Integrable approximation of regular regions with a nonlinear resonance chain

Kullig, Julius; Löbner, Clemens; Mertig, Normann; Bäcker, Arnd; Ketzmerick, Roland

doi:10.1103/physreve.90.052906

Phys. Rev. E

2014

DOI: 10.1103/physreve.90.052906

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Integrable approximation of regular regions with a nonlinear resonance chain

Julius Kullig

Clemens Löbner

Normann Mertig

et al.

Abstract: Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the non-integrable system. In … Show more

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Cited by 6 publications

(16 citation statements)

References 39 publications

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“…Semiclassical theories exist only for time-domain quantities [23,24], cases when resonances are irrelevant [39], and near-integrable systems [22,46,47]. On the other hand, quantitatively accurate predictions of γ m [43,48] explicitely require integrable approximations [31,37,38,49] which needs some numerical effort.In this paper we establish an intuitive, semiclassical,Re q |Im p| Re p −0.4 0.4 0.1 0.9 I m I I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat I rat p rat p A 2 T γ rat γ d 1/h γ 0 FIG. 1.…”

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confidence: 99%

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“…Overview -Our method is based on a semiclassical evaluation of a recently developed non-perturbative prediction of γ m [43]. At its heart is an integrable approximation of the regular region, which includes the relevant nonlinear resonance chain [49]. In that, we justify and generalize the use of semiclassical techniques developed for near-integrable systems [22,46,47] in the wider class of generic systems with a mixed phase space.…”

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confidence: 99%

“…Integrable approximation -The key tool for deriving our prediction is an integrable approximation. It is a one degree of freedom time-independent Hamiltonian, which resembles the regular dynamics of the original system [49]. It is based on the universal description of the classical dynamics in the vicinity of a r:s resonance by the pendulum Hamiltonian [21,22,46,49]…”

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confidence: 99%

“…r:s (θ, I) = H 0 (I) + 2V r:s I I r:s r/2 cos (rθ),(2)using action-angle coordinates of H 0 (I). It is determined by the frequencies ω 0 (I) of tori in the co-rotating frame of the resonance, as ω 0 (I) = ∂ I H 0 (I)[49], where H 0 (I) ≈ (I − I r:s ) 2 /(2M r:s ) close to the resonant torus I r:s . The quantities I r:s , V r:s , and M r:s can be computed from the position and the size of the resonance chain and the linearized dynamics of its central orbit[27,49].…”

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Complex-path prediction of resonance-assisted tunneling in mixed systems

Fritzsch¹,

Bäcker²,

Ketzmerick³

et al. 2017

Phys. Rev. E

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We present a semiclassical prediction of regular-to-chaotic tunneling in systems with a mixed phase space, including the effect of a nonlinear resonance chain. We identify complex paths for direct and resonance-assisted tunneling in the phase space of an integrable approximation with one nonlinear resonance chain. We evaluate the resonance-assisted contribution analytically and give a prediction based on just a few properties of the classical phase space. For the standard map excellent agreement with numerically determined tunneling rates is observed. The results should similarly apply to ionization rates and quality factors.Tunneling through energetic barriers is a textbook paradigm of quantum mechanics. While classically motion is confined to either side of the barrier, wave functions exhibit contributions on both sides. In contrast, nature often exhibits confinement on dynamically disjoint regions of regular and chaotic motion in a mixed phase space, see Fig. 1(a). Here, a classical particle follows a trajectory of regular motion while the correponding wave function admits an exponentially small contribution on the chaotic region. This phenomenon is called dynamical tunneling [1,2].Until today dynamical tunneling has emerged in many fields of physics. It determines the vibrational spectrum of molecules [1], ionization rates of atoms in laser fields [3,4], and chaos-assisted tunneling oscillations [5,6] in cold atom systems [7,8]. In optics dynamical tunneling is experimentally explored in microwave resonators [9][10][11][12] as well as microlasers [13][14][15][16][17][18][19][20], where it determines the quality factor of lasing modes. Here, a recent experimental breakthrough [12,17] is the measured enhancement of dynamical tunneling due to nonlinear resonance chains [21,22].To reveal the universal features of dynamical tunneling it is extensively studied theoretically [10,13, mainly in model systems. A central object is the tunneling rate γ m , which describes the transition from a state on the mth quantizing torus of the regular region into the chaotic sea. Qualitatively γ m can be understood from the theory of resonance-assisted tunneling [21,22,27,44], see dashed line in Fig. 1(b): On average γ m decreases exponentially for decreasing wavelength or decreasing effective Planck constant, i.e. Plancks constant scaled to some typical action of the system. In addition a drastic enhancement of γ m is observed for some values of h. This is due to resonant coupling of regular states, induced by a nonlinear resonance chain [45] within the regular region, see Fig. 1 In this paper we establish an intuitive, semiclassical,

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confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Complex-path prediction of resonance-assisted tunneling in mixed systems

Fritzsch¹,

Bäcker²,

Ketzmerick³

et al. 2017

Phys. Rev. E

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Frequency splittings in deformed optical microdisk cavities

et al. 2017

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Qspoiling in deformed optical microdisks due to resonance-assisted tunneling

Kullig

Wiersig

2016

Phys. Rev. E

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A recent experiment by Kwak et al. [Sci. Rep. 5, 9010 (2015)10.1038/srep09010] demonstrated the relevance of resonance-assisted tunneling for optical microcavities where resonance chains emerge in phase space due to boundary deformations. In this paper we adapt the perturbative description of resonance-assisted tunneling to calculate optical modes and the imaginary part of their complex wavenumber which determines the lifetime of the mode. We demonstrate our method at three example cavity shapes and compare our results to numerical data and perturbation theory for weakly deformed microdisk cavities.

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Integrable approximation of regular regions with a nonlinear resonance chain

Cited by 6 publications

References 39 publications

Complex-path prediction of resonance-assisted tunneling in mixed systems

Complex-path prediction of resonance-assisted tunneling in mixed systems

Frequency splittings in deformed optical microdisk cavities

Qspoiling in deformed optical microdisks due to resonance-assisted tunneling

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