Mapping out the quasicondensate transition through the dimensional crossover from one to three dimensions

By measuring the density fluctuations in a highly elongated weakly interacting Bose gas, we observe and quantify the transition from the ideal gas to a quasicondensate regime throughout the dimensional crossover from a purely one-dimensional (1D) to an almost three-dimensional (3D) gas. We show that that the entire transition region and the dimensional crossover are described surprisingly well by the modified Yang-Yang model. Furthermore, we find that at low temperatures the linear density at the quasicondensate transition scales according to an interaction-driven scenario of a longitudinally uniform 1D Bose gas, whereas at high temperatures it scales according to the degeneracy-driven critical scenario of transverse condensation of a 3D ideal gas.

Figure 1

Density fluctuations across the quasicondensate transition, for $k_{B}T/\hslash {\omega }_{\perp }$ values of (a) 3.6, (b) 1.0, and (c) 0.09. The measured atom-number variances in individual pixels, $\langle \delta N^2\rangle$, as a function of the mean atom number $\langle N\rangle$ are shown as circles, with the error bars representing the statistical uncertainty. Different curves denote the predictions from theoretical models, rescaled by the resolution factor $\kappa =0.53$, $0.55,$ and $0.4$ for (a), (b), and (c), respectively (see text): modified Yang-Yang model (solid), ideal Bose-gas (dashed-dotted), quasicondensate (dashed), and the Poissonian shot-noise level (dotted). The two vertical gray lines give the atom numbers $N_1=\Delta n_1$ and $N_2=\Delta n_2$ (see text). The insets show the average density profiles, together with the predictions from the same models (with different curves as in the main graphs). The chemical potential ${\mu }_0$ in the trap center is deduced from the fit of the wings of the density profile to the ideal Bose-gas EoS for (a), and from the peak density and the quasicondensate EoS for (b) and (c). The trap oscillation frequencies are ${\omega }_{\perp }/2\pi =3.0$ kHz, ${\omega }_{\parallel }/2\pi =8.0(5)$ Hz for (a) and (b) and ${\omega }_{\perp }/2\pi =3.9$ kHz, ${\omega }_{\parallel }/2\pi =4.0(5)$ Hz for (c). The absolute temperatures $T$ are (a) $510$ nK, (b) 160 nK, and (c) 18 nK.

Figure 2

Phase diagram for the 1D–3D crossover. The data at $k_{B}T/\hslash {\omega }_{\perp }<0.8$ ($>0.8$ kHz) are taken with ${\omega }_{\perp }/2\pi =3.9$ kHz ($=3.0$ kHz). The measured transition boundaries $n_1$ (circles) and $n_2$ (squares) are plotted along with the predictions from the MYYM (solid gray curves). The dashed-dotted curve is the perturbative calculation of $n_1$; all other lines are as labeled (see text). The inset shows the dimensional crossover temperature $T_{\text{DC}}$ vs $a/l_{\perp }$ (crosses), fitted with a power law $(a/l_{\perp }){}^{-2/11}$ (dashed line).