Dynamics of noise-induced heating in atom traps

A Fokker-Planck equation is derived for the energy distribution of atoms in a three-dimensional harmonic trap with fluctuations in the spring constant and the equilibrium position. Using this model, we predict trap lifetimes based on the measurable noise spectra of the fluctuations. The energy distributions evolve into a single eigenmode where the apparent temperature of the distribution remains constant while the population decays as a consequence of the energy input. The method of analysis and the corresponding results are applicable to any optical, magnetic, or ion trap that is approximately harmonic, and offer useful insights into both noise-induced and optical heating processes. @S1050-2947~98! 05211-1# PACS number~s! : 32.80. Pj I. INTRODUCTION Stable atom traps have diverse applications in quantum optics ranging from studies of Bose-Einstein condensates ~BEC’s ! @ 1‐3# in magnetic traps to quantum computing @4# in ion traps. Optical far-off-resonance traps have been demonstrated for confining arbitrary atomic ground states in nearly identical potentials @5,6# and for optical lattices @7#. Multiple spin condensates have been stored in a shallow faroff-resonance trap to study interactions in the quantum degenerate regime @8#. These traps also offer an attractive possibility for investigating weakly interacting atomic fermions where multiple atomic states are required for S-wave scattering @9#. In all of these applications, achievement of long storage times is of great importance. For some time it has been appreciated that fluctuations in the trap parameters can cause atom heating, and that the resulting trap loss limits the maximum storage period. This problem has been circumvented in BEC experiments by using magnetic traps with relatively low trap resonance frequencies, so that fluctuations in the potential are only weakly transmitted to the atoms. Consequently, adequate mechanical and power supply stability are not too difficult to achieve. In contrast, far-off-resonance optical traps with high resonance frequencies can be very sensitive to laser intensity fluctuations and beam-pointing noise which cause trap fluctuations and subsequent heating. However, the conditions on the trap stability needed to achieve long storage times have not been carefully studied. In a recent paper, we considered fluctuations in a far-offresonance optical trap using a simple harmonic-oscillator model @10#. Heating rates were estimated in terms of the intensity and position noise power spectra measured for an argon ion laser. Intensity noise causes fluctuations in the spring constant and results in exponential heating, while beam-pointing noise causes fluctuations in the center of the trap and leads to heating at a constant rate. It was shown that achieving heating time constants well beyond 10 sec imposes stringent requirements on the trap stability. Since this model is applicable to any harmonic trap, the results apply equally well to magnetic, optical, and ion traps, and provide estimates of the expected heating time scales. In this paper, we apply the fluctuating harmonic-oscillator model to noise-induced trap dynamics by deriving a FokkerPlanck equation for the energy distribution of atoms in a three-dimensional fluctuating trap. Numerical modeling for a trap with a finite depth yields estimates of the atom loss rate and the average energy for a number of loading conditions. We begin by reviewing the results of our previous paper for the heating rates, and then derive an approximate FokkerPlanck equation that describes the evolution of the atom energy distribution in the trap.