Multimode Dynamics and Emergence of a Characteristic Length Scale in a One-Dimensional Quantum System

We study the nonequilibrium dynamics of a coherently split one-dimensional Bose gas by measuring the full probability distribution functions of matter-wave interference. Observing the system on different length scales allows us to probe the dynamics of excitations on different energy scales, revealing two distinct length-scale-dependent regimes of relaxation. We measure the crossover length scale separating these two regimes and identify it with the prethermalized phase-correlation length of the system. Our approach enables a direct observation of the multimode dynamics characterizing one-dimensional quantum systems.

Figure 1

Matter-wave interferometry and correlation measurements. (a) The experiment is initiated by coherently splitting a single quasi-1D Bose gas into two uncoupled gases using a horizontal double-well potential. The system is then held to evolve for a variable evolution time $t_e$ before being released from the trapping potential. The contrast of the resultant interference pattern is a direct measure of the relative phase fluctuations (b). Integrating the interference pattern over a length $L$ and fitting the resulting line profile (c) gives the phase $\Phi (L)$ and the contrast $C(L)$, which are plotted as circular statistics (d). (e) The whole process is repeated many times (typically 150) to map the FDF for a particular length scale $L$ and evolution time $t_e$. The point plot in (e) is then converted into a density plot (f).

Figure 2

Multimode dynamics revealed by FDFs of matter-wave interference. The probability density of contrast $C(L)$ and phase $\Phi (L)$ of interference patterns is measured for each integration length $L$ (horizontal axis) and evolution time $t_e$ (vertical axis). Red and blue denote high and low probability, respectively, and the color map is rescaled for each plot. For each value of $L$, the right (left) columns correspond to the experimental data (theoretical calculations) and full cloud to $L=100\mu \mathrm{m}$ [33]. At $t_e=1.5\mathrm{ms}$, the high contrasts and small phase spreads demonstrate the coherence of the splitting process. As time evolves, a steady state emerges and two distinct length-scale-dependent regimes appear: the phase-diffusion regime (A) and the contrast-decay regime (B). For short (long) $L$ values, the phase is random and the probability of observing a high contrast is high (low), resulting in a ring (disk) shape in the density plot. The theoretical calculations take into account the technical noise on the relative atom number between the two wells (standard deviation measured to be 5% of the total atom number) and the error associated with the fitting of the interference patterns.

Figure 3

Contrast dynamics. Measured values of the mean squared contrast for various integration lengths, corrected for the contrast reduction factor due to the imperfections of our imaging system (see Supplemental Material [24]). From top to bottom: $L=18$, 40, 60, $100\mu \mathrm{m}$. The lines show the results of the Luttinger liquid calculations for these integration lengths. The theory data for all $L$ values have been rescaled by the same factor of $r=0.74$ [34]. Apart from this factor, no fitting parameter is used. We observe a relaxation process in which a steady state is established on a time scale depending on $L$ and corresponding to the dephasing of the different excitations probed within that $L$.

Figure 4

Comparison of the relative-phase correlation length of the prethermalized system (${\lambda }_{\mathrm{eff}}$) and of a system at thermal equilibrium (${\lambda }_{\varphi }$). Blue squares: values of ${\lambda }_{\mathrm{eff}}$ obtained by varying the temperature $T_{\mathrm{init}}$ before splitting the quasicondensate (see main text). Error bars denote 1 standard deviation and are obtained by a bootstrapping method (see Supplemental Material [24]). The red shaded area corresponds to the 95% confidence interval around the theoretical calculation (line) accounting for the uncertainty on the experimental parameters. Green circles: correlation length ${\lambda }_{\varphi }$ of a system of two quasicondensates in thermal equilibrium, normalized to the mean of the different densities. Gray shaded area: 95% confidence interval around the theory (dashed line). Inset: estimation of the correlation length. The filled circles are the values of $⟨{\widehat{C}}^2(L)⟩$ in the prethermalized state (data of Fig. 3 for $t_e>10\mathrm{ms}$), normalized to the value at $L=6\mu \mathrm{m}$. The solid line is a fit to an effective thermal equilibrium theory with the correlation length being the only free parameter.