Weakly Interacting Bose Gas in the One-Dimensional Limit

We prepare a chemically and thermally one-dimensional (1D) quantum degenerate Bose gas in a single microtrap. We introduce a new interferometric method to distinguish the quasicondensate fraction of the gas from the thermal cloud at finite temperature. We reach temperatures down to $kT\approx 0.5\hslash {\omega }_{\perp }$ (transverse oscillator eigenfrequency ${\omega }_{\perp }$) when collisional thermalization slows down as expected in 1D. At the lowest temperatures the transverse-momentum distribution exhibits a residual dependence on the line density $n_{1\mathrm{D}}$, characteristic for 1D systems. For very low densities the approach to the transverse single-particle ground state is linear in $n_{1\mathrm{D}}$.

Figure 1

Interference of two 1D quasicondensates at relatively low (top left) and high (bottom left) temperatures. The absorption images were taken along the long direction ($z$) of the clouds after they had expanded from a double well trap in potential free time-of-flight. The absolute value of the Fourier transform (in arbitrary units) along the direction ($x$) of modulated density (fringes) is also depicted. The central panels show the density profiles at the interference fringe period [dashed line (blue online)] and at zero spatial frequency [solid line (red online)] and Gaussian fits (bars indicate the respective standard deviations $\sigma$). The panel on the right illustrates the increase in ratio between the widths of the cloud and the part of it that interferes as a function of final trap depth after the evaporative cooling process.

Figure 2

Collision rate divided by density as a function of temperature across the 1D–3D crossover. The lowest achieved temperatures ($kT\lesssim 0.5\hslash {\omega }_{\perp }$) appear to be limited by slow rethermalization due to inhibited elastic collisions. The increase in collision rate as temperature is increased does not follow the square root law (solid line shows best fit) that is valid in 3D, but is compatible with an exponential suppression as predicted in 1D (dashed line shows best fit for data within $kT<2\hslash {\omega }_{\perp }$) [29].

Figure 3

Transverse width (Gaussian standard deviation, normalized to the single-particle ground state expanded size) of the expanded cloud after time-of-flight as a function of longitudinal line density of atoms throughout the quasi-one-dimensional regime of $\mu \ll \hslash {\omega }_{\perp }$ and into the 1D–3D crossover. The points represent our data, measured in two different traps (blue squares and red circles, example data see upper inset). The line shows the theory without any adjustable parameters [Eq. (1) [34] ]. The lower insets illustrate that a Gaussian distribution models the data very well in the 1D regime (left, $n_{1\mathrm{D}}=100\mu {\mathrm{m}}^{-1}$), while systematic deviations become discernible in the 1D–3D crossover regime (right, $n_{1\mathrm{D}}=250\mu {\mathrm{m}}^{-1}$). Data, best Gaussian fits, and residuals (magnified by a factor of 3) are shown.