o}dinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics. Mapping function \(\rho\) is obtained for each potential case.

http://arxiv.org/abs/1406.3298

Quantum Physics (quant-ph)

]]>We have obtained the metric operator \(\Theta=\exp T\) for the non-Hermitian Hamiltonian model \(H=\omega(a^{\dag}a+1/2)+\alpha(a^{2}-a^{\dag^{2}})\). We have also found the intertwining operator which connects the Hamiltonian to the adjoint of its pseudo-supersymmetric partner Hamiltonian for the model of hyperbolic Rosen-Morse II potential.

http://arxiv.org/abs/1406.3179

Mathematical Physics (math-ph)

In this work, we explore a prototypical example of a genuine continuum breather (i.e., not a standing wave) and the conditions under which it can persist in a \(\mathcal{P T}\)-symmetric medium. As our model of interest, we will explore the sine-Gordon equation in the presence of a \(\mathcal{P T}\)- symmetric perturbation. Our main finding is that the breather of the sine-Gordon model will only persist at the interface between gain and loss that \(\mathcal{P T}\)-symmetry imposes but will not be preserved if centered at the lossy or at the gain side. The latter dynamics is found to be interesting in its own right giving rise to kink-antikink pairs on the gain side and complete decay of the breather on the lossy side. Lastly, the stability of the breathers centered at the interface is studied. As may be anticipated on the basis of their “delicate” existence properties such breathers are found to be destabilized through a Hopf bifurcation in the corresponding Floquet analysis.

http://arxiv.org/abs/1406.3082

Pattern Formation and Solitons (nlin.PS)

The modified Dirac-Pauli equations, which are introduced by means of \({\gamma_5}\)-mass factorization of the ordinary Klein-Gordon operator, are considered. We also take into account the interaction of fermions with the intensive homogenous magnetic field focusing attention to their (g-2) gyromagnetic factor. The basis of this approach is developing of methods for study of the structure of regions of unbroken \(\cal PT\) symmetry of Non-Hermitian Hamiltonians which be no studied earlier. For that, without the use of perturbation theory in the external field the exact energy spectra are deduced with regard to spin effects of fermions. We also investigate the unique possible of experimental observability the non-Hermitian restrictions in the spectrum of mass consistent with the conjecture Markov about Maximal Mass. This, in principal will may allow to find out the existence of an upper limit value in spectrum masses of elementary particles and confirm or deny the significance of the Planck mass.

http://arxiv.org/abs/1406.0383

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

Two theoretical methods of finding resonant states in open quantum systems, namely the approach of the Siegert boundary condition and the Feshbach formalism, are reviewed and shown to be algebraically equivalent to each other for a simple model of the T-type quantum dot. It is stressed that the seemingly Hermitian Hamiltonian of an open quantum system is implicitly non-Hermitian outside the Hilbert space. The two theoretical approaches extract an explicitly non-Hermitian effective Hamiltonian in a contracted space out of the seemingly Hermitian (but implicitly non-Hermitian) full Hamiltonian in the infinite-dimensional state space of an open quantum system.

http://arxiv.org/abs/1405.7021

Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Mathematical Physics (math-ph); Nuclear Theory (nucl-th)

We study theoretically a PT-symmetric saturable balanced gain-loss system in a ring cavity configuration. The saturable gain and loss are modeled by two-level medium with or without population inversion. We show that the specifics of the spectral singularity can be fully controlled by the cavity and the atomic detuning parameters. The theory is based on the mean-field approximation as in standard theory of optical bistability. Further, in the linear regime we demonstrate the regularization of the singularity in detuned systems, while larger input power levels are shown to be adequate to limit the infinite growth in absence of detuning

http://arxiv.org/abs/1405.6812

Optics (physics.optics)

For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrodinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only \(\rm{Dn}(x,m)\) and \(\rm{Cn}(x,m)\) but even their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lame polynomials of order 1, but it also admits solutions in terms of Lame polynomials of order 2, even though they are not the solutions of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.

http://arxiv.org/abs/1405.5267

Pattern Formation and Solitons (nlin.PS)

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)

We study the parity- and time-reversal PT symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials \(\pm i\gamma\) at two end sites. The SSH model is known as one of the simplest two-band topological models which has topologically trivial and nontrivial phases. We find that the non-Hermitian terms can lead to different effects on the properties of the eigenvalues spectrum in topologically trivial and nontrivial phases. In the topologically trivial phase, the system undergos an abrupt transition from unbroken PT-symmetry region to spontaneously broken \(\mathcal{PT}\)-symmetry region at a certain \(\gamma_{c}\), and a second transition occurs at another transition point \(\gamma_{c^{‘}}\) when further increasing the strength of the imaginary potential \(\gamma\). But in the topologically nontrivial phase, the zero-mode edge states become unstable for arbitrary nonzero \(\gamma\) and the \(\mathcal{PT}\)-symmetry of the system is spontaneously broken, which is characterized by the emergence of a pair of conjugated imaginary modes.

http://arxiv.org/abs/1405.5591

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

We investigate the Hermitian and the non-Hermitian dynamics of the mode entanglement in two identical optical cavities coupled by a chiral mirror. By employing the non-Hermitian quantum evolution, we calculate the logarithmic negativity measure of entanglement for initially Fock, coherent and squeezed states, separately. We verify the non-conservation of mean spin for the initially coherent and squeezed states when the coupling is non-reciprocal and report the associated spin noise for each case. We examine the effects of non-conserved symmetries on the mode correlations and determine the degree of non-reciprocal coupling to establish robust quantum entanglement.

http://arxiv.org/abs/1405.5079

Quantum Physics (quant-ph)

Parity-time (PT) symmetry is a fundamental notion in quantum field theories1,2. It has opened a new paradigm for non-Hermitian Hamiltonians ranging from quantum mechanics, electronics, to optics. In the realm of optics, optical loss is responsible for power dissipation, therefore typically degrading device performance such as attenuation of a laser beam. By carefully exploiting optical loss in the complex dielectric permittivity, however, recent exploration of PT symmetry revolutionizes our understandings in fundamental physics and intriguing optical phenomena such as exceptional points and phase transition that are critical for high-speed optical modulators3-9. The interplay between optical gain and loss in photonic PT synthetic matters offers a new criterion of positively utilizing loss to efficiently manipulate gain and its associated optical properties10-19. Instead of simply compensating optical loss in conventional lasers, for example, it is theoretically proposed that judiciously designed delicate modulation of optical loss and gain can lead to PT synthetic lasing20,21 that fundamentally broadens laser physics. Here, we report the first experimental demonstration of PT synthetic lasers. By carefully exploiting the interplay between gain and loss, we achieve degenerate eigen modes at the same frequency but with complex conjugate gain and loss coefficients. In contrast to conventional ring cavity lasers with multiple modes, the PT synthetic micro-ring laser exhibits an intrinsic single mode lasing: the non-threshold PT broken phase inherently associated in such a photonic system squeezes broadband optical gain into a single lasing mode regardless of the gain spectral bandwidth. This chip-scale semiconductor platform provides a unique route towards fundamental explorations of PT physics and next generation of optoelectronic devices for optical communications and computing.

http://arxiv.org/abs/1405.2863

Optics (physics.optics)

We discuss space-time symmetric Hamiltonian operators of the form \(H=H_{0}+igH^{\prime}\), where \(H_{0}\) is Hermitian and \(g\) real. \(H_0\) is invariant under the unitary operations of a point group \(G\) while \(H^\prime\) is invariant under transformation by elements of a subgroup \(G^\prime\) of \(G\). If \(G\) exhibits irreducible representations of dimension greater than unity, then it is possible that \(H\) has complex eigenvalues for sufficiently small nonzero values of \(g\). In the particular case that \(H\) is parity-time symmetric then it appears to exhibit real eigenvalues for all \(0<g<g_c\), where \(g_{c}\) is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether \(H\) may exhibit real or complex eigenvalues for \(g>0\). We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.

http://arxiv.org/abs/1405.5234

Quantum Physics (quant-ph)

We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Mobius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.

http://arxiv.org/abs/1405.4791

Quantum Physics (quant-ph)

]]>We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, \(R^{l/r}(k)\) and \(T(k)\), of general \({\cal PT}\)-symmetric scattering potentials. We use these identities to give a general proof of the relations, \(|T(-k)|=|T(k)|\) and \(|R^r(-k)|=|R^l(k)|\), conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: \(R^{l/r}(k)R^{l/r}(-k)+|T(k)|^2=1\), and show that it is a common property of both real and complex \({\cal PT}\)-symmetric potentials. The same holds for \(T(-k)=T(k)^*\) and \(|R^r(-k)|=|R^l(k)|\).

http://arxiv.org/abs/1405.4212

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

Solitons in one-dimensional parity-time (PT)-symmetric periodic potentials are studied using exponential asymptotics. The new feature of this exponential asymptotics is that, unlike conservative periodic potentials, the inner and outer integral equations arising in this analysis are both coupled systems due to complex-valued solitons. Solving these coupled systems, we show that two soliton families bifurcate out from each Bloch-band edge for either self-focusing or self-defocusing nonlinearity. An asymptotic expression for the eigenvalues associated with the linear stability of these soliton families is also derived. This formula shows that one of these two soliton families near band edges is always unstable, while the other can be stable. In addition, infinite families of PT-symmetric multi-soliton bound states are constructed by matching the exponentially small tails from two neighboring solitons. These analytical predictions are compared with numerics. Overall agreements are observed, and minor differences explained.

http://arxiv.org/abs/1405.2827

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

We carry out a modulation instability (MI) analysis in nonlinear complex parity-time (PT) symmetric periodic structures. All the three regimes defined by the PT-symmetry breaking point or threshold, namely, below threshold, at threshold and above threshold are discussed. It is found that MI exists even beyond the PT-symmetry threshold indicating the possible existence of solitons or solitary waves, in conformity with some recent reports. We find that MI does not exist at the PT-symmetry breaking point in the case of normal dispersion below a certain nonlinear threshold. However, in the case of anomalous dispersion regime, MI does exist even at the PT-symmetry breaking point.

http://arxiv.org/abs/1405.2706

Optics (physics.optics)

Quantum mechanical spreading of a particle hopping on tight binding lattices can be suppressed by the application of an external ac force, leading to periodic wave packet reconstruction. Such a phenomenon, referred to as dynamic localization (DL), occurs for certain magic values of the ratio \(\Gamma=F_0/\omega\) between the amplitude F0 and frequency ω of the ac force. It is generally believed that in the low-frequency limit \((\omega\to0)\) DL can be achieved for an infinitesimally small value of the force F0, i.e. at finite values of \(\Gamma\). Such a normal behavior is found in homogeneous lattices as well as in inhomogeneous lattices of Glauber-Fock type. Here we introduce a tight-binding lattice model with inhomogeneous hopping rates, referred to as pseudo Glauber-Fock lattice, which shows DL but fails to reproduce the normal low-frequency behavior of homogeneous and Glauber-Fock lattices. In pseudo Glauber-Fock lattices, DL can be exactly realized, however at the DL condition the force amplitude \(F_0\) remains finite as \(\omega\to0\). Such an anomalous behavior is explained in terms of a PT symmetry breaking transition of an associated two-level non-Hermitian Hamiltonian that effectively describes the dynamics of the Hermitian lattice model.

http://arxiv.org/abs/1405.2549

Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el)

]]>We demonstrate experimentally that stable single longitudinal mode operation can be readily achieved in PT-symmetric arrangements of coupled microring resonators. Whereas any active resonator is in principle capable of displaying single-wavelength operation, selective breaking of PT-symmetry can be utilized to systematically enhance the maximum achievable gain of this mode, even if a large number of competing longitudinal or transverse resonator modes fall within the amplification bandwidth of the inhomogeneously broadened active medium. This concept is robust with respect to fabrication tolerances, and its mode selectivity is established without the need for additional components or specifically designed filters. Our results may pave the way for a new generation of versatile cavities lasing at a desired longitudinal resonance. Along these lines, traditionally highly multi-moded microring resonator configurations can be fashioned to suppress all but one longitudinal mode.

http://arxiv.org/abs/1405.2103

Optics (physics.optics)

]]>We study interaction of a soliton in a parity-time (PT) symmetric coupler which has local perturbation of the coupling constant. Such a defect does not change the PT-symmetry of the system, but locally can achieve the exceptional point. We found that the symmetric solitons after interaction with the defect either transform into breathers or blow up. The dynamics of anti-symmetric solitons is more complex, showing domains of successive broadening of the beam and of the beam splitting in two outwards propagating solitons, in addition to the single breather generation and blow up. All the effects are preserved when the coupling strength in the center of the defect deviates from the exceptional point. If the coupling is strong enough the only observable outcome of the soliton-defect interaction is the generation of the breather.

http://arxiv.org/abs/1405.1829

Optics (physics.optics)

We address the existence and stability of spatial localized modes supported by a parity-time-symmetric complex potential in the presence of power-law nonlinearity. The analytical expressions of the localized modes, which are Gaussian in nature, are obtained in both (1+1) and (2+1) dimensions. A linear stability analysis corroborated by the direct numerical simulations reveals that these analytical localized modes can propagate stably for a wide range of the potential parameters and for various order nonlinearities. Some dynamical characteristics of these solutions, such as the power and the transverse power-flow density, are also examined.

http://arxiv.org/abs/1404.7322

Quantum Physics (quant-ph); Pattern Formation and Solitons (nlin.PS); Exactly Solvable and Integrable Systems (nlin.SI)

We present a technique based on extended Lax Pairs to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies. As illustrative examples, we consider generalizations of KdV equations, three variants of generalized MKdV equations, and a recently-considered nonlocal PT-symmetric NLS equation. It is demonstrated that the technique yields Lax- or S-integrable NLPDEs with both time- AND space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. Employing the Painleve singular manifold method, some solutions are also presented for the generalized variable-coefficient integrable KdV and MKdV equations derived here. Current and future work is centered on generalizing other integrable hierarchies of NLPDEs similarly, and deriving various integrability properties such as solutions, Backlund Transformations, and hierarchies of conservation laws for these new integrable systems with variable coefficients.

http://arxiv.org/abs/1404.4602

Mathematical Physics (math-ph)

]]>The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open) and for the ring lattice geometries the spectrum is complex, lattice truncation makes the spectrum real. However, an exceptional point of order equal to the number of lattice sites emerges. When a homogeneous dc force is applied to the lattice, in all cases an equally-spaced real Wannier-Stark ladder spectrum is obtained, corresponding to periodic oscillatory dynamics in real space. Possible physical realizations of non-Hermitian lattices with unidirectional hopping are briefly discussed.

http://arxiv.org/abs/1404.3662

Quantum Physics (quant-ph)

When two resonant modes in a system with gain or loss coalesce in both their resonance position and their width, a so-called “Exceptional Point” occurs which acts as a source of non-trivial physics in a diverse range of systems. Lasers provide a natural setting to study such “non-Hermitian degeneracies”, since they feature resonant modes and a gain material as their basic constituents. Here we show that Exceptional Points can be conveniently induced in a photonic molecule laser by a suitable variation of the applied pump. Using a pair of coupled micro-disk quantum cascade lasers, we demonstrate that in the vicinity of these Exceptional Points the laser shows a characteristic reversal of its pump-dependence, including a strongly decreasing intensity of the emitted laser light for increasing pump power. This result establishes photonic molecule lasers as promising tools for exploring many further fascinating aspects of Exceptional Points, like a strong line-width enhancement and the coherent perfect absorption of light in their vicinity as well as non-trivial mode-switching and the accumulation of a geometric phase when encircling an Exceptional Point parametrically.

http://arxiv.org/abs/1404.1837

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

We study the coherent scattering from complex potentials to find that the coherent perfect absorption (CPA) without lasing is not possible in the PT-symmetric domain as the s-matrix is such that \(|\det S(k)|=1\). We confirm that in the domain of broken PT-symmetry\(|\det S(k)|\) can become indeterminate 0/0 at the spectral singularity (SS), k=k∗, of the potential signifying CPA with lasing at threshold gain. We also find that in the domain of unbroken symmetry (when the potential has real discrete spectrum) neither SS nor CPA can occur. In this, regard, we find that exactly solvable Scarf II potential is the unique model that can exhibit these novel phenomena and their subtleties analytically and explicitly. However, we show that the other numerically solved models also behave similarly.

http://arxiv.org/abs/1404.1679

Quantum Physics (quant-ph)

]]>We study optical spectral singularities of a weakly nonlinear PT-symmetric bilinear planar slab of optically active material. In particular, we derive the lasing threshold condition and calculate the laser output intensity. These reveal the following unexpected features of the system: 1. For the case that the real part of the refractive index η of the layers are equal to unity, the presence of the lossy layer decreases the threshold gain; 2. For the more commonly encountered situations when η−1 is much larger than the magnitude of the imaginary part of the refractive index, the threshold gain coefficient is a function of η that has a local minimum. The latter is in sharp contrast to the threshold gain coefficient of a homogeneous slab of gain material which is a decreasing function of η. We use these results to comment on the effect of nonlinearity on the prospects of using this system as a CPA-laser.

http://arxiv.org/abs/1404.1737

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

We investigate the rich physics of photonic molecule lasers using a non-Hermitian dimer model. We show that several interesting features, predicted recently using a rigorous steady state ab-initio laser theory (SALT), can be captured by this toy model. In particular, we demonstrate the central role played by exceptional points in both pump-selective lasing and laser self-terminations phenomena. Due to its transparent mathematical structure, our model provides a lucid understanding for how different physical parameters (optical loss, modal coupling between microcavities and pump profiles) affect the lasing action. Interestingly, our analysis also confirms that, for frequency mismatched cavities, operation in the proximity of exceptional points (without actually crossing the square root singularities) can still lead to laser self-termination. We confirm this latter prediction for two coupled slab cavities using scattering matrix analysis and SALT technique. In addition, we employ our model to investigate the pump-controlled lasing action and we show that emission patterns are governed by the locations of exceptional points in the gain parameter space. Finally we extend these results to multi-cavity photonic molecules, where we found the existence of higher-order EPs and pump-induced localization.

http://arxiv.org/abs/1404.1242

Optics (physics.optics)

]]>Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, C-selfadjoint extensions of C-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of C-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables.

http://arxiv.org/abs/1404.1304

Functional Analysis (math.FA); Other Condensed Matter (cond-mat.other); Mathematical Physics (math-ph); Operator Algebras (math.OA); Spectral Theory (math.SP)

It is shown that the toy-model-based considerations of loc. cit. (see also arXiv:1312.3395) are based on an incorrect, manifestly unphysical choice of the Hilbert space of admissible quantum states. A two-parametric family of all of the eligible correct and potentially physical Hilbert spaces of the model is then constructed. The implications of this construction are discussed. In particular, it is emphasized that contrary to the conclusions of loc. cit. there is no reason to believe that the current form of the PT-symmetric quantum theory should be false as a fundamental theory.

http://arxiv.org/abs/1404.1555

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

Starting with the modified Dirac equations for free massive particles with the γ5-extension of the physical mass \(m\to m_1+\gamma_5m_2\), we consider equations of relativistic quantum mechanics in the presence of an external electromagnetic field. The new approach is developing on the basis of existing methods for study the unbroken PT symmetry of Non-Hermitian Hamiltonians. The paper shows that this modified model contains the definition of the mass parameter, which may use as the determination of the magnitude scaling of energy M. Obviously that the transition to the standard approach is valid when small in comparison with M energies and momenta. Formally, this limit is performed when \(M\to\infty\), which simultaneously should correspond to the transition to a *Hermitian limit* \(m2\to0\). Inequality \(m\leq M\) may be considered and as the restriction of the mass spectrum of fermions considered in the model. Within of this approach, the effects of possible *observability* mass parameters: \(m_1, m_2, M\) are investigated taking into account the interaction of the magnetic field with charged fermions together with the accounting of their anomalous magnetic moments.

http://arxiv.org/abs/1404.0503

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

We investigate dipolar Bose-Einstein condensates in a complex external double-well potential that features a combined parity and time-reversal symmetry. On the basis of the Gross-Pitaevskii equation we study the effects of the long-ranged anisotropic dipole-dipole interaction on ground and excited states by the use of a time-dependent variational approach. We show that the property of a similar non-dipolar condensate to possess real energy eigenvalues in certain parameter ranges is preserved despite the inclusion of this nonlinear interaction. Furthermore, we present states that break the PT symmetry and investigate the stability of the distinct stationary solutions. In our dynamical simulations we reveal a complex stabilization mechanism for PT-symmetric, as well as for PT-broken states which are, in principle, unstable with respect to small perturbations.

http://arxiv.org/abs/1403.6742

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)

We overview recent theoretical studies of the dynamics of one- and two-dimensional spatial dissipative solitons in models based on the complex Ginzburg-Landau equations with the cubic-quintic combination of loss and gain terms, which include imaginary, real, or complex spatially periodic potentials. The imaginary potential represents periodic modulation of the local loss and gain. It is shown that the effective gradient force, induced by the inhomogeneous loss distribution, gives rise to three generic propagation scenarios for one-dimensional (1D) dissipative solitons: transverse drift, persistent swing motion, and damped oscillations. When the lattice-average loss/gain value is zero, and the real potential has spatial parity opposite to that of the imaginary component, the respective complex potential is a realization of the parity-time symmetry. Under the action of lattice potentials of the latter type, 1D solitons feature unique motion regimes in the form of transverse drift and persistent swing. In the 2D geometry, three types of axisymmetric radial lattices are considered, viz., ones based solely on the refractive-index modulation, or solely on the linear-loss modulation, or on a combination of both. The rotary motion of solitons in such axisymmetric potentials can be effectively controlled by varying the strength of the initial tangential kick.

http://arxiv.org/abs/1403.5436

Optics (physics.optics)

We study the connection between the cumulants of a time-integrated observable of a quantum system and the PT-symmetry properties of the non-Hermitian deformation of the Hamiltonian from which the generating function of these cumulants is obtained. This non-Hermitian Hamiltonian can display regimes of broken and of unbroken PT-symmetry, depending on the parameters of the problem and on the counting field that sets the strength of the non-Hermitian perturbation. This in turn determines the analytic structure of the long-time cumulant generating function (CGF) for the time-integrated observable. We consider in particular the case of the time-integrated (longitudinal) magnetisation in the one-dimensional Ising model in a transverse field. We show that its long-time CGF is singular on a curve in the magnetic field/counting field plane that delimits a regime where PT-symmetry is spontaneously broken (which includes the static ferromagnetic phase), from one where it is preserved (which includes the static paramagnetic phase). In the paramagnetic phase, conservation of PT -symmetry implies that all cumulants are sub-linear in time, a behaviour usually associated to the absence of decorrelation.

http://arxiv.org/abs/1403.4538

Statistical Mechanics (cond-mat.stat-mech); Quantum Physics (quant-ph)

We demonstrate that in-bulk vortex localized modes, and their surface half-vortex (“horseshoe”) counterparts (which were not reported before in truncated settings) self-trap in two-dimensional (2D) nonlinear optical systems with PT-symmetric photonic lattices (PLs). The respective stability regions are identified in the underlying parameter space. The in-bulk states are related to truncated nonlinear Bloch waves in gaps of the PL-induced spectrum. The basic vortex and horseshoe modes are built, severally, of four and three beams with appropriate phase shifts between them. Their stable complex counterparts, built of up to 12 beams, are reported too.

http://arxiv.org/abs/1403.4745

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

In a thin multidimensional layer we consider a second order differential PT-symmetric operator. The operator is of rather general form and its coefficients are arbitrary functions depending both on slow and fast variables. The PT-symmetry of the operator is ensured by the boundary conditions of Robin type with pure imaginary coefficient. In the work we determine the limiting operator, prove the uniform resolvent convergence of the perturbed operator to the limiting one, and derive the estimates for the rates of convergence. We establish the convergence of the spectrum of perturbed operator to that of the limiting one. For the perturbed eigenvalues converging to the limiting discrete ones we prove that they are real and construct their complete asymptotic expansions. We also obtain the complete asymptotic expansions for the associated eigenfunctions.

http://arxiv.org/abs/1403.4524

Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

]]>We investigate the parity- and time-reversal (PT)-symmetry breaking in lattice models in the presence of long-ranged, non-hermitian, PT-symmetric potentials that remain finite or become divergent in the continuum limit. By scaling analysis of the fragile PT threshold for an open finite lattice, we show that continuum loss-gain potentials \(V_a(x)\sim i|x|^a{\rm sign}(x)\) have a positive PT-breaking threshold for \(\alpha>−2\), and a zero threshold for α≤−2. When α<0 localized states with complex (conjugate) energies in the continuum energy-band occur at higher loss-gain strengths. We investigate the signatures of PT-symmetry breaking in coupled waveguides, and show that the emergence of localized states dramatically shortens the relevant time-scale in the PT-symmetry broken region.

http://arxiv.org/abs/1403.4204

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

The parity-time-symmetric structure was experimentally accessible very recently in coupled optical resonators with which, for normal or non-PT-symmetric cases, a phonon laser device had also been realized. Here we study cavity optomechanics of this system now with tunable gain-loss ratio. We find that nonlinear behaviors emerge for cavity-photon populations around balanced point, resulting giant enhancement of both optical pressure and phonon-lasing action. Potential applications range from enhancing mechanical cooling to designing highly-efficient phonon-laser amplifier.

http://arxiv.org/abs/1403.0657

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

Three classes of finite-dimensional models of quantum systems exhibiting spectral degeneracies called quantum catastrophes are described in detail. Computer-assisted symbolic manipulation techniques are shown unexpectedly efficient for the purpose.

http://arxiv.org/abs/1403.0723

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

We consider a non-Hermitian medium with a gain and loss symmetric, exponentially damped potential distribution to demonstrate different scattering features analytically. The condition for critical coupling (CC) for unidirectional wave and coherent perfect absorption (CPA) for bidirectional waves are obtained analytically for this system. The energy points at which total absorption occurs are shown to be the spectral singular points for the time reversed system. The possible energies at which CC occurs for left and right incidence are different. We further obtain periodic intervals with increasing periodicity of energy for CC and CPA to occur in this system.

http://arxiv.org/abs/1403.0539

Quantum Physics (quant-ph)

]]>We propose to observe mechanical PT symmetry in the coupled optomechanical systems. In order to provide gain to one mechanical resonator and equivalent amount of damp to another, we drive the two optical cavities with a blue and a red detuned laser fields respectively. After adiabatically eliminating the freedom of the cavity modes, we develop a formalism for describing mechanical PT-symmetric system. Moreover, we discuss the experimental feasibility of our scheme and show that the observation of mechanical PT-symmetric transition in the coupled optomechanical systems is within the reach of resent experiments.

http://arxiv.org/abs/1402.7222

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

]]>Systems governed by the Non-linear Schroedinger Equation (NLSE) with various external PT-symmetric potentials are considered. Exact solutions have been obtained for the same through the method of ansatz, some of them being solitonic in nature. It is found that only the unbroken PT-symmetric phase is realized in these systems, characterized by real energies.

http://arxiv.org/abs/1402.5762

Quantum Physics (quant-ph)

We propose a new type of laser resonator based on imaginary “energy-level splitting” (imaginary coupling, or quality factor Q splitting) in a pair of coupled microcavities. A particularly advantageous arrangement involves two microring cavities with different free-spectral ranges (FSRs) in a configuration wherein they are coupled by “far-field” interference in a shared radiation channel. A novel Vernier-like effect for laser resonators is designed where only one longitudinal resonant mode has a lower loss than the small signal gain and can achieve lasing while all other modes are suppressed. This configuration enables ultra-widely tunable single-frequency lasers based on either homogeneously or inhomogeneously broadened gain media. The concept is an alternative to the common external cavity configurations for achieving tunable single-mode operation in a laser. The proposed laser concept builds on a high-Q “dark state” that is established by radiative interference coupling and bears a direct analogy to parity-time (PT) symmetric Hamiltonians in optical systems. Variants of this concept should be extendable to parametric-gain based oscillators, enabling use of ultrabroadband parametric gain for widely tunable single-frequency light sources.

http://arxiv.org/abs/1402.4767

Optics (physics.optics)

The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the classical and quantum-mechanical interpretations of the optical phenomenon are exactly the same. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.

http://arxiv.org/abs/1402.4473

Quantum Physics (quant-ph)

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)

The spectral, dynamical and topological properties of physical systems described by non-Hermitian (including PT-symmetric) Hamiltonians are deeply modified by the appearance of exceptional points and spectral singularities. Here we show that exceptional points in the continuum can arise in non-Hermitian (yet admitting and entirely real-valued energy spectrum) optical lattices with engineered defects. At an exceptional point, the lattice sustains a bound state with an energy embedded in the spectrum of scattered states, similar to the von-Neumann Wigner bound states in the continuum of Hermitian lattices. However, the dynamical and scattering properties of the bound state at an exceptional point are deeply different from those of ordinary von-Neumann Wigner bound states in an Hermitian system. In particular, the bound state in the continuum at an exceptional point is an unstable state that can secularly grow by an infinitesimal perturbation. Such properties are discussed in details for transport of discretized light in a PT-symmetric array of coupled optical waveguides, which could provide an experimentally accessible system to observe exceptional points in the continuum.

http://arxiv.org/abs/1402.3764

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

]]>Bound states in the continuum (BIC), i.e. normalizable modes with an energy embedded in the continuous spectrum of scattered states, are shown to exist in certain optical waveguide lattices with PT-symmetric defects. Two distinct types of BIC modes are found: BIC states that exist in the broken PT phase, corresponding to exponentially-localized modes with either exponentially damped or amplified optical power; and BIC modes with sub-exponential spatial localization that can exist in the unbroken PT phase as well. The two types of BIC modes at the PT symmetry breaking point behave rather differently: while in the former case spatial localization is lost and the defect coherently radiates outgoing waves with an optical power that linearly increases with the propagation distance, in the latter case localization is maintained and the optical power increase is quadratic.

http://arxiv.org/abs/1402.3761

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

]]>The spectral and localization properties of PT-symmetric optical superlattices, either infinitely extended or truncated at one side, are theoretically investigated, and the criteria that ensure the unbroken PT phase are derived. The analysis is applied to the case of superlattices describing a complex (PT-symmetric) extension of the Harper Hamiltonian in the rational case.

http://arxiv.org/abs/1402.3165

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

Scattering experiments with microwave cavities were performed and the effects of broken time-reversal invariance (TRI), induced by means of a magnetized ferrite placed inside the cavity, on an isolated doublet of nearly degenerate resonances were investigated. All elements of the effective Hamiltonian of this two-level system were extracted. As a function of two experimental parameters, the doublet and also the associated eigenvectors could be tuned to coalesce at a so-called exceptional point (EP). The behavior of the eigenvalues and eigenvectors when encircling the EP in parameter space was studied, including the geometric amplitude that builds up in the case of broken TRI. A one-dimensional subspace of parameters was found where the differences of the eigenvalues are either real or purely imaginary. There, the Hamiltonians were found PT-invariant under the combined operation of parity (P) and time reversal (T) in a generalized sense. The EP is the point of transition between both regions. There a spontaneous breaking of PT occurs.

http://arxiv.org/abs/1402.3537

Chaotic Dynamics (nlin.CD)

In the present work we consider the introduction of PT-symmetric terms in the context of classical Klein-Gordon field theories. We explore the implication of such terms on the spectral stability of coherent structures, namely kinks. We find that the conclusion critically depends on the location of the kink center relative to the center of the PT-symmetric term. The main result is that if these two points coincide, the kink’s spectrum remains on the imaginary axis and the wave is spectrally stable. If the kink is centered on the “lossy side” of the medium, then it becomes stabilized. On the other hand, if it becomes centered on the “gain side” of the medium, then it is destabilized. The consequences of these two possibilities on the linearization (point and essential) spectrum are discussed in some detail.

http://arxiv.org/abs/1402.2942

Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph)

PT symmetric Aubry-Andre model describes an array of N coupled optical waveguides with position dependent gain and loss. We show that the reality of the spectrum depends sensitively on the degree of disorder for small number of lattice sites. We obtain the Hofstadter Butterfly spectrum and discuss the existence of the phase transition from extended to localized states. We show that rapidly changing periodical gain/loss materials almost conserves the total intensity.

http://arxiv.org/abs/1402.2749

Quantum Physics (quant-ph)

In systems with “balanced loss and gain”, the PT-symmetry is broken by increasing the non-hermiticity or the loss-gain strength. We show that finite lattices with oscillatory, PT-symmetric potentials exhibit a new class of PT-symmetry breaking and restoration. We obtain the PT phase diagram as a function of potential periodicity, which also controls the location complex eigenvalues in the lattice spectrum. We show that the sum of PT-potentials with nearby periodicities leads to PT-symmetry restoration, where the system goes from a PT-broken state to a PT-symmetric state as the average loss-gain strength is increased. We discuss the implications of this novel transition for the propagation of a light in an array of coupled waveguides.

http://arxiv.org/abs/1402.2544

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

The correspondence between exotic quantum holonomy that occurs in families of Hermitian cycles, and exceptional points (EPs) for non-Hermitian quantum theory is examined in quantum kicked tops. Under a suitable condition, an explicit expressions of the adiabatic parameter dependencies of quasienergies and stationary states, which exhibit anholonomies, are obtained. It is also shown that the quantum kicked tops with the complexified adiabatic parameter have a higher order EP, which is broken into lower order EPs with the application of small perturbations. The stability of exotic holonomy against such bifurcation is demonstrated.

http://arxiv.org/abs/1402.1634

Quantum Physics (quant-ph)

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.

http://arxiv.org/abs/1402.1082

Spectral Theory (math.SP); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

]]>We show how to construct, out of a certain basis invariant under the action of one or more unitary operators, a second biorthogonal set with similar properties. In particular, we discuss conditions for this new set to be also a basis of the Hilbert space, and we apply the procedure to coherent states. We conclude the paper considering a simple application of our construction to pseudo-hermitian quantum mechanics.

http://arxiv.org/abs/1402.0425

Mathematical Physics (math-ph)

We consider the role of degeneracy in Parity-Time (PT) symmetry breaking for non-hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of T-breaking, and T is broken in a generic manner (without preserving other discrete symmetries), then the standard PT-symmetry breaking transition does not occur, meaning that the spectrum is complex even for infinitesimal strength of gain and loss. However the reality of the entire spectrum can be preserved over a finite interval if additional discrete symmetries X are imposed when T is broken, if X decouple all degenerate modes. When this is true only for a subset of the degenerate spectrum, there can be a partial PT transition in which this subset remains real over a finite interval of T-breaking. If the spectrum has odd-degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even without imposing additional discrete symmetries, and they are analogous to dark states in atomic physics. These results are illustrated by the example of different T-breaking perturbations of a uniform dielectric disk and sphere. Finally, we show that multimode coupling is capable of restoring the PT-symmetric phase at finite T-breaking. We also analyze these questions when the parity operator is replaced by another spatial symmetry operator and find that the behavior can be qualitatively different.

http://arxiv.org/abs/1402.0428

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

We introduce a stochastic PT-symmetric coupler, which is based on dual-core waveguides with fluctuating parameters, such that the gain and the losses are exactly balanced in average. We consider different parametric regimes which correspond to the broken and unbroken PT symmetry, as well as to the exceptional point of the underlying deterministic system. We demonstrate that in all the cases the statistically averaged intensity of the field grows. This result holds for either linear or nonlinear coupler and is independent on the type of fluctuations.

http://arxiv.org/abs/1401.6352

Optics (physics.optics)

We consider a model of planar PT-symmetric waveguide and study the phenomenon of the eigenvalues collision under the perturbation of boundary conditions. This phenomenon was discovered numerically in previous works. The main result of this work is an analytic explanation of this phenomenon.

http://arxiv.org/abs/1401.6316

Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

The spectrum of a one-dimensional Hamiltonian with potential V(x)=ix2 for negative x and \(V(x)=−ix^2\) for positive x is analyzed. The Schrodinger equation is algebraically solvable and the eigenvalues are obtained as the zeros of an expression explicitly given in terms of Gamma functions. The spectrum consists of one real eigenvalue and an infinite set of pairs of complex conjugate eigenvalues.

http://arxiv.org/abs/1401.5937

Quantum Physics (quant-ph)

We investigate some questions on the construction of \(\eta\) operators for pseudo-Hermitian Hamiltonians. We give a sufficient condition which can be exploited to systematically generate a sequence of \(\eta\) operators starting from a known one, thereby proving the non-uniqueness of \(\eta\) for a particular pseudo-Hermitian Hamiltonian. We also study perturbed Hamiltonians for which \(\eta\)’s corresponding to the original Hamiltonian still work.

http://arxiv.org/abs/1401.5255

Quantum Physics (quant-ph)

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schroedinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy eigenspectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.

http://arxiv.org/abs/1401.4426

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

The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the S-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(t). We show that the application of the adiabatic approximation to H(t) corresponds to the semiclassical description of the original scattering problem. In particular, the geometric part of the phase of the evolving eigenvectors of H(t) gives the pre-exponential factor of the WKB wave functions. We use these observations to give an explicit semiclassical expression for the transfer matrix. This allows for a detailed study of the semiclassical unidirectional reflectionlessness and invisibility. We examine concrete realizations of the latter in the realm of optics.

http://arxiv.org/abs/1401.4315

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

We discuss phase transitions in PT-symmetric optical systems. We show that due to frequency dispersion of the dielectric permittivity, an optical system can have PT-symmetry at isolated frequency points only. An assumption of the existence of a PT-symmetric system in a continuous frequency interval violates the causality principle. Therefore, the ideal symmetry-breaking transition cannot be observed by simply varying the frequency.

http://arxiv.org/abs/1401.4043

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

]]>We consider the linear and nonlinear Schrodinger equation for a Bose-Einstein condensate in a harmonic trap with PT-symmetric double-delta function loss and gain terms. We verify that the conditions for the applicability of a recent proposition by Mityagin and Siegl on singular perturbations of harmonic oscillator type self-adjoint operators are fulfilled. In both the linear and nonlinear case we calculate numerically the shifts of the unperturbed levels with quantum numbers n of up to 89 in dependence on the strength of the non-Hermiticity and compare with rigorous estimates derived by those authors. We confirm that the predicted 1/n1/2 estimate provides a valid upper bound on the the shrink rate of the numerical eigenvalues. Moreover, we find that a more recent estimate of log(n)/n3/2 is in excellent agreement with the numerical results. With nonlinearity the shrink rates are found to be smaller than without nonlinearity, and the rigorous estimates, derived only for the linear case, are no longer applicable.

http://arxiv.org/abs/1401.2896

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Spectral Theory (math.SP)

]]>A Bose-Einstein condensate trapped in a double-well potential, where atoms are incoupled to one side and extracted from the other, can in the mean-field limit be described by the nonlinear Gross-Pitaevskii equation (GPE) with a \({\mathcal PT}\)-symmetric external potential. If the strength of the in- and outcoupling is increased two \({\mathcal PT}\) broken states bifurcate from the \({\mathcal PT}\)-symmetric ground state. At this bifurcation point a stability change of the ground state is expected. However, it is observed that this stability change does not occur exactly at the bifurcation but at a slightly different strength of the in-/outcoupling effect. We investigate a Bose-Einstein condensate in a \({\mathcal PT}\)-symmetric double-\(\delta\) potential and calculate the stationary states. The ground state’s stability is analysed by means of the Bogoliubov-de Gennes equations and it is shown that the difference in the strength of the in-/outcoupling between the bifurcation and the stability change can be completely explained by the norm-dependency of the nonlinear term in the Gross-Pitaevskii equation.

http://arxiv.org/abs/1401.2354

Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Chaotic Dynamics (nlin.CD)

The concept of quasi-PT symmetry in optical wave guiding system is elaborated by comparing the evolution dynamics of a PT-symmetric directional coupler and a passive directional coupler. In particular we show that in the low loss regime, apart for an overall exponentially damping factor that can be compensated via a dynamical renormalization of the power flow in the system along the propagation direction, the dynamics of the passive coupler fully reproduce the one of the PT-symmetric system.

http://arxiv.org/abs/1401.2299

Optics (physics.optics)

]]>We find that real and complex Bohmian quantum trajectories resulting from well-localized Klauder coherent states in the quasi-Poissonian regime possess qualitatively the same type of trajectories as those obtained from a purely classical analysis of the corresponding Hamilton-Jacobi equation. In the complex cases treated the quantum potential results to a constant, such that the agreement is exact. For the real cases we provide conjectures for analytical solutions for the trajectories as well as the corresponding quantum potentials. The overall qualitative behaviour is governed by the Mandel parameter determining the regime in which the wavefunctions evolve as soliton like structures. We demonstrate these features explicitly for the harmonic oscillator and the Poeschl-Teller potential.

http://arxiv.org/abs/1305.4619

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

We show that obliquely-incident, transversely-magnetic-polarized plane waves can be totally transmitted (with zero reflection) through epsilon-near-zero (ENZ) bi-layers characterized by balanced loss and gain with parity-time (PT) symmetry. This tunneling phenomenon is mediated by the excitation of a surface-wave localized at the interface separating the loss and gain regions. We determine the parameter configurations for which the phenomenon may occur and, in particular, the relationship between the incidence direction and the electrical thickness. We show that, below a critical threshold of gain and loss, there always exists a tunneling angle which, for moderately thick (wavelength-sized) structures, approaches a critical value dictated by the surface-wave phase-matching condition. We also investigate the unidirectional character of the tunneling phenomenon, as well as the possible onset of spontaneous symmetry breaking, typical of PT-symmetric systems. Our results constitute an interesting example of a PT-symmetry-induced tunneling phenomenon, and may open up intriguing venues in the applications of ENZ materials featuring loss and gain.

http://arxiv.org/abs/1401.1619

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

The elementary quadratic plus inverse sextic interaction containing a strongly singular repulsive core in the origin is made regular by a complex shift of coordinate \(x=s−i\epsilon\). The shift \(\epsilon>0\) is fixed while the value of s is kept real and potentially observable, \(s∈(−\infty,\infty)\). The low-lying energies of bound states are found in closed form for the large couplings g. Within the asymptotically vanishing \(\mathcal{O}(g^{−1/4})\) error bars these energies are real so that the time-evolution of the system may be expected unitary in an {\em ad hoc} physical Hilbert space.

http://arxiv.org/abs/1401.1435

Quantum Physics (quant-ph)

The author introduces a different kind of Hermtian quantum mechanics, called J-Hermitian quantum mechanics. He shows that PT-symmetric quantum mechanics is indeed J-Hermitian quantum mechanics, and that temporal evolution is unitary if and only if Hamiltonian is PT-symmetric.

http://arxiv.org/abs/1312.7738

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

We show that a recently developed semiclassical expansion for the eigenvalues of PT-symmetric oscillators of the form \(V(x)=(ix)^{2N+1}+bix\) does not agree with an earlier WKB expression for \(V(x)=−(ix)^{2N+1}\) the case \(b=0\). The reason is due to the choice of different paths in the complex plane for the calculation of the WKB integrals. We compare the Stokes and anti-Stokes lines that apply to each case for the quintic oscillator and derive a general WKB expression that contains the two earlier ones.

http://arxiv.org/abs/1312.5117

Quantum Physics (quant-ph)

We investigate the spectral and dynamical properties of a quantum particle constrained on a ring threaded by a magnetic flux in presence of a complex (non-Hermitian) potential. For a static magnetic flux, the quantum states of the particle on the ring can be mapped into the Bloch states of a complex crystal, and magnetic flux tuning enables to probe the spectral features of the complex crystal, including the appearance of exceptional points. For a time-varying (linearly-ramped) magnetic flux, Zener tunneling among energy states is realized owing to the induced electromotive force. As compared to the Hermitian case, striking effects are observed in the non-Hermitian case, such as a highly asymmetric behavior of particle motion when reversing the direction of the magnetic flux and field-induced delayed transparency.

http://arxiv.org/abs/1312.4693

Quantum Physics (quant-ph)

We study the generic interaction of a monochromatic electromagnetic field with bi-isotropic nanoparticles. Such an interaction is described by dipole-coupling terms associated with the breaking of dual, P- and T-symmetries, including the chirality and the nonreciprocal magnetoelectric effect. We calculate absorption rates, radiation forces, and radiation torques for the nanoparticles and introduce novel characteristics of the field quantifying the transfer of energy, momentum, and angular-momentum in these interactions. In particular, we put forward the concept of ‘magnetoelectric energy density’, quantifying the local PT-symmetry of the field. Akin to the ‘super-chiral’ light suggested recently for sensitive local probing of molecular chirality [Phys. Rev. Lett. 104, 163901 (2010); Science 332, 333 (2011)], here we describe a complex field for sensitive probing of the nonreciprocal magnetoelectric effect in nanoparticles or molecules.

http://arxiv.org/abs/1312.4325

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

]]>Multi-dimensional complex optical potentials with partial parity-time (PT) symmetry are proposed. The usual PT symmetry requires that the potential is invariant under complex conjugation and simultaneous reflection in all spatial directions. However, we show that if the potential is only partially PT-symmetric, i.e., it is invariant under complex conjugation and reflection in a single spatial direction, then it can also possess all-real spectra and continuous families of solitons. These results are established analytically and corroborated numerically.

http://arxiv.org/abs/1312.3660

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

The subject of the work are pairs of linearly coupled PT-symmetric dimers. Two different settings are introduced, namely, straight-coupled dimers, where each gain site is linearly coupled to one gain and one loss site, and cross-coupled dimers, with each gain site coupled to two lossy ones. The latter pair with equal coupling coefficients represents a “PT-hypersymmetric” quadrimer. We find symmetric and antisymmetric solutions in these systems, chiefly in an analytical form, and explore the existence, stability and dynamical behavior of such solutions by means of numerical methods. We thus identify bifurcations occurring in the systems, including spontaneous symmetry breaking and saddle-center bifurcations. Simulations demonstrate that evolution of unstable branches typically leads to blowup. However, in some cases unstable modes rearrange into stable ones.

http://arxiv.org/abs/1312.3376

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Bender et al. have developed PT-symmetric quantum theory as an extension of quantum theory to non-Hermitian Hamiltonians. We show that when this model has a local PT symmetry acting on composite systems it violates the non-signaling principle of relativity. Since the case of global PT symmetry is known to reduce to standard quantum mechanics, this shows that the PT-symmetric theory is either a trivial extension or likely false as a fundamental theory.

http://arxiv.org/abs/1312.3395

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

We consider a Parity-time (PT) invariant non-Hermitian quasi-exactly solvable (QES) potential which exhibits PT phase transition. We numerically study this potential in a complex plane classically to demonstrate different quantum effects. The particle with real energy makes closed orbits around one of the periodic wells of the complex potential depending on the initial condition. However interestingly the particle can have open orbits even with real energy if it is initially placed in certain region between the two wells on the same side of the imaginary axis. On the other hand when the particle energy is complex the trajectory is open and the particle tunnels back and forth between two wells which are separated by a classically forbidden path. The tunneling time is calculated for different pair of wells and is shown to vary inversely with the imaginary component of energy. At the classical level unlike the analogous quantum situation we do not see any qualitative differences in the features of the particle dynamics for PT symmetry broken and unbroken phases.

http://arxiv.org/abs/1312.0757

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

Exceptional points (EPs) determine the dynamics of open quantum systems and cause also PT symmetry breaking in PT symmetric systems. From a mathematical point of view, this is caused by the fact that the phases of the wavefunctions (eigenfunctions of a non-Hermitian Hamiltonian) relative to one another are not rigid when an EP is approached. The system is therefore able to align with the environment to which it is coupled and, consequently, rigorous changes of the system properties may occur. We compare analytically as well as numerically the eigenvalues and eigenfunctions of a 2×2 matrix that is characteristic of either open quantum systems at high level density or of PT symmetric optical lattices. In both cases, the results show clearly the influence of the environment onto the system in the neighborhood of EPs. Although the systems are very different from one another, the eigenvalues and eigenfunctions indicate the same characteristic features.

http://arxiv.org/abs/1311.6320

Dynamical Systems (math.DS); Quantum Physics (quant-ph)

We investigate the evolution of a state which is dominated by a finite-dimensional non-Hermitian time-dependent Hamiltonian operator with a nondegenerate spectrum by using a biorthonormal approach. The geometric phase between any two states, biorthogonal or not, are generally derived by employing the generalized interference method. The counterpart of Manini-Pistolesi non-diagonal geometric phase in the non-Hermitian setting is taken as a typical example.

http://arxiv.org/abs/1311.5631

Quantum Physics (quant-ph)

We consider parity-time (PT) symmetric arrays formed by N optical waveguides with gain and N waveguides with loss. When the gain-loss coefficient exceeds a critical value γc, the PT-symmetry becomes spontaneously broken. We calculate γc(N) and prove that γc→0 as N→∞. In the symmetric phase, the periodic array is shown to support 2N solitons with different frequencies and polarisations.

http://arxiv.org/abs/1311.4123

Optics (physics.optics); Mathematical Physics (math-ph); Pattern Formation and Solitons (nlin.PS)

We introduce one- and two-dimensional (1D and 2D) models of parity-time PT-symmetric couplers with the mutually balanced linear gain and loss applied to the two cores, and cubic-quintic (CQ) nonlinearity acting in each one. The 2D and 1D models may be realized in dual-core optical waveguides, in the spatiotemporal and spatial domains, respectively. Stationary solutions for PT-symmetric solitons in these systems reduce to their counterparts in the usual coupler. The most essential problem is the stability of the solitons, which become unstable against symmetry breaking with the increase of the energy (norm), and retrieve the stability at still larger energies. The boundary value of the intercore-coupling constant, above which the solitons are completely stable, is found by means of an analytical approximation, based on the CW (zero-dimensional) counterpart of the system. The approximation demonstrates good agreement with numerical findings for the 1D and 2D solitons. Numerical results for the stability limits of the 2D solitons are obtained by means of the computation of eigenvalues for small perturbations, and verified in direct simulations. Although large parts of the solitons families are unstable, the instability is quite weak. Collisions between 2D solitons in the PT-symmetric coupler are studied by means of simulations. Outcomes of the collisions are inelastic but not destructive, as they do not break the PT symmetry.

http://arxiv.org/abs/1311.3677

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

We report a diversity of stable gap solitons in a spin-orbit coupled Bose-Einstein condensate subject to a spatially periodic Zeeman field. It is shown that the solitons, can be classified by the main physical symmetries they obey, i.e. symmetries with respect to parity (P), time (T), and internal degree of freedom, i.e. spin, (C) inversions. The conventional gap and gap-stripe solitons are obtained in lattices with different parameters. It is shown that solitons of the same type but obeying different symmetries can exist in the same lattice at different spatial locations. PT and CPT symmetric solitons have anti-ferromagnetic structure and are characterized respectively by nonzero and zero total magnetizations.

http://arxiv.org/abs/1310.8517

Quantum Gases (cond-mat.quant-gas); Pattern Formation and Solitons (nlin.PS)

We study the nonlinear Schro¨dinger equation with a PT-symmetric potential. Using a hydrodynamic formulation and connecting the phase gradient to the field amplitude, allows for a reduction of the model to a Duffing or a generalized Duffing equation. This way, we can obtain exact soliton solutions existing in the presence of suitable PT-symmetric potentials, and study their stability and dynamics. We report interesting new features, including oscillatory instabilities of solitons and (nonlinear) PT-symmetry breaking transitions, for focusing and defocusing nonlinearities.

http://arxiv.org/abs/1310.7635

Pattern Formation and Solitons (nlin.PS)

In natural light harvesting systems, the sequential quantum events of photon absorption by specialized biological antenna complexes, charge separation, exciton formation and energy transfer to localized reaction centers culminates in the conversion of solar to chemical energy. A notable feature in these processes is the exceptionally high efficiencies (> 95 %) at which excitation is transferred from the illuminated protein complex site to the reaction centers. Such high exciton propagation rates within a system of interwoven biomolecular network structures, is yet to be replicated in artificial light harvesting complexes. A clue to unraveling the quantum puzzles of nature may lie in the observation of long lived coherences lasting several picoseconds in the electronic spectra of photosynthetic complexes, even in noisy environmental baths. A number of experimental and theoretical studies have been devoted to unlocking the links between quantum processes and information protocols, in the hope of finding answers to nature’s puzzling mode of energy propagation. This review presents developments in quantum theories, and links information-theoretic aspects with photosynthetic light-harvesting processes in biomolecular systems. There is examination of various attempts to pinpoint the processes that underpin coherence features arising from the light harvesting activities of biomolecular systems, with particular emphasis on the effects that factors such non-Markovianity, zeno mechanisms, teleportation, quantum predictability and the role of multipartite states have on the quantum dynamics of biomolecular systems. A discussion of how quantum thermodynamical principles and agent-based modeling and simulation approaches can improve our understanding of natural photosynthetic systems is included.

http://arxiv.org/abs/1310.7761

Quantum Physics (quant-ph); Biological Physics (physics.bio-ph); Chemical Physics (physics.chem-ph)

]]>Spectral singularities are ubiquitous with PT-symmetry leading to infinite transmission and reflection coefficients. Such infinities imply the divergence of the fields in the medium thereby breaking the very assumption of the linearity of the medium used to obtain such singularities. We identify saturable nonlinearity retaining contributions from all orders of the field to limit the infinite growth and regularize the spectral singularity. We present explicit numerical results to demonstrate regularization. The all order nonlinear PT-symmetric device is shown to exhibit very effective isolation or optical diode action, since transmission through such a system is nonreciprocal. In contrast, a linear system or a system with Kerr nonlinearity is known to have only reciprocal transmission. Further we demonstrate optical bistability for such a system with high contrast.

http://arxiv.org/abs/1310.8282

Optics (physics.optics)

]]>We investigate the dynamical behavior of continuous and discrete Schr\”odinger systems exhibiting parity-time (PT) invariant nonlinearities. We show that such equations behave in a fundamentally different fashion than their nonlinear Schr\”odinger counterparts. In particular, the PT-symmetric nonlinear Schr\”odinger equation can simultaneously support both bright and dark soliton solutions. In addition, we study a two-element discretized version of this PT nonlinear Schr\”odinger equation. By obtaining the underlying invariants, we show that this system is fully integrable and we identify the PT-symmetry breaking conditions. This arrangement is unique in the sense that the exceptional points are fully dictated by the nonlinearity itself.

http://arxiv.org/abs/1310.7399

Pattern Formation and Solitons (nlin.PS)

We give a partially alternate proof of the reality of the spectrum of the imaginary cubic oscillator in quantum mechanics.

http://arxiv.org/abs/1310.7767

Mathematical Physics (math-ph)

We have investigated the interaction between a strong soliton and a weak probe with certain configurations that allow optical trapping in gas-filled hollow-core photonic crystal fibers in the presence of the shock effect. We have shown theoretically and numerically that the shock term can lead to an unbroken parity-time (PT) symmetry potential in these kinds of fibers. Reciprocity breaking, a remarkable feature of the PT symmetry, is also demonstrated numerically. Our results will open different configurations and avenues for observing PT-symmetry breaking in optical fibers, without the need to resort to cumbersome dissipative structures.

http://arxiv.org/abs/1310.7497

Optics (physics.optics)

]]>We complexify a 1-d potential which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters becomes imaginary. For one PT-symmetric case we have entire real bound state spectrum. Explicit scattering states are constructed to show reciprocity at certain discrete values of energy even though the potential is not parity symmetric. Coexistence of deep energy minima of transmissivity with the multiple spectral singularities (MSS) is observed. We further show that this potential becomes invisible from left (or right) at certain discrete energies. The penetrating states in the other PT-symmetric configuration are always reciprocal even though it is PT-invariant and no spectral singularity (SS) is present in this case. Presence of MSS and reflectionlessness are also discussed for the free states in the later case.

http://arxiv.org/abs/1310.7752

Quantum Physics (quant-ph)

]]>We study the nonlinear Schrodinger equation with a PT-symmetric potential. Using a hydrodynamic formulation and connecting the phase gradient to the field amplitude, allows for a reduction of the model to a Duffing or a generalized Duffing equation. This way, we can obtain exact soliton solutions existing in the presence of suitable PT-symmetric potentials, and study their stability and dynamics. We report interesting new features, including oscillatory instabilities of solitons and (nonlinear) PT-symmetry breaking transitions, for focusing and defocusing nonlinearities.

http://arxiv.org/abs/1310.7635

Pattern Formation and Solitons (nlin.PS)

The spectrum of the quantum Ising chain can be found by expressing the spins in terms of free fermions. An analogous transformation exists for clock chains with Zn symmetry, but is of less use because the resulting parafermionic operators remain interacting. Nonetheless, Baxter showed that a certain non-hermitian (but PT-symmetric) clock Hamiltonian is “free”, in the sense that the entire spectrum is found in terms of independent energy levels, with the striking feature that there are n possibilities for occupying each level. Here I show this directly explicitly finding shift operators obeying a Zn generalization of the Clifford algebra. I also find higher Hamiltonians that commute with Baxter’s and prove their spectrum comes from the same set of energy levels. This thus provides an explicit notion of a “free parafermion”. A byproduct is an elegant method for the solution of the Ising/Kitaev chain with spatially varying couplings.

http://arxiv.org/abs/1310.6049

Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

We generalize a finite parity-time (PT-)symmetric network of the discrete nonlinear Schrodinger type and obtain general results on linear stability of the zero equilibrium, on the nonlinear dynamics of the dimer model, as well as on the existence and stability of large-amplitude stationary nonlinear modes. A result of particular importance and novelty is the classification of all possible stationary modes in the limit of large amplitudes. We also discover a new integrable configuration of a PT-symmetric dimer.

http://arxiv.org/abs/1310.5651

Pattern Formation and Solitons (nlin.PS); Dynamical Systems (math.DS)

A paradigm model of modern atom optics is studied, strongly interacting ultracold bosons in an optical lattice. This many-body system can be artificially opened in a controlled manner by modern experimental techniques. We present results based on a non-hermitian effective Hamiltonian whose quantum spectrum is analyzed. The direct access to the spectrum of the metastable many-body system allows us to easily identify relatively stable quantum states, corresponding to previously predicted solitonic many-body structures.

http://arxiv.org/abs/1310.5937

Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas); Quantum Physics (quant-ph)

Stable discrete compactons in arrays of inter-connected three-line waveguide arrays are found in linear and nonlinear limits in conservative and in parity-time PT symmetric models. The compactons result from the interference of the fields in the two lines of waveguides ensuring that the third (middle) line caries no energy. PT-symmetric compactons require not only the presence of gain and losses in the two lines of the waveguides but also complex coupling, i.e. gain and losses in the coupling between the lines carrying the energy and the third line with zero field. The obtained compactons can be stable and their branches can cross the branches of the dissipative solitons. Unusual bifurcations of branches of solitons from linear compactons are described.

http://arxiv.org/abs/1310.5328

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

We investigate the onset of parity-time (PT) symmetry breaking in non-Hermitian tight-binding lattices with spatially-extended loss/gain regions in presence of an advective term. Similarly to the instability properties of hydrodynamic open flows, it is shown that PT-symmetry breaking can be either absolute or convective. In the former case, an initially-localized wave packet shows a secular growth with time at any given spatial position, whereas in the latter case the growth is observed in a reference frame moving at some drift velocity while decay occurs at any fixed spatial position. In the convective unstable regime, PT-symmetry is restored when the spatial region of gain/loss in the lattice is limited (rather than extended). We consider specifically a non-Hermitian extension of the Rice-Mele tight binding lattice model, and show the existence of a transition from absolute to convective symmetry breaking when the advective term is large enough. An extension of the analysis to ac-dc-driven lattices is also presented, and an optical implementation of the non-Hermitian Rice-Mele model is suggested, which is based on light transport in an array of evanescently-coupled optical waveguides with a periodically-bent axis and alternating regions of optical gain and loss.

http://arxiv.org/abs/1310.5004

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

We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space \(\Lc^2(\Bbb R^2)\), but instead only D-quasi bases. As recently proved by one of us (FB), this is sufficient to deduce several interesting consequences.

http://arxiv.org/abs/1310.4775

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

]]>For the one-dimensional nonlinear Schroedinger equation with a complex potential, it is shown that if this potential is not parity-time (PT) symmetric, then no continuous families of solitons can bifurcate out from linear guided modes, even if the linear spectrum of this potential is all real. Both localized and periodic non-PT-symmetric potentials are considered, and the analytical conclusion is corroborated by explicit examples. Based on this result, it is argued that PT-symmetry of a one-dimensional complex potential is a necessary condition for the existence of soliton families.

http://arxiv.org/abs/1310.4490

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

We introduce a new type of Fano resonances, realized in a photonic circuit which consists of two nonlinear PT-symmetric micro-resonators side-coupled to a waveguide, which have line-shape and resonance position that depends on the direction of the incident light. We utilize these features in order to induce asymmetric transport up to 47 dBs in the optical C-window. Our set-up requires low input power and does not compromise the power and frequency characteristics of the output signal.

http://arxiv.org/abs/1310.2313

Optics (physics.optics)

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrodinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, anti-lasing, and unidirectional invisibility.

http://arxiv.org/abs/1310.0592

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

We show how some recent models of PT-quantum mechanics perfectly fit into the settings of \(\cal D\) pseudo-bosons, as introduced by one of us. Among the others, we also consider a model of non-commutative quantum mechanics, and we show that this model too can be described in terms of \(\cal D\) pseudo-bosons.

http://arxiv.org/abs/1310.0359

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

The resonant scattering of gap solitons (GS) of the periodic nonlinear Schr\”odinger equation with a localized defect which is symmetric under the parity and the time-reversal (PT) symmetry, is investigated. It is shown that for suitable amplitudes ratios of the real and imaginary parts of the defect potential the resonant transmission of the GS through the defect becomes possible. The resonances occur for potential parameters which allow the existence of localized defect modes with the same energy and norm of the incoming GS. Scattering properties of GSs of different band-gaps with effective masses of opposite sign are investigated. The possibility of unidirectional transmission and blockage of GSs by PT defect, as well as, amplification and destruction induced by multiple reflections from two PT defects, are also discussed.

http://arxiv.org/abs/1309.7655

Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas); Atomic Physics (physics.atom-ph); Optics (physics.optics)

]]>We investigate the properties of PT-symmetric tight-binding models by considering both bounded and unbounded models. For the bounded case, we obtain closed form expressions for the corresponding energy spectra and we analyze the structure of eigenstates as well as their dependence on the gain/loss contrast parameter. For unbounded PT-lattices, we explore their scattering properties through the development of analytical models. Based on our approach we identify a mechanism that is responsible to the emergence of localized states that are entirely due to the presence of gain and loss. The derived expressions for the transmission and reflection coefficients allow one to better understand the role of PT-symmetry in energy transport problems occurring in such PT-symmetric tight-binding settings. Our analytical results are further exemplified via pertinent examples.

http://arxiv.org/abs/1309.6708

]]>Non-self-adjoint Schrodinger operators A which correspond to non-symmetric zero-range potentials are investigated. For a given A, the description of non-real eigenvalues, spectral singularities and exceptional points are obtained; the possibility of interpretation of A as a self-adjoint operator in a Krein space is studied, the problem of similarity of A to a self-adjoint operator in a Hilbert space is solved.

http://arxiv.org/abs/1309.5482

Mathematical Physics (math-ph); Spectral Theory (math.SP); Quantum Physics (quant-ph)

The quantization of higher order time derivative theories including interactions is unclear. In this paper in order to solve this problem, we propose to consider a complex version of the higher order derivative theory and map this theory to a real first order theory. To achieve this relationship, the higher order derivative formulation must be complex since there is not a real canonical transformation from this theory to a real first order theory with stable interactions. In this manner, we work with a non-Hermitian higher order time derivative theory. To quantize this complex theory, we introduce reality conditions that allow us to map the complex higher order theory to a real one, and we show that the resulting theory is regularizable and renormalizable for a class of interactions.

http://arxiv.org/abs/1309.2928

High Energy Physics – Theory (hep-th)

We propose a scheme of creating a tunable highly nonlinear defect in a one-dimensional photonic crystal. The defect consists of an atomic cell filled in with two isotopes of three-level atoms. The probe-field refractive index of the defect can be made parity-time (PT) symmetric, which is achieved by proper combination of a control field and of Stark shifts induced by a far-off-resonance field. In the PT-symmetric system families of stable nonlinear defect modes can be formed by the probe field.

http://arxiv.org/abs/1309.2839

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

]]>For the one-dimensional nonlinear Schroedinger equations with parity-time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT-symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT-symmetric potential, symmetry breaking of PT-symmetric solitary waves do not occur. Based on these findings, it is conjectured that one-dimensional PT-symmetric potentials cannot support continuous families of non-PT-symmetric solitary waves.

http://arxiv.org/abs/1309.1652

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

]]>We analyse several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. We show that PT-symmetric Hamiltonians with point-group symmetry \(C_{2v}\) exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit.

http://arxiv.org/abs/1309.0808

Quantum Physics (quant-ph)

]]>The modified Dirac equations for the massive particles with the replacement of the physical mass \(m\) with the help of the relation \(m\rightarrow m_1 + \gamma_5 m_2\) are investigated. It is shown that for a free fermion theory with a \(\gamma_5\) mass term, the finiteness of the mass spectrum at the value \(m_{max}= {m_1}^2/2m_2\) takes place. In this case the region of the unbroken \(\cal PT\)-symmetry may be expressed by means of the simple restriction of the physical mass \(m\leq m_{max}\). Furthermore, we have that the areas of unbroken \(\cal PT\)-symmetry \(m_1\geq m_2\geq 0\), which guarantees the reality values of the physical mass \(m\), consists of three different parametric subregions: i) \(0\leq m_2 < m_1/\sqrt{2}\), \,\,ii) \(m_2=m_1/\sqrt{2}=m_{max},\) \,\,(iii) \(m_1/\sqrt{2}< m_2 \leq m_1\). It is vary important, that only the first subregion (i) defined mass values \(m_1,m_2,\) which correspond to the description of traditional particles in the modified models, because this area contain the possibility transform the modified model to the ordinary Dirac theory. The second condition (ii) is defined the “maximon” – the particle with maximal mass \(m=m_{max}\). In the case (iii) we have to do with the unusual or “exotic” particles for description of which Hamiltonians and equations of motion have no a Hermitian limit. The formulated criterions may be used as a major test in the process of the division of considered models into ordinary and “exotic fermion theories”.

http://arxiv.org/abs/1309.0231

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

We discuss a PT-symmetric Hamiltonian with complex eigenvalues. It is based on the dimensionless Schrodinger equation for a particle in a square box with the PT-symmetric potential \(V(x,y)=iaxy\). Perturbation theory clearly shows that some of the eigenvalues are complex for sufficiently small values of \(|a|\). Point-group symmetry proves useful to guess if some of the eigenvalues may already be complex for all values of the coupling constant. We confirm those conclusions by means of an accurate numerical calculation based on the diagonalization method. On the other hand, the Schrodinger equation with the potential \(V(x,y)=iaxy^{2}\) exhibits real eigenvalues for sufficiently small values of \(|a|\). Point group symmetry suggests that PT-symmetry may be broken in the former case and unbroken in the latter one.

http://arxiv.org/abs/1308.6179

Quantum Physics (quant-ph)

We present results on the dynamics of split-ring dimers having both gain and loss in one dimensional nonlinear parity-time- (PT-)symmetric magnetic metamaterials. For the longwave (continuum) limit approximation and in the weakly nonlinear limit, we show analytic results on the existence of gap soliton solutions and on symmetry breaking phenomenon at a critical value of the gain/loss term.

http://arxiv.org/abs/1308.5362

Optics (physics.optics)

Optical systems combining balanced loss and gain profiles provide a unique platform to implement classical analogues of quantum systems described by non-Hermitian parity-time- (PT-) symmetric Hamiltonians and to originate new synthetic materials with novel properties. To date, experimental works on PT-symmetric optical systems have been limited to waveguides in which resonances do not play a role. Here we report the first demonstration of PT-symmetry breaking in optical resonator systems by using two directly coupled on-chip optical whispering-gallery-mode (WGM) microtoroid silica resonators. Gain in one of the resonators is provided by optically pumping Erbium (Er3+) ions embedded in the silica matrix; the other resonator exhibits passive loss. The coupling strength between the resonators is adjusted by using nanopositioning stages to tune their distance. We have observed reciprocal behavior of the PT-symmetric system in the linear regime, as well as a transition to nonreciprocity in the PT symmetry-breaking phase transition due to the significant enhancement of nonlinearity in the broken-symmetry phase. Our results represent a significant advance towards a new generation of synthetic optical systems enabling on-chip manipulation and control of light propagation.

http://arxiv.org/abs/1308.4564

Optics (physics.optics); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Classical Physics (physics.class-ph); Quantum Physics (quant-ph)

We study the effects of management of the PT-symmetric part of the potential within the setting of Schrodinger dimer and trimer oligomer systems. This is done by rapidly modulating in time the gain/loss profile. This gives rise to a number of interesting properties of the system, which are explored at the level of an averaged equation approach. Remarkably, this rapid modulation provides for a controllable expansion of the region of exact PT-symmetry, depending on the strength and frequency of the imposed modulation. The resulting averaged models are analyzed theoretically and their exact stationary solutions are translated into time-periodic solutions through the averaging reduction. These are, in turn, compared with the exact periodic solutions of the full non-autonomous PT-symmetry managed problem and very good agreement is found between the two.

http://arxiv.org/abs/1308.3738

Pattern Formation and Solitons (nlin.PS)

We study the geometric phase for the ground state of a generalized one-dimensional non-Hermitian quantum XY model, which has transverse-field-dependent intrinsic rotation-time reversal symmetry. Based on the exact solution, this model is shown to have full real spectrum in multiple regions for the finite size system. The result indicates that the phase diagram or exceptional boundary, which separates the unbroken and broken symmetry regions corresponds to the divergence of the Berry curvature. The scaling behaviors of the groundstate energy and Berry curvature are obtained in an analytical manner for a concrete system.

http://arxiv.org/abs/1308.4057

Quantum Physics (quant-ph)

The Hermiticity condition in quantum mechanics required for the characterisation of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called ‘biorthogonal quantum mechanics’, is developed here in some detail in the case for which the Hilbert space dimensionality is finite. Specifically, characterisations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.

http://arxiv.org/abs/1308.2609

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

It is known that the perfect absorption of two identical waves incident on a complex potential from left and right can occur at a fixed real energy and that the time-reversed setting of this system would act as a laser at threshold at the same energy. Here, we argue and show that PT-symmetric potentials are exceptional in this regard which do not allow Coherent Perfect Absorption without lasing as the modulus of the determinant of the \(S\)-matrix, \(|\det S|\), becomes 1, for all positive energies. Next we show that in the parametric regimes where the PT-symmetry is unbroken, the eigenvalues, \(s_\pm\) of \(S\), can become unitary (uni-modular) for all energies. Then the potential becomes coherent perfect emitter on both sides for any energy of coherent injection. We call this property Transparency.

http://arxiv.org/abs/1308.1270

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

We introduce the simplest one-dimensional nonlinear model with the parity-time (PT) symmetry, which makes it possible to find exact analytical solutions for localized modes (“solitons”). The PT-symmetric element is represented by a point-like (delta-functional) gain-loss dipole \(\delta^\prime(x)\), combined with the usual attractive potential \(\delta\)(x). The nonlinearity is represented by self-focusing (SF) or self-defocusing (SDF) Kerr terms, both spatially uniform and localized ones. The system can be implemented in planar optical waveguides. For the sake of comparison, also introduced is a model with separated \(\delta\)-functional gain and loss, embedded into the linear medium and combined with the \(\delta\)-localized Kerr nonlinearity and attractive potential. Full analytical solutions for pinned modes are found in both models. The exact solutions are compared with numerical counterparts, which are obtained in the gain-loss-dipole model with the \(\delta^\prime\)- and \(\delta\)- functions replaced by their Lorentzian regularization. With the increase of the dipole’s strength, \(\gamma\), the single-peak shape of the numerically found mode, supported by the uniform SF nonlinearity, transforms into a double-peak one. This transition coincides with the onset of the escape instability of the pinned soliton. In the case of the SDF uniform nonlinearity, the pinned modes are stable, keeping the single-peak shape.

http://arxiv.org/abs/1308.0426

Pattern Formation and Solitons (nlin.PS)

We reveal previously unnoticed Type II perfect absorption and amplification modes of optical scattering system. These modes, in contrast to the counterparts in recent works [Phys. Rev. A 82, 031801 (2010); Phys. Rev. Lett. 106, 093902 (2011).] which could be referred as Type I modes, appear at a continuous region along the real frequency axis with any frequency. The Type II modes can be demonstrated in the PT-symmetric/traditional Bragg grating combined structures. A series of exotic nonreciprocal absorption and amplification behaviours are observed in the combined structures, making them have potential for versatile devices acting simultaneously as a perfect absorber, an amplifier, and an ultra-narrowband filter. Based on the properties of Type II modes, we also propose structures with controllable perfect absorption and amplification bandwidth at any single or multiple wavelengths.

http://arxiv.org/abs/1307.7583

Optics (physics.optics)

Using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrodinger equation (NLSE) with space and time modulated nonlinearity and in the presence of an external potential depending on space and time. In particular we obtain exact solutions of NLSE in the presence of a number of non Hermitian \(\mathcal{PT}\) symmetric external potentials.

http://arxiv.org/abs/1307.7591

Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph)

]]>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)

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)

]]>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)

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)

]]>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 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)

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)

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)

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)

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)

]]>I study vector solitons involving two incoherently-coupled field components in periodic PT-symmetric optical lattices. The specific symmetry of the lattice imposes the restrictions on the symmetry of available vector soliton states. While all configurations with asymmetric intensity distributions are prohibited, such lattices support multi-hump solitons with equal number of “in-phase” or “out-of-phase” spots in two components, residing on neighboring lattice channels. In the focusing medium only the solitons containing out-of-phase spots in at least one component can be stable, while in the defocusing medium stability is achieved for structures consisting of in-phase spots. Mixed-gap vector solitons with components emerging from different gaps in the lattice spectrum also exist and can be stable in the PT-symmetric lattice.

http://arxiv.org/abs/1306.6743

Optics (physics.optics); Quantum Gases (cond-mat.quant-gas); Pattern Formation and Solitons (nlin.PS)

The existence and stability of the nonlinear spatial localized modes are investigated in parity-time symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with competing gain and loss profile. The exact analytical expressions of the localized modes are found for all values of the competing parameter and in the presence of both the self-focusing and self-defocusing Kerr nonlinearity. The effect of competing gain/loss profile on the stability structure of these localized modes are discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. The spatial localized modes in two-dimensional geometry as well as the transverse power-flow density associated with these localized modes are also examined.

http://arxiv.org/abs/1306.5983

Optics (physics.optics)

]]>We investigate bifurcations of nonlinear modes in parity-time (PT-) symmetric discrete systems. We consider a general class of nonlinearities allowing for existence of the nonlinear modes and address situations when the underlying linear problem is characterized by the presence of degenerate eigenvalues or exceptional-point singularity. In each of the cases we construct formal expansions for small-amplitude nonlinear modes. We also report a class of nonlinearities allowing for a system to admit one or several integrals of motion, which turn out to be determined by the pseudo-Hermiticity of the nonlinearity.

http://arxiv.org/abs/1306.5286

Pattern Formation and Solitons (nlin.PS)

We study a Bose-Einstein condensate in a PT-symmetric double-well potential where particles are coherently injected in one well and removed from the other well. In mean-field approximation the condensate is described by the Gross-Pitaevskii equation thus falling into the category of nonlinear non-Hermitian quantum systems. After extending the concept of PT symmetry to such systems, we apply an analytic continuation to the Gross-Pitaevskii equation from complex to bicomplex numbers and show a thorough numerical investigation of the four-dimensional bicomplex eigenvalue spectrum. The continuation introduces additional symmetries to the system which are confirmed by the numerical calculations and furthermore allows us to analyze the bifurcation scenarios and exceptional points of the system. We present a linear matrix model and show the excellent agreement with our numerical results. The matrix model includes both exceptional points found in the double-well potential, namely an EP2 at the tangent bifurcation and an EP3 at the pitchfork bifurcation. When the two bifurcation points coincide the matrix model possesses four degenerate eigenvectors. Close to that point we observe the characteristic features of four interacting modes in both the matrix model and the numerical calculations, which provides clear evidence for the existence of an EP4.

http://arxiv.org/abs/1306.3871

Quantum Physics (quant-ph)

]]>In this paper, we revisit one of the prototypical PT-symmetric oligomers, namely the trimer. We find all the relevant branches of “regular” solutions and analyze the bifurcations and instabilities thereof. Our work generalizes the formulation that was proposed recently in the case of dimers for the so-called “ghost states” of trimers, which we also identify and connect to symmetry-breaking bifurcations from the regular states. We also examine the dynamics of unstable trimers, as well as those of the ghost states in the parametric regime where the latter are found to exist. Finally, we present the current state of the art for optical experiments in PT-symmetric trimers, as well as experimental results in a gain-loss-gain three channel waveguide structure.

http://arxiv.org/abs/1306.2255

Quantum Physics (quant-ph)

We observe that the reflection and transmission coefficients of a particle within a double, PT-symmetric heterojunction with spatially varying mass, show interesting features, depending on the degree of non Hermiticity, although there is no spontaneous breakdown of PT symmetry. The potential profile in the intermediate layer is considered such that it has a non vanishing imaginary part near the heterojunctions. Exact analytical solutions for the wave function are obtained, and the reflection and transmission coefficients are plotted as a function of energy, for both left as well as right incidence. As expected, the spatial dependence on mass changes the nature of the scattering solutions within the heterojunctions, and the space-time (PT) symmetry is responsible for the left-right asymmetry in the reflection and transmission coefficients. However, the non vanishing imaginary component of the potential near the heterojunctions gives new and interesting results.

http://arxiv.org/abs/1306.2226

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

]]>We theoretically study the non-Hermitian systems, the non-Hermiticity of which arises from the unequal hopping amplitude (UHA) dimers. The distinguishing features of these models are that they have full real spectra if all of the eigenvectors are time-reversal (T) symmetric rather than parity-time-reversal (PT) symmetric, and that their Hermitian counterparts are shown to be an experimentally accessible system, which have the same topological structures as that of the original ones but modulated hopping amplitudes within the unbroken region. Under the reflectionless transmission condition, the scattering behavior of momentum-independent reflectionless transmission (RT) can be achieved in the concerned non-Hermitian system. This peculiar feature indicates that, for a certain class of non-Hermitian systems with a balanced combination of the RT dimers, the defects can appear fully invisible to an outside observer.

http://arxiv.org/abs/1306.1969

Quantum Physics (quant-ph)

]]>Lattice field theories with complex actions are not easily studied using conventional analytic or simulation methods. However, a large class of these models are invariant under CT, where C is charge conjugation and T is time reversal, including models with non-zero chemical potential. For Abelian models in this class, lattice duality maps models with complex actions into dual models with real actions. For extended regions of parameter space, calculable for each model, duality resolves the sign problem for both analytic methods and computer simulations. Explicit duality relations are given for models for spin and gauge models based on Z(N) and U(1) symmetry groups. The dual forms are generalizations of the Z(N) chiral clock model and the lattice Frenkel-Kontorova model, respectively. From these equivalences, rich sets of spatially-modulated phases are found in the strong-coupling region of the original models.

http://arxiv.org/abs/1306.1495

High Energy Physics – Lattice (hep-lat)

We investigate the spectral properties and dynamical features of a time-periodic PT-symmetric Hamiltonian on a one-dimensional tight-binding lattice. It is shown that a high-frequency modulation can drive the system under a transition between the broken-PT and the unbroken-PT phases. The time-periodic modulation in the unbroken-PT regime results in a significant broadening of the quasi-energy spectrum, leading to a hyper-ballistic transport regime. Also, near the PT-symmetry breaking the dispersion curve of the lattice band becomes linear, with a strong reduction of quantum wave packet spreading.

http://arxiv.org/abs/1306.1048

Quantum Physics (quant-ph)

We study both analytically and numerically the spectrum of inhomogeneous strings with \(\mathcal{PT}\)-symmetric density. We discuss an exactly solvable model of \(\mathcal{PT}\)-symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules \(Z(p) \equiv \sum_{n=1}^\infty 1/E_n^p\), with \(p=1,2,\dots\) and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex.

http://arxiv.org/abs/1306.1419

Mathematical Physics (math-ph)

A non-Hermitian shortcut to adiabaticity is introduced. By adding an imaginary term in the diagonal elements of the Hamiltonian of a two state quantum system, we show how one can cancel the nonadiabatic losses and perform an arbitrarily fast population transfer, without the need to increase the coupling. We apply this technique to two popular level-crossing models: the Landau-Zener model and the Allen-Eberly model.

http://arxiv.org/abs/1306.0698

Quantum Physics (quant-ph)

Scattering of a quantum particle from an oscillating barrier or well does not generally conserve the particle energy owing to energy exchange with the photon field, and an incoming particle-free state is scattered into a set of outgoing (transmitted and reflected) free states according to Floquet scattering theory. Here we introduce two families of oscillating non-Hermitian potential wells in which Floquet scattering is fully suppressed for any energy of the incident particle. The scattering-free oscillating potentials are synthesized by application of the Darboux transformation to the time-dependent Schr\”{o}dinger equation. For one of the two families of scattering-free potentials, the oscillating potential turns out to be fully invisible.

http://arxiv.org/abs/1306.0675

Quantum Physics (quant-ph)

We show that invisible localized defects, i.e. defects that can not be detected by an outside observer, can be realized in a crystal with an engineered imaginary potential at the defect site. The invisible defects are synthesized by means of supersymmetric (Darboux) transformations of an ordinary crystal using band-edge wave functions to construct the superpotential. The complex crystal has an entire real-valued energy spectrum and Bragg scattering is not influenced by the defects. An example of complex crystal synthesis is presented for the Mathieu potential.

http://arxiv.org/abs/1306.0667

Quantum Physics (quant-ph)

We theoretically investigate the optical properties of parity-time (PT)-symmetric three dimensional metamaterials composed of strongly-coupled planar plasmonic waveguides. By tuning the loss-gain balance, we show how the initially isotropic material becomes both asymmetric and unidirectional. The highly tunable optical dispersion of PT -symmetric metamaterials provides a foundation for designing an entirely new class of three-dimensional bulk synthetic media, with applications ranging from sub-diffraction-limited optical lenses to non-reciprocal nanophotonic devices.

http://arxiv.org/abs/1306.0059

Optics (physics.optics)

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)

The inspiration for this theoretical paper comes from recent experiments on a PT-symmetric system of two coupled optical whispering galleries (optical resonators). The optical system can be modeled as a pair of coupled linear oscillators, one with gain and the other with loss. If the coupled oscillators have a balanced loss and gain, the system is described by a Hamiltonian and the energy is conserved. This theoretical model exhibits two PT transitions depending on the size of the coupling parameter \epsilon. For small \epsilon the PT symmetry is broken and the system is not in equilibrium, but when \epsilon becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large \(\epsilon\) the system undergoes a second transition and is no longer in equilibrium. The classical and the quantized versions of the system exhibit transitions at exactly the same values of \(\epsilon\).

http://arxiv.org/abs/1305.7107

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

We demonstrate that a coherently-prepared four-level atomic medium can provide a versatile platform for realizing parity-time (PT) symmetric optical potentials. Different types of PT-symmetric potentials are proposed by appropriately tuning the exciting optical fields and the pertinent atomic parameters. Such reconfigurable and controllable systems can open up new avenues in observing PT-related phenomena with appreciable gain/loss contrast in coherent atomic media.

http://arxiv.org/abs/1305.4908

Optics (physics.optics)

The practical use of non-Hermitian (i.e., typically, PT-symmetric) phenomenological quantum Hamiltonians is discussed as requiring an explicit reconstruction of the ad hoc Hilbert-space metrics which would render the time-evolution unitary. Just the N-dimensional matrix toy models Hamiltonians are considered, therefore. For them, the matrix elements of alternative metrics are constructed via solution of a coupled set of polynomial equations, using the computer-assisted symbolic manipulations for the purpose. The feasibility and some consequences of such a model-construction strategy are illustrated via a discrete square well model endowed with multi-parametric close-to-the-boundary real bidiagonal-matrix interaction. The degenerate exceptional points marking the phase transitions are then studied numerically. A way towards classification of their unfoldings in topologically non-equivalent dynamical scenarios is outlined.

http://arxiv.org/abs/1305.4822

Quantum Physics (quant-ph)

Over the last two decades, advances in fabrication have led to significant progress in creating patterned heterostructures that support either carriers, such as electrons or holes, with specific band structure or electromagnetic waves with a given mode structure and dispersion. In this article, we review the properties of light in coupled optical waveguides that support specific energy spectra, with or without the effects of disorder, that are well-described by a Hermitian tight-binding model. We show that with a judicious choice of the initial wave packet, this system displays the characteristics of a quantum particle, including transverse photonic transport and localization, and that of a classical particle. We extend the analysis to non-Hermitian, parity and time-reversal (\(\mathcal{PT}\)) symmetric Hamiltonians which physically represent waveguide arrays with spatially separated, balanced absorption or amplification. We show that coupled waveguides are an ideal candidate to simulate \(\mathcal{PT}\)-symmetric Hamiltonians and the transition from a purely real energy spectrum to a spectrum with complex conjugate eigenvalues that occurs in them.

http://arxiv.org/abs/1305.3565

Optics (physics.optics); Disordered Systems and Neural Networks (cond-mat.dis-nn); Quantum Physics (quant-ph)