Quasiclassical surface of section perturbation theory

Prange, R. E.; Narevich, R.; Zaitsev, Oleg

doi:10.1103/physreve.59.1694

Cited by 18 publications

(27 citation statements)

References 57 publications

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“…This model has rich and complex dynamics and finds various applications, e.g. for dynamical localization in billiards [8][9][10]. Our algorithm, based on the Quantum Fourier Transform (QFT) [11], simulates the dynamics of a system with N levels in O((log 2 N )…”

Section: Efficient Quantum Computing Of Complex Dynamicsmentioning

confidence: 99%

Efficient Quantum Computing of Complex Dynamics

et al. 2001

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We propose a quantum algorithm which uses the number of qubits in an optimal way and efficiently simulates a physical model with rich and complex dynamics described by the quantum sawtooth map. The numerical study of the effect of static imperfections in the quantum computer hardware shows that the main elements of the phase space structures are accurately reproduced up to a time scale which is polynomial in the number of qubits. The errors generated by these imperfections are more dangerous than the errors of random noise in gate operations.PACS numbers: 03.67. Lx, 05.45.Mt, 24.10.Cn When applied to computation, quantum mechanics opens completely new perspectives: a quantum computer, if constructed, could perform certain computations faster than classical computers, exploiting quantum mechanical features such as entanglement or superposition [1]. Shor discovered a quantum algorithm [2] which factorizes large integers exponentially faster than any known classical algorithm. It was also shown by Grover [3] that the search of an item in an unstructured database can be done with a square root speed up over any classical algorithm. However, at present only a few quantum algorithms are known which simulate physical systems with exponential efficiency. Such systems include certain many-body problems [4], spin lattices [5], and models of quantum chaos [6,7].In this Letter, we present a quantum algorithm which computes the time evolution of the quantum sawtooth map, exponentially faster than any classical computation. This model has rich and complex dynamics and finds various applications, e.g. for dynamical localization in billiards [8][9][10]. Our algorithm, based on the Quantum Fourier Transform (QFT) [11], simulates the dynamics of a system with N levels in O((log 2 N )2 ) operations per map iteration, while a classical computer, which performs Fast Fourier Transforms (FFT), requires O(N log 2 N ) operations. A further striking advantage of the algorithm is the optimum utilization of qubits: one needs only n q = log 2 N qubits (without any extra work space). We demonstrate that complex phase space structures can be simulated with less that 10 qubits, while about 40 qubits would allow one to make computations inaccessible to present-day supercomputers. This is particularly important, since experiments with few qubits are being performed at present [12,13], in particular the QFT was implemented on a three qubit nuclear magnetic resonance quantum computer [14]. For this reason the investigation of this interesting physical system will be accessible to first quantum computers, operating with few qubits and for which large-scale computations like integer factoring are not possible.The classical sawtooth map is given bywhere (n, θ) are conjugated action-angle variables (0 ≤ θ < 2π), and the bars denote the variables after one map iteration. Introducing the rescaled variable p = T n, one can see that the classical dynamics depends only on the single parameter K = kT , so that the motion is stable for −4 < K < 0 and com...

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Section: Efficient Quantum Computing Of Complex Dynamicsmentioning

confidence: 99%

Efficient Quantum Computing of Complex Dynamics

et al. 2001

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“…[25]. We now adapt this procedure to dielectric cavities and in particular to the complex setting occupied by evanescent escape.…”

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“…We leave discussion of this more complicated case to a future presentation [26] and concentrate here on the leading contribution (5), noting that it correctly captures the main qualitative behavior even where amplitude terms are needed for detailed quantitative agreement. For (1) to hold, g(θ ) should satisfy [25,26] g…”

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Differences between emission patterns and internal modes of optical resonators

Creagh

White

2012

Phys. Rev. E

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The evanescent wave field outside an optical resonator is typically strongly directional when the shape deviates even very slightly from being perfectly circular or spherical. In this Rapid Communication we show that the tunneling mechanism underlying escape from such weakly deformed resonators can lead to emission patterns that look quite different to the corresponding internal mode. A direct short-wavelength analysis is not possible because the required complex ray data cannot be found due to the presence of natural boundaries. We show, however, that an approach based on perturbative approximation of the ray families successfully describes this phenomenon. In appropriate limits, the emission pattern allows one effectively to perform a direct observation of solutions of the one-dimensional Schrödinger equation in the complex plane.

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“…Whether any non-perturbative structure in the ray dynamics can be resolved by the wave field, depends on kR [16]: a directionality measurement will be able to distinguish the peaks at φ = 90 • ± 45 • if the conjugate angular momentum m satisfies the uncertainty relation ∆φ ∆m ≥ 1/2 with ∆φ ≈ π/4. This implies ∆m > 2/π ≈ 1, which in our spheroids translates to a fluctuating angle of incidence of ∆ sin χ = ∆m/(nkR) ≈ 10 −3 .…”

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Directional Tunneling Escape from Nearly Spherical Optical Resonators

et al. 2003

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We report the surprising observation of directional tunneling escape from nearly spherical fused silica optical resonators, in which most of the phase space is filled with non-chaotic regular trajectories. Experimental and theoretical studies of the dependence of the far-field emission pattern on both the degree of deformation and the excitation condition show that non-perturbative phase space structures in the internal ray dynamics profoundly affect tunneling leakage of the whispering gallery modes.Electromagnetic fields in a uniform dielectric sphere can be calculated in much the same way as quantum mechanical wave functions in a spherically symmetric potential. The sphere exhibits whispering-gallery (WG) modes, which are long-lived resonances with electromagnetic fields confined near the surface [1]. For small deviations from the spherical shape, ǫ ≡ (R max − R min )/(R max + R min ) ≪ 1 (where R max , R min are the maximal and minimal radii), perturbative treatments [2] are routinely employed to infer deformation parameters from the splitting of azimuthal degeneracy of the WG modes [3]. Certain strongly non-spherical resonators, on the other hand, can be analyzed with methods from quantum chaos such as random-matrix theory or periodicorbit expansions [4]. Many generic resonator shapes, however, fall into a transition regime in which none of these known methods apply globally. When entering this regime from the perturbative side, calculations may encounter singularities and undefined limits [5]. Experimental studies in this regime can thus provide insight into how nature resolves the competition between perturbative and non-perturbative physics, here with the resonator shape as a control parameter.In this paper we present studies of far-field emission patterns and resonance lifetimes of deformed fused-silica microspheres. The experimental and theoretical studies of the dependence of the far-field emission pattern on both the degree of deformation and the excitation condition show that highly directional tunneling escape can occur in microspheres with small deformation (ǫ ≈ 1%) and with a large size parameter (kR = 2πR/λ ≈ 785, where λ ≈ 800nm, R ≈ 100µm). These results are completely unexpected in the ray optics limit or in the earlier perturbative wave treatment with ǫ as a small parameter. The observed emission pattern shows that on one hand, the ray model breaks down in predicting the escape mechanism of WG modes for resonators with small deformation. On the other hand, the nonperturbative phase space structures predicted by the ray model can profoundly affect tunneling leakage of the WG modes. Even at extremely small deformation, nonperturbative phase space structures still persist and are directly responsible for directional tunneling escape. This intricate interplay between ray and wave dynamics provides essential physical insights into properties of weakly deformed WG resonators, especially, for those in which the phase space is filled with non-chaotic regular trajectories. Directional emission pattern...

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Quasiclassical surface of section perturbation theory

Cited by 18 publications

References 57 publications

Efficient Quantum Computing of Complex Dynamics

Efficient Quantum Computing of Complex Dynamics

Differences between emission patterns and internal modes of optical resonators

Directional Tunneling Escape from Nearly Spherical Optical Resonators

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