Quench-Induced Breathing Mode of One-Dimensional Bose Gases

We measure the position- and momentum-space breathing dynamics of trapped one-dimensional Bose gases at finite temperature. The profile in real space reveals sinusoidal width oscillations whose frequency varies continuously through the quasicondensate to ideal Bose gas crossover. A comparison with theoretical models taking temperature into account is provided. In momentum space, we report the first observation of a frequency doubling in the quasicondensate regime, corresponding to a self-reflection mechanism due to the repulsive interactions. Such a mechanism is predicted for a fermionized system, and has not been observed to date. The disappearance of the frequency doubling through the crossover is mapped out experimentally, giving insights into the dynamics of the breathing evolution.

Figure 1

Quench sequence and subsequent density evolution in position and momentum space for a QBEC (data $A$). (a) Longitudinal trapping frequency ${\omega }_z$ as a function of time $\tau$. (b) $\rho (z,\tau )$, (c) $n(p,\tau )$, experimental data (in atoms/pixel). (d),(e), corresponding plots from ab initio scaling calculation. (f) and (g) show instantaneous $\rho (z)$, $n(p)$ at $\tau =1$ and 21 ms. Experimental data (dots) are compared with the scaling solutions (solid lines). QMC calculation of $n(p,\tau )$ (dashed line) is shown in (f). $z(p)$ is in units of pixels ${\mathrm{\Delta }}_z({\mathrm{\Delta }}_p)$, with ${\mathrm{\Delta }}_z=2.7\mu \mathrm{m}$, and ${\mathrm{\Delta }}_p=0.14\hslash /\mu \mathrm{m}$.

Figure 2

Time evolution of the in situ mean-square width $⟨z^2⟩$ (top) and the momentum HWHM $w_p$ (bottom) for 3 data sets. The solid lines show the fit: a damped sinusoid in real space (top), and a two-harmonic model according to Eq. (3) in momentum space (bottom). Statistical error of the widths are shown for the first 100 ms. Data $A$, $B$, and $C$ differ in thermodynamic regime. See Fig. 3 and text for details.

Figure 3

Breathing mode through the crossover. (a) Breathing frequency in units of dipole frequency, as a function of ${\gamma }_0$. Our theory evaluated at $t=1100$ (dashed line) and $t=400$ (dash-dotted line) are shown. (b) Relative power of the first (square) and second (diamond) harmonics of $w_p$. Data labels ($A–C$) correspond to those in Fig. 2. The error bars account for fitting error only. The dotted lines are the asymptotic limits. The shaded region shows the crossover criteria ${\gamma }_{\mathrm{co}}=t^{-2/3}$ (the dashed line in the inset), at $t\simeq 1100$ given by the in situ profile with 20% uncertainly. Inset: the phase diagram of the LL model [29], where all lines represent smooth crossovers. The horizontal line segment shows the region explored by the data [20].