Nonlinear resonance-assisted tunneling induced by microcavity deformation

Noncircular two-dimensional microcavities support directional output and strong confinement of light, making them suitable for various photonics applications. It is now of primary interest to control the interactions among the cavity modes since novel functionality and enhanced light-matter coupling can be realized through intermode interactions. However, the interaction Hamiltonian induced by cavity deformation is basically unknown, limiting practical utilization of intermode interactions. Here we present the first experimental observation of resonance-assisted tunneling in a deformed two-dimensional microcavity. It is this tunneling mechanism that induces strong inter-mode interactions in mixed phase space as their strength can be directly obtained from a separatrix area in the phase space of intracavity ray dynamics. A selection rule for strong interactions is also found in terms of angular quantum numbers. Our findings, applicable to other physical systems in mixed phase space, make the interaction control more accessible.


I. RESONANCE-ASSISTED TUNNELING IN A 2D SYSTEM
In this section, we recapitulate the resonance-assisted tunneling (RAT) theory for a twodimensional (2D) system in the framework of the standard secular perturbation theory and discuss its applicability to a 2D optical microcavity.
The Hamiltonian dynamics near nonlinear resonances can be described by using the secular perturbation theory [1,2]. For a 2D system, the Hamiltonian can be decomposed as H = H 0 (I 1 , I 2 ) + V (I 1 , I 2 , θ 1 , θ 2 ), (S1) in terms of action-angle variables. Here H 0 is an integrable Hamiltonian and V , which is a small perturbation, may contain nonintegrable terms. A resonance arises when the condition is satisfied for co-prime positive integers p and q. With a choice of a generating function we obtain canonical transformation from (I 1 , I 2 , θ 1 , θ 2 ) to (Î 1 ,Î 2 ,θ 1 ,θ 2 ) as For this transformation,θ 1 remains constant at the p : q resonance and varies slowly near the resonance whereasθ 2 varies rapidly there. It is then possible to "average" the Hamiltonian over θ 2 in order to obtain a transformed Hamiltonian to the first-order, SinceV is a 2π p -periodic function ofθ 1 , it can be expanded in series asV Note V (i) p:q generally falls off rapidly as i increases, so we can take only the leading terms with i = 0, 1. In addition, without loss of generality we can take ξ 1 = 0 by trivial constant shift inθ 1 .
By expanding H 0 around I p:q , the action at the resonance, to the non-vanishing lowest power of (I − I p:q ), ignoring the constant terms in H 0 andV and keeping the lowest-order terms in the perturbation, we can derive a pendulum-like effective Hamiltonian near the p : q resonance as where we have substituted I =Î 1 , θ =θ 1 and V p:q = V (1) p:q . We have also defined For two-dimensional billiard systems, the Hamiltonian can be described in the polar coordinate (r, θ). For a circular boundary shape, the angular coordinate θ and its conjugate momentum ka sin χ are the action-angular (θ 1 , I 1 ) whereas the radial coordinate r and its conjugate momentum are not action-angle variables. However, the radial coordinate can be transformed to an angle variable θ 2 in principle [3], because the motion in circular billiard is "periodic" in the radial coordinate, although we do not have to find that transformation for deriving Eq. (S7). Therefore, the formulation presented above can be readily applied to 2D billiard systems.  whereH ≡ H/( kc),Ṽ p:q ≡ V p:q /( kc),Ĩ p:q ≡ I p:q /( ka) andM p:q ≡ M p:q /( ka 2 /c). In this (s −Ĩ) phase space, the relation Eq. (S10) is scaled to bẽ

II. UNPERTURBED-BASIS MODES
The areaS p:q is readily obtained from the Poincarè surface of section in the Birkhoff coordinates.
In our experiment the gap of avoided crossing (AC) is measured in terms of size parameter ka. For high Q modes, the AC gap is approximately given by twice of the coupling strength g between two UBM's. Since an UBM |m with angular quantum number m has the angular dependence e i2πms , the coupling strength g between m and m + p modes is given by g = m|V p:q cos 2πps|m + p V p:q 1 0 e −i2πms cos 2πps e i2π(m+p)s ds The AC gap is then given by (AC gap) 2g V p:q = kcṼ p:q . (S13) In ka unit, we let c = a, and thus Exact calculation ofM p:q requires knowledge on the integrable Hamiltonian H 0 . Unfortunately, the exact form of H 0 is not known in general. In our work,M p:q is obtained from the data fitting with Eq. (S14). It shows that the AC gaps of modes associated with the same resonance structure (p : q) differ only by their ka values since they share the sameM p:q and S p:q . This feature elucidates the semiclassical nature of the resonance-assisted tunneling.
The interaction Hamiltonian has higher-order cosine terms like cos(ipθ) as shown in Eq.
(S6), so the UBM |m can couple to |m + ip mode with a coupling constant proportional toṼ (i) p:q or (Ṽ p:q ) i . Both amplitudes are much smaller thanṼ p:q . This is why the AC gap between l = 2 and 4 modes with ∆m = 12 = 2p (second order) in Fig. 1 in the text appear much weaker than that of l = 2 and 3 (or l = 3 and 4) with ∆m = p = 6 (first order). We solved the wave equation for the same size and shape as our liquid-jet microcavity by employing the boundary element method (BEM) [4]. The resulting quasi-eigenvalues or resonance-mode frequencies are presented in terms of the size parameter ka with k = 2π/λ the wavevector and a the mean radius of the cavity. Spatial intensity plots of the l=1, 2, 3 and 4 modes marked by (i), (ii), (iii) and (iv) in Fig. 2 are shown in Fig. S1. They are far from each other in resonance frequency, i.e., located in an uncoupled region. Their husimi functions are shown in Fig. S2. The angular mode number m is just the half of the number of antinodes in Fig. S1. The radial mode number l is the same as the number of anti-nodes in the radial direction. We can identify the resonance chain (thus p) involved in the interaction l = 1 mode is located above the p = 8 resonance chain. (ii) l = 2 mode is located below the p = 8 resonance chain but above the p = 6 resonance chain. (iii) l = 3 mode is located at or just below p = 6 resonance chain. (iv) l = 4 mode is located much below p = 6 resonance chain. As ka increases, the position of these modes shift upward gradually so that l = 3 mode become located just above p = 6 resonance chain when 150 < ka < 180 within the ka range of our investigation. by comparing the PSOS and the husimi functions of the involved modes in Fig. S2.