A spectroscopic determination of scattering lengths for sodium atom collisions - Statistical Data Included

Journal of Research of the National Institute of Standards and Technology, Jan, 2002 by Eite Tiesinga, Carl J. Williams, Paul S. Julienne, Kevin M. Jones, Paul D. Lett, William D. Phillips

We report a preliminary value for the zero magnetic field Na [S.sup.2](f = 1, m = -1) Na [S.sup.2](f = 1, m = -1) scattering length. [a.sub.1,-1]. This parameter describes the low-energy elastic two-body processes in a dilute gas of composite bosoas and determines, to a large extent, the macroscopic wavefunction of a Bose condensate in a trap. Our scattering length is obtained from photoassociative spectroscopy with samples of uncundensed atoms. The temperature of the atoms is sufficiently law that contributions from the three lowest partial waves dominate the spectrum. The observed lineshapes for the purely long-range [0.sup.-.sub.g], molecular state enable as to establish key features of the ground state scattering wavefunction. The fortuitous occurrence of a p-wave node near the deepest point ([R.sub.e], = 72 [a.sub.0]) of the [0.sup.-.sub.g] potential curve is instrumental in determining [a.sub.1,-1] = (52[ or -]5) [a.sub.0] and [a.sub.2,2] = (85[ or -]3) [a.sub.0], where the latter is for a collision of two Na [S.sup.2](f = 2, m = 2) atoms.

Key words: laser cooling; photoassociation speetroscopy: scattering length; spectral line shapes; ultracold sodium atom collisions.

1. Introduction

Last year two groups reported the observation of Bose-Einstein Condensation (BEC) in dilute gasses of ultra-cold [Rb.sup.87] and [Na.sup.23] (1,2), and another reported evidence for reaching the quantum degenerate regime in [Li.sup.7] (3) but without observing BEC (4). The observation of BEC in a weakly-interacting gas opens up a whole range of possibilities, from fundamental studies of the coherent atomic samples produced, to the construction of the atom-analog of a laser. Theoretical descriptions of the weakly interacting Bose condensate are only now being developed and experimental techniques to probe the condensate are just beginning to be explored.

One of the fundamental parameters required to understand the approach to BEC and the properties of the condensate is the s-wave scattering length. This scattering length determines the low energy elastic scattering rate and thus the evaporative cooling rate as well as the nonlinear coupling parameter in the Gross-Pitaevski equation (5) for the condensate wavefunction, It is not necessary to produce a condensate to measure the s-wave scattering length: temperatures in a magneto-optic trap (MOT) are sufficiently low ([approximately equal to] 1 mK) to limit scattering to a few partial waves and thus permit a determination of the s-wave scattering length.

We probe the scattering wavefunction using the technique of photoassoeiation spectroscopy (6-10). Two Na atoms colliding along the ground state 3[S.sup.2] 3[S.sup.2] potential can absorb a photon to produce a bound molecule, in our case to vibrational levels with energy near the 3[S.sup.2] 3[P.sup.2.sub.3/2] asymptote. We detect the formation of molecules by sending in a second photon which excites the molecule to an autoionizing state, thereby producing an easily detected [Na.sup. .sub.2] ion. The relative intensities of the molecular photoassociation lines carry information about the ground state wavefunction. In particular, we find that two specific rovibrational lines that arise from p-wave scattering are significantly weaker than the corresponding lines for other nearby vibrational levels. This indicates that the former rovibrational state is centered at an internuclear separation near a node in the p-wave ground state wavefunction. With the location of this node established, the intensities and lineshapes of other rovibrational lines allow us to constrain the location of the corresponding s-wave node, and thus to determine the scattering length.

The transitions which we use are from two colliding Na 3[S.sup.2](f = 1) atoms to the [Na.sub.2] [0.sup.-.sub.g] "purely long range" molecular state which asymptotically correlates to a 3[S.sup.2] and a 3[P.sup.2.sub.3/2] atom (11-15). The wavefunctions of the lowest vibrational levels in this potential are localized at distances between 50 [a.sub.0] and 100 [a.sub.0], as shown in Fig. 1. (The Bohr radius [a.sub.0] = 0.0529177 nm.) This molecular potential is determined almost entirely by the known long range forces between atoms and the magnitude of the atomic spin-orbit splitting, and thus may be calculated to high precision. The transition rate depends on the overlap between the ground state wavefunction for a low energy collision and the excited bound state wavefunction. It is a fortuitous coincidence that there is a node in the p-wave scattering wavefunction that is nearly centered on the minimum of the [0.sup.-.sub.g] potential. This leads to an almost complete cancellation of the overlap integr al between the Na [S.sup.2](f = 1, m = -1) Na [S.sup.2](f = 1, m = -1) p-wave scattering wavefunction and the symmetric v - 0 vibrational wavefunction, resulting in a striking and characteristic absence of p-wave features in the spectrum of the v - 0 level of the [0.sup.-.sub.g] state in our experiments. We are able to construct a family of ground state potentials consistent with the known spectroscopy of the molecular ground states that also reproduce the p-wave node near the minimum of the [0.sup.-.sub.g] state. We obtain further constraints on the acceptable potentials from the width and the relative heights of the rotational features in the spectrum. This, in turn, places constraints on the position of the corresponding s-wave node. Finally, we relate the s-wave nodal position to the scattering length.