Visualization of Branch Points inPT-Symmetric Waveguides

Abstract

The visualization of an exceptional point in a PT symmetric directional coupler(DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schrödinger perturbation theory we prove that the spectrum of a PT symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a PT symmetric DC to measure the radius of convergence for non PT symmetric structures. For such systems the physical meaning of the rather mathematical term: radius of convergence, is exemplified.In the past several years, following the seminal paper by Bender and Boettcher [1], non-hermitian PTsymmetric Hamiltonians have caught a lot of attention (see [2] and references therein). Under certain conditions PT -symmetric Hamiltonians can have a completely real spectrum and thus can serve, under the appropriate inner products, as the Hamiltonians for unitary quantum systems [3].Recently, the realization of PT -symmetric "Hamiltonians" has been studied using optical waveguides with complex refractive indices [4,5]. The equivalence of the Maxwell and Schrödinger equations in certain regimes provides a physical system in which the properties of PT -symmetric operators can be studied and exemplified.An extremely interesting property of PT -symmetric operators stems from the anti-linearity of the time symmetry operator. Consider a PT -symmetric operatorĤ, i.e., [PT ,Ĥ] = 0. Due to the non-linearity of T one cannot in general choose simultaneous eigenfunctions of the operators PT andĤ. However, if an eigenvalue of thê H is real then it's corresponding eigenfunction is a also an eigenfunction of the PT operator. This property has come to be known as exact/spontaneously-broken PTsymmetry. Exact PT -symmetry refers to the case when every eigenfunction of the PT symmetric operator is also an eigenfunction of the PT operator. In any other case the PT -symmetry is said to be broken.Usually, the transition between exact and spontaneously-broken PT symmetry can be controlled by a parameter in the Hamiltonian. This parameter serves as a measure of the non-hermiticity. An important class of PT -symmetric Hamiltonians are of the form: H(λ) = H 0 + iλV . Where H 0 (and V ) are real and symmetric(anti-symmetric) with respect to parity so that [PT ,Ĥ] = 0. When λ = 0 the Hamiltonian is hermitian and the entire spectrum is real. The spectrum remains real even when λ = 0 as long as λ < λ c . At this critical value and beyond, pairs of eigenvalues collide and become complex, see for example [6]. Bender et al. [7] showed that the reality of the spectrum is explained by the real secular equations one can write for PT -symmetric matrices. These secular equations will depend on the non-hermiticity parameter and, consequently, yield either real or complex solutions. Delabaere et al. [8] showed for the one-parameter family of complex cubic oscillators that pairs of eigenvalues cross each other at Bender and Wu branch points. Dorey et al. [9] after proving ...

Unidirectional nonlinearPT-symmetric optical structures

“…A surprising result associated with this class of problems is the possibility that such a PT symmetric Hamiltonian H can have an entirely real energy spectrum, despite the fact that, in general, they are non-Hermitian [7,8,[11][12][13][14][15][16][17][18][19][20]. For the specific example of the non-hermitian Hamiltonian of Eq.…”

Optical mesh lattices withPTsymmetry

“…In other words the index distribution must be an even function of position whereas the gain/loss must be anti-symmetric. Thus far, several works have pointed out that PTsymmetry can lead to altogether new optical dynamics which are otherwise impossible in standard passive optical arrangements [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. These may include for example the occurrence of abrupt phase transitions along with the appearance of the so-called exceptional points [24][25][26], power oscillations [18], breaking left-right symmetry and the occurrence of secondary emissions [18].…”

Graphene-assisted control of coupling between optical waveguides

“…The possibility of tuning losses in each waveguide by acting on a thin loss element permits symmetry of the system of coupled waveguides to be broken without introducing a strong perturbation of the single waveguide. Remarkably we demonstrate that tunable losses induced by graphene and a careful design of ridge waveguides allows to probe passive PT-symmetry breaking in directional couplers [11][12][13]. Moreover we prove that the exceptional point of the proposed structure can be dynamically reached by varying the applied voltage.…”

“…Beyond the critical loss supermodes coalesce, indeed they are characterized by equal propagation constant β and different losses. In particular, one supermode experiences increasing losses with increasing α, whereas the other one is characterized by the opposite trend [11,12]. When α is much larger than C, one supermode is characterized by losses which are close to those of a single lossy waveguide, whereas the other one tends to be loss-free (see next section).…”