Category Lancaster University

Nonhermitian transport effects in coupled-resonator optical waveguides

Henning Schomerus, Jan Wiersig

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)

Selective enhancement of topologically induced interface states

C. Poli, M. Bellec, U.Kuhl, F. Mortessagne, H. Schomerus

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)

Topologically protected midgap states in complex photonic lattices

Henning Schomerus

One of the principal goals in the design of photonic crystals is the engineering of band gaps and defect states. Drawing on the concepts of band-structure topology, I here describe the formation of exponentially localized, topologically protected midgap states in photonic systems with spatially distributed gain and loss. When gain and loss are suitably arranged these states maintain their topological protection and then acquire a selectively tunable amplification rate. This finds applications in the beam dynamics along a photonic lattice and in the lasing of quasi-one-dimensional photonic crystals.

http://arxiv.org/abs/1301.0777
Optics (physics.optics); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

Parity anomaly and Landau-level lasing in strained photonic honeycomb lattices

Henning Schomerus, Nicole Yunger Halpern

We describe the formation of highly degenerate, Landau-level-like amplified states in a strained photonic honeycomb lattice in which amplification breaks the sublattice symmetry. As a consequence of the parity anomaly, the zeroth Landau level is localized on a single sublattice and possesses an enhanced or reduced amplification rate. The spectral properties of the higher Landau levels are constrained by a generalized time-reversal symmetry. In the setting of two-dimensional photonic crystal lasers, the anomaly directly affects the mode selection and lasing threshold while in three-dimensional photonic lattices it can be probed via beam dynamics.

http://arxiv.org/abs/1208.2901

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

Random-matrix theory of amplifying and absorbing resonators with PT or PTT’ symmetry

Christopher Birchall, Henning Schomerus

We formulate gaussian and circular random-matrix models representing a coupled system consisting of an absorbing and an amplifying resonator, which are mutually related by a generalized time-reversal symmetry. Motivated by optical realizations of such systems we consider a PT or a PTT’ time-reversal symmetry, which impose different constraints on magneto-optical effects, and then focus on five common settings. For each of these, we determine the eigenvalue distribution in the complex plane in the short-wavelength limit, which reveals that the fraction of real eigenvalues among all eigenvalues in the spectrum vanishes if all classical scales are kept fixed. Numerically, we find that the transition from real to complex eigenvalues in the various ensembles display a different dependence on the coupling strength between the two resonators. These differences can be linked to the level spacing statistics in the hermitian limit of the considered models.

http://arxiv.org/abs/1208.2575
Quantum Physics (quant-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn); Optics (physics.optics)

Fractal Weyl laws for amplified states in PT-symmetric resonators

Christopher Birchall, Henning Schomerus

We find that in nonhermitian PT-symmetric systems (as realized in resonators with balanced absorption and amplification), a mechanism based on quantum-to-classical correspondence reduces the occurrence of strongly amplified states. The reduction arises from semiclassically emerging hierarchical phase-space structures that are associated with the coupling of the amplifying and absorbing regions (forward and backward-trapped sets and their complements), and amounts to a generalization of the fractal Weyl law, earlier proposed for ballistically open systems. In the context of the recently introduced class of PT-symmetric laser-absorbers, this phenomenon reduces the number of states participating in the mode competition.

http://arxiv.org/abs/1208.2259
Quantum Physics (quant-ph); Optics (physics.optics)

From scattering theory to complex wave dynamics in non-hermitian PT-symmetric resonators

Henning Schomerus

I review how methods from mesoscopic physics can be applied to describe the multiple wave scattering and complex wave dynamics in non-hermitian PT-symmetric resonators, where an absorbing region is coupled symmetrically to an amplifying region. Scattering theory serves as a convenient tool to classify the symmetries beyond the single-channel case and leads to effective descriptions which can be formulated in the energy domain (via Hamiltonians) and in the time domain (via time evolution operators). These models can then be used to identify the mesoscopic time and energy scales which govern the spectral transition from real to complex eigenvalues. The possible presence of magneto-optical effects (a finite vector potential) in multichannel systems leads to a variant (termed PTT’ symmetry) which imposes the same spectral constraints as PT symmetry. I also provide multichannel versions of generalized flux-conservation laws.

http://arxiv.org/abs/1207.1454
Quantum Physics (quant-ph); Optics (physics.optics)

Quantum noise and mode nonorthogonality in nonhermitian PT-symmetric optical resonators

Gwangsu Yoo, H.-S. Sim, Henning Schomerus

PT-symmetric optical resonators combine absorbing regions with active, amplifying regions. The latter are the source of radiation generated via spontaneous and stimulated emission, which embodies quantum noise and can result in lasing. We calculate the frequency-resolved output radiation intensity of such systems and relate it to a suitable measure of excess noise and mode nonorthogonality. The lineshape differs depending on whether the emission lines are isolated (as for weakly amplifying, almost hermitian systems) or overlapping (as for the almost degenerate resonances in the vicinity of exceptional points associated to spontaneous PT-symmetry breaking). The calculations are carried out in the scattering input-output formalism, and are illustrated for a quasi one-dimensional resonator set-up. In our derivations we also allow for the more general case of a resonator in which the amplifying and absorbing regions are not related by symmetry.

http://arxiv.org/abs/1109.4932
Optics (physics.optics)

Universal routes to spontaneous PT-symmetry breaking in non-hermitian quantum systems

Henning Schomerus

(a) Sketch of a nonhermitian PT-symmetric system, where a region with absorption rate μ (and mean level spacing, left) is coupled symmetrically via a tunnel barrier (supporting N channels with transmission probability T) to an amplifying region with a matching amplification rate (right). Below this, the scattering description of the system. (b) Two routes to spontaneous PT-symmetry breaking, depending on whether the hermitian limit μ = 0 is T -symmetric (orthogonal class displaying level crossings, left) or not (unitary class displaying avoided crossings, right). Shown are real eigenvalues of a random Hamiltonian H [Eq. (4)] as function of T for fixed μ = 0 (left of dashed line), and then as a function of μ for fixed T = 1 (right of dashed line). Complex-valued levels (formed by level coalescence at μ > 0) are not shown. Here μ0 = √N/2, and we set N = 10.PT-symmetric systems can have a real spectrum even when their Hamiltonian is non-hermitian, but develop a complex spectrum when the degree of non-hermiticity increases. Here we utilize random-matrix theory to show that this spontaneous PT-symmetry breaking can occur via two distinct mechanisms, whose predominance is associated to different universality classes. Present optical experiments fall into the orthogonal class, where symmetry-induced level crossings render the characteristic absorption rate independent of the coupling strength between the symmetry-related parts of the system.

http://arxiv.org/abs/1011.1385
Subjects: Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Optics (physics.optics)