Here we demonstrate the impacts of emission mechanisms on the light confinements in open systems. Taking the oval-shaped cavities as examples, we show that the enhancements in quality (Q) factors are usually associated with the universal emissions. When the coupled resonances have similar far field patterns, the Q factor of the long-lived resonance has the possibility to be enhanced by the coherent destruction at the decay channels. Otherwise, the Q factors of long-lived resonances are usually reduced around the level crossings.
© 2014 Optical Society of America
When a quantum system interacts with the environment, the energies of the states will not be conserved and can vary in time due to the exchange with the external environment . In this case, the eigenvalues of quantum states can be written as complex-valued numbers En = ωn ± iγn, where γn determines the ability of confinement (or lifetime τn ∝ 1/γn) and ωn/2|γn| defines the Q factors. Understanding the decreasing of γn in open systems is of central importance for many research subjects, e.g. microcavities , quantum dots , nanodevices , quantum chaos , and cavity quantum electrodynamics . Some open systems are chaotic or partially chaotic. In such systems, most of the chaotic states have very short lifetimes. The relative long-lived (with small γn) chaotic modes are usually scarred along the unstable periodic orbits using wave localization . The regular resonances are typically well confined along the regular states such as stable islands and connect to the environment through chaotic assisted tunneling [8–10]. The enhancement in lifetime (decreasing of γn) is usually realized by suppressing the tunneling rate via dynamical localization  or destructive interference [11–13].
Very recently, the avoided resonance crossings (ARCs) in open systems have attracted enormous research interest [14, 15]. ARC is a very interesting phenomenon around the exceptional point [14, 16, 17] and has been widely studied in atoms or molecules in magnetic field , microwave cavities , and optical microcavities [4, 15, 17, 20, 21]. Quantum mechanically, the interaction between two quasi-bound states can be understood with a 2 × 2 non-Hermitian Hamiltonian matrix 
The eigenvalues E± are quite similar to Ej when the quasistates are far away from the ARC. The difference only happens in a narrow parameter region where |E1(Δ) − E2(Δ)| is smaller or on the size of . Usually two types of ARCs can be distinguished. One is strong coupling which consists of a frequency repulsion and a lifetime crossing. The other one is weak coupling with frequency crossing and lifetime repulsion. As the coupling constant is a complex number in open system, it is easy to find from Eq. (2) that the imaginary parts of E± can be tailored by VW, indicating the possibility to control the confinement of open systems [4, 15]. Introducing new decay channels through mode coupling [20,22] has been applied to form directional emissions of long-lived quasi-bound states. Most importantly, ARCs can be used to enhance the lifetimes of modes in microcavities [12, 15, 23] and nanocavities  through the destructive interference between coupled resonances.
Taking E1 = Δ − 0.01i, E2 = −0.001i, and VW = 0.00001 + 0.000026i as an example, we can easily find a level crossing and a repulsion in real [Fig. 1(a)] and imaginary [Fig. 1(b)] parts of eigenvalues respectively, clearly demonstrating the weak coupling . The enhancement of lifetime happens around Δ ∼ 0, where Im(E2) is significantly increased. Defining the enhancement as max(Im(E1,2))/max(im(E±)), we obtain an EF ∼ 10. Figure 1(c) shows the enhancement factor as a function of Im(VW). We can see that the enhancement factor EF can be increased several orders of magnitude at larger Im(VW), indicating the importance of the phases of coupling constant . Similar effects also hold true for strong coupling [see Figs. 1(d)–1(f)].
Compared with the mathematical definition in the toy model, constructing the imaginary part dominated is quite challenging in real systems. The influence of field distributions has been thoroughly studied and the mode numbers of resonances have been key parameters to change the field distribution to get the lifetime enhancements. In open systems, the decay channels, which determine the far field properties, are equally important as the field distributions. However, their influence on the lifetime enhancement has not been thoroughly studied.
Microdisks are attractive systems that confine light along the cavity boundary with total internal reflection. The resonances and their frequencies in microdisks play the roles of states and their energies [10, 24]. The applications of microdisks range from passive add-drop filters to cavity quantum electrodynamics. Moreover, due to their additional information at far field, microdisks have also been nice platforms to study quantum effects, e.g. dynamical tunneling [8, 9, 12, 22, 25], dynamical localzation , scars [26, 27], and local perturabtion . Interestingly, the recent studies on the ARCs in microdisks provided several valuable hints to the influences of emission mechanisms. In cases of lifetime increasing, all the resonances have similar far field patterns, e.g. the boundary wave leakages at the corners [15,23] or the refractive escape along the unstable manifolds [12, 29–31]. On the contrary, the reductions of lifetimes with ARCs are usually associated with diverse emission mechanisms [20, 22]. The long-lived resonances are confined along the boundary and emit evanescently, whereas the short-lived ones refractively escape caused by the reflection at the inner surface  or leakage along the unstable manifolds . In this paper, we will take the chaotic cavities as examples to thoroughly explore the influences of decay channels on the lifetime enhancement.
2. Numerical calculations and discussion
The selected microdisk has oval shaped cross section [see Fig. 2(a)], which is defined in polar coordinates ρ = R(1 + ε1cos2ϕ + ε2cos4ϕ + ε3cos6ϕ), where R and ε1–3 are the size and deformation parameters. Features of the internal ray dynamics in oval cavity have been thoroughly studied and can be described by using the Poincaré surface of section (SOS). The Poincaré SOS is a reduction of ray dynamics to a two-dimensional map describing successive bounces of rays with the bouncing positions (ϕ) and angles of incidences (χ). We show one example of Poincaré SOS in Fig. 2(b). Due to the large deformation in cavity shape, the Poincaré SOS of oval cavity is mostly chaotic, except for several stable islands. The dominant islands are period-2 islands around sinχ ∼ 0, period-4 islands around sinχ ∼ 0.15, and period-4 islands around sinχ ∼ 0.7. One important feature of oval cavity is there is only one set of period-4 islands at large χ [sinχ > 0.25] and it is surrounded by chaotic sea. The period-4 islands correspond to rectangle orbits, which can confine light with total internal reflection in most of materials ranging from polymer to GaAs . Thus the oval cavity can be a good platform to study the energy transportation between the stable islands and chaotic sea [8,9]. It is worth to note that the whispering gallery like orbits do exist but usually at very larger sinχ > 0.99. In real experiments, the resonances along such states can be simply excluded with selective pumping .
By applying effective refractive index n, microidisks can be simply treated as two-dimensional objects. Then the wave equations for transverse electric (TE, E is in plane) polarized modes Hz(x, y, t) = ψ(x, y)e−iωt can be replaced by the scalar wave equation2, 4]. We first assume n = 3.5 for silicon or GaAs. The calculated results are plotted in Fig. 3(a). Within a kR range from 63 to 66, three types of long-lived resonances with equal mode spacings can be clearly observed. Figures 3(b)–3(d) show the field distributions of the resonances marked as i – iii in Fig. 3(a) as examples. The longest-lived resonances (open squares) are the fundamental rectangle modes [see Fig. 3(b) for an example]. Most of the Q factors are above 106 and some particular value reaches 109. These high Q factors are consistent with the light confinement along the period-4 stable islands by total internal reflection. The second long-lived resonances (open triangles) are also confined by total internal reflections along period-4 stable islands [see Fig. 3(d)]. Their relative low Q factors are caused by the shorter tunneling distance  of higher order rectangle modes. The open circles represent the diamond resonances in chaotic sea that are confined along unstable periodic orbits in chaotic sea by wave localizations .
As mentioned above, the rectangle resonances are well confined by the total internal reflection. Their decay channels are usually dominated by the assistance from chaotic states . Thus coupling to a chaotic mode can be a nice way to tailor the light confinement. Below, we take the rectangle and diamond resonances as examples to study the suppression of γn with ARCs. As the rectangle resonance and the diamond resonance are confined within different places, their dependencies on the cavity shape are different. Then two resonances can approach one another with the variation of shape deformation. The interaction with resonance-3 is neglected here because resonance-3 is far away from the rectangle and diamond modes in the interesting range of shape deformation. Figures 4(a) and 4(b) show the dependence of their frequencies and Q factors on ε3. With the increasing of ε3 from 0.0108 to 0.0113, the frequencies of rectangle resonance and diamond modes changes reversly and cross at ε3 ∼ 0.01096. Figure 4(c) shows the field distributions (|Hz|) of modes 1–6 marked in Fig. 4(b). The modes on rectangle (or diamond) branch have very similar field patterns and no identities exchange has been observed at the crossing point, indicating the occurrence of weak coupling. One very important feature happens in Fig. 4(b), where the Q factor of rectangle resonance is enhanced around the crossing point. The maximum Q factor is ∼ 4×109, which is more than two orders of magnitude higher than the resonance at ε3 = 0.0108.
Associated with the enhancement in Q factors, we find that the far field patterns of rectangle resonances also change. When the resonances are far away from the crossing points, their far field patterns are mainly along ϕFF = ±90°. When the resonances approach the crossing point, the situation changes gradually. the far field patterns will transit from bi-directional emission [Fig. 5(b)] to multiple directional outputs [Fig. 5(f)]. The emissions along ϕFF = ±45°, ±135° become comparable to the leakages along unstable manifolds.
The bi-directional outputs of rectangle resonances are understandable. For the light confined within stable islands, it has two type of decay channels. One is the direct tunneling which is the evanescent escape along the tangential lines at four bouncing points of the rectangle orbits [10, 29]; the other one is the dynamical tunneling to the chaotic sea which is followed by the refractive escape along the unstable manifolds. As the later one is usually orders of magnitude larger due to its smaller tunneling distance, it usually dominates the far field emissions along ϕFF = ±90° [see Fig. 5(b) for an example]. The emissions along ϕFF = ±45°, ±135° are similar to the evanescent leakages at the bouncing points. However, this cannot be confirmed by from the far field patterns.
To get clear view of the emission mechanisms, we have projected the wave functions on the cavity boundary to phase space via the Husimi function [32, 33]. The corresponding Husimi maps of rectangle and diamond modes are plotted in Figs. 5(a), 5(c), and 5(e). The horizontal and vertical axes correspond to the bouncing positions and incident angles of resonances. Figure 5(a) shows the phase space structure of mode-1 in Fig. 4(b). The light is well confined in the stable islands. The leakages below the critical lines happen at ϕ = 0 and ϕ = π, which are marked as α and β. The connections between the leakages and the stable islands are the well know chaotic layer  and the unstable manifolds . When the resonance approach the crossing point, its phase space structure changes [see Fig. 5(e)]. While the main light localization is still similar to Fig. 5(a), the emission mechanism is significantly changed. The leakages at positions i – iv increase and become comparable to the emissions following the unstable manifolds. As the positions i – iv are right below the stable islands, they are consistent with the far field pattern in Fig. 5(f) and clearly demonstrate the appearance of direct tunneling. It is well known that dynamical tunneling is usually orders of magnitude higher than the direct tunneling [8, 9]. Then the changes in emission mechanism means that the dynamical tunneling is suppressed by orders of magnitude, similar to the coherent destruction of dynamical tunneling [11–13].
Here we can recall the previous studied on ARCs in optical microcavity. In deformed limaçon microdisk, long-lived resonances are whispering-gallery like and emit at the positions with largest curvatures; the low Q modes fall in chaotic sea and follow the unstable manifolds . In this case, the Q factors of high Q modes are reduced around the ARCs. In case of resonances with universal outputs, e.g. the square  and hexagon  where the emissions are dominated by the boundary wave leakage and pseudointegrable leakage, the Q factors of high Q modes are enhanced, just like the reports in this paper. In this study, the Husimi map of far field pattern in Figs. 5(c) and 5(d) show that the emission mechanism of diamond mode away from crossing point is quite similar to rectangle mode. Based on the previous reports and this research, we thus conjugate that the universal outputs are important for the enhancement in Q factors.
To further support this hypothesis, we constructed two coupled resonance with similar field distributions but different far field patterns. Following the analysis, the Q factor at the ARCs should be reduced due to their different emission mechanisms. As the field distributions and their ray dynamics are both sensitive to the cavity shape, we obtain these modes by changing the refractive index while keeping the oval shape and the mode orders. Here we set the refractive index n = 2.0 for ZnO as an example. The field distributions of two resonances are shown in Figs. 6(c) and 6(d). They are also the rectangle resonance and diamond resonance with the same mode orders as Figs. 3(b) and 3(c). Figures 6(a) and 6(b) show the behaviors of resonant frequencies and Q factors of two resonances as a function of shape deformation ε3. Similar to the results in Fig. 4, the frequencies of rectangle resonance and diamond mode are directly and inversely proportional to the deformation parameter ε3, respectively. Two resonances approach each other and cross at ε3 ∼ 0.011. The important phenomenon also happens in the Q factors. In contrast to the enhancement of lifetime in Fig. 4(b), the Q factor in Fig. 6(b) is obviously reduced around the crossing point, supporting our guess very well. As the mode orders of rectangle and diamond resonances are similar to Figs. 3(b) and 3(c), there must be other mechanism that dominates light confinement in stable islands.
To reveal such mechanism, we have also calculated the Husimi maps of resonances in oval cavity with ε3 = 0.0108 and compared them with Figs. 5(a) and 5(c). The computed results are plotted them in Figs. 7(a) and 7(c). For the rectangle modes, the light is confined by total internal reflection along the period-4 stable islands. Similar to Fig. 5(c), the wave localization along the unstable periodic orbit can also be observed in Fig. 7(c). The significant difference happens in the far field emissions. As the critical line is increased to sinχ = 0.5, it cuts the chaotic layer directly. Thus the tunneled light escapes quickly at points i – iv in Fig. 7(a) instead of following the unstable manifolds. In case of the diamond resonance, the bouncing points around ϕ = 0 and ϕ = π are also cut by the critical line. The differences in the emission mechanisms can also be observed in far field patterns. In Figs. 7(b) and 7(d), we can clearly see that the directionality of outputs beams of rectangle resonance is quite different from those of the diamond modes. Then we know that both the starting points on the cavity boundary and the directionality at far field are different at low refractive index such as n = 2.0. Thus the interference between the leakages are very tiny both inside (see the Husimi map in Figs. 7(a) and 7(b)) and outside (see the far field patterns in Figs. 7(c) and 7(d)) the cavity.
Then the enhancement in lifetime can be qualitatively explained. In the open systems, the increasing in lifetimes means the decay channel to the environment is partially blocked. Such blocking is usually formed by the coherent destruction between two wavefunctions at the decay channels. In general, the interaction between two wavefunctions requires both spatial and spectral overlaps. The spectral overlap can be easily fulfilled around the crossing point. The spatial overlap at the decay channels is determined by their emission mechanisms. Therefore, the emission mechanism not only affects the far field patterns, but also has strong impact on the confinement around the crossing point. When the emission mechanisms of two states are different as Fig. 6, while the frequency and phases can be matched, the spatial overlapping, which can be retrieved by using the time reversal symmetry, is too small to generate significant coherent destruction. Once the emission mechanisms are similar, two wavefunctions have enough spatial overlapping right at the decay channel and thus can coherently reduce the decay channels if the phase difference is also matched .
The above analysis can naturally give the following extensions in chaotic cavities. At high refractive index, as the decay of rectangle resonance is dominated by dynamical tunneling and the same unstable manifolds as diamond resonance, its Q factor can be enhanced by the destructive interference between two resonances along the unstable manifolds. At low refractive index, as the decay channel of rectangle resonance turns to direct tunneling and to be quite different from unstable manifolds, the mode coupling will reduce the Q factor of rectangle resonance. The transition from enhancement to reduction in Q factors should happen at the refractive index where the universal emission mechanism has been destroyed.
To verify this analysis, we have calculated the behaviors of mode couplings at different refractive indices. The results are summarized in Fig. 8(a). At high refractive index n ∼ 2.7 – 3.5, the Q factors of rectangle resonances around the crossing point are increased. At low refractive index n < 2.3, the Q factor is reduced. The transition from enhancement to reduction happens at n ∼ 2.3 – 2.5, where no obvious changes can be found in Q factors. Figure 8(b) shows the far field patterns of resonance away from the crossing point. Here the U is defined as . When n is ∼ 2.7 – 3.5, U > 0 and it means the main emissions away from crossing point are following ϕFF = ±90°; at small refractive index, U < 0 and it means the main outputs along ϕFF = ±45°, ±135°. Consistence with Fig. 8(a), here the transition from bi-directional outputs to four directional outputs also happens at n ∼ 2.5, where no obvious change can be found in Q factors. At n ∼ 2.5, the direct tunneling and dynamical tunneling of rectangle resonance are comparable. This can be confirmed by the far field pattern of rectangle resonance in Fig. 8(c), where the emission along ϕFF = ±90° is suppressed and becomes to be similar to the intensity along ϕFF = ±45°, ±135°, confirming the coexistence of two types of leakages. As two emission mechanisms have reversed effects on the light confinement, their total effect on the Q factor can be roughly cancelled and no obvious Q change can be found.
Overall, in additional to the simple definition of coupling constant, we have studied the importance of emission mechanism on the formation of extremely long-lived resonances. In examples of oval-shaped chaotic cavities, we have shown that the enhancements of Q factors at ARCs are associated with the universal emissions along the unstable manifolds. The corresponding Husimi maps and far field patterns show that the destructive interference happens in the decay channels and inhibits the leakages. For a comparison, we show that the Q factor of long-lived resonance is reduced with the same cavity shape and mode orders once the emission emissions are broken. All these results clearly demonstrate the importance of universal emissions on the formation of long-lived resonances in open systems. We believe that our research should be important for the pursuit of long-lived resonances in open systems. It will also be interesting for the fundamental researches such as coherent destruction of tunneling.
This work is supported by NSFC 11204055, 61222507, 11374078, NCET-11-0809, KQCX2012080709143322, and KQCX20130627094615410.
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