Formation of Long-Lived, Scarlike Modes near Avoided Resonance Crossings in Optical Microcavities

Abstract

We study the formation of long-lived states near avoided resonance crossings in open systems. For three different optical microcavities (rectangle, ellipse, and semi-stadium) we provide numerical evidence that these states are localized along periodic rays, resembling scarred states in closed systems. Our results shed light on the morphology of long-lived states in open mesoscopic systems. 05.45.Mt, 32.80.Rm Understanding the properties of long-lived, quasi-bound states in open mesoscopic systems is of central importance for many research subjects, e.g. semiconductor ballistic quantum dots [1,2,3,4], photoionization of Rydberg atoms [5], microwave systems [6,7], quantum chaos [8], and optical microcavities [9,10,11,12,13,14,15]. In several of these studies the long-lived states are scarred. The original scarring phenomenon has been discovered for closed systems in the field of quantum chaos [16]. It refers to the existence of a small fraction of eigenstates with strong localization along unstable periodic orbits of the underlying classical system. In open systems, however, scarred states seem to be the rule rather than the exception. The nature of the mechanism behind this scarlike phenomenon is not yet understood.Avoided level crossings in closed or conservative systems are discussed in textbooks on quantum mechanics. They occur when the curves of two energy eigenvalues, as function of a real parameter ∆, come near to crossing but then repel each other [17]. This behaviour can be understood in terms of a 2 × 2 Hamiltonian matrixFor a closed system this matrix is Hermitian, thus the energies E j are real and the complex off-diagonal elements are related by W = V * . The eigenvalues of the coupled system,differ from the energies of the uncoupled system E j only in a narrow parameter region where the detuning from resonance, |E 1 (∆) − E 2 (∆)|, is smaller or of the size of the coupling strength √ V W . The parameter dependence of V and W can often be safely ignored.The matrix (1) also captures features of avoided resonance crossings (ARCs) in open or dissipative systems if one allows for complex-valued energies E j . The imaginary part determines the lifetime τ j ∝ 1/Im(E j ) of the quasi-bound state far away from the avoided crossing |E 1 −E 2 | 2 ≫ V W , where the off-diagonal coupling can be neglected. Keeping the restriction W = V * allows for two different kinds of ARCs [18]. For 2|V | > |Im(E 1 ) − Im(E 2 )|, there is an avoided crossing in the real part of the energy and a crossing in the imaginary part. At resonance Re(E 1 ) = Re(E 2 ) the eigenvectors of the matrix (1) are symmetric and antisymmetric superpositions of the eigenvectors of the uncoupled system. If one of the latter corresponds to a localized state then such an ARC leads to delocalization and lifetime shortening [19]. For 2|V | < |Im(E 1 ) − Im(E 2 )|, there is a crossing in the real part and an avoided crossing in the imaginary part. This kind of ARC has been exploited to design optical microcavities with unidirectional light emission ...

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…Numerical simulations indicate that scarring in optical microcavities with strongly chaotic ray dynamics is rather the rule than the exception; see, e.g., Fang et al, , 2005bHarayama et al, 2003;Lee et al, 2002Lee et al, , 2007cLee et al, , 2004Lee et al, , 2005Rex et al, 2002;Wiersig, 2006;Wiersig et al, 2010). This conclusion can also be drawn from studies of open quantum maps (Ermann et al, 2009;Wisniacki and Carlo, 2008).…”

“…In the strong coupling regime one can observe in the near-field of the cavity, beside the conventional hybridization of modes (Carmon et al, 2008), a surprising phenomenon, the localization of intensity along special marginal stable periodic rays (Unterhinninghofen et al, 2008;Wiersig, 2006;Yi et al, 2011); see Fig. 28.…”

“…While Wiersig (2003b) and Kudo et al (2013) concluded that rounding the corners increases the Q-factor, Dietrich et al Other dielectric polygonal cavities have been studied as well, such as squares (Chen et al, 2006;Guo et al, 2003;Lohmeyer, 2002;Poon et al, 2001), rectangles (Wiersig, 2006), regular pentagons (Bogomolny et al, 2011;Lebental et al, 2007), equilateral triangles (Lai et al, 2007;Wysin, 2005), regular dodecagons (Nobis et al, 2007), and octahedrons (Korthout et al, 2009).…”

Lasing action from photonic bound states in continuum

“…These eigenvalues exist in a multidimensional space (hyperspace), but in a given frequency range, the investigation can be reduced to a finite number of variables, limiting the complexity of the effective Hamiltonian [7]. When eigenvalues come close to crossing as a function of a geometrical parameter that modifies the system, avoided resonance crossing (ARC) occurs, i.e., eigenvalues repel each other in the entire complex plane [17][18]. Friedrich and Wintgen showed that resonantly trapped BICs represent a particular type of ARC for which coupling occurs predominantly in the far field.…”

Short periodic orbit approach to resonances and the fractal Weyl law

“…This fractal Weyl law has been investigated in several systems [3][4][5][6][7][8][9]. One the other hand, the (right) eigenfunctions are supported by the unstable manifold of this repeller [10,11], and its scarring properties have also been under investigation [12][13][14].…”