Trapping of particles by lasers: the quantum Kapitza pendulum

Gilary, Ido; Moiseyev, Nimrod; Rahav, Saar; Fishman, Shmuel

doi:10.1088/0305-4470/36/25/101

Cited by 44 publications

(60 citation statements)

References 19 publications

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“…IV assumes high frequencies. A more complete study of this specific system and a discussion regarding the physical implications of this resonance are given in [41].…”

Section: Scattering From An Oscillating Gaussian Potentialmentioning

confidence: 99%

Effective Hamiltonians for periodically driven systems

2003

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The dynamics of classical and quantum systems which are driven by a high frequency (ω) field is investigated. For classical systems the motion is separated into a slow part and a fast part. The motion for the slow part is computed perturbatively in powers of ω −1 to order ω −4 and the corresponding time independent Hamiltonian is calculated. Such an effective Hamiltonian for the corresponding quantum problem is computed to order ω −4 in a high frequency expansion. Its spectrum is the quasienergy spectrum of the time dependent quantum system. The classical limit of this effective Hamiltonian is the classical effective time independent Hamiltonian. It is demonstrated that this effective Hamiltonian gives the exact quasienergies and quasienergy states of some simple examples as well as the lowest resonance of a non trivial model for an atom trap. The theory that is developed in the paper is useful for the analysis of atomic motion in atom traps of various shapes.

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“…IV assumes high frequencies. A more complete study of this specific system and a discussion regarding the physical implications of this resonance are given in [41].…”

Section: Scattering From An Oscillating Gaussian Potentialmentioning

confidence: 99%

Effective Hamiltonians for periodically driven systems

2003

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“…Similarly to the classical case, in the high modulation regime such a potential can be approximated by an effective time-independent potential [5,6] …”

Section: Imaginary Kapitza Pendulummentioning

confidence: 99%

“…It was first explained and demonstrated in classical physics by Pyotr Kapitza in 1951, who showed that an inverted pendulum can be stabilized by the addition of a vertical vibration [2]. Later nonlinear and quantum analogues of this phenomenon were studied in several papers [3][4][5][6] and found important applications, for example in Paul traps for charged particles [7], in driven bosonic Josephson junctions [8], and in nonlinear dispersion management and diffraction control of light in optics [3,9]. The main result underlying Kapitza stabilization is that the motion of a classical or quantum particle in an external rapidly oscillating potential can be described at leading order by an effective time-independent potential, which shows a local minimum (a well) while the non-oscillating potential did not.…”

Section: Introductionmentioning

confidence: 99%

Imaginary Kapitza pendulum

2013

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We extend the theory of Kapitza stabilization within the complex domain, i.e. for the case of an imaginary oscillating potential. At a high oscillation frequency, the quasi-energy spectrum is found to be entirely real-valued, however a substantial difference with respect to a real potential emerges, that is the formation of a truly bound state instead of a resonance. The predictions of the Kapitza averaging method and the transition from a complex to an entirely real-valued quasi-energy spectrum at high frequencies are confirmed by numerical simulations of the Schrödinger equation for an oscillating Gaussian potential. An application and a physical implementation of the imaginary Kapitza pendulum to the stability of optical resonators with variable reflectivity is discussed.

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“…The unique feature of Kapitza pendulum is that the vibrating suspension can cause it to balance stably in an inverted position, with the bob above the suspension point and can be employed to different problems in applied sciences. For instance, in Quantum Mechanics, the trapping of particles [1][2][3], or in Mechanical engineering, the control of robotic devices [4,5]. Kapitza oscillator has different structure of its stable points under the influence of rapidly oscillating periodical force.The most traditional way to find out the stable points, is the method of effective potential energy proposed by Kapitza himself [6].…”

Section: Introductionmentioning

confidence: 99%

“…Now since the eigenvalues are real, unequal, and of opposite signs so the origin is a saddle point and therefore the fixed point is unstable for nonlinear equation (4). If …”

Section: Introductionmentioning

confidence: 99%

Lyapunov Spectra for Kapitza Oscillator

Iqbal

Ahmad²,

Hussain³

2012

JIMS

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Abstract.Here we purpose a simple but realistic model of one dimensional nonlinear Kapitza oscillator driven by sin-or cos-rapidly external oscillating periodical force. The model has a parameter 2gl/a 2 γ 2 of dimension one, depending on the amplitude a and frequency of modulation γ. Changing its value we construct phase portraits of the system in the neighbourhood of fixed points and demonstrate the changing in Lyapunov spectrum. Our purpose is to observe the behavior of system at fixed points due to the different structures of the Lyapunov spectra.Key words and Phrases: Kapitza oscillator, Lyapunov spectrum, Lyapunov spectra. Abstrak. Pada paper ini disajikan sebuah model sederhana tapi realistis dariosilator Kapitza nonlinear dimensi satu yang dikendalikan oleh gaya luar berosilasi periodik sin-dan cos-dengan cepat. Model ini mempunyai parameter 2gl/a 2 γ 2 satu dimensi, bergantung pada amplitudo a dan frekuensi modulasi γ. Dengan merubah nilai parameternya, kami mengkonstruksi diagram fasa dari sistim pada ketetanggaan titik tetap dan mendemonstrasikan perubahan spektrum Lyapunov.Kemudian, kami menyajikan pengamatan dari tingkah laku sistim pada titik tetap dikaitkan dengan struktur dari spektral Lyapunov.

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Trapping of particles by lasers: the quantum Kapitza pendulum

Cited by 44 publications

References 19 publications

Effective Hamiltonians for periodically driven systems

Effective Hamiltonians for periodically driven systems

Imaginary Kapitza pendulum

Lyapunov Spectra for Kapitza Oscillator

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