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Physical Review A

Observation of the Hopf bifurcation in parametrically driven trapped atoms

Kihwan Kim, Heung‐Ryoul Noh, Young-Hee Yeon, Wonho Jhe

Published Sep 25, 2003• 23 citations
Publisher

Abstract

We have made a nonlinear dynamics study of a parametrically driven magneto-optical trap. When the trapping potential is modulated at around twice the trap frequency above a threshold, the limit cycle is manifested as two separate collective atomic packets oscillating with opposite phases and finite amplitudes, determined by the balance of the nonlinearity and the damping of the trap. In particular, we directly observe in situ the subcritical Hopf bifurcation, which suddenly converts the combined motions of a limit cycle and a stable attractor into a limit cycle motion. The experimental results are in quantitative agreement with the numerical results based on the simple Doppler cooling theory.

DOI
10.1103/physreva.68.031403

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We have made a nonlinear dynamics study of a parametrically driven magneto-optical trap. When the trapping potential is modulated at around twice the trap frequency above a threshold, the limit cycle is manifested as two separate collective atomic packets oscillating with opposite phases and finite amplitudes, determined by the balance of the nonlinearity and the damping of the trap. In particular, we directly observe in situ the subcritical Hopf bifurcation, which suddenly converts the combined motions of a limit cycle and a stable attractor into a limit cycle motion. The experimental results are in quantitative agreement with the numerical results based on the simple Doppler cooling theory.
We demonstrate parametric resonance in a magneto-optical trap. When we modulate the intensity of the cooling laser at about twice the resonant frequency of the trap, the atoms in the trap are divided into two parts and oscillate with 180 degree phase difference with the finite length due to nonlinearity of the trap potential. These are the effects of general nonlinear dynamics, called the Hopf bifurcation, or limit cycle motion. The amplitude and the phase of the oscillations are measured and compared with the theoretical calculations based on simple Doppler cooling theory. The experimental results are in excellent agreement with the simulation results based on the simple Doppler cooling theory.
Abstract We report the first observation of motional excitation induced by parametric resonance of cold 133^{133}133Cs atoms in a dual-species magneto-optical trap (MOT) simultaneously trapping 133 Cs and 85 Rb atoms. By modulating the cooling laser’s intensity at approximately twice the resonant frequency of the trap, the atoms inside the trap separate into two atomic clouds, oscillating with a π phase difference due to the trap potential’s nonlinearity, resulting in a finite oscillation length. This behavior reflects general nonlinear dynamics and is described as a Hopf bifurcation or limit cycle motion. The measured oscillation amplitude was compared with theoretical calculations derived from simple Doppler cooling theory, changing modulation amplitude, magnetic field gradient, and saturation parameter. In addition, we investigate the noise-induced switching dynamics by measuring the temporal evolution of atomic populations localized in each of the two attractors under resonant symmetry lifting induced by an effective bias field. Notably, this dual-species MOT system is expected to provide a new platform for studying nonlinear and nonequilibrium dynamics between cold rubidium and cesium atomic ensembles. PACS numbers: 37.10.Vz, 37.10.De, 37.10.Gh
The double structure potential predicted in the atomic cloud of a magneto-optical trap by one- (Doppler part) and two-photon process (sub-Doppler part) is experimentally observed and studied through the parametric excitation with the modulation of cooling laser intensities. Modulating the intensity of the cooling laser at around twice its resonant frequency of Doppler part, the atoms in Doppler part are divided and they oscillate reciprocally with finite amplitude due to the nonlinearity of the trap, and the others in sub-Doppler part are not influenced. By the modulation, we can investigate the sub-Doppler part spatially without Doppler part and also measure the temperature of sub-Doppler part independently.
We present a method to measure the trap frequency or the spring constant of a magneto-optical atom trap via the parametric resonance. When the intensity of the cooling laser is modulated at around twice its resonant trap frequency, the trapped atoms are parametrically excited and divided into two groups oscillating with the motional phase difference of $\ensuremath{\pi}$. We find that one can measure the natural trap frequency as the modulation amplitude is decreased down to its threshold value. The experimental results are in good agreement with the three-dimensional Doppler cooling theory based on the rate equation model.
We investigate the scaling behavior of the relaxation process for an unstable state near a subcritical Hopf bifurcation point. When the parametric modulation is applied to a magneto-optical trap, the atomic cloud becomes unstable and decays to the dynamic bistable states. Near the subcritical Hopf bifurcation point, we experimentally show that the relaxation process exhibits the scaling behavior; the relaxation time shows a scaling exponent of -1.002 (±0.024). We also present the passage time distribution for the statistical interpretation of the escape process associated with the relaxation of the unstable state. We compare the experimental results to the numerical and analytic results, demonstrating the good agreement between them.
We have investigated noise-induced transition of atoms between double or triple phase-space attractors that are produced in the parametrically driven magneto-optical trap. The transition rates between two or three dynamic attractors, measured for various modulation frequencies and amplitudes, are in good agreement with theoretical calculations and Monte-Carlo simulations based on the Langevin equations. Our experiment may be useful to study nonlinear dynamic problems involving transitions between states far from equilibrium.