Three-Body Correlation Functions and Recombination Rates for Bosons in Three Dimensions and One Dimension

We investigate local three-body correlations for bosonic particles in three dimensions and one dimension as a function of the interaction strength. The three-body correlation function $g^{(3)}$ is determined by measuring the three-body recombination rate in an ultracold gas of Cs atoms. In three dimensions, we measure the dependence of $g^{(3)}$ on the gas parameter in a BEC, finding good agreement with the theoretical prediction accounting for beyond-mean-field effects. In one dimension, we observe a reduction of $g^{(3)}$ by several orders of magnitude upon increasing interactions from the weakly interacting BEC to the strongly interacting Tonks-Girardeau regime, in good agreement with predictions from the Lieb-Liniger model for all strengths of interaction.

Figure 1

(a) Relative atom number $N(t)/N(0)$ vs hold time $t$ in 3D geometry: BEC (squares and circles) and thermal gas (diamonds) for $a_{3\mathrm{D}}=101(2)a_0$, $386(3)a_0$, and $386(3)a_0$, respectively. The dashed lines are fits to the data based on the loss equation (see text). The solid lines are linear fits that include the data from 100% to 85%. (b) The ratio of correlation functions $g_{\mathrm{th}}^{(3)}/g_{\mathrm{BEC}}^{(3)}$ as a function of $a_{3\mathrm{D}}$ (experimental data: circles; prediction [20]: squares). All error bars reflect the $1\sigma$ statistical uncertainty.

Figure 2

(a) Sketch of the experimental setup: A 2D optical lattice traps atoms in an array of 1D tubes. (b) Example of a computed atom number distribution $N_{i,j}$ (see text). (c) The relative atom number $N(t)/N(0)$ as a function of time $t$ in 1D geometry: squares and circles correspond to $a_{3\mathrm{D}}=23(1)a_0$ and $568(3)a_0$ with initial densities of $4.5\mu {\mathrm{m}}^{-1}$ and $1.7\mu {\mathrm{m}}^{-1}$ at the center of the center tube, respectively. The solid lines are linear fits to the initial slopes. (d) The relative atom number $N(t)/N(0)$ in 1D for fixed $a_{3\mathrm{D}}=568(3)a_0$ and for various values of $\gamma$ as the 1D density is changed: $\gamma =12$ (circles), $\gamma =13$ (triangles), and $\gamma =14$ (squares). The solid (dashed) lines are linear fits to the data points for short (large) times to guide the eye.

Figure 3

(a) Three-body loss coefficient $K^{(3)}{g}^{(3)}$ vs $a_{3\mathrm{D}}$ for a quantum degenerate gas in 3D (squares) and in 1D (circles). The line gives the $K^{(3)}=C\hslash a_{3\mathrm{D}}^4/m$ scaling in the universal regime in 3D with $C=67.9$ [31, 34, 35, 36, 37]. The error bars of $K^{(3)}{g}^{(3)}$ reflect the $1\sigma$ statistical uncertainty of the linear fit. The increased relative error for the larger values of $a_{3\mathrm{D}}$ is the result of greatly reduced atom loss, giving smaller slopes for the decay curves at roughly the same shot-to-shot particle variations, primarily due to the behavior of $g^{(3)}$, but also due to a reduction of the density (for details, see text). (b) The measured correlation function $g_{1\mathrm{D}}^{(3)}$ vs $\gamma$ in 1D geometry (circles). The crosses indicate the values for $g_{1\mathrm{D}}^{(3)}$ corrected for the variations of $g_{3\mathrm{D}}^{(3)}$ with $a_{3\mathrm{D}}$ as in Eq. (1) (see text). The dashed and dotted lines are approximate analytical solutions for $\gamma \ll 1$ and $\gamma \gg 1$ from Ref. 25. The solid line is the prediction from Refs. 27, 28.