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.

OCIS codes: 020.7010; 300.6210

1. E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, “Trapping of Neutral Sodium Atoms with Radiation Pressure,” Phys. Rev. Lett., vol. 59, no. 23, pp. 2631-2634, 1987.
   
2. J. Dalibard and C. C.-Tannoudji, “Laser Cooling Below the Doppler Limit by Polarization Gradients - Simple Theoretical Models,” J. Opt. Soc. Am. B, vol. 6, no. 11, pp. 2023-2045, 1989.
   
3. P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, “Optical Molasses,” J. Opt. Soc. Am. B, vol. 6, no. 11, pp. 2084- 2107, 1989.
   
4. C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-Space Density in the Magnetooptical Trap,” Phys. Rev. A, vol. 52, no. 2, pp. 1423-1440, 1995.
   
5. D. Wilkowski, J. Ringot, D. Hennequin, and J. C. Garreau, “Instabilities in a Magneto-Optical Trap: Noise-Induced Dynamics in an Atomic System,” Phys. Rev. Lett., vol. 85, no. 9, pp. 1839-1842, 2000.
   
6. T. Walker, D. Sesko, and C. Wieman, “Collective Behavior of Optically Trapped Neutral Atoms,” Phys. Rev. Lett., vol. 64, no. 4, pp. 408-411, 1990.
   
7. S. Friebel, C. D"Andrea, J.Walz, M.Weitz, and T.W. H¨ansch, “$CO_2$-laser optical lattice with cold rubidium atoms,” Phys. Rev. A, vol. 57, no. 1, pp. R20-R23, 1998.
   
8. J. Tan and G. Gabrielse, “Synchronization of Parametrically Pumped Electron Oscillators with Phase Bistability,” Phys. Rev. Lett., vol. 67, no. 22, pp. 3090-3093, 1991
   
9. J. Tan and G. Gabrielse, ”Parametrically Pumped Electron Oscillators,” Phys. Rev. A, vol. 48, no. 4, pp. 3105-3121, 1993.
   
10. C. H. Tseng, D. Enzer, G. Gabrielse, and F. L.Walls, “1-bit memory using one electron: Parametric oscillations in a Penning trap,” Phys. Rev. A, vol. 59, no. 3, pp. 2094-2104, 1999.
    
11. L. J. Lapidus, D. Enzer, and G. Gabrielse, “Stochastic Phase Switching of a Parametrically Driven Electron in a Penning Trap,” Phys. Rev. Lett., vol. 83, no. 5, pp. 899-902, 1999.
    
12. L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, London, UK, 1976).
13. N. H. Nayfeh and D. T. Moore, Nonlinear Oscillations (Wiley, New York, USA, 1979).
14. S. H. Strogatz, Nonlinear Dynamics and Chaos (Perseus, New York, USA, 2001).
15. S. Chang and V. Minogin, “Density-matrix approach to dynamics of multilevel atoms in laser fields,” Phys. Rep., vol. 365, pp. 65-143, 2002.