Momentum distribution of one-dimensional Bose gases at the quasicondensation crossover: Theoretical and experimental investigation

We investigate the momentum distribution of weakly interacting one-dimensional (1D) Bose gases at thermal equilibrium both experimentally and theoretically. Momentum distribution of 1D Bose gases is measured using a focusing technique, whose resolution we improve via a guiding scheme. The momentum distribution compares very well with quantum Monte Carlo calculations for the Lieb-Liniger model at finite temperature, allowing for an accurate thermometry of the gas that agrees with (and improves upon) the thermometry based on in situ density fluctuation measurements. The quasicondensation crossover is investigated via two different experimental parameter sets, corresponding to the two different sides of the crossover. Classical field theory is expected to correctly describe the quasicondensation crossover of weakly interacting gases. We derive the condition of validity of the classical field theory, and find that, in typical experiments, interactions are too strong for this theory to be accurate. This is confirmed by a comparison between the classical field predictions and the numerically exact quantum Monte Carlo calculations.

Figure 1

Phase diagram of the 1D Bose gas in the parameter space $(\gamma ,t)$. The ideal Bose gas to quasicondensation crossover occurs for $\gamma \simeq {\gamma }_{co}=t^{-2/3}$ (solid line). Within the ideal Bose gas regime (light gray area), the crossover from the classical to the degenerate subregime occurs for $\gamma \simeq {\gamma }_d=t^{-1/2}$ (dashed line). The region $\gamma ,t\gg 1$ (dark gray area) is the fermionized regime. The segments correspond to the regions explored by the two sets of experimental data (see Sec. 4). Note that the effect of quantum fluctuations on the $g^{\left(1\right)}$ function in the quasicondensate regime are noticeable only for extremely low temperature [21], which is experimentally irrelevant.

Figure 2

Momentum distribution of a homogeneous gas at $t=1000$. (a) FWHM according to the classical field theory (dashed line), ideal Bose gas theory (solid line), Maxwell-Boltzmann prediction (dash-dotted line), and QMC calculations (stars). The arrows show the interaction parameter at the quasicondensation crossover (${\gamma }_{co}=t^{-2/3}$) and at the degeneracy crossover (${\gamma }_d=t^{-1/2}$). (b) Momentum distribution in log scale for three different value of $\gamma$. Shown are QMC results (solid lines), classical field predictions (dashed lines), and ideal Bose gas prediction (dotted lines). Error bars, shown for a single point on the side of the profile, correspond to one standard deviation of the QMC calculation.

Figure 3

(a) Wire configuration: currents $I_1$, $I_2$, $I_3$, and $I_4$ realize the longitudinal trapping potential of frequency $f_z$. The transverse confinement of frequency $f_{\perp }$ is ensured by the three wires carrying a current modulated at 200 kHz. (b),(c) Focusing sequences. The focusing potential, harmonic to order 5 in $z$, is applied during $t_{\mathrm{kick}}$ before the longitudinal potential is turned off. After a free evolution during the focusing time $\tau$, an absorption image is taken. Without guiding, see (b), the transverse confinement is removed during the whole focusing time, though a 1 ms ramp down to $f_{\perp }=700$ Hz is carried out to decrease the transverse expansion. In the presence of guiding, see (c), $f_{\perp }$ is kept at 700 Hz for a time $t_g$, during which the cloud does not fall. Numerical values in (b) and (c) correspond to data of Figs. 5 and 4, respectively.

Figure 4

Momentum distribution of a gas of about 7000 atoms at thermal equilibrium, in a $7.5$ Hz longitudinal trap, and $4.6$ kHz transverse trap. The solid distribution is obtained without guiding and with a free fall of $24.7$ ms. The circles are the momentum distribution of the same sample, but with a guiding time of $20.6$ ms followed by a free fall of $22.7$ ms.

Figure 5

Data for atoms confined in a harmonic trap of transverse oscillation frequency of (a) 6.4 kHz and (b) 2.1 kHz, and a longitudinal oscillation frequency of (a) 8.3 Hz and (b) 7.6 Hz. The pixel size, in the object plane, for in situ data is $\Delta =2.7\mu$m. (1),(2) Momentum distribution in linear and log scales. QMC fits are shown in solid lines, yielding (a) $T=72$ nK and (b) $T=84$ nK. The dashed line in (a,1) is the prediction using the LDA and the Lorentzian momentum distribution in the quasicondensate limit. In (2), the dashed-dotted lines give the $1/p^2$ behavior, while the momentum associated with the healing length $\xi =\hslash /\sqrt{m\rho g}$ indicates that sample (a) is more in the quasicondensate, with $\Delta p\ll \hslash /\xi$, and sample (b) is more in the ideal Bose gas, with $\Delta p\gg \hslash /\xi$. In addition, the dotted lines indicate the contribution from the transverse excited states, which is insignificant for data (a) and hence neglected in our analysis. (3) In situ density fluctuations, with Yang-Yang prediction in solid lines, evaluated at the above-mentioned temperatures. The gray dashed and dotted lines show Yang-Yang predictions with 30$\%$ temperature deviations. The dash-dotted lines indicate the Poissonian fluctuation level. (4) In situ profile, with the QMC profile in dashed lines. The dotted lines again show the contribution from transverse excited states. We also plot in solid lines the density profile fitted with Yang-Yang calculation, which yields (a) $T=111$ nK and (b) $T=76$ nK.