Ramy El-Ganainy, Mercedeh Khajavikhan, Li Ge
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)
Stephan Ramon Garcia, Emil Prodan, Mihai Putinar
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)
Miloslav Znojil
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)
V.N.Rodionov
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)
Rüdiger Fortanier, Dennis Dast, Daniel Haag, Holger Cartarius, Jörg Main, Günter Wunner
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)
Yingji He, Boris A. Malomed, Dumitru Mihalache
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)
James M. Hickey, Emanuele Levi, Juan P. Garrahan
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)
Huagang Li, Xing Zhu, Zhiwei Shi, Boris A. Malomed, Tianshu Lai, Chaohong Lee
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)
Denis Borisov
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)
Yogesh N. Joglekar, Derek D. Scott, Avadh Saxena
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)